XenReference - User contributions [en]

Tuning recommendations for beginners

2026-10-26T13:42:16Z

Unque:

'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition. 12edo is the perfect tuning for people who don't want to have to think about tuning theory, but there are some close options.

To provide a simple overview, consonance of a musical interval is based on approximating [[just intonation|simple frequency ratios]], where you multiply the frequency of one note by the ratio to get the frequency of the other note. "Subgroups" are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.
* 2.3.5: Known as "pental", widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.
* 2.3.7: Known as "septal", more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.
* 2.3.5.7 (2...7): Known as "septimal", the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.
* 2.3.5.7.11.13 (2...13): Known as "tridecimal", an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).

The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:
* Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Renaissance and (especially earlier) Baroque music. Intervals involving 7 are small alterations of diatonic.
* Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.
* Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.

== Ground's list ==
{{UTBox|g_|Ground|7766ff}}

I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).

{{adv|My favorite edos are not my best edos. 25edo is one of my favorites, but I can't reasonably recommend it to most people. This is an optimization of several specific tuning properties I'm interested in, being 7+18, 9+16, and 10+15 (summing them combines the properties they have in common). If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following. This is the philosophical distinction that I have to make in order to stop thinking so much about tuning theory.}}

This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:
* Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).
* Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).

I intend to add sound examples to make the selection process easier.

{{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}

Tuning descriptions (incomplete):
* 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.
* 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.
* 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.
* 26edo: Debatable 2...13.
* 27edo: Good 2...7.13, all tuned sharp.
* 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.
* 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.
* 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. I had to invent [[Straddle primes|straddle theory]] and the trackdye scale structure (aberrated compressed diatonic) just to be satisfied with it. I classify it as straddle-3, and the only edos I've found to be even more confusing are straddle-3-5 (42, 57).
* 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.

These are the main types of MOS-based (rank-2) composition theory that I recommend for the edos on this list:
* [[Diatonic]]: Already known from 12edo, affects everything
* [[Dicoid]]: 17, 24, 27, 31, 37, 41
* [[Slendric]]: 26, 31, 41
* [[Semiquartal]]: 19, 24, 29
* [[Porcupine]]: 22, 29, 37
I add rank-3 [[Aberrisma|aberrismic theory]] for all edos above 19. Aberrismic is a broad category where other scale theories combine. For example, blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on. 37 is the only edo on this list that I think requires an understanding of straddle theory to be used effectively.

{{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}

'''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.

=== Small non-diatonic edos ===

I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.

This is every edo between 5 and 19 that I haven't listed yet:
* [[5edo]] (2), [[7edo]] (1): Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.
* [[6edo]] (2): A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.
* [[8edo]] (3), [[9edo]] (3), [[10edo]] (3): 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.
* [[11edo]] (1), [[13edo]] (1): Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.
* [[14edo]] (2), [[15edo]] (2): Supersets of 7 and 5edo with a crude approximation of 2...7.
* [[16edo]] (1), [[18edo]] (1): Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.

== Inthar's list ==
{{UTBox|KC|Inthar|000000}}

=== Edos ===
Note: I specialized in oneirotonic and have a pro-13edo bias. '''This list is for those who are not content with the standard list of "beginner edos" 5, 7, 17, 19, 22, and 31''':

# Straddle-fifth edos: I particularly recommend [[11edo]], [[13edo]], and [[18edo]] to beginners for melodic familiarity (whole tones and semitones) and harmonic unfamiliarity (lack of diatonic or low-complexity JI). 18edo in particular provides a [[straddle primes|straddle-fifth]] 12edo approximation and [[smitonic]] (4L3s) and [[oneirotonic]] (5L3s) MOS scales, both melodically familiar but warped. However, if you just want to specialize in one of 4L3s and 5L3s, you can't go wrong with 11edo (for 4L3s) or 13edo (for 5L3s).
#* Like diatonic, oneirotonic is also a segment of a "circle of fifths", albeit a very sharp fifth (between 720c and 750c). This leads to a somewhat familiar but warped form of functional harmony.
# JI approximation: [[41edo]] is the first structurally accurate 7-limit edo in that it observes both 81/80 and 64/63, and additionally approximates 11-limit intervals. 41edo has an advantage over the comparable [[46edo]] and [[53edo]] in size and in that it supports the Kite guitar. Smaller edos sacrifice approximation of either prime 5 or 7, or they temper at least one of 81/80 and 64/63.
#* 41 and 46 both have acceptable tunings of 11/8 and support [[Slendric]] and [[penslen]]. 53edo does none of these things but is extremely accurate in the 5-limit, has a sharp tuning of 7/4 and approximates 13/8 more accurately via the high-accuracy 2.3.5.13 temperament [[Kleismic]].
# Armotonic edos: Flat-of-7edo but sharp-of-9edo fifths generate the [[armotonic]] (7L2s) MOS scale, which contains easier-to-use tunings of the antidiatonic (2L5s) MOS scale compared to the balzano (2L7s) MOS which is generated by an even flatter fifth. The smallest armotonic edos are [[16edo]], [[23edo]], and [[25edo]]. Somewhat familiar triads — major and minor triads are just swapped in the 2L5s armotonic MOS — but melodically more exotic. [[9edo]] is a degenerate version of armotonic. See Ground's list above.
# Multiples of 5 and 7: I recommend [[15edo]] if crude 7-limit approximations and melodically challenging 160c and 240c "whole tones" are your thing. [[21edo]] is a multiple of 7edo with conventional-sounding major and minor thirds like 15edo, but less challenging from a melodic perspective. [[14edo]] has a semiquartal (5L4s) scale (which doesn't sound that "[[interordinal]]" unlike 19edo or 24edo's semiquartal but you might prefer it melodically due to its larger semitones). [[10edo]], like 7edo, has simplified neutral harmony but has more melodic variation, providing a 3L4s scale. [[20edo]] and [[28edo]] are not very commonly used; 20edo is the first edo with a balzano (2L7s) MOS scale.
# [[8edo]]: 8edo is the only edo between 7 and 23 I haven't mentioned. It ''is'' uniquely strange — it has no familiar melodic steps, and major/minor triadic harmony doesn't work the same way — though in my opinion that is all the more reason to be somewhat partial to 8edo. You might prefer [[Ammonite]] (e.g. 29edo) or [[Sensi]] (e.g. 19edo or 27edo) for quasi-8edo but more JI-interpretable harmonic structures.

=== JI scales ===
On the JI side, I recommend
# modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation (like maqam). Try various modes of these scales that have 3/2 perfect fifths, such as 12:13:14:16:18:20:22:24.
#* 16:18:20:21:24:27:28:32 is a JI quasi-diatonic scale, a mode of so-called "septimalydian". It is notably both 7-prime-limit and harmonic-series-oriented.
# [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo, allowing for a kind of functional harmony in 2.3.7 just intonation.

== Vector's list ==
{{UTBox|SS|Vector|cebaff}}

=== Edos ===
The set of three "beginner EDOs" provided by Leriendil are as follows:

'''- To retain a composite structure and JI-agnostic characteristic similar to 12edo, while changing the relationships between intervals:''' 15edo

Harmonic unfamiliarity: ★★★

Stylistic unfamiliarity: ★

'''- To retain a 5-limit heptatonic MOS with recognizable but inverted relationships between intervals, while using an unusual tuning:''' 16edo

Harmonic unfamiliarity: ★★

Stylistic unfamiliarity: ★★

'''- To retain diatonic and familiar interval categories, while approximating something that isn't the 5-limit and having a higher resolution including neutral intervals:''' 17edo

Harmonic unfamiliarity: ★

Stylistic unfamiliarity: ★★★

Additionally, 22edo and 24edo provide complementary ways of elaborating on 12edo's interval qualities: 22edo splits each quality into two, while 24edo adds new qualities in between each 12edo quality. (An edo that provides both, while large, is 34edo.) Thus they are good introductions to the more general variety of intervals found in xenharmonic music.

== Unque's list ==
{{UTBox|UQ|Unque|00aaff}}
While my [[User:Unque|user page]] contains more in-depth propaganda for the tuning systems listed here and more, I wanted to make this list more compact and perhaps more beginner-friendly. My list specifically caters to someone whose approach is focused on expanding the familiar tonal world into something more diverse and colorful; as such, I've prioritized a tuning's ability to integrate its new contents smoothly into familiar or conventional melodic and harmonic frameworks, rather than appending them on top or completely abandoning those frameworks.

* '''15edo''': The "[[Diatonic#Greek diatonic scales|Nicetone]]" scale from 15edo approximates a diatonic scale, but distinguishes two different types of wholetones, and thus contains "wolf" intervals. This Nicetone scale is also a subset of the ten-note [[Blackwood]] scale, which simplifies the scale to contain only one type of wholetone and one type of semitone; because the Blackwood scale is a [[Glossary#Mode of limited transposition|mode of limited transposition]], the diatonic scale it contains is repeated on every other step, and it can thus be used to smoothly transition between keys that in 12edo would seem rather unrelated. Finally, 15edo can [[User:Unque/Porcupine|split the perfect fourth into three equal parts]], allowing you to construct a [[tetrachord]] where all intervals are the same size, and a scale made of these equal tetrachords can also be considered analogous to diatonic in a way.
* '''17edo''': This is one of the most common choices for a beginner tuning, and for good reason. 17edo adds neutral thirds on top of the regular diatonic paradigm, which means there are three types of tonalities available (major, neutral, and minor) rather than the classicalist two. The "neutral" or "[[Mosh]]" scale provides a contrast with the classic diatonic scale, as its small step is the diatonic [[Chromatic semitone|chroma]] and vice versa; because of their close relationship, borrowing chords between these two scales or transitioning between them can be done much more cleanly than in tunings such as 24edo where diatonic is less closely tied to the neutral scale.
* '''29edo''': In spite of its larger size, I believe 29edo is the absolute best tuning in this category. Its perfect fifth is even closer to JI than 12edo's, and it contains a single exemplary interval of each [[interordinal]] space. Unlike in 24edo, these interordinals are placed on the chain of fifths, allowing them to be reached by an extension of conventional chromatic harmony. The perfect fourth is additionally highly divisible, making 29edo an excellent introduction to divisibility properties of important intervals.
* '''36edo''': The largest interval in this category, 36edo actually contains 12edo as a subset. In spite of this, the new content that it offers fit much more cleanly into the structure than in 24edo, and additionally offer excellent approximations of septal harmonies. 36edo contains a whopping six main types of tonalities: major and minor from 12edo, supermajor and subminor approximating the septal ratios, and submajor and supraminor representing the more complex 17/14 and 21/17.

[[Category:Core knowledge]]

17edo

2026-10-25T00:02:06Z

Unque:

{{problematic}}

'''17edo''', or 17 equal divisions of the octave, is the equal tuning featuring steps of (1200/17) ~= 70.6 cents, 17 of which stack to the octave 2/1.

17edo is the smallest edo that has a sharper-than-just diatonic fifth (705.8c, compared to just 3/2 = 702.0c), as well as the second edo containing a diatonic fifth, not counting degenerate cases [[5edo]] and [[7edo]]. 17edo has somewhat sharp diatonic major thirds and provides no good approximation of 5/4, which encourages many composers to use nonfunctional and modal harmony instead of standard functional harmony, although standard functional harmony can still justify itself via listeners' general familiarity with 12edo, whose major third is only 1/3 of an edostep off of 17edo's, compared to 5/4 being a half-edostep away. It is also notable for having neutral intervals.

Since 17edo's perfect eleventh is highly divisible (24 steps), you can do 2\17 x 12, 3\17 x 8, 4\17 x 6, 6\17 x 4, 8\17 x 3, or 12\17 x 2 movements to pump this "comma".

== Tuning theory ==

=== Intervals ===
17edo can be notated with [[diatonic notation]] with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too. These can be written in Unicode-supported text as t/d, or with ^/v; occasionally ‡ is used for a semisharp.
{| class="wikitable"
!Edostep
!Cents
!Name (Neutral diatonic / ADIN)
!Notation
!JI interpretation (2.3.25.(11/7).13, 68edo subset)
|-
|0
|0
|Unison
|A
|1/1
|-
|1
|70.6
|(Far)minor second
|B♭, At
|25/24
|-
|2
|141.2
|Neutral second
|A♯, C♭ Bd
|13/12
|-
|3
|211.8
|(Far)major second
|B, Cd
|9/8
|-
|4
|282.4
|(Far)minor third
|C, Bt
|33/28, 32/27
|-
|5
|352.9
|Neutral third
|B♯, D♭, Ct
|16/13
|-
|6
|423.5
|(Far)major third
|C♯, Dd
|14/11
|-
|7
|494.1
|Perfect fourth
|D
|4/3
|-
|8
|564.7
|Semiaugmented/neutral fourth
|E♭, Dt
|25/18, 18/13
|-
|9
|635.3
|Semidiminished/neutral fifth
|D♯, F♭, Ed
|36/25, 13/9
|-
|10
|705.9
|Perfect fifth
|E, Fd
|3/2
|-
|11
|776.5
|(Far)minor sixth
|F, Et
|11/7
|-
|12
|847.1
|Neutral sixth
|E♯, G♭, F
|13/8
|-
|13
|917.6
|(Far)major sixth
|F♯, Gd
|56/33, 27/16
|-
|14
|988.2
|(Far)minor seventh
|G
|16/9
|-
|15
|1058.9
|Neutral seventh
|A♭, Gt
|24/13
|-
|16
|1129.4
|(Far)major seventh
|G♯, Ad
|48/25
|-
|17
|1200
|Octave
|A
|2/1
|}
The major third is a farmajor third, interpreted as 14/11 either with 17edo's patent val or as a subset of 68edo; 17edo is at the upper edge of [[gentle tuning]]. While 17edo has reasonable approximations of 7 and 11, they are not shared with 68edo, although their ratio 14/11 is.

The edostep is a near-just 25/24, sometimes seen as the optimal size for a leading tone.

=== Prime approximations ===
{{Harmonics in ED|17|23}}

=== Notation ===
Because a sharp is 2 steps in 17edo, it may be notated with neutral [[diatonic notation]] (semisharps and semiflats).

=== Tuning properties ===

== Scales ==
Some scales:
* [[Diatonic#MOS diatonic|MOS diatonic]] (5L2s, 3331331)
* [[Mosh]] (3L4s, 2323232)
* Lissotonic1 (3223322): modes of the ascending mode of Maqam Rast
* Trachytonic1 or screamapillar2 (3313322)

1 CellularAutomaton's names

2 Xen Wiki's name

{{Navbox EDO}}

{{cat|Edos}}

Tuning recommendations for beginners

2026-10-24T22:41:19Z

Unque:

'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition. 12edo is the perfect tuning for people who don't want to have to think about tuning theory, but there are some close options.

To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. "Subgroups" are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.
* 2.3.5: Known as "pental", widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.
* 2.3.7: Known as "septal", more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.
* 2.3.5.7 (2...7): Known as "septimal", the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.
* 2.3.5.7.11.13 (2...13): Known as "tridecimal", an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).

The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:
* Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.
* Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.
* Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.

== Ground's list ==
{{UTBox|g_|Ground|7766ff}}

I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).

{{adv|My favorite edos are not my best edos. 25edo is one of my favorites, but I can't reasonably recommend it to most people. This is an optimization of several specific tuning properties I'm interested in, being 7+18, 9+16, and 10+15 (summing them combines the properties they have in common). If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following. This is the philosophical distinction that I have to make in order to stop thinking so much about tuning theory.}}

This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:
* Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).
* Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).

I intend to add sound examples to make the selection process easier.

{{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}

Tuning descriptions (incomplete):
* 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.
* 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.
* 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.
* 26edo: Debatable 2...13.
* 27edo: Good 2...7.13, all tuned sharp.
* 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.
* 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.
* 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. I had to invent [[Straddle primes|straddle theory]] and the trackdye scale structure (aberrated compressed diatonic) just to be satisfied with it. I classify it as straddle-3, and the only edos I've found to be even more confusing are straddle-3-5 (42, 57).
* 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.

These are the main types of MOS-based (rank-2) composition theory that I recommend for the edos on this list:
* [[Diatonic]]: Already known from 12edo, affects everything
* [[Dicoid]]: 17, 24, 27, 31, 37, 41
* [[Slendric]]: 26, 31, 41
* [[Semiquartal]]: 19, 24, 29
* [[Porcupine]]: 22, 29, 37
I add rank-3 [[Aberrisma|aberrismic theory]] for all edos above 19. Aberrismic is a broad category where other scale theories combine. For example, blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on. 37 is the only edo on this list that I think requires an understanding of straddle theory to be used effectively.

{{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}

'''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.

=== Small non-diatonic edos ===

I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.

This is every edo between 5 and 19 that I haven't listed yet:
* [[5edo]] (2), [[7edo]] (1): Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.
* [[6edo]] (2): A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.
* [[8edo]] (3), [[9edo]] (3), [[10edo]] (3): 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.
* [[11edo]] (1), [[13edo]] (1): Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.
* [[14edo]] (2), [[15edo]] (2): Supersets of 7 and 5edo with a crude approximation of 2...7.
* [[16edo]] (1), [[18edo]] (1): Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.

== Inthar's list ==
{{UTBox|KC|Inthar|000000}}

=== Edos ===
Note: I have a pro-13edo bias. '''This list is for those who are not content with the conventional "beginner edos" 5, 7, 17, 19, 22, and 31''':

* Straddle-fifth edos: I particularly recommend [[11edo]], [[13edo]], and [[18edo]] to beginners for melodic familiarity (whole tones and semitones) and harmonic unfamiliarity (lack of diatonic or low-complexity JI). 18edo in particular provides a [[straddle primes|straddle-fifth]] 12edo approximation and [[smitonic]] (4L3s) and [[oneirotonic]] (5L3s) MOS scales, both melodically familiar but warped. However, if you just want to specialize in one of 4L3s and 5L3s, you can't go wrong with 11edo (for 4L3s) or 13edo (for 5L3s).
* JI approximation: [[41edo]] is the first structurally accurate 7-limit edo in that it observes both 81/80 and 64/63. 41edo has an advantage over the comparable [[46edo]] and [[53edo]] in size and in that it supports the Kite guitar. Smaller edos sacrifice approximation of either prime 5 or 7, or they temper at least one of 81/80 and 64/63.
** 41 and 46 have [[Slendric]] and [[penslen]]. 53edo does not.
* Armotonic edos: Flat-of-7edo but sharp-of-9edo fifths generate the [[armotonic]] (7L2s) MOS scale, which contains easier-to-use tunings of the antidiatonic (2L5s) MOS scale compared to the balzano (2L7s) MOS which is generated by an even flatter fifth. The smallest armotonic edos are [[16edo]], [[23edo]], and [[25edo]]. Somewhat familiar triads — major and minor triads are just swapped in the 2L5s armotonic MOS — but melodically more exotic. [[9edo]] is a degenerate version of armotonic. See Ground's list above.
* Multiples of 5 and 7: I recommend [[15edo]] if crude 7-limit approximations and melodically challenging 160c and 240c "whole tones" are your thing. [[21edo]] is a multiple of 7edo with conventional-sounding major and minor thirds like 15edo, but less challenging from a melodic perspective. [[14edo]] has a semiquartal (5L4s) scale (which doesn't sound that "[[interordinal]]" unlike 19edo or 24edo's semiquartal but you might prefer it melodically due to its larger semitones). [[10edo]], like 7edo, has simplified neutral harmony but has more melodic variation, providing a 3L4s scale. [[20edo]] and [[28edo]] are not very commonly used; 20edo is the first edo with a balzano (2L7s) MOS scale.
* [[8edo]]: 8edo is the only edo between 7 and 21 I haven't mentioned. I myself am somewhat partial to 8edo, but you might prefer [[Ammonite]] (e.g. 29edo) or [[Sensi]] (e.g. 19edo or 27edo) for quasi-8edo but more JI-interpretable harmonic structures.

=== JI scales ===
On the JI side, I recommend
* modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation (like maqam)
* [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo, allowing for a kind of functional harmony in 2.3.7 just intonation.

== Vector's list ==
{{UTBox|SS|Vector|cebaff}}

=== Edos ===
The set of three "beginner EDOs" provided by Leriendil are as follows:

'''- To retain a composite structure and JI-agnostic characteristic similar to 12edo, while changing the relationships between intervals:''' 15edo

Harmonic unfamiliarity: ★★★

Stylistic unfamiliarity: ★

'''- To retain a 5-limit heptatonic MOS with recognizable but inverted relationships between intervals, while using an unusual tuning:''' 16edo

Harmonic unfamiliarity: ★★

Stylistic unfamiliarity: ★★

'''- To retain diatonic and familiar interval categories, while approximating something that isn't the 5-limit and having a higher resolution:''' 17edo

Harmonic unfamiliarity: ★

Stylistic unfamiliarity: ★★★

Additionally, 22edo and 24edo provide complementary ways of elaborating on 12edo's interval qualities: 22edo splits each quality into two, while 24edo adds new qualities in between each 12edo quality. (An edo that provides both, while large, is 34edo.) Thus they are good introductions to the more general variety of intervals found in xenharmonic music.

== Unque's List ==
{{UTBox|UQ|Unque|00aaff}}
While my [[User:Unque|user page]] contains more in-depth propaganda for the tuning systems listed here and more, I wanted to make this list more compact and perhaps more beginner-friendly. My list specifically caters to someone whose approach is focused on expanding the familiar tonal world into something more diverse and colorful; as such, I've prioritized a tuning's ability to accommodate familiar harmonic and melodic ideas, while fitting the new contents smoothly into these structures rather than appending them on top.

* '''15edo''': The "[[Diatonic#Greek diatonic scales|Nicetone]]" scale from 15edo approximates a diatonic scale, but distinguishes two different types of wholetones, and thus contains "wolf" intervals. This Nicetone scale is also a subset of the ten-note [[Blackwood]] scale, which simplifies the scale to contain only one type of wholetone and one type of semitone; because the Blackwood scale is a [[Glossary#Mode of limited transposition|mode of limited transposition]], the diatonic scale it contains is repeated on every other step, and it can thus be used to smoothly transition between keys that in 12edo would seem rather unrelated. Finally, 15edo can [[User:Unque/Porcupine|split the perfect fourth into three equal parts]], allowing you to construct a [[tetrachord]] where all intervals are the same size, and a scale made of these equal tetrachords can also be considered analogous to diatonic in a way.
* '''17edo''': This is one of the most common choices for a beginner tuning, and for good reason. 17edo adds neutral thirds on top of the regular diatonic paradigm, which means there are three types of tonalities available (major, neutral, and minor) rather than the classicalist two. The "neutral" or "[[Mosh]]" scale provides a contrast with the classic diatonic scale, as its small step is the diatonic [[Chromatic semitone|chroma]] and vice versa; because of their close relationship, borrowing chords between these two scales or transitioning between them can be done much more cleanly than in tunings such as 24edo where diatonic is less closely tied to the neutral scale.
* '''29edo''': In spite of its larger size, I believe 29edo is the absolute best tuning in this category. Its perfect fifth is even closer to JI than 12edo's, and it contains a single exemplary interval of each [[interordinal]] space. Unlike in 24edo, these interordinals are placed on the chain of fifths, allowing them to be reached by an extension of conventional chromatic harmony. The perfect fourth is additionally highly divisible, making 29edo an excellent introduction to divisibility properties of important intervals.
* '''36edo''': The largest interval in this category, 36edo actually contains 12edo as a subset. In spite of this, the new content that it offers fit much more cleanly into the structure than in 24edo, and additionally offer excellent approximations of septal harmonies. 36edo contains a whopping six main types of tonalities: major and minor from 12edo, supermajor and subminor approximating the septal ratios, and submajor and supraminor representing the more complex 17/14 and 21/17.

17edo

2026-10-08T00:32:55Z

Unque:

{{problematic}}

'''17edo''', or 17 equal divisions of the octave, is the equal tuning featuring steps of (1200/17) ~= 70.6 cents, 17 of which stack to the octave 2/1.

17edo is the smallest edo that has a sharper-than-just diatonic fifth (705.8c, compared to just 3/2 = 702.0c), not counting the degenerate case 5edo. 17edo has discordant diatonic major thirds and provides no good approximation of 5/4, which to some extent forces one to use nonfunctional and modal harmony instead of standard functional harmony. It is also notable for having neutral intervals.

Since 17edo's perfect eleventh is highly divisible (24 steps), you can do 2\17 x 12, 3\17 x 8, 4\17 x 6, 6\17 x 4, 8\17 x 3, or 12\17 x 2 movements to pump this "comma".

== Tuning theory ==

=== Intervals ===
17edo can be notated with [[diatonic notation]] with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too. These can be written in Unicode-supported text as t/d, or with ^/v; occasionally ‡ is used for a semisharp.
{| class="wikitable"
!Edostep
!Cents
!Name (Neutral diatonic / ADIN)
!Notation
!JI interpretation (2.3.25.(11/7).13, 68edo subset)
|-
|0
|0
|Unison
|A
|1/1
|-
|1
|70.6
|(Far)minor second
|B♭, At
|25/24
|-
|2
|141.2
|Neutral second
|A♯, C♭ Bd
|13/12
|-
|3
|211.8
|(Far)major second
|B, Cd
|9/8
|-
|4
|282.4
|(Far)minor third
|C, Bt
|33/28, 32/27
|-
|5
|352.9
|Neutral third
|B♯, D♭, Ct
|16/13
|-
|6
|423.5
|(Far)major third
|C♯, Dd
|14/11
|-
|7
|494.1
|Perfect fourth
|D
|4/3
|-
|8
|564.7
|Semiaugmented/neutral fourth
|E♭, Dt
|25/18, 18/13
|-
|9
|635.3
|Semidiminished/neutral fifth
|D♯, F♭, Ed
|36/25, 13/9
|-
|10
|705.9
|Perfect fifth
|E, Fd
|3/2
|-
|11
|776.5
|(Far)minor sixth
|F, Et
|11/7
|-
|12
|847.1
|Neutral sixth
|E♯, G♭, F
|13/8
|-
|13
|917.6
|(Far)major sixth
|F♯, Gd
|56/33, 27/16
|-
|14
|988.2
|(Far)minor seventh
|G
|16/9
|-
|15
|1058.9
|Neutral seventh
|A♭, Gt
|24/13
|-
|16
|1129.4
|(Far)major seventh
|G♯, Ad
|48/25
|-
|17
|1200
|Octave
|A
|2/1
|}
The major third is a farmajor third, interpreted as 14/11 either with 17edo's patent val or as a subset of 68edo; 17edo is at the upper edge of [[gentle tuning]]. While 17edo has reasonable approximations of 7 and 11, they are not shared with 68edo, although their ratio 14/11 is.

The edostep is a near-just 25/24, sometimes seen as the optimal size for a leading tone.

=== Prime approximations ===
{{Harmonics in ED|17|23}}

=== Notation ===
Because a sharp is 2 steps in 17edo, it may be notated with neutral [[diatonic notation]] (semisharps and semiflats).

=== Tuning properties ===

== Scales ==
Some scales:
* [[Diatonic#MOS diatonic|MOS diatonic]] (5L2s, 3331331)
* [[Mosh]] (3L4s, 2323232)
* Lissotonic (3223322)
* Trachytonic (3313322)

{{Navbox EDO}}

{{cat|Edos}}

Chthonic harmony

2026-08-31T16:43:15Z

Unque:

'''Chthonic harmony''', sometimes called '''semiquartal''' or '''latal harmony''' (''latal''/''latus'', pl. ''lati'', means "1-step of the 5-form", and is borrowed from Latin ''lātus'' (1st declension) "carried"), is a type of chord structure useful in many styles of xenharmonic music. It serves as an analog for tertian harmony that splits the [[perfect fourth]] rather than the [[Perfect fifth|fifth]]. Thus, a chord can be constructed with the chthonic interval serving the same intrachord function as the third would in tertian harmony.

== Chthonic intervals ==
The term "chthonic" is derived from the name of an [[interordinal]] interval which splits the perfect fourth into two roughly-equal parts. When this division is not equal, one may call the larger chthonic "major" and the smaller one "minor," by analog to diatonic ordinals; when it is equal or nearly so, such as the case with [[15/13]], we may call it a "neutral" or "perfect" chthonic. Where further types of chthonics exist, one may use [[ups and downs]] alteration (in an EDO) or [[ADIN]] (in any tuning) to describe the qualities.

=== In just intonation ===
In JI structures, chthonic intervals will usually have either a numerator or denominator divisible by 3, which creates a relatively concordant triad bounded by the perfect fourth.
{| class="wikitable"
|+Chthonic intervals in JI
!Ratio
!Cents
!ADIN
!Triad
!Complement
|-
|15/13
|247.7
|Neutral/Perfect
|39:45:52
|52/45
|-
|7/6
|266.9
|Submajor
|6:7:8
|8/7
|-
|32/27
|294.1
|Nearmajor
|27:32:36
|9/8
|-
|6/5
|315.6
|Farmajor
|15:18:20
|10/9
|-
|11/9
|347.4
|Ultramajor
|9:11:12
|12/11
|}

== On notation and stacking ==
Diatonic ordinals are typically notated by a symbol (P, M, m, etc.) to display its quality, plus a 1-indexed number (5, 3, 7, etc.) to display its place in the scale. The quality symbols can be maintained for describing interordinals. However, because these numbers are conventionally integers and notation has developed around that, it is difficult to slot interordinal intervals in between.

One solution uses decimal points (or equivalently, the symbol ½) and allows for intuitive placement of the intervals without much memorization. However, this comes with a number of issues when applied to staff notation, primarily with figured bass (where slashes already have a meaning and decimal points could be confused for two separate entries) and other aspects of notation where elements would need to be widened to fit a two- or three-character symbol, or symbols other than single-digit whole numbers could be ambiguous, such as reading and writing chords, or notating scale degrees.

An alternate notation system used on this page solves the issues with decimal notation, and is intended to be compatible with the constraints of figured bass and other conventional notations and avoid ambiguity by making all symbols single characters. It uses an alternate set of ordered symbols, specifically the Greek alphabet, to represent ordinals and interordinals together, based on the degrees (1-indexed) of the 14-form; odd-numbered degrees will therefore be equivalent to ordinals of the 7-form, and as such can be notated with either Greek or Arabic numerals. One potential problem with this notation, however, is that the ordering of the intervals is not immediately obvious to someone who does not know the Greek alphabet and may require memorization.

Just as there is a clear circle created by a stack of fifths or thirds, so too is there one created by a stack of chthonics; two chthonics will create some type of fourth, three will create a [[cocytic]] (fifth-inter-sixth), four will create a seventh, etc.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+Interordinal notation
!Chthonics up
!Interval
!Arabic
!Greek
!14-form
|-
|0
|Unison
|1
|α
|0\14
|-
|1
|Chthonic
|2.5
|δ
|3\14
|-
|2
|Fourth
|4
|ζ
|6\14
|-
|3
|Cocytic
|5.5
|ι
|9\14
|-
|4
|Seventh
|7
|μ
|12\14
|-
|5
|Interizer†
|1.5
|β
|1\14
|-
|6
|Third
|3
|ε
|4\14
|-
|7
|Tritone
|4.5
|η
|7\14
|-
|8
|Sixth
|6
|κ
|10\14
|-
|9
|Antiïnterizer†
|7.5
|ν
|13\14
|-
|10
|Second
|2
|γ
|2\14
|-
|11
|Naiadic
|3.5
|ϛ
|5\14
|-
|12
|Fifth
|5
|θ
|8\14
|-
|13
|Ouranic
|6.5
|λ
|11\14
|}
†The term "interizer" is coined by [[User:Inthar|Inthar]], as it is the interval which separates a diatonic ordinal from an interordinal; thus it can be used to "inter-ize" those ordinals. Because it is also half of a diatonic minor second, it may also be considered a [[quartertone]], though this name may not be preferred, as four of them do not necessarily make a wholetone.

== Chthonic chords ==
Chords can be constructed most intuitively by alternating a stack of a chthonic with its fourth complement, just as tertian chords tend to follow an alternating pattern of a third with its fifth complement.

Where tertian harmony is most practical with chords of three and four notes, chthonic harmony is most practical with chords of four or five notes. Because a chthonic added to a fourth will always yield a cocytic, chords can be named by the type of chthonic and the type of cocytic, just as tertian chords can be named for the quality of their third and seventh. Unless otherwise specified, the fourth is always assumed to be perfect, and the seventh minor.

To display how these qualities propagate, this example will use only two types of chthonics.
{| class="wikitable"
|+Cocytic (four-note) Chords
!Quality
!Intervals (Greek)
!Intervals (Decimal)
!Formula (δ = 2½)
|-
|Major Cocytic
|Mδ, Pζ, Mι
|M2½, P4, M5½
|Mδ + mδ + Mδ
|-
|Minor Cocytic
|mδ, Pζ, mι
|m2½, P4, m5½
|mδ + Mδ + mδ
|-
|Major Minor Cocytic
|Mδ, Pζ, mι
|M2½, P4, m5½
|Mδ + mδ + mδ
|-
|Minor Major Cocytic
|mδ, Pζ, Mι
|m2½, P4, M5½
|mδ + Mδ + Mδ
|-
|Half-Augmented Cocytic
|Mδ, Aζ, Mι
|M2½, A4, M5½
|Mδ + Mδ + mδ
|-
|Augmented Cocytic
|Mδ, Aζ, Aι
|M2½, A4, A5½
|Mδ + Mδ + Mδ
|-
|Half-Diminished Cocytic
|mδ, dζ, mι
|m2½, d4, m5½
|mδ + mδ + Mδ
|-
|Diminished Cocytic
|mδ, dζ, dι
|m2½, d4, d5½
|mδ + mδ + mδ
|}
== Chthonic scales ==
The scales which most prominently feature chthonic harmony are, as one might suspect, those which are generated by chthonic intervals, just as the diatonic scale can be generated by an alternating stack of major and minor thirds. Also just like the diatonic scale, many chthonic scales have alternative constructions as well, analogous to the diatonic circle of fifths construction.

=== Diasem ===
''See also: [[Aberrisma#Quasi-diatonic aberrismic scales|Quasi-diatonic aberrismic scales]]''

The most fundamental of chthonic scales is Diasem, a 9-note scale created by alternating a stack of major and minor chthonic intervals, such as 7/6 with 8/7. This construction ensures a maximum number of major and minor cocytic chords.

Diasem has two possible chiralities, based on whether the major or minor chthonic was stacked first; each chirality has nine unique modes.
{| class="wikitable sortable"
|+Left-Hand Diasem Modes
! rowspan="2" |Pattern
! colspan="8" |Degrees
! rowspan="2" |Cocytic Chord
|-
!1\9
!2\9
!3\9
!4\9
!5\9
!6\9
!7\9
!8\9
|-
|LsLmLsLmL
|9/8
|8/7
|9/7
|4/3
|3/2
|32/21
|12/7
|16/9
|Minor
|-
|sLmLsLmLL
|64/63
|8/7
|32/27
|4/3
|256/189
|32/21
|128/81
|16/9
|Minor
|-
|LmLsLmLLs
|9/8
|7/6
|21/16
|4/3
|3/2
|14/9
|7/4
|63/32
|Major
|-
|mLsLmLLsL
|28/27
|7/6
|32/27
|4/3
|112/81
|14/9
|7/4
|16/9
|Major
|-
|LsLmLLsLm
|9/8
|8/7
|9/7
|4/3
|3/2
|27/16
|12/7
|27/14
|Minor (wolf cocytic)
|-
|sLmLLsLmL
|64/63
|8/7
|32/27
|4/3
|3/2
|32/21
|12/7
|16/9
|Minor
|-
|LmLLsLmLs
|9/8
|7/6
|21/16
|189/128
|3/2
|27/16
|7/4
|63/32
|Half-Augmented (wolf cocytic)
|-
|mLLsLmLsL
|28/27
|7/6
|21/16
|4/3
|3/2
|14/9
|7/4
|16/9
|Major
|-
|LLsLmLsLm
|9/8
|81/64
|9/7
|81/56
|3/2
|27/16
|12/7
|27/14
|Wolf Major
|}
{| class="wikitable sortable"
|+Right-Hand Diasem Modes
! rowspan="2" |Pattern
! colspan="8" |Degrees
! rowspan="2" |Cocytic Chord
|-
!1\9
!2\9
!3\9
!4\9
!5\9
!6\9
!7\9
!8\9
|-
|LmLsLmLsL
|9/8
|7/6
|21/16
|4/3
|3/2
|14/9
|7/4
|16/9
|Major
|-
|mLsLmLsLL
|28/27
|7/6
|32/27
|4/3
|112/81
|14/9
|128/81
|16/9
|Major
|-
|LsLmLsLLm
|9/8
|8/7
|9/7
|4/3
|3/2
|32/21
|12/7
|27/14
|Minor
|-
|sLmLsLLmL
|64/63
|8/7
|32/27
|4/3
|256/189
|32/21
|12/7
|16/9
|Minor
|-
|LmLsLLmLs
|9/8
|7/6
|21/16
|4/3
|3/2
|27/16
|7/4
|63/32
|Major (wolf cocytic)
|-
|mLsLLmLsL
|28/27
|7/6
|32/27
|4/3
|3/2
|14/9
|7/4
|16/9
|Major
|-
|LsLLmLsLm
|9/8
|8/7
|9/7
|81/56
|3/2
|27/16
|12/7
|27/14
|Minor (wolf fourth)
|-
|sLLmLsLmL
|64/63
|8/7
|9/7
|4/3
|3/2
|32/21
|12/7
|16/9
|Minor
|-
|LLmLsLmLs
|9/8
|81/64
|21/16
|189/128
|3/2
|27/16
|7/4
|63/32
|Wolf Half-Augmented
|}

=== Superdiatonic ===
One may note that the Diasem scale pattern produces an imperfect "wolf" interval on each step, which creates an awkward and discordant quality in some of the chords. One potential way to fix this issue is by equating the large wolf chthonic with the regular major chthonic; this may be seen as analogous to how Meantone temperament equates the wolf intervals of the Zarlino pattern with the more frequent forms of the ordinals. This is at the cost of severely sharpening both the major chthonic and the fourth.

This scale is sometimes known as "armotonic," or more often "superdiatonic," as it can be constructed by pentachords of LLLs analogous to the diatonic scale's construction by [[Tetrachord|tetrachords]] of LLs.
{| class="wikitable sortable"
|+Modes of Superdiatonic
! rowspan="2" |Pattern
! rowspan="2" |Brightness
! colspan="8" |Degrees
! rowspan="2" |Cocytic Chord
|-
!1\9
!2\9
!3\9
!4\9
!5\9
!6\9
!7\9
!8\9
|-
|LLLsLLLLs
| +3
|Maj.
|Maj.
|Maj.
|Perf.
|Perf.
|Maj.
|Maj.
|Maj.
|Major
|-
|LLsLLLLsL
| +1
|Maj.
|Maj.
|Min.
|Perf.
|Perf.
|Maj.
|Maj.
|Min.
|Major
|-
|LsLLLLsLL
| -1
|Maj.
|Min.
|Min.
|Perf.
|Perf.
|Maj.
|Min.
|Min.
|Minmaj
|-
|sLLLLsLLL
| -3
|Min.
|Min.
|Min.
|Perf.
|Perf.
|Min.
|Min.
|Min.
|Minor
|-
|LLLLsLLLs
| +4
|Maj.
|Maj.
|Maj.
|Aug.
|Perf.
|Maj.
|Maj.
|Maj.
|Half-Augmented
|-
|LLLsLLLsL
| +2
|Maj.
|Maj.
|Maj.
|Perf.
|Perf.
|Maj.
|Maj.
|Min.
|Major
|-
|LLsLLLsLL
|0
|Maj.
|Maj.
|Min.
|Perf.
|Perf.
|Maj.
|Min.
|Min.
|Major
|-
|LsLLLsLLL
| -2
|Maj.
|Min.
|Min.
|Perf.
|Perf.
|Min.
|Min.
|Min.
|Minor
|-
|sLLLsLLLL
| -4
|Min.
|Min.
|Min.
|Perf.
|Dim.
|Min.
|Min.
|Min.
|Minor
|}

=== Semiquartal ===
Another way to simplify the structure of Diasem is to use a single type of neutral/perfect chthonic rather than the two alternating major and minor chthonics. As such, the chord qualities need not be specified for this scale.

'''Todo: table'''

=== Chthonic chromatic ===
The Chthonic analog to the chromatic scale can be made by continuing the stack of neutral chthonics until you reach a scale of 14 notes. This scale has the pattern 5L 9s; because the step size is the interizer, the 14 notes of the scale can be considered to represent the seven diatonic ordinals plus the seven interordinal intervals. As a result, the degrees of this scale correspond exactly to the fourteen Greek numerals, and provide "canonical" representations for the major and minor qualities of each degree given a tuning system which contains this scale, just as diatonic provides such a representation for the major and minor qualities of the 7-form degrees.

For reference, the corresponding diatonic modes will also be noted here; every diatonic mode corresponds to two chthonic chromatic modes.
{| class="wikitable sortable"
|+Modes of Chthonic Chromatic
! rowspan="2" |Pattern
! rowspan="2" |Brightness
! colspan="13" |Degrees
! rowspan="2" |Diamode
|-
!β (1.5)
!γ (2)
!δ (2.5)
!ε (3)
!ϛ (3.5)
!ζ (4)
!η (4.5)
!θ (5)
!ι (5.5)
!κ (6)
!λ (6.5)
!μ (7)
!ν (7.5)
|-
|LssLssLsLssLss
| +5
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Ionian
|-
|ssLssLsLssLssL
| -5
|Min.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Phrygian
|-
|sLssLsLssLssLs
| +1
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Dorian
|-
|LssLsLssLssLss
| +6
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Aug.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Lydian
|-
|ssLsLssLssLssL
| -4
|Min.
|Min.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Phrygian
|-
|sLsLssLssLssLs
| +2
|Min.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Mixolydian
|-
|LsLssLssLssLss
| +7
|Maj.
|Maj.
|Aug.
|Maj.
|Maj.
|Aug.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Lydian
|-
|sLssLssLssLssL
| -3
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Aeolian
|-
|LssLssLssLssLs
| +3
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Mixolydian
|-
|ssLssLssLssLsL
| -7
|Min.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Dim.
|Min.
|Min.
|Dim.
|Min.
|Min.
|Locrian
|-
|sLssLssLssLsLs
| -2
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Maj.
|Aeolian
|-
|LssLssLssLsLss
| +4
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Maj.
|Maj.
|Ionian
|-
|ssLssLssLsLssL
| -6
|Min.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Dim.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Locrian
|-
|sLssLssLsLssLs
| -1
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Dorian
|}
''todo: pajara''

Semiquartal

2026-08-31T16:30:46Z

Unque: Deploying new navbox

'''Semiquartal''', also known as 5L 4s, is a [[MOS]] pattern consisting of 9 steps, with pattern LLsLsLsLs. It is supported by [[14edo]], [[19edo]], and [[24edo]], among others. Like [[Mosh|mosh]], semiquartal contains many intervals almost exactly between 12edo intervals, which improves [[Xenness|xenness]]. Unlike mosh, semiquartal triads have vaguely major or minor qualities.
{| class="wikitable"
|+
!Generator
!Notes
|-
|1\5
|Collapsed.
|-
|<nowiki>||||||| 9\44</nowiki>
|Notable archy. Better as a larger scale.
|-
|<nowiki>|||||| 8\39</nowiki>
|Notable archy. Better as a larger scale.
|-
|<nowiki>||||| 7\34</nowiki>
|Good approximation of Immunity temperament.
|-
|<nowiki>|||| 6\29</nowiki>
|
|-
|<nowiki>||||| 11\53</nowiki>
|
|-
|<nowiki>||| 5\24</nowiki>
|Often acceptable approximation of [[Intergan]] by a smaller edo.
|-
|<nowiki>||||| 14\67</nowiki>
|Nearly optimal [[Intergan]].
|-
|<nowiki>|||| 9\43</nowiki>
|Often acceptable approximation of [[Intergan]] with a notable meantone.
|-
|<nowiki>||||| 13\62</nowiki>
|Notable meantone.
|-
|<nowiki>|| 4\19</nowiki>
|Smallest semiquartal edo to generate a [[diatonic fifth]]. A passable representation of meantone-range semiquartal. The [[patent val]] makes Semaphore temperament viable.
|-
|<nowiki>||| 7\33</nowiki>
|Semisixth is very close to 9/7.
|-
|<nowiki>| 3\14</nowiki>
|
|-
|<nowiki>|| 5\23</nowiki>
|Good approximation of Bug temperament.
|-
|2\9
|Equalized.
|}

== See also ==
* [[Interordinal]]
* [[Chthonic harmony]]
* [[Semifourth-generated scales]]
{{Stub}}{{Navbox MOS}}

Tcherepnin

2026-08-31T16:30:08Z

Unque: Deploying new navbox

The '''Tcherepnin''' scale, or '''3L 6s''', is a common structural scale which repeats at one third of the octave; as such, it is often associated with the [[Augmented (temperament)|Augmented]] temperament which splits the octave into three major thirds. The scale is named after the music theorist Alexander Tcherepnin, who utilized its 12edo tuning to create a double-tonal sound with two different types of thirds; the pattern was also noted by Olivier Messiaen, who listed it as the third scale in his list of [[Glossary#Mode of Limited Transposition|modes of limited transposition]].

Because the Tcherepnin scale was popularized by progressive 20th century musicians such as Allan Holdsworth and Toru Takemitsu, a great deal of composition theory can be attained through analysis of these pieces, which can be applied to composition in xenharmonic systems which include it.

== General scale theory ==
The Tcherepnin scale is a mode of limited transposition, which means that certain rotations of this scale are identical to one another; while the scale has nine notes per octave, there are only three unique rotations: a bright mode, a dark mode, and a symmetrical mode. The symmetrical mode was the one primarily used by Tcherepnin, and the bright mode was the one documented by Messiaen.
{| class="wikitable"
|+Modes of 3L 6s
!Mode
!Pattern
!Degree Qualities
|-
|Dark
|ssLssLssL
|mmPmmPmmP
|-
|Symmetrical
|sLssLssLs
|mMPmMPmMP
|-
| Bright
|LssLssLss
|MMPMMPMMP
|}

=== Degrees, intervals, and notation ===
The degrees and intervals of the Tcherepnin scale can be classed in two main ways: diamond MOS notation, where ordinal numbers correspond to the precise number of steps, or quasi-diatonic notation, where interval names are based on diatonic categories with the unchanging 400c and 800c intervals being dubbed the naiadic and cocytic degrees. The diamond MOS system yields more convenient methods of interval arithmetic in a vacuum, while the quasi-diatonic system yields familiar categories and preserves octave complements and chord constructions from diatonic music.

Degrees can be said to tend bright or dark based on whether their quality in the symmetrical mode is equivalent to that in the bright mode or the dark mode; in quasi-diatonic notation, odd numbers tend bright while even numbers tend dark, and the degrees labeled with Greek symbols do not have more than one quality.

The interval just below the cocytic (the "major sixth" in Tcherepnin ordinals, or "major fifth" in quasidiatonic ordinals) clusters around the size of a [[Pentic|pentatonic]] generator, or an [[antipentic]] generator in hard tunings; 15edo, with a step size ratio of 3:1, represents the midpoint where the fifth generates 5edo. With this in mind, the intervals and degrees of the Tcherepnin scale can be compared to those of the scales that this fifth yields.
{| class="wikitable"
|+Tcherepnin vs. Diatonic
!Degree
(Quasi)
!Degree
(Tcher)
!Quality
!Cent Range
!Diatonic
|-
| rowspan="2" |2
| rowspan="2" |1-step
|minor
|0 - 133
|m2
|-
|major
|133 - 400
|M2
|-
| rowspan="2" |3
| rowspan="2" |2-step
|minor
|0 - 266
|d3
|-
|major
|266 - 400
|m3
|-
|ς
|3-step
|perfect
|400
|M3~d4
|-
| rowspan="2" |4
| rowspan="2" |4-step
|minor
|400 - 533
|P4
|-
|major
|533 - 800
|A4
|-
| rowspan="2" |5
| rowspan="2" |5-step
|minor
|400 - 666
|d5
|-
|major
|666 - 800
|P5
|-
|ι
|6-step
|perfect
|800
|A5~m6
|-
| rowspan="2" |6
| rowspan="2" |7-step
|minor
|800 - 933
|M6
|-
|major
|933 - 1200
|A6
|-
| rowspan="2" |7
| rowspan="2" |8-step
|minor
|800 - 1066
|m7
|-
|major
|1066 - 1200
|M7
|}
Where one of the two notations must be chosen to avoid ambiguity, this page will prefer Quasi-diatonic notation. When quality must be specified for a degree, this page will use ♯/♭ to represent raising or lowering by a MOS chroma, and the symmetrical mode will be taken as the default qualities. Intervals between two pitches will use M or m with the Arabic numerals, or P with the Greek symbols, to specify in-scale qualities.

== Chords and tonality ==
Alexander Tcherepnin considered the most characteristic part of the scale to be its separation of the diatonic minor and major third onto different degrees, allowing both to be reached within the same mode. If we consider the 3 and ς degrees to be the main building blocks of chords, there are four types of triad constructions: chthonics (quasi-diatonic degrees 1 - 3 - 4), minor (degrees 1 - 3 - 5), major (1 - ς - 5), and augmented (1 - ς - ι). This leads to eight total chords, with three qualities of chthonic, two qualities of minor, two qualities of major, and one quality of augmented.

Each degree of the scale has one of each type of triad:
{| class="wikitable"
|+Modes and chords
!Mode
!Chthonic
!Minor
!Major
!Augmented
|-
|Bright
|M3, M4
| rowspan="2" |M3, M5
| rowspan="2" |Pς, M5
| rowspan="3" |Pς, Pι
|-
|Symmetrical
|M3, m4
|-
|Dark
|m3, m4
|m3, m5
|Pς, m5
|}
Where the qualities of these triads need be distinguished, they can be described with their characteristic mode and construction type, such as "bright major" for {Pς, M5} or "dark chthonic" for m3, m4.

It is notable that in 12edo tuning, the bright chthonic triad is enharmonically equivalent to the diminished chord from the diatonic scale. Many contemporary musicians who use the Tcherepnin scale take advantage of this fact, and curious xen composers might treat their usage of it as a model for how to fit chthonic chords and diatonic-like tertian chords into the same harmonic framework.

=== Harmonic systems ===
There are several possibilities for establishing a tonal framework in the Tcherepnin scale, based both on analysis of usage of the scale in music both xen and otherwise, and on applying fundamental principles to the scale as a self-standing system.

A common thread among contemporary treatment of the scale is how to work around the limited transposition; the two major types of approaches to this are to establish an [[axis system]] where there are multiple tonics, or to establish a modal system where there is no tonic at all.

==== Tcherepnin's axis system ====
An axis system is a system whereby functions repeat at some "axis" that subdivides the octave. In the case of the Tcherepnin scale, the axis occurs between degrees 1, ς, and ι, making them all tonics under this analysis; this is the primary framework seen in the works of Alexander Tcherepnin himself.

Because the m4 and M5 degrees resemble a diatonic generator, a significant amount of diatonic harmony can be transferred to this scale; Alexander Tcherepnin himself took advantage of this similarity, using the major triads on the 4 and 5 degrees for the functions of diatonic IV and V. Thus degrees 1, ς, and ι have tonic functions; 2, 4, and 6 have subdominant functions; and degrees 3, 5, and 7 have dominant functions. Note also that these degree functions correspond to brightness tendencies, with tonic degrees being the same in all modes, subdominant degrees tending dark, and dominant degrees tending bright.

Tcherepnin primarily employed major and minor chords in his axis-based harmony, with augmented chords being occasionally used, and chthonics only consisting of the bright quality; however, all eight chord types can be applied to this system. Note that chthonic chords, the main type of chord that Tcherepnin did not use, necessarily contain one degree for each of the functions; for instance, the chthonic chord on the 1 degree contains 1 (a tonic), 3 (a dominant), and 4 (a subdominant). This tonal ambiguity makes it a very useful tension with a plethora of resolution options, acting as a nexus that is limited only by voice leading.

==== Holdsworth's harmonic system ====
Jazz guitarist Allan Holdsworth made usage of the Tcherepnin scale in several pieces, and used unconventional constructions of chords to create a flowing modal sound. Unlike Tcherepnin's axis system, however, Holdsworth prioritized the qualities and relative intervals within the chords rather than the absolute degrees which they contained. Because the limited transposition provides a very unclear sense of tonality, Holdsworth's approach appears to favor tension and release paradigms that explore the scale modally, rather than establishing any clear center of gravity.

{{Navbox MOS}}

Oneirotonic

2026-08-31T16:29:43Z

Unque: Deploying new navbox

'''Oneirotonic''' is the 5L3s [[MOS]] pattern LLsLLsLs. Its generator ranges from equalized [[8edo|3\8]] (450c) to collapsed 2\5 (480c) and the generator's basic tuning (L/s = 2/1) is [[13edo|5\13]] (461.5c). It is notable for being a compressed diatonic with one extra small step.

The basic idea of oneirotonic is sharpening the fifth beyond 5edo: the fifth becomes so sharp that two of them make a subminor/flat minor third and one more note is required to form a MOS.
== Notation ==
This article uses KISS notation, where the Celephaisian mode is 123456781. Chroma up is denoted #, chroma down as b. Alternatively, there is ground's oneirotonic notation, which preserves the diatonic order of letter names when stacking fourths (with one extra nominal): BEADGCFX, again with the Celephaisian mode as ACBDFEGXA. Note that C and B are swapped, and F and E are also swapped.

== Structural theory ==
In the all-natural Celephaisian mode, the generators are arranged in the order 3-6-1-4-7-2-5-8.
=== Modes ===
The mode names were originally given by Cryptic Ruse, but they have disavowed them since.
{| class="wikitable"
|-
! Mode name !!Gens up !! Pattern !! 1-step !! 2-step !! 3-step !! 4-step !! 5-step !! 6-step
!7-step
|-
| Sarnathian || 0|| sLsLLsLL || style="background-color:#d66" |m || style="background-color:#d66" |m || style="background-color:#d66" |d || style="background-color:#d66" |m || style="background-color:#d66" |P || style="background-color:#d66" |m || style="background-color:#d66" |m
|-
|Hlanithian
|1
|sLLsLsLL
| style="background-color:#d66"|m
| style="background-color:#d66"|m
| style="background-color:#66d"|P
| style="background-color:#d66"|m
| style="background-color:#d66"|P
| style="background-color:#d66"|m
| style="background-color:#d66"|m
|-
| Kadathian
|2
| sLLsLLsL
| style="background-color:#d66" |m
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#d66" |m
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |m
|-
| Mnarian
|3|| LsLsLLsL
| style="background-color:#66d" |M
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#d66" |m
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |m
|-
| Ultharian
|4|| LsLLsLsL
| style="background-color:#66d" |M
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |m
|-
| Celephaisian
|5|| LsLLsLLs
| style="background-color:#66d" |M
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#66d" |M
|-
| Ilarnekian
|6|| LLsLsLLs
| style="background-color:#66d" |M
| style="background-color:#66d" |M
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#66d" |M
|-
| Dylathian
|7|| LLsLLsLs
| style="background-color:#66d" |M
| style="background-color:#66d" |M
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#66d" |A
| style="background-color:#66d" |M
| style="background-color:#66d" |M
|}

=== Modes of melodic Mnarian ===
The ''melodic Mnarian'' (LsLsLLLs ascending, LsLsLsLL descending) scale has an "LLL" and recreates the characteristic of diatonic modes such as Phrygian and Locrian which have LLL prominently. It's the unique binary MODMOS of oneirotonic that doesn't have consecutive s steps. The mode names of melodic Mnarian are portmanteaus of the oneirotonic mode and the diatonic mode the mode sounds most similar to.
{| class="wikitable"
|-
! Mode name !! Pattern !! 1-step !! 2-step !! 3-step !! 4-step !! 5-step !! 6-step
!7-step
|-
| Sarlocrian || sLsLsLLL
| style="background-color:#d66" |m
| style="background-color:#d66" |m
| style="background-color:#d66" |d
| style="background-color:#d66" |m
| style="background-color:#e33" |d
| style="background-color:#d66" |m
| style="background-color:#d66" |m
|-
| Sardorian ||sLsLLLsL
| style="background-color:#d66"|m
| style="background-color:#d66"|m
| style="background-color:#d66"|d
| style="background-color:#d66"|m
| style="background-color:#d66"|P
| style="background-color:#66d"|M
| style="background-color:#d66"|m
|-
| Mnaeolian || LsLsLsLL
| style="background-color:#66d" |M
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#d66" |m
| style="background-color:#d66" |P
| style="background-color:#d66" |m
| style="background-color:#d66" |m
|-
| Mnionian || LsLsLLLs
| style="background-color:#66d" |M
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#d66" |m
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#66d" |M
|-
| Ulphrygian || sLLLsLsL
| style="background-color:#d66" |m
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |m
|-
| Celdorian || LsLLLsLs
| style="background-color:#66d" |M
| style="background-color:#d66" |m
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#66d" |A
| style="background-color:#66d" |M
| style="background-color:#66d" |M
|-
| Ilarmixian || LLsLsLsL
| style="background-color:#66d" |M
| style="background-color:#66d" |M
| style="background-color:#66d" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |P
| style="background-color:#66d" |M
| style="background-color:#d66" |m
|-
| Dylydian || LLLsLsLs
| style="background-color:#66d" |M
| style="background-color:#66d" |M
| style="background-color:#33e" |A
| style="background-color:#66d" |M
| style="background-color:#66d" |A
| style="background-color:#66d" |M
| style="background-color:#66d" |M
|}

== Notable tunings and tuning ranges ==
* 37edo (hardness 5/4) to 29edo (hardness 4/3): support the 2.7/5.11/5.13/5 temperament [[Ammonite]] where the generator is 13/10, the large step is 11/10, and the minor 4-step is 7/5.
* 21edo (hardness 3/2)
* 13edo (hardness 2/1)
* 31edo (hardness 5/2): [[A-Team]] tuning with an excellent 5/4. Approximates 13:17:19.
* 18edo (hardness 3/1): Minor 2-step is an excellent 7/6; 12edo whole tone. Practically optimal for 2.9.21 A-Team.
* 23edo (hardness 4/1): Supports A-Team with diminished 3-step = ~6/5.
* 28edo (hardness 5/1) and harder support 2.3.7 [[Buzzard]] which equates four perfect 3-oneirosteps with a 3/1.

== Chords of oneirotonic ==
{{proposed}}
These chord names have been proposed by [[User:Ground|ground]] and [[User:Inthar|Inthar]].
The names have been selected to avoid overloading diatonic chord names and symbols.
=== Fifth-bounded triads ===
The prefix "tract" is used to denote compressions of diatonic chords, and symbolically, <code>>...<</code> is used to denote tract-diatonic chords.

First and second inversions of triads are denoted <code>triad₁</code> and <code>triad₂</code> in chord symbols, for example <code>>maj<₁</code> = the first inversion of the tract-major triad.
{| class="wikitable"
|+ Perfect fifth (sharp fifth) bounded triads
|-
!|Name (notation)
!|Description
!|In degrees (TAMNAMS)
!|13edo tuning
!Listen
!|18edo tuning
!Listen
|-
!|Supertaphric (<code>suptph</code>)
||
||0–m4d–P5d
||0–554–738
|[[File:Oneirodemossixeightofthirteen.wav|thumb]]
||0–533–733
|[[File:Oneirodemoszeroeightelevenundereighteen.wav|thumb]]
|-
!|Taphric (<code>tph</code>)
||From τάφρος "trench", because the fifth generators go down–up.
||0–P3d–P5d
||0–462–738
|[[File:Oneirodemothirteenfiveeight.wav|thumb]]
||0–467–733
|[[File:Oneirodemoszerosevenelevenundereighteen.wav|thumb]]
|-
!|Subtaphric (<code>subtph</code>)
||
||0–M2d–P5d
||0–369–738
|
||0–400–733
|[[File:Oneirodemoszerosixelevenundereighteen.wav|thumb]]
|-
!|Neutral (<code>neu</code>)
||Splits the sharp fifth in half.
||
||0–369–738
|[[File:Oneirodemothirteenffoureight.wav|thumb]]
||
|
|-
!|Suprasimic (<code>supsim</code>)
||
||0–d3d–P5d
||0–369–738
|
||0–333–733
|[[File:Oneirodemoszerofiveelevenundereighteen.wav|thumb]]
|-
!|Simic (<code>sim</code>)
||From Modern Greek σημείο "point", because the fifth generators go up–down.
||0–m2d–P5d
||0–277–738
|[[File:Oneirodemothirteenthreeeight.wav|thumb]]
||0–267–733
|[[File:Oneirodemoszerofoureleven.wav|thumb]]
|-
!|Subsimic (<code>subsim</code>)
||
||0–M1d–P5d
||0–185–738
|[[File:Oneirodemothirteentwoeight.wav|thumb]]
||0–200–733
|[[File:Oneirodemoszerothreeeleven.wav|thumb]]
|}

{| class="wikitable"
|+ Major tritone (flat fifth) bounded triads
|-
!|Name (notation)
!|Description
!|In degrees (TAMNAMS)
!|13edo tuning
!Listen
!|18edo tuning
!Listen
|-
!|Tract-sus4 (<code>>sus4<</code>)
||
||0–P3d–M4d
||0–462–646
|[[File:Oneirodemothirteenfiveseven.wav|thumb]]
||0–467–667
|[[File:Oneirodemosseventenofeighteen.wav|thumb]]
|-
!|Tract-major (<code>>maj<</code>)
||
||0–M2d–M4d
||0–369–646
|[[File:Oneirodemothirteenfourseven.wav|thumb]]
||0–400–667
|[[File:Oneirodemossixtenofeighteen.wav|thumb]]
|-
!|Tract-neutral (<code>>neu<</code>)
||Splits the flat fifth in half.
||0–n2d–M4d
||
|
||0–333–667
|[[File:OneirodemosNEWfivesevenofeighteen.wav|thumb]]
|-
!|Tract-minor (<code>>min<</code>)
||
||0–m2d–M4d
||0–277–646
|[[File:Oneirodemonewthirteenthreeseven.wav|thumb]]
||0–267–667
|[[File:OneirodemosNEWfoursevenofeighteen.wav|thumb]]
|-
!|Tract-sus2 (<code>>sus2<</code>)
||
||0–M1d–M4d
||0–185–646
|[[File:Oneirodemothirteentwoseven.wav|thumb]]
||0–200–667
|[[File:Oneirodemosthreetenofeighteen.wav|thumb]]
|}

{{Cat|MOS patterns}}

=== Other triads ===
{| class="wikitable"
|+ Other triads
|-
!|Name (notation)
!|Description
!|In degrees (TAMNAMS)
!|13edo tuning
!Listen
!|18edo tuning
!Listen
|-
!|Grammic (<code>grm</code>)
||From "line", since the generators form a line.
||0-P3-m6
||0-462-923
|[[File:Oneirodemothirteensixth.wav|thumb]]
||0-467-933
|[[File:Oneirodemongrammic.wav|thumb]]
|}

=== Notes ===
Taphric, simic, and grammic are all antipentic terms but otherwise do not reflect specific tuning ranges. Subtaphric/suprasimic/etc. chords in grammic form can just be called inversions the same way diatonic chords are. Grammic may also be known as sensic in checkertonic.

== Jaimbee and Inthar's oneirotonic functional system ==
{{Proposed}}

Todo: convert to KISS notation

The following system has been developed by Jaimbee and Inthar. It is meant for 13edo and tunings close to 13edo or somewhat harder (like 31edo or 18edo). 18edo oneirotonic has a discordant >maj< chord and so has a more stable tonal center on minor keys.

For a DR-forward framework like this, prefer mellow timbres to bright ones to bring out the DR effect.
=== Note on symbols ===
Todo: use Ilarnekian as default mode and indicate alterations therefrom

Degrees are indicated with Arabic numerals (or "t" for tritone/4-mosstep) with hats with the corresponding intervals' qualities placed before them; they're just intervals with hats. In order:

<math>\hat{1}, \hat{2}, \hat{3}, \hat{4}, \hat{\mathrm{t}}, \hat{5}, \hat{6}, \hat{7}.</math>

For now, degrees are conflated with functions. This isn't a proper functional tonality notation — it doesn't specify the default key for unaltered functions — but it generalizes better to modal music. (While we could choose one of oneiromajor and oneirominor as the default tonality, there are arguments for both and we're choosing not to privilege one of them.)

When quality is not explicitly indicated, the quality is from the current mode being discussed. Degrees <math>\hat{2}</math> (second from root), <math>\hat{3}</math> (third from root), <math>\hat{\mathrm{t}}</math> (tritone from root), <math>\hat{6}</math> (sixth from root), and <math>\hat{7}</math> (seventh from root) may be written as follows to explicitly indicate the quality of the interval from the root: <math>\mathrm{m}\hat{2}, \mathrm{m}\hat{3}, \mathrm{m}\hat{\mathrm{t}}, \mathrm{m}\hat{6}, \mathrm{m}\hat{7}</math> for minor intervals from the root and <math>\mathrm{M}\hat{2}, \mathrm{M}\hat{3}, \mathrm{M}\hat{\mathrm{t}}, \mathrm{M}\hat{6}, \mathrm{M}\hat{7}</math> for major intervals from the root. Degrees <math>\hat{1}</math> (root), <math>\hat{4}</math> (fourth from root), and <math>\hat{5}</math> (fifth from root) may use <math>\mathrm{P}\hat{1}, \mathrm{P}\hat{4}, \mathrm{P}\hat{5}</math> for perfect, <math>\mathrm{A}\hat{1}, \mathrm{A}\hat{4}, \mathrm{A}\hat{5}</math> for augmented, and <math>\mathrm{d}\hat{1}, \mathrm{d}\hat{4}, \mathrm{d}\hat{5}</math> for diminished.

The chord symbols used are ground's system, described above.

=== Basic chords ===
The most basic chords in this functional harmony system are:
* Tract-major triad 0-4-7\13 (<code>>maj<</code>): A compressed major triad that sounds desaturated and somewhat bittersweet. Somewhat dubiously +1+1. Oneirotonic provides only two of these triads, so alterations are somewhat frequently used to get this triad. The tract-major triad has the following important tetrad supersets:
** 0-2-4-7\13 (<code>>majsus2<</code>): Reinforces the quasi-DR effect with an extra tone; approximately +1+1+2.
** 0-4-7-10\13 (<code>>dom7<</code>): A compressed dominant tetrad; approximately +1+?+1.
** 0-4-7-12\13 (<code>>maj<7</code>): Approximately +1+1+2.
* The simic triad 0-3-8\13 (<code>sim</code>): A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26 or +1+2. The important supersets are:
** 0-3-8-10 (<code>sim6</code>): Approximately +1+2+1.
** 0-3-8-12 (<code>simmaj7</code>): Approximately +1+2+2.
** 0-3-8-11 (<code>simmin7</code>): Something like a minor 7th tetrad.
** 0-3-8-15 (<code>simadd9</code>)
** 0-3-8-12-15 (<code>simmaj7add9</code>): A concatenation of the minor +1+2 and major +1+1 triads.
* 0-5-9\13 (<code>>IV/I<</code>): A +1+1 triad and a compressed 2nd inversion major triad. Approximately 13:17:21.
** 0-5-7-9: Approximately +2+1+1.
** 0-5-9-12: A tract-major triad on top of a subfourth.
** 0-5-9-12-15
** 0-5-7-9-12-15-17
* 0-5-7\13 (<code>>sus4<</code>): Compressed sus4. Approximately +2+1.
* 0-4-8\13 (<code>>aug<</code>): "Submajor augmented" triad.
* 0-3-6\13 (<code>>dim<</code>): The most diminished-like triad.

=== Functional patterns ===
{{UserTag|KC|Inthar|000000|13edo oneirotonic enjoys two main (rooted) delta-rational sonorities analogous to major and minor triads: 0-(185)-369-646 ("tract-major triad" or just ">maj<") and 0-277-738-923 ("simic sixth" or "sim6"). One of these chords are on the root in the 6 brightest modes of oneirotonic. In the two darkest modes, I think 0-277-738-1015 or 0-738-1015-277 works well. The chord 0-277-738 will be called "simic", and 0-277-646-923 will be called "tract-minor 7th".

If you wish to avoid the dissonance of the fifths, just using 10\13 (e.g. [0 4 10 (12)]\13, [0 3 10 (12)]\13) works well.}}

=== Progressions ===
{{UserTag|KC|Inthar|000000|Common motions:
* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{M}\hat{2}</math> sim6
* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{P}\hat{4}</math> sim6 (when ending on <math>\hat{1}</math> this sounds like diatonic V to I)
* <math>\hat{1}</math> (>maj< or sim6) → <math>\mathrm{m}\hat{6}</math> sim6
* <math>\hat{1}</math> >maj< → <math>\mathrm{P}\hat{5}</math> >maj<, <math>\hat{1}</math> sim6 → <math>\mathrm{P}\hat{5}</math> (>maj< or sim6) (when ending on <math>\hat{1}</math> this is a "dominant to tonic" motion)
* <math>\hat{1}</math> >maj< → <math>\mathrm{M}\hat{3}</math> >maj<
* <math>\mathrm{M}\hat{2}</math> sim6 → <math>\mathrm{m}\hat{2}</math> >maj< → <math>\hat{1}</math> >maj<
}}

=== Functional harmony ===
Like classical functional harmony, our functional harmony system is based on two tonalities:
# oneiromajor, with tonic chord tract-major and the pitches of Dylathian and Ilarnekian modes
# oneirominor, with tonic chord simic 6th and the pitches of Celephaisian and Ultharian modes
==== Oneiromajor (Dylathian) ====
{{UserTag|bz|Jaimbee|DAA520|In the below, interval names are in ADIN.

In Dylathian, we find tract-major chords on the <math>\hat{1}</math> and <math>\hat{4}</math> degrees, while simic chords appear on the degrees <math>\hat{2}, \hat{\mathrm{t}}, \hat{6}, \hat{7}.</math> For the <math>\hat{3}</math> and <math>\hat{5}</math> degrees, you get a chord of edo steps 0-3-9-11, which is the third type of DR tetrad, which could be viewed as an inversion of the tract-dominant tetrad. Alternatively, you could also play a DR chord of scale degrees 0-5-9 on the third degree, and in some contexts it may be favorable (see below).

For each of these chords, we can associate functions with them. The simplest of these relationships is between the root tract-major chord and the tract-major chord on the perfect fourth. By adding octaves on certain notes, we can recreate the familiar dominant cadence from diatonic, only now on the <math>\hat{4}</math> rather than the <math>\hat{5}.</math> The most simple of these progressions would look something like this (in ground's notation):
# B-D-F-G-F
# E-G-X-C-D
# B-D-F-G-F
Or, in 13edo steps:
# 0-2-4-7-17
# 5-7-9-12-18
#0-2-4-7-17
In this cadence, the fifth on E is so narrow that it creates a leading tone relative to the root, and by playing the octaves above E, you can create a minor tritone that wants to resolve inwards to the tract-major chord on the root. The presence of the octaves above the major third helps drive this resolution, but can be omitted. Another neat effect is that given the dominant is now on the <math>\hat{4},</math> then the <math>\hat{2}</math> simic sixth chord would be exactly halfway to the dominant, making it the mediant. It also has a much nicer simic sixth chord on it compared to the <math>\hat{3}</math> 1st inversion tract-major chord, making it more akin to how the mediant works in diatonic.

We can also relate other chords to the dominant, mediant and tonic. The relative minor is more or less exactly analogous to diatonic, being a minor third below the tonic (in the case of B Dylathian, it would be A Celephaïsian, the <math>\hat{6}</math> with the "minor" tonality). The mediant can also function as a secondary dominant for resolutions to the relative minor; the highest note in the <math>\hat{2}</math> minor chord is one semitone below the minor third of the <math>\hat{6}</math> minor chord. By playing the octave above certain notes, resolving between the two modes is pretty simple.
# B-D-F-G-B
# `D-`E-`A-`C-C
# `A-`B-`E-`G-B
In 13edo steps:
# 0-2-4-7-13
# `2-`5-`10-`12-12
#`10-0-5-7-13
The ` denotes playing an octave lower than the root.

The <math>\hat{2}</math> (D minor) and <math>\hat{6}</math> (A minor) would probably sound the smoothest when played in a lower register than the tonic (B major) as notated, but if you want to move upwards from the root it still works. Resolving from the relative minor (A minor) to the tonic (B major) is a pretty weak but still usable resolution. Another neat resolution is moving from the <math>\hat{3}</math> (inverted major) to the dominant <math>\hat{4}.</math>

If you play the 0-5-9 chord on <math>\hat{3},</math> the lowest note will be a semitone lower than the lowest in the dominant, and the highest note will be a semitone higher than the highest in the dominant. By either extending the 0-5-9 chord to 0-3-5-9, or simplifying the dominant chord to a 0-4-7 chord, you can drive this resolution very powerfully, and this could either create a chain of strong resolutions going <math>\hat{3}</math> (inverted major) - <math>\hat{4}</math> - <math>\hat{1},</math> or it could help drive resolutions to the Ilarnekian mode above (in this case, E Ilarnekian).

Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels somewhat empty. All in all, using this technique you could probably simplify all the tetrads down to 0-4-7 and 0-3-8 for major and minor, respectively. These would help since the 0-3-5-9 chord doesn't have much of a DR effect, while the simplified major and minor still have a DR effect, though a bit weaker than the tetrads. The <math>\hat{6}</math> inverted major (9-12-18-20) chord also has some neat features, as it functions as an inversion of the dominant <math>\hat{4}</math> chord. It also doesn't need any extensions with octaves to work well unlike the dominant chord, so it could be seen as a more tense version of dominant. Since it also drives the resolution up by a minor third, the same tetrad on the <math>\hat{3}</math> could be used to drive a resolution to a major <math>\hat{5},</math> helping to shift the key center from B to G#. If done twice, this resolution can shift your key center up a minor third from B -> G# -> F#, which gives the progression a really jazzy feel.

The only chord we haven't covered now would be the minor <math>\hat{\mathrm{t}}</math> (namely 7-10-15-17). This chord has a much weaker relationship to the other chords, so it doesn't have any strong directionality. However, it does share some notes with a few important chords, notably the <math>\hat{1}</math> chord and the relative minor on the <math>\hat{6}.</math> A resolution to either of these will be similarly strong, that is to say, not very strong. In this case it could also be seen as a secondary mediant which is not the relative minor, about halfway between the <math>\hat{1}</math> chord and the <math>\hat{7}</math> chord an octave above it, and either of these resolutions would probably sound fine in most contexts. This gives it a role completely unlike any of the functions in traditional diatonic. It also works pretty well as a setup for the <math>\hat{5}</math> inverted major, so in a progression it can help add some flair or beef to the resolution.
}}
==== Oneiromajor (Ilarnekian) ====
{{UserTag|bz|Jaimbee|DAA520|To start with the basics, Ilarnekian is just Dylathian with a flattened 6th.
In E Ilarnekian, you'd get:
E G X A C B D F E

With Ilarnekian being the second major mode (after Dylathian), we'd get the same <math>\hat{1}</math> chord, E>maj< One of the most immediate effects we'd see, however, is that the dominant <math>\hat{4}</math> is now IV>min7<. It still can function as a dominant, though only with the added octave above the 4th, and slightly weaker than the Dylathian dominant cadence. Another interesting thing is that the <math>\hat{4}-\hat{1}</math> cadence is now simultaneously a minor plagal and a dominant cadence, radically different from anything in diatonic.

The <math>\hat{2}</math> chord is again simic sixth, with it now driving a secondary resolution to D Ultharian instead of D Celephaïsian. The <math>\hat{3}</math> inverted major chord is also still first inversion tract-major, and also still drives a pretty good resolution up to the simic sixth <math>\hat{4},</math> though admittedly weaker than Dylathian.

The <math>\hat{\mathrm{t}}</math> chord is still simic sixth, and still functions as a secondary mediant.

It gets interesting again when looking at the tract-major <math>\hat{5}</math> chord.

By playing the <math>\hat{5}</math> lower than the tonic and playing the octave above the root of the <math>\hat{5}</math> chord, you get an entirely new approach to a dominant chord. The third of the <math>\hat{5}</math> is one semitone below the tonic, and the octave above the root is one semitone above the fifth of the tonic chord. This drives a very strong inward resolution that resembles dominant in diatonic slightly more than the dominant <math>\hat{4}</math> chord in Dylathian, and a lot more than the tract-minor <math>\hat{4}</math> chord in Ilarnekian.
It would look something like this:
* E G X C
* `B `D `F G B
* E G X C

In 13edo steps:
* 0-2-4-7
* `8-`10-`12-2-8
*0-2-4-7
with ` again notating playing an octave lower than the starting chord.

Moving on, the <math>\hat{6}</math> chord would be simic sixth driving the resolution to the <math>\hat{4}</math> simic sixth, in the same way Dylathian's <math>\hat{2}</math> simic sixth drives the resolution to the relative minor. It would also function as the Ilarnekian relative minor, in this case D Ultharian. The <math>\hat{7}</math> chord would be an inverted major chord, and would drive a resolution to the tonic pretty well. This comes from the fact that the 0-3-9-11 chord would have the minor third become the major second of the tonic, the root move up a semitone to the tonic, and the perfect fifth move down a semitone to become the tritone of the tonic chord. The minor sixth in the chord could be omitted to make the resolution stronger, but the chord would sound much more dissonant.
}}
==== Oneirominor ====
{{UserTag|KC|Inthar|000000|The pitches of the oneirominor tonality are usually that of the Celephaisian mode, but the 7 degree may be either <math>\mathrm{m}\hat 7</math> (for its relationship to other degrees) or <math>\mathrm{M}\hat 7</math> (mainly but not exclusively for a leading tone function).

The degrees/functions of oneirominor are listed below in generator order. The Celephaisian mode is assumed for purposes of notation.
}}
===== Degree 3 =====
{{UserTag|KC|Inthar|000000|Similar to diatonic, degree 3 serves as the "relative major".

Alternatively you can use degree 3 to set up the 5 to 1 cadence using one of the following progressions:
* <math>\hat{3}</math>>maj< <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim
* <math>\hat{3}</math>>maj< <math>\hat 4</math>sim <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim
}}
===== Degree 5 =====
{{UserTag|KC|Inthar|000000|Degree 5, which has a major third on it, has the strongest tendency to go to degree 1 in oneirominor. Alternatively serves as degree 4 of the relative oneiromajor on degree 3, serving to tonicize the relative oneiromajor.

The main cadence in oneirominor is <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim or <math>\hat{5}</math>>dom7< <math>\hat{1}</math>sim because of the leading tone.
}}
===== Degree 1 =====
{{UserTag|KC|Inthar|000000|The tonic.}}
===== Degree 4 =====
{{UserTag|KC|Inthar|000000|<math>\hat{4}</math>sim <math>\hat{1}</math>sim is a weaker cadence than <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim, in fact weaker than iv -> i in diatonic because there is no descending semitone to the fifth. For a stronger resolution <math>\hat{4}</math>sim <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim is preferable.}}

===== Degree 6 =====
{{UserTag|KC|Inthar|000000|Degree 6 is an important function in oneirominor with no analogue in diatonic. It can be used in the repeating <math>\hat 1</math>sim6 -> <math>\hat 6</math>sim6 -> <math>\hat 1</math>sim6 -> <math>\hat 6</math>sim6 -> ... vamp. It can also be used in conjunction with other degrees that are closely related:
* <math>\hat 4</math>sim <math>\hat 6</math>sim or <math>\hat 6</math>sim <math>\hat 4</math>sim
* <math>\hat 2</math>sim <math>\hat 6</math>sim or <math>\hat 6</math>sim <math>\hat 2</math>sim
}}
===== Degree 2 =====
{{UserTag|KC|Inthar|000000|Similar to degree 3, degree 2 can be used to set up the 5 to 1 cadence:

<math>\hat{2}</math>sim <math>\hat 4</math>sim <math>\hat{5}</math>>maj< <math>\hat{1}</math>sim

Or it can be used in conjunction with degree 4 or 6:

* <math>\hat 1</math>sim <math>\hat 2</math>sim <math>\hat 4</math>sim
* <math>\hat 1</math>sim <math>\hat 4</math>sim <math>\hat 2</math>sim
* <math>\hat 1</math>sim <math>\hat 2</math>sim <math>\hat 6</math>sim
* <math>\hat 1</math>sim <math>\hat 6</math>sim <math>\hat 2</math>sim
}}
===== Degree t =====
{{UserTag|KC|Inthar|000000|Degree t is an oneirotonic function with no analogue in diatonic. It serves several secondary roles, being related to more central functions in oneirominor, <math>\hat 6,</math> <math>\hat 4</math> and <math>\hat 2</math>:
* <math>\hat 6</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 6</math>sim
* <math>\hat 4</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 4</math>sim
* <math>\hat 2</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\hat 2</math>sim
}}
===== Degree 7 =====
{{UserTag|KC|Inthar|000000|Leading tone is the main function of <math>\mathrm{M}\hat 7</math>. It can also be related to degrees <math>\hat 2,</math> <math>\hat \mathrm{t}</math> and <math>\hat 6</math>:
* <math>\hat 2</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 2</math>sim
* <math>\hat 6</math>sim <math>\mathrm{M}\hat 7</math>sim or <math>\mathrm{M}\hat 7</math>sim <math>\hat 6</math>sim
* <math>\mathrm{M}\hat 7</math>sim <math>\hat \mathrm{t}</math>sim or <math>\hat \mathrm{t}</math>sim <math>\mathrm{M}\hat 7</math>sim
However, these are fairly weak because <math>\mathrm{M}\hat 7</math> is fairly distant from the tonic in generator space.

The 7th degree may be flattened to give it a stronger functional relationship to the tonic or other degrees; the flattened 7th degree (relative to the Celephaisian mode) provides a <math>\mathrm{P}\hat{5}</math> "dominant" for the degree <math>\hat{3}</math>.
}}{{Navbox MOS}}

Diatonic

2026-08-31T16:29:09Z

Unque: Deploying new navbox

[[File:Piano.png|thumb|417x417px|A diatonic musical keyboard]]
[[File:MOS diatonic.png|thumb|418x418px|The generation of a MOS diatonic scale from the chain of fifths]]
'''Diatonic''' is a kind of scale characterized by the division of the [[octave]] into 5 large steps (''L'') and 2 small steps (''s'') in an ''LLsLLLs'' pattern, or any [[rotation]] thereof. The most widespread diatonic is the [[MOS]] form, ''5L 2s'', where the large steps are equally sized and the small steps are also equally sized; in general, there can be multiple sizes of large or small step. When ''diatonic'' is used as an adjective on the wiki to describe something other than a scale, the MOS is what it usually refers to (for example, the [[diatonic major third]] is the major third in the MOS diatonic scale). However, other similar scales may be called diatonic as well.
[[File:Diatonic scales high-rank.png|thumb|390x390px|Various Greek diatonic scales.]]

== MOS diatonic ==

The MOS diatonic scale is ''5L 2s'', tuned most simply in [[12edo]]. It is the basic scale for a number of rank-2 temperaments, such as [[meantone]], [[archy]], and [[Pythagorean tuning#Schismic temperament|schismic]]. It can be formed by stacking seven [[perfect fifth]]s and octave-reducing.
MOS diatonic has seven modes, from classical music theory:
{| class="wikitable"
|-
! Mode name !! Pattern !! 2nd !! 3rd !! 4th !! 5th !! 6th !! 7th
|-
| Locrian || sLLsLLL || style="background-color:#d66"|m || style="background-color:#d66"|m || style="background-color:#d66"|P || style="background-color:#d66"|d || style="background-color:#d66"|m || style="background-color:#d66"|m
|-
| Phrygian || sLLLsLL || style="background-color:#d66"|m || style="background-color:#d66"|m || style="background-color:#d66"|P || style="background-color:#66d"|P || style="background-color:#d66"|m || style="background-color:#d66"|m
|-
| Aeolian (Minor) || LsLLsLL ||style="background-color:#66d"| M || style="background-color:#d66"|m || style="background-color:#d66"|P || style="background-color:#66d"|P || style="background-color:#d66"|m || style="background-color:#d66"|m
|-
| Dorian || LsLLLsL ||style="background-color:#66d"| M || style="background-color:#d66"|m || style="background-color:#d66"|P || style="background-color:#66d"|P || style="background-color:#66d"|M || style="background-color:#d66"|m
|-
| Mixolydian || LLsLLsL || style="background-color:#66d"|M || style="background-color:#66d"|M || style="background-color:#d66"|P || style="background-color:#66d"|P || style="background-color:#66d"|M || style="background-color:#d66"|m
|-
| Ionian (Major) || LLsLLLs || style="background-color:#66d"|M || style="background-color:#66d"|M || style="background-color:#d66"|P || style="background-color:#66d"|P || style="background-color:#66d"|M || style="background-color:#66d"|M
|-
| Lydian || LLLsLLs || style="background-color:#66d"|M || style="background-color:#66d"|M || style="background-color:#66d"|A || style="background-color:#66d"|P || style="background-color:#66d"|M || style="background-color:#66d"|M
|}
{{Navbox MOS}}[[File:The swry.png|thumb|362x362px|The space of 4/3-spanning tetrachords, with Greek genera highlighted]]

== Greek diatonic scales ==
''Main article: [[Tetrachord]]''

Diatonic may be defined more broadly according to the Greek diatonic genus. In Greek music, a diatonic scale is constructed from identical [[tetrachord]]s spanning a [[perfect fourth]] each, and separated by a [[9/8|Pythagorean major second]], such that

a) the largest interval in the tetrachord is at most half of its total span

b) the middle interval in the tetrachord is not the smallest.

As such, the scale pattern for a generalized diatonic (in its Phrygian mode) may be ''sABCsAB'', where ''C'' is an approximation of 9/8 and ''A'' and ''B'' are both intervals that are larger than ''s'' but no larger than half the tetrachord's span.

An example of a Greek diatonic is Ptolemy's intense diatonic, where ''C'' and ''A'' are equated to 9/8, ''B'' is 10/9, and ''s'' is 16/15. This results in the scale pattern ''sLMLsLM'', which may more abstractly be called 'zarlino' or 'nicetone', and has fourteen modes rather than seven due to its chirality (for example, mosdiatonic Ionian ''LLsLLLs'' becomes two modes: LH-Ionian (''MLsLMLs'') and RH-Ionian (''LMsLMLs'').)

Another example of a Greek diatonic is the equable diatonic, which is a tuning where ''C'' is 9/8, ''B'' is 10/9, ''A'' is 11/10, and ''s'' is 12/11. This has the property of being easily expressible as a [[Otonality and utonality|utonal]] sequence 1/(18:20:22:24:27:30:33:36).

The simplest rank-3 diatonic tunings are those found in [[14edo]] (3-2-1-2-3-2-1) and [[15edo]] (3-2-1-3-2-3-1). Note that in both cases, the largest interval in the tetrachord is exactly half of its width. This is similar to the edge-case found with the tuning of MOS diatonic in 5edo or 7edo.

Similarly, the MOS scales [[onyx]] and [[mosh]] with the large step between 4/3 and 3/2 are other edge cases of Greek diatonic scales.

== Straddle-fifth diatonic scales ==

A [[straddle-fifth]] system has a sharp and a flat fifth, which combine to create a 9/8 wholetone. So, in those systems, a diatonic scale analogous to the MOS diatonic may be generated by alternating the two fifths. This creates an ''LLmLLLs'' pattern, notably meaning there are multiple varieties of small step, rather than multiple varieties of large step. Again, there are fourteen modes, because, for example, there are again two Ionian modes: ''LLsLLLm'' and ''LLmLLLs''. Note that the perfect fourth is always based on one variety of fifth, and the perfect fifth is always the other.

Mosh

2026-08-31T16:28:13Z

Unque: Deploying new navbox

The '''Mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range.

== General scale theory ==
The Mosh scale has seven modes, whose names are given by Andrew Heathwaite.
{| class="wikitable"
|+Modes of Mosh
!Mode Name
!Pattern
!2nd
!3rd
!4th
!5th
!6th
!7th
|-
|Dril
|LsLsLss
|M
|P
|M
|M
|A
|M
|-
|Gil
|LsLssLs
|M
|P
|M
|M
|P
|M
|-
|Kleeth
|LssLsLs
|M
|P
|m
|M
|P
|M
|-
|Bish
|sLsLsLs
|m
|P
|m
|M
|P
|M
|-
|Fish
|sLsLssL
|m
|P
|m
|M
|P
|m
|-
|Jwl
|sLssLsL
|m
|P
|m
|m
|P
|m
|-
|Led
|ssLsLsL
|m
|d
|m
|m
|P
|m
|}
Like all [[MOS]] scales, tunings of Mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\3 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings create an [[MOS#Table of MOS scales|antipentic]] generator; this creates a clear divide in how these two types of tunings are handled.

The qualities of scale degrees in soft and hard Mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:
{| class="wikitable"
|+Mosh vs. Diatonic
!Mosh Degree
!Mosh Quality
!Soft
!Hard
|-
| rowspan="2" |2
|minor
|dia n2
|m2
|-
|major
|dia M2
|A2
|-
| rowspan="2" |3
|diminished
|dia m3
|d3
|-
|perfect
|dia n3
|dia M3
|-
| rowspan="2" |4
|minor
|dia P4
|dia d4
|-
|major
|dia A4
|dia AA4
|-
| rowspan="2" |5
|minor
|dia d5
|dia dd5
|-
|major
|dia P5
|dia A5
|-
| rowspan="2" |6
|perfect
|dia n6
|dia m6
|-
|augmented
|dia M6
|dia A6
|-
| rowspan="2" |7
|minor
|dia m7
|dia d7
|-
|major
|dia n7
|dia M7
|}

== Chords and tendency tones ==
{{Wip}}

=== Soft Mosh ===
In the soft tuning of Mosh, we may notice that two generators create an approximation of 3/2, and thus the generator sequence creates useful tertian chords; on most degrees, these chords will be "neutral," splitting that fifth in half. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9). For clarity's sake, these chords will not be given new arcane names, and will simply be referred to with the names of their modes.

Unlike in diatonic, soft Mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this Mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of Mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.

Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the "diminished third" interval of the Mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.

From all this, we can see a presently emergent paradigm for functional harmony in Soft Mosh, a "neutral tonality" which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.
{| class="wikitable"
|+Chords of Kleeth
!Degrees
!Quality
!Functional description
|-
|1 - 3 - 5
|Neutral
|Tonic triad.
|-
|'''2''' - '''4''' - 6
|Led
|Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.
|-
|3 - 5 - '''7'''
|Neutral
|Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.
|-
|'''4''' - 6 - 1
|Neutral
|Expansion of subdominant area. Creates motion from tonic to elsewhere.
|-
|5 - '''7''' - '''2'''
|Neutral
|Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).
|-
|6 - 1 - 3
|Neutral
|Primary subdominant chord, "opposite" of III. Creates motion from tonic to elsewhere.
|-
|7 - '''2''' - '''4'''
|Jwl
|Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.
|}
Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.

The functional paradigm for tonal music in Mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.

{{Navbox MOS}}

Template:Navbox MOS

2026-08-31T16:27:14Z

Unque: Created page with " {{Navbox | name = Navbox MOS | Title = Moment of Symmetry scales | Is Collapsible = true | Header 1 = 5-note | Data 1 = 1L 4s • 2L 3s • 3L 2s • 4L 1s | Header 2 = 6-note | Data 2 = 1L 5s • 2L_4s • 3L 3s • 4L 2s • 5L 1s | Header 3 = 7-note | Data 3 = 1L 6s • 2L 5s • 3L 4s • 4L 3s • 5L 2s • 6L 1s | Header 4 = 8-note |..."

{{Navbox
| name = Navbox MOS
| Title = Moment of Symmetry scales
| Is Collapsible = true
| Header 1 = 5-note
| Data 1 = [[1L 4s]] • [[2L 3s]] • [[3L 2s]] • [[4L 1s]]
| Header 2 = 6-note
| Data 2 = [[1L 5s]] • [[2L_4s]] • [[3L 3s]] • [[4L 2s]] • [[5L 1s]]
| Header 3 = 7-note
| Data 3 = [[1L 6s]] • [[2L 5s]] • [[Mosh|3L 4s]] • [[4L 3s]] • [[Diatonic|5L 2s]] • [[6L 1s]]
| Header 4 = 8-note
| Data 4 = [[1L 7s]] • [[2L 6s]] • [[3L 5s]] • [[4L 4s]] • [[5L 3s]] • [[6L 2s]] • [[7L 1s]]
| Header 5 = 9-note
| Data 5 = [[1L 8s]] • [[2L 7s]] • [[tcherepnin|3L 6s]] • [[4L 5s]] • [[semiquartal|5L 4s]] • [[6L 3s]] • [[7L 2s]] • [[8L 1s]]
}}

User:Unque

2026-08-21T12:55:54Z

Unque:

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Diatonic#Greek diatonic scales|Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[17edo]] ===
While it's a common choice, it is so for good reason. It's the second edo to support diatonic, the first to distinguish enharmonic notes, and the first to contain nondiatonic intervals alongside diatonic ones, in the form of neutral intervals. It additionally has multiple 7-form MOS scales, including diatonic, mosh, and antidiatonic, allowing for modal interchange between these scales to create more complex chromatic harmonies.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely halfway between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for composers who want to skip the frivolities and pick up and play with a tuning, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot (temperament)|Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Chthonic harmony]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]
* [[Blackwood#Blackwood tonality|Blackwood tonality]]
* [[Buzzard]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, sketches for page replacements, and other stuff I write that I don't believe is fit to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]] (disorganized thoughts)
* [[User:Unque/Porcupine|Porcupine]] (rewrite proposed to replace main page)

L'Antica Musica

2026-07-29T16:09:56Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament; see section on '''tuning''' for details.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent discussing the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct character by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and (hypo-) Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
The enharmonic modes are the most complex of the scales, with half of them having ten steps to the octave, and the other half having twelve. There are three step sizes from which the enharmonic modes are constructed: the incomposite third of 10\31, the chromatic semitone of 2\31, and the diesis of 1\31, which can be labelled L, m, and s like before; in addition to the aforementioned structural elements, the L step occurs precisely twice in each enharmonic mode.

The first mode is made by taking the diesis above and below the tonic, plus the chroma above the tonic; then including the fourth and fifth above each degree. This mode has twelve steps per octave, the most of any mode Vicentino writes about. The second mode (hypofirst), of course, is found by rotating the scale to begin on the perfect fifth.

The third mode is formed by stepping down from the tonic by a diesis, then lower by a chromatic semitone; the fourth and fifth are added above each of these degrees. Because the steps of this scale do not adhere to the step sizes available in the enharmonic modes, the 15\31 tritone is removed and replaced with the nearby intervals of 14\31 and 16\31, yielding a final count of ten notes per octave. The fourth mode (hypothird) can again be found via starting on the perfect fifth.

Where the other enharmonic modes used the fourth and fifth as spacers, the fifth mode uses the second and fifth instead. From each degree, step up by a diesis and then by a chromatic semitone, and thus can be considered a sort of counterpart to the third mode which steps by this same pattern downwards. Because this scale provides two consecutive semitone steps, Vicentino offers that the latter be split into two dieses, in order to divide up this disorientating segment; this rounds out the scale to ten notes per octave.

The seventh mode, representing final pair of modes, uses 16\31 as a generator to yield a four-note non-MOS scale, and then adds notes a perfect fourth above and a perfect fifth above each degree. This rounds out the mode to twelve notes just like the first pair of modes.
{| class="wikitable"
|+Enharmonic Modes
!Number
!Pattern
!Comments
|-
|1
|sLsssmssLss
|
|-
|2
|LsssLsssmss
|
|-
|3
|LmssmssLms
|Nearly the inverse of mode 6
|-
|4
|LmsLmssmss
|Nearly the inverse of mode 5
|-
|5
|smsssmLsmL
|Nearly the inverse of mode 4
|-
|6
|smLsmsssmL
|Nearly the inverse of mode 3
|-
|7
|smLsmsssLss
|
|-
|8
|sLsssmLsmss
|
|}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' describes basic notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

=== Archicembalo Layout ===
The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size was likely variable throughout. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯.

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; so that if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Tuning ===

==== Vicentino's potential preferences ====
In the first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around [[Meantone#Quarter-comma Meantone|1/4-comma meantone]] would be preferred, providing a justly tuned [[5/4]] as the major third. In the 31-note MOS scale of this tuning, the wolf fifth between G♭-^C♯ is approximately the size of a Pythagorean fifth; this would ensure that the wolf forms of intervals would fall between the sizes of the Pythagorean tuning and the meantone tuning, making it a favorable tuning for providing consonant tunings for those wolves. Additionally, very little size adjustment from 1/4-comma is needed to provide a wolf fifth which is precisely the same size as the generating fifth, thus closing the circle by yielding precisely 31edo.

In the second paradigm, with a fraction of a comma between C-^C, the most likely preferred tuning would have been [[Meantone#1/3-comma Meantone|1/3-comma meantone]], providing a justly tuned 6/5 as the minor third. Since the two keyboards represent two 19-note circles that don't overlap, the wolf fifths are between B♯-G♭ and their comma-altered counterparts, and 1/3-comma meantone tunes this wolf fifth to a size almost precisely the same as the generating fifth (thus approximately [[19edo]] per keyboard). Additionally, Vicentino offers that a root on the lower keyboard may be paired with a fifth and major third from the upper keyboard, a suggestion that would only make sense if the major third were significantly flat of 5/4; in 1/3-comma tuning, we find a major third of approximately 379 cents. Sharpening this third by 1/3 of a syntonic comma in fact yields a tuning which is almost precisely justly tuned 5/4.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo's first paradigm:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated, can be taken as 7/6 based on relations to other intervals
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

== Modern Application ==
While Vicentino's ideas about modes and intervals can be applied to composition in 31edo, many more ideas in other tunings may be considered through his work.

=== Diatonic third categorization ===
Vicentino states that the archicembalo offers four types of thirds that are unheard of in the traditional [[wikipedia:Musical_keyboard|Halberstadt]] layout, totaling six types of thirds; however, he only states five of these six qualities: minimal, minor, proximate (supraminor), major, and supermajor. Because the interval at 6\31 is a major tone and not a third, and because the interval at 7\31 is described as minimal, we may deduce that the sixth type of third is above the other five, hence a "maximal" third; indeed, the interval at 12\31 is the only interval not analyzed in Vicentino's treatise, and falls between the supermajor third and the perfect fourth. Because it is the fifth complement of the major tone, we may assign this maximal third the ratio 21/16.

The descriptions given for the melodic and harmonic properties of these thirds may be transferred to other tunings as well, especially tunings with comparable third organization, such as [[29edo]] or [[34edo]].

=== 38-tone tuning ===
By noticing that 1/3-comma meantone roughly closes at 19 steps along the circle of fifths, we can create a more rotationally-symmetrical version of the 38-note paradigm using the MOS scale 19L 19s, which is two instances of 19edo with their roots offset by a tiny amount. A near-pure 3/2 will yield a hardness of around 8:1, thus allowing us to cleanly approximate this paradigm by considering it a subset of 171edo. While the number is unwieldy, it provides an additional means of extending the paradigm beyond the two keyboards, such that nine keyboards (twenty-seven ranks) would yield a closed circle of 1/3-commas. This would be incredibly inefficient on an analog keyboard instrument of the 16th century, but could be quite simple and useful on isomorphic keyboards of the modern era, such as the Lumatone.

=== 30edo Enharmonic modes ===
With the exception of modes seven and eight, the enharmonic modes all contain either 15\31 or 16\31 over the root, never both; in fact, mode 3 actively replaces a 15\31 degree with 16\31 while making no adjustment to accommodate how this impacts the structure of the mode. If we take 15\31 and 16\31 as representing a singular semioctave rather than two separate types of tritones, we might translate the first six enharmonic modes cleanly into [[30edo]] rather than 31.
{| class="wikitable"
|+30edo enharmonic modes
!Number
!Pattern
!Comments
|-
|1
|sLsssssssLss
| rowspan="2" |The singular m step is now s, making the scale binary (though not MOS)
|-
|2
|LsssLsssssss
|-
|3
|LmsssssLms
| rowspan="2" |Contains two Lms tetrachords
|-
|4
|LmsLmsssss
|-
|5
|smssmLsmL
| rowspan="2" |Contains two smL tetrachords and one sms tetrachord
|-
|6
|smLsmssmL
|}

=== Valentine temperament ===
Vicentino notes "major" and "minor" forms of the wholetone; these intervals are separated from each other by a chroma, just as major and minor thirds, sixths, and sevenths, but neither of them are the diatonic major second. However, the major tone differs from the minor third by a chroma as well, as does the minor tone differ from the chroma itself by a chroma, hence we may use a chain of chromatic semitones to represent a singular system which categorizes Vicentino's primary intervals within the tetrachord. The temperament which produces this chain -- that is, equally spacing 25/24 - 35/32 - 8/7 - 6/5 - 5/4 -- is known as [[Valentine]]. It is also quite comparable to the "Alpha Scale" of Wendy Carlos.

Buzzard

2026-07-28T21:19:42Z

Unque:

{{Infobox regtemp
| Title = Buzzard
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11.13
| Comma basis = [[65536/64827]] (2.3.7); [[1728/1715]], [[5120/5103]] (7-limit); [[176/175]], [[351/350]], [[540/539]], [[676/675]] (13-limit)
| Edo join 1 = 53 | Edo join 2 = 58
| Mapping = 1; 4 21 -3 39 27
| Generators = 21/16
| Generators tuning = 475.7
| Optimization method = CWE
| MOS scales = 3L 2s
| Ploidacot = alpha-tetracot
| Pergen =
| Color name =
| Odd limit 1 = 2.3.7 9 | Mistuning 1 = 3.42 | Complexity 1 = 13
| Odd limit 2 = 15 | Mistuning 2 = 4.09 | Complexity 2 = 43
}}
'''Buzzard''', 2.3.7[{{e|48}} & {{e|53}}], is a 2.3.7 temperament that splits 3/1 into four sharpened 21/16 subfourths. It can be considered the "simplest 2.3.7 temperament after [[Slendric]]" since it equates two (not one) 64/63's with one 49/48, exaggerating the difference between 21/16 and 64/49. This 64/63 interval is primarily used as an inflection comma, as it separates the simple 3-limit intervals from 7-limit consonances; as a result of this "spine and comma" arrangement of the consonances, the MOS scales generated by Buzzard are not always the primary type of scale considered in the temperament.

Since Buzzard sharpens 147/128, equating it to an exact semifourth at -2 generators, it can be extended to 2.3.7.13/5 by equating 147/128 to 15/13, thus tempering out the huntsma, 640/637.

== List of patent vals ==
{| class="wikitable"
|+
!Edo
!Fifth tuning
!7/4 tuning
!Generator tuning
|-
|23
|678.26
|991.30
|469.57
|-
|28
|685.71
|985.71
|471.43
|-
|33
|690.91
|981.82
|472.73
|-
|38
|694.74
|978.95
|473.68
|-
|43
|697.67
|976.74
|474.42
|-
|48
|700.00
|975.00
|475.00
|-
|101
|700.99
|974.26
|475.25
|-
|53
|701.89
|973.58
|475.47
|-
|111
|702.70
|972.97
|475.68
|-
|58
|703.45
|972.41
|475.86
|-
|179
|703.91
|972.07
|475.98
|-
|121
|704.13
|971.90
|476.03
|-
|184
|704.35
|971.74
|476.09
|-
|63
|704.76
|971.43
|476.19
|-
|131
|705.34
|970.99
|476.34
|-
|68
|705.88
|970.59
|476.47
|-
|73
|706.85
|969.86
|476.71
|-
|78
|707.69
|969.23
|476.92
|-
|83
|708.43
|968.67
|477.11
|-
|5
|720.00
|960.00
|480.00
|}

== Extensions ==
While 13/5 has an obvious mapping via the 15/13 semifourth at -2 generators, 13 and 5 themselves are less obvious, as is 11.

=== Aberschismic Buzzard ===
The most common extension of Buzzard is Aberschismic -- that is, tempering out 5120/5103. This equates the 64/63 inflection comma with 81/80 and 105/104, allowing the 5-limit and 7-limit consonances to each be reached by one inflection away from a 3-limit consonance. This also provides a corresponding mapping of prime 13 as the difference between 15/8 and 15/13.

==== Valinorsmic Buzzard ====
The "canonical" mapping of prime 11 is to equate the inflection comma with 100/99; this equally spaces the sequences 7/6 - 13/11 - 6/5 - 40/33, as well as 21/16 - 4/3 - 27/20 - 15/11.

==== Telepathic Buzzard ("Buteo") ====
An alternative mapping of prime 11 is to conflate the inflection comma with 55/54 instead. This equally spaces the 7/6 - 32/27 - 6/5 - 11/9 sequence, as well as 21/16 - 4/3 - 27/20 - 11/8. This tuning is sometimes called Buteo, to distinguish from the Valinormic tuning that is considered the canonical extension of Buzzard.

=== Sensamagic Buzzard ("Lemongrass") ===
An alternative way to reach prime 5 is to equate 49/48 with 81/80, thus equating the sequence 5/4 - 80/63 - 81/64 - 9/7. Prime 13 is still the difference between 15/8 and 15/13, but in this case the tuning of 15/8 is much flatter than in Aberschismic Buzzard, thus providing a sharper tuning of 13/8. This extension is sometimes called Lemongrass, to distinguish from the Aberschismic tuning that is considered canonical.

== Scales ==
{{Wip}}{{Navbox regtemp}}
{{Cat| temperaments}}

L'Antica Musica

2026-07-28T20:58:10Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament; see section on '''tuning''' for details.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent discussing the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct character by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and (hypo-) Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' describes banal notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

=== Archicembalo Layout ===
The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size was likely variable throughout. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯.

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; so that if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Tuning ===

==== Vicentino's potential preferences ====
In the first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around [[Meantone#Quarter-comma Meantone|1/4-comma meantone]] would be preferred, providing a justly tuned [[5/4]] as the major third. In the 31-note MOS scale of this tuning, the wolf fifth between G♭-^C♯ is approximately the size of a Pythagorean fifth; this would ensure that the wolf forms of intervals would fall between the sizes of the Pythagorean tuning and the meantone tuning, making it a favorable tuning for providing consonant tunings for those wolves. Additionally, very little size adjustment from 1/4-comma is needed to provide a wolf fifth which is precisely the same size as the generating fifth, thus closing the circle by yielding precisely 31edo.

In the second paradigm, with a fraction of a comma between C-^C, the most likely preferred tuning would have been [[Meantone#1/3-comma Meantone|1/3-comma meantone]], providing a justly tuned 6/5 as the minor third. Since the two keyboards represent two 19-note circles that don't overlap, the wolf fifths are between B♯-G♭ and their comma-altered counterparts, and 1/3-comma meantone tunes this wolf fifth to a size almost precisely the same as the generating fifth (thus approximately [[19edo]] per keyboard). Additionally, Vicentino offers that a root on the lower keyboard may be paired with a fifth and major third from the upper keyboard, a suggestion that would only make sense if the major third were significantly flat of 5/4; in 1/3-comma tuning, we find a major third of approximately 379 cents. Sharpening this third by 1/3 of a syntonic comma in fact yields a tuning which is almost precisely justly tuned 5/4.

==== Modern tuning implications ====
By noticing that 1/3-comma meantone roughly closes at 19 steps along the circle of fifths, we can create a more rotationally-symmetrical version of the 38-note paradigm using the MOS scale 19L 19s, which is two instances of 19edo with their roots offset by a tiny amount. A near-pure 3/2 will yield a hardness of around 8:1, thus allowing us to cleanly approximate this paradigm by considering it a subset of 171edo. While the number is unwieldy, it provides an additional means of extending the paradigm beyond the two keyboards, such that nine keyboards (twenty-seven ranks) would yield a closed circle of 1/3-commas. This would be incredibly inefficient on an analog keyboard instrument of the 16th century, but could be quite simple and useful on isomorphic keyboards of the modern era, such as the Lumatone.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo's first paradigm:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated, can be taken as 7/6 based on relations to other intervals
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

User:Unque

2026-07-21T19:13:23Z

Unque:

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Diatonic#Greek diatonic scales|Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely half way between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for composers who want to skip the frivolities and pick up and play with a tuning, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot (temperament)|Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Chthonic harmony]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]
* [[Blackwood#Blackwood tonality|Blackwood tonality]]
* [[Buzzard]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, sketches for page replacements, and other stuff I write that I don't believe is fit to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]] (disorganized thoughts)
* [[User:Unque/Porcupine|Porcupine]] (rewrite proposed to replace main page)

Buzzard

2026-07-21T19:06:55Z

Unque:

{{Infobox regtemp
| Title = Buzzard
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11.13
| Comma basis = [[65536/64827]] (2.3.7); [[1728/1715]], [[5120/5103]] (7-limit); [[176/175]], [[351/350]], [[540/539]], [[676/675]] (13-limit)
| Edo join 1 = 53 | Edo join 2 = 58
| Mapping = 1; 4 21 -3 39 27
| Generators = 21/16
| Generators tuning = 475.7
| Optimization method = CWE
| MOS scales = 3L 2s
| Ploidacot = alpha-tetracot
| Pergen =
| Color name =
| Odd limit 1 = 2.3.7 9 | Mistuning 1 = 3.42 | Complexity 1 = 13
| Odd limit 2 = 15 | Mistuning 2 = 4.09 | Complexity 2 = 43
}}
'''Buzzard''', 2.3.7[{{e|48}} & {{e|53}}], is a 2.3.7 temperament that splits 3/1 into four sharpened 21/16 subfourths. It can be considered the "simplest 2.3.7 temperament after [[Slendric]]" since it equates two (not one) 64/63's with one 49/48, exaggerating the difference between 21/16 and 64/49. This 64/63 interval is primarily used as an inflection comma, as it separates the simple 3-limit intervals from 7-limit consonances; as a result of this "spine and comma" arrangement of the consonances, the MOS scales generated by Buzzard are not always the primary type of scale considered in the temperament.

Since Buzzard sharpens 147/128, equating it to an exact semifourth at -2 generators, it can be extended to 2.3.7.13/5 by equating 147/128 to 15/13, thus tempering out the huntsma, 640/637.

== List of patent vals ==
{| class="wikitable"
|+
!Edo
!Fifth tuning
!7/4 tuning
!Generator tuning
|-
|23
|678.26
|991.30
|469.57
|-
|28
|685.71
|985.71
|471.43
|-
|33
|690.91
|981.82
|472.73
|-
|38
|694.74
|978.95
|473.68
|-
|43
|697.67
|976.74
|474.42
|-
|48
|700.00
|975.00
|475.00
|-
|101
|700.99
|974.26
|475.25
|-
|53
|701.89
|973.58
|475.47
|-
|111
|702.70
|972.97
|475.68
|-
|58
|703.45
|972.41
|475.86
|-
|179
|703.91
|972.07
|475.98
|-
|121
|704.13
|971.90
|476.03
|-
|184
|704.35
|971.74
|476.09
|-
|63
|704.76
|971.43
|476.19
|-
|131
|705.34
|970.99
|476.34
|-
|68
|705.88
|970.59
|476.47
|-
|73
|706.85
|969.86
|476.71
|-
|78
|707.69
|969.23
|476.92
|-
|83
|708.43
|968.67
|477.11
|-
|5
|720.00
|960.00
|480.00
|}

== Extensions ==
While 13/5 has an obvious mapping via the 15/13 semifourth at -2 generators, 13 and 5 themselves are less obvious, as is 11.

=== Aberschismic Buzzard ===
The most common extension of Buzzard is Aberschismic -- that is, tempering out 5120/5103. This equates the 64/63 inflection comma with 81/80 and 105/104, allowing the 5-limit and 7-limit consonances to each be reached by one inflection away from a 3-limit consonance. This also provides a corresponding mapping of prime 13 as the difference between 15/8 and 15/13.

==== Valinorsmic Buzzard ====
The "canonical" mapping of prime 11 is to equate the inflection comma with 100/99; this equally spaces the sequence 11/9 - 5/4 - 81/64 - 9/7, as well as 21/16 - 4/3 - 27/20 - 15/11. One detriment of this extension is that the tunings of 11/8 and 13/8 are rather close together, with their difference in most tunings being closer to 250c rather than the 290c that we get in just intonation; in fact, this difference is the semifourth at -2 generators, which makes 11/8 equivalent to 169/120.

==== Telepathic Buzzard ("Buteo") ====
An alternative mapping of prime 11 is to conflate the inflection comma with 55/54 instead. This equally spaces the 7/6 - 32/27 - 6/5 - 11/9 sequence, as well as 21/16 - 4/3 - 27/20 - 11/8. This tuning is sometimes called Buteo, to distinguish from the Valinormic tuning that is considered the canonical extension of Buzzard.

=== Sensamagic Buzzard ("Lemongrass") ===
An alternative way to reach prime 5 is to equate 49/48 with 81/80, thus equating the sequence 5/4 - 80/63 - 81/64 - 9/7. Prime 13 is still the difference between 15/8 and 15/13, but in this case the tuning of 15/8 is much flatter than in Aberschismic Buzzard, thus providing a sharper tuning of 13/8. This extension is sometimes called Lemongrass, to distinguish from the Aberschismic tuning that is considered canonical.

== Scales ==
{{Wip}}{{Navbox regtemp}}
{{Cat| temperaments}}

User:Unque

2026-07-18T15:53:39Z

Unque:

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Diatonic#Greek diatonic scales|Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely half way between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for composers who want to skip the frivolities and pick up and play with a tuning, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot (temperament)|Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Chthonic harmony]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]
* [[Blackwood#Blackwood tonality|Blackwood tonality]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, sketches for page replacements, and other stuff I write that I don't believe is fit to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]] (disorganized thoughts)
* [[User:Unque/Porcupine|Porcupine]] (rewrite proposed to replace main page)

Blackwood

2026-07-18T15:51:28Z

Unque:

[[File:Vector Blackwood.mp3|thumb|An example of Blackwood temperament in a song by Vector]]
'''Blackwood''' is a regular temperament primarily supported by 15edo but in technicality by any EDO with 5edo's fifth (such as 25edo or 10edo) which takes 5edo as its 2.3.7, and treats 5 as an independent generator. Its 10-note scale (LsLsLsLsLs, '''pentawood''') is unique among scales of its complexity for making a perfect fifth available on every note of the scale, at the cost of a ~18c detuned fifth. This means that every step of the scale has either a major or a minor triad built on it. Blackwood has [[Diatonic#Greek diatonic scales|Zarlino]] as a subset, specifically tunings wherein the difference between the large and medium steps is the same size as the small step. This overlaps with [[Porcupine]] tunings of Zarlino only at [[15edo]].

There is also a 15-note, 20-note, etc. Blackwood scale, but these are much less common than the 10-note version which will be the main version discussed here.

== Name ==
Blackwood is named after Easley Blackwood Jr., a microtonal composer and theorist who extensively used what we now know as the Blackwood[10] scale.

The 7-limit version of Blackwood was called "Blacksmith" originally; this name is largely obsolete.

== Generator chain ==
''The provided tuning is DR 4:5:6.''

Cells highlighted in gray are intervals found in 5edo, which Blackwood expands. Yellow intervals are found in major blackwood[10], and green intervals in minor blackwood[10]. Blackwood[15] can thus be considered the "chromatic scale" of blackwood, and generally approaches equal temperament in most useful ranges.
{| class="wikitable"
|+
!
! colspan="2" rowspan="1" |Period 0
! colspan="2" rowspan="1" |Period 1
! colspan="2" rowspan="1" |Period 2
! colspan="2" rowspan="1" |Period 3
! colspan="2" rowspan="1" |Period 4
|-
!Generators
!Cents
!JI
!Cents
!JI
!Cents
!JI
!Cents
!JI
!Cents
!JI
|-
!-2
|'''1034.4'''
|50/27
|'''74.4'''
|25/24
|'''314.4'''
|25/21
|'''554.4'''
|25/18
|'''794.4'''
|100/63
|-
!-1
| class="prime11" |'''1117.2'''
| class="prime11" |15/8, 40/21
| class="prime11" |'''157.2'''
| class="prime11" |10/9
| class="prime11" |'''397.2'''
| class="prime11" |5/4
| class="prime11" |'''637.2'''
| class="prime11" |10/7
| class="prime11" |'''877.2'''
| class="prime11" |5/3
|-
!0
| class="prime2" |'''0'''
| class="prime2" |1/1
| class="prime2" |'''240'''
| class="prime2" |8/7
| class="prime2" |'''480'''
| class="prime2" |4/3
| class="prime2" |'''720'''
| class="prime2" |3/2
| class="prime2" |'''960'''
| class="prime2" |7/4
|-
!1
| class="prime5" |'''82.8'''
| class="prime5" |21/20, 16/15
| class="prime5" |'''322.8'''
| class="prime5" |6/5
| class="prime5" |'''562.8'''
| class="prime5" |7/5
| class="prime5" |'''802.8'''
| class="prime5" |8/5
| class="prime5" |'''1042.8'''
| class="prime5" |10/9
|-
!2
|'''165.6'''
|27/25
|'''405.6'''
|63/50
|'''645.6'''
|36/25
|'''885.6'''
|42/25
|'''1125.6'''
|48/25
|}

== Blackwood[10] modes ==
Blackwood[10] has only two modes: major (LsLsLsLsLs) and minor (sLsLsLsLsL). The unilatus, fourth, fifth, and antilatus, under [[10-form]] classification, are always perfect. The second, third, tritone, sixth, and seventh are all major in the major scale, and all minor in the minor scale.

{| class="wikitable"
|+ style="font-size: 105%;" | Blackwood[10] in 15edo
! Mode
! Step 0
! Step 1
! Step 2
! Step 3
! Step 4
! Step 5
! Step 6
! Step 7
! Step 8
! Step 9
! Step 10
|-
! Major
| 0c
| 160c
| 240c
| 400c
| 480c
| 640c
| 720c
| 880c
| 960c
| 1120c
| 1200c
|-
! Minor
| 0c
| 80c
| 240c
| 320c
| 480c
| 560c
| 720c
| 800c
| 960c
| 1040c
| 1200c
|}

== Blackwood tonality ==

=== Tendency tones ===
Like in diatonic, the fourth and fifth may be considered stable tones; another consideration is consider the unilatus and antilatus, and thus all perfect intervals, also stable. However, note that the unilatus creates a tendency tone towards the minor third, much as the perfect fourth creates a tendency tone towards the major third. So they may be considered dissonant in that context. While the tritone of blackwood is not a perfect antitonic, the sharp tritone regardless leads up to the fifth, and the flat tritone down to the fourth. The flat tritone also resolves inward to a major third.

=== Chord functions ===
In Blackwood[10], chords can be built by taking every third step of the scale; thus, the tonic chord contains steps 0, 3, and 6 (the root, third, and fifth of 10-form notation). Because the relative fifth interval over every degree is necessarily perfect, these chords come exclusively in major and minor forms.

The functional paradigm for the major tonality can be decompiled as tonic, mediant, subdominant, predominant, dominant, and tonic; note that the subdominant and predominant are considered different functions. The mediant function is almost exclusively built on the major third degree (step 3 of the scale), and the subdominant on the perfect fourth (step 5). The predominant, however, can be variously built from the perfect latus or the major tritone degrees (both chords containing the major tritone tendency tone), and the dominant built on either the perfect fifth or major seventh (both chords containing the major seventh tendency tone). Thus the basal chord paradigm is thus I - iv - V - {III, vi} - {VII, x} - I, using 1-indexed roman numerals.

The primary difference between the major and minor mode is that the minor intervals lead downward into the perfect intervals below, whereas their major forms lead upward to the perfect intervals above. As such, the minor mode paradigm can be seen as an inversion of the major paradigm, with the mediant function being taken by the chord built on the minor tritone, and the dominant can be built on the minor second, both functions making use of the downwards lead. The minor paradigm can be regularized i - bVI - v - {iii, bVIII} - bII - i.

{{Navbox regtemp}}
{{Cat|temperaments}}

User:Unque

2026-07-06T15:47:10Z

Unque:

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Diatonic#Greek diatonic scales|Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely half way between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for composers who want to skip the frivolities and pick up and play with a tuning, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot (temperament)|Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Chthonic harmony]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, sketches for page replacements, and other stuff I write that I don't believe is fit to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]] (disorganized thoughts)
* [[User:Unque/Porcupine|Porcupine]] (rewrite proposed to replace main page)

Tcherepnin

2026-07-06T15:45:22Z

Unque:

Tcherepnin

2026-06-14T23:25:00Z

Unque: Created page with "The '''Tcherepnin''' scale, or ''3L 6s'', is a common structural scale which repeats at one third of the octave; as such, it is often associated with the augmented temperament which splits the octave into three major thirds. The scale is named after the music theorist Alexander Tcherepnin, who utilized its 12edo tuning to create a double-tonal sound with two different types of thirds; the pattern was also noted by Olivier Messiaen, who listed..."

The '''Tcherepnin''' scale, or ''3L 6s'', is a common structural scale which repeats at one third of the octave; as such, it is often associated with the [[Augmented (temperament)|augmented]] temperament which splits the octave into three major thirds. The scale is named after the music theorist Alexander Tcherepnin, who utilized its 12edo tuning to create a double-tonal sound with two different types of thirds; the pattern was also noted by Olivier Messiaen, who listed it as the third scale in his list of [[modes of limited transposition]].

Because the Tcherepnin scale was popularized by progressive 20th century musicians such as Allan Holdsworth and Toru Takemitsu, a great deal of composition theory can be attained through analysis of these pieces, which can be applied to composition in xenharmonic systems which include it.

== General scale theory ==
The Tcherepnin scale is a mode of limited transposition, which means that certain rotations of this scale are identical to one another; while the scale has nine notes per octave, there are only three unique rotations: a bright mode, a dark mode, and a symmetrical mode. The symmetrical mode was the one used by Tcherepnin, and the bright mode was the one documented by Messiaen.
{| class="wikitable"
|+Modes of 3L 6s
!Brightness
!Pattern
!Degree Qualities
|-
| +
|LssLssLss
|MMPMMPMMP
|-
|0
|sLssLssLs
|mMPmMPmMP
|-
| -
|ssLssLssL
|mmPmmPmmP
|}

=== Degrees, intervals, and notation ===
The degrees and intervals of the Tcherepnin scale can be classed in two main ways: diamond MOS notation, where ordinal numbers correspond to the precise number of steps, or quasi-diatonic notation, where interval names are based on diatonic categories with the unchanging 400c and 800c intervals being dubbed the naiadic and cocytic degrees. The diamond MOS system yields more convenient methods of interval arithmetic in a vacuum, while the quasi-diatonic system yields familiar categories and preserves octave complements and chord constructions from diatonic music.

Degrees can be said to tend bright or dark based on whether their quality in the symmetrical mode is equivalent to that in the bright mode or the dark mode; in quasi-diatonic notation, odd numbers tend bright while even numbers tend dark, and the degrees labeled with Greek symbols do not have more than one quality.

The interval just below the cocytic (the "major sixth" in Tcherepnin ordinals, or "major fifth" in quasidiatonic ordinals) clusters around the size of a [[Pentic|pentatonic]] generator, or an [[antipentic]] generator in hard tunings; 15edo, with a step size ratio of 3:1, represents the midpoint where the fifth generates 5edo. With this in mind, the intervals and degrees of the Tcherepnin scale can be compared to those of the scales that this fifth yields.
{| class="wikitable"
|+Tcherepnin vs. Diatonic
!Degree
(Quasi)
!Degree
(Tcher)
!Quality
!Cent Range
!Diatonic
|-
| rowspan="2" |2
| rowspan="2" |1-step
|minor
|0 - 133
|m2
|-
|major
|133 - 400
|M2
|-
| rowspan="2" |3
| rowspan="2" |2-step
|minor
|0 - 266
|d3
|-
|major
|266 - 400
|m3
|-
|ς
|3-step
|perfect
|400
|M3~d4
|-
| rowspan="2" |4
| rowspan="2" |4-step
|minor
|400 - 533
|P4
|-
|major
|533 - 800
|A4
|-
| rowspan="2" |5
| rowspan="2" |5-step
|minor
|400 - 666
|d5
|-
|major
|666 - 800
|P5
|-
|ι
|6-step
|perfect
|800
|A5~m6
|-
| rowspan="2" |6
| rowspan="2" |7-step
|minor
|800 - 933
|M6
|-
|major
|933 - 1200
|A6
|-
| rowspan="2" |7
| rowspan="2" |8-step
|minor
|800 - 1066
|m7
|-
|major
|1066 - 1200
|M7
|}
Where one of the two notations must be chosen to avoid ambiguity, this page will prefer Quasi-diatonic notation. When quality must be specified for a degree, this page will use ♯/♭ to represent raising or lowering by a MOS chroma, and the symmetrical mode will be taken as the default qualities. Intervals between two pitches will use M or m with the Arabic numerals, or P with the Greek symbols, to specify in-scale qualities.

== Chords and tonality ==
Alexander Tcherepnin considered the most characteristic part of the scale to be its separation of the diatonic minor and major third onto different degrees, allowing both to be reached within the same mode. If we consider the 3 and ς degrees to be the main building blocks of chords, there are four types of triad constructions: chthonics (quasi-diatonic degrees 1 - 3 - 4), minor (degrees 1 - 3 - 5), major (1 - ς - 5), and augmented (1 - ς - ι). This leads to eight total chords, with three qualities of chthonic, two qualities of minor, two qualities of major, and one quality of augmented.

Each degree of the scale has one of each type of triad, with the augmented triad being the same in every mode:
{| class="wikitable"
|+Modes and chords
!Mode
!Chthonic
!Minor
!Major
|-
|Bright
|M3, M4
| rowspan="2" |M3, M5
| rowspan="2" |Pς, M5
|-
|Symmetrical
|M3, m4
|-
|Dark
|m3, m4
|m3, m5
|Pς, m5
|}
Where the qualities of these triads need be distinguished, they can be described with their characteristic mode and construction type, such as "bright major" for {Pς, M5} or "dark chthonic" for m3, m4.

It is notable that in 12edo tuning, the bright chthonic triad is enharmonically equivalent to the diminished chord from the diatonic scale. Many contemporary musicians who use the Tcherepnin scale take advantage of this fact, and curious xen composers might treat their usage of it as a model for how to fit chthonic chords and diatonic-like tertian chords into the same harmonic framework.

=== Harmonic systems ===
There are several possibilities for establishing a tonal framework in the Tcherepnin scale, based both on analysis of usage of the scale in music both xen and otherwise, and on applying fundamental principles to the scale as a self-standing system.

A common thread among contemporary treatment of the scale is how to work around the limited transposition; the two major types of approaches to this are to establish an [[axis system]] where there are multiple tonics, or to establish a modal system where there is no tonic at all.

==== Tcherepnin's axis system ====
An axis system is a system whereby functions repeat at some "axis" that subdivides the octave. In the case of the Tcherepnin scale, the axis occurs between degrees 1, ς, and ι, making them all tonics under this analysis; this is the primary framework seen in the works of Alexander Tcherepnin himself.

Because the m4 and M5 degrees resemble a diatonic generator, a significant amount of diatonic harmony can be transferred to this scale; Alexander Tcherepnin himself took advantage of this similarity, using the major triads on the 4 and 5 degrees for the functions of diatonic IV and V. Thus degrees 1, ς, and ι have tonic functions; 2, 4, and 6 have subdominant functions; and degrees 3, 5, and 7 have dominant functions. Note also that these degree functions correspond to brightness tendencies, with tonic degrees being the same in all modes, subdominant degrees tending dark, and dominant degrees tending bright.

Tcherepnin primarily employed major and minor chords in his axis-based harmony, with augmented chords being occasionally used, and chthonics only consisting of the bright quality; however, all eight chord types can be applied to this system. Note that chthonic chords, the main type of chord that Tcherepnin did not use, necessarily contain one degree for each of the functions; for instance, the chthonic chord on the 1 degree contains 1 (a tonic), 3 (a dominant), and 4 (a subdominant). This tonal ambiguity makes it a very useful tension with a plethora of resolution options, acting as a nexus that is limited only by voice leading.

==== Holdsworth's harmonic system ====
Jazz guitarist Allan Holdsworth made usage of the Tcherepnin scale in several pieces, and used unconventional constructions of chords to create a flowing modal sound. Unlike Tcherepnin's axis system, however, Holdsworth prioritized the qualities and relative intervals within the chords rather than the absolute degrees which they contained. Because the limited transposition provides a very unclear sense of tonality, Holdsworth's approach appears to favor tension and release paradigms that explore the scale modally, rather than establishing any clear center of gravity.

Mosh

2026-06-08T21:34:42Z

Unque:

User:Unque/Porcupine

2026-06-06T01:10:14Z

Unque:

'''Porcupine''' is the '''15 & 22''' [[temperament]] generated by an interval of approximately 160 - 165 cents known as a quill; a stack of two quills represents 6/5, and a stack of three represents 4/3, thus making the quill represent 10/9 in the 5-limit.

== Terminology, notation, and intervals ==
The most important property of the porcupine temperament is splitting 4/3 into three equal parts, and similarly splitting the [[apotome]] into three equal parts. Each third of 4/3 is the quill, representing 10/9~11/10~12/11, and each third of the apotome is the Porcupine chroma, representing 25/24~81/80.

The easiest way to notate Porcupine using a staff is to include a set of accidentals representing the Porcupine chroma. One can repurpose ♯/♭ for these purposes, but this may get confusing as these symbols traditionally represent the apotome rather than one-third of it, so the most common option is to use ^/v for the Porcupine chroma. The quill is thus a diatonic vM2 (read as "downmajor second" or "submajor second"), and 6/5 is a diatonic ^m3 (read as "upminor third" or "supraminor third").

The other possible paradigm is to redefine the ordinals to represent the equable scale (see scales) rather than the 3-limit diatonic scale, thus requiring fewer accidentals to effectively notate Porcupine structures. This requires composers and performers to unlearn familiar interval arithmetic, but it is ultimately more conducive to working with Porcupine as a system long-term. On this page, the equable labels will be used, however the conversion between equable labels and diatonic labels will be given in the interval table below.

The interval chain is shown below (for information on "Archy" and "Aberschismic," see extensions). Octave complements for diatonic and equable labels can be found using standard arithmetic (2-7, 3-6, 4-5), where major corresponds with minor, augmented corresponds with diminished, and ^ corresponds with v. Octave-reduced harmonics and subharmonics are shown in '''bold'''.
{| class="wikitable"
|+Interval Chain
!Quills up
!5-limit ratios
!11-limit ratios
!Archy ratios
!Aberschismic ratios
!Diatonic Label
!Equable Label
|-
|1
|10/9, 27/25
|11/10, 12/11
|
|
|vM2
|P2
|-
|2
|6/5
|11/9
|'''77/64'''
|
|^m3
|m3
|-
|3
|'''4/3'''
|
|'''21/16'''
|
|P4
|m4
|-
|4
|36/25, 40/27
|'''16/11''', 22/15
|35/24, 63/44
|
|vP5
|m5
|-
|5
|8/5
|44/27
|35/22
|
|^M6
|m6
|-
|6
|'''16/9'''
|
|'''7/4'''
|'''105/64'''
|m7
|d7
|-
|7
|48/25, 160/81
|'''64/33''', 88/45
|21/11, 35/18
|63/32
|vP8 / vP1
|d8
|-
|8
|'''16/15'''
|132/125
|21/20
|
|^m2
|d2
|-
|9
|'''32/27'''
|
|7/6
|'''77/64'''
|m3
|d3
|-
|10
|'''32/25'''
|'''128/99'''
|14/11
|'''21/16'''
|vP4
|d4
|-
|11
|'''64/45'''
|176/125
|7/5
|35/24, 63/44
|^d5
|d5
|-
|12
|'''128/81'''
|
|14/9
|35/22
|M6
|d6
|-
|13
|'''128/75'''
|
|56/33, 77/45
|'''7/4'''
|vm7
|dd7
|-
|14
|'''256/135'''
|
|28/15
|21/11, 36/35
|^d8 / ^d1
|dd8 / dd1
|-
|15
|'''256/243''', 156/125
|
|28/27
|21/20
|m2
|dd2
|-
|16
|104/75
|
|
|7/6
|vm3
|dd3
|}

== Supporting systems and extensions ==
Porcupine in the 5-limit is supported by 7, 8, 15, 22, 29, and 37edo; it also has a reasonable tuning in 36edo if we use the flat tuning for prime 5 rather than the sharp tuning borrowed from 12edo. All of these tunings additionally find prime 11 by equating the quill with 11/10~12/11, and equating the minor third with 11/9, making this extension quite ubiquitous. This also equalizes the harmonic segment 9:10:11:12 as a stack of quills, a feature which is relevant to scale construction.

=== Prime 7 ===
Prime 7 is slightly more contentious than 11 and 13 in this respect, as its mapping is not so ubiquitous across these supporting tunings.

==== Archy Porcupine (15 & 22) ====
The most commonly-accepted extension to add prime 7 is via Archy tuning, where 16/9 (found at six quills up) is equated with 7/4, a range which is reasonably supported by 15, 22, and 37edo. Note that all of these tunings have a quill which is on the narrower end of the range, around 160 cents; this provides a somewhat more accurate tuning of 6/5, but a very flat tuning of 4/3.

In this tuning, the Porcupine chroma is equated with 36/35, making the 7/6 - 6/5 - 5/4 - 9/7 segment equidistant.

==== Aberschismic Porcupine (29 & 36c) ====
Another reasonable extension to prime 7 is via Aberschismic tuning, where the Porcupine chroma is equated with 64/63, making the 7/6 - 32/27 - 6/5 segment equidistant; this range is supported primarily by 29edo and 36c. Both of these tunings have a quill around 165 cents, providing near-just 4/3 but a high-error 6/5; this property makes this extension act as something of a foil to the Archy extension, which has the opposite tendency.

Because 5/4 and 11/8 are both relatively high-error in this range, but their difference 11/10 is not, these tunings may be analyzed under the fractional subgroup 2.3.7.11/5 instead; however, this analysis has the detriment of providing no clear low-complexity interpretation for the thirds of the equable scale, which are 6/5~11/9 and 5/4~27/22 in the 2.3.5.7.11 subgroup.

=== Prime 13 ===
Because perfect fifth and the 5-limit thirds of porcupine tend to be tuned quite sharp, 16/13 tends to fall near the middle of 5/4 and 6/5 rather than leaning towards the sharp end; this is especially prevalent in 15edo, where 13/8 has almost precisely 50% error. This makes it rather difficult to convincingly map prime 13 to the porcupine generator chain.

==== Wilsormatic Porcupine (22 & 29) ====
Because 6/5 is equated with 11/9, making a third which is rather sharp of the former ratio and flat of the latter, we may map prime 13 by equating 16/13 with 5/4. Despite its higher error, this mapping for prime 13 is supported by 22, 29, and 36edo, but notably not 37edo.

This range has the property of making 13/8 and 5/4 into octave complements, equating the otonal chord 8:10:13:16 with its own utonal retroversion and making it symmetrical across the octave.

==== Harmoneutral Porcupine (30 & 37) ====
37edo does not support the above mapping, instead splitting the Porcupine chroma into two 65/64 intervals, which we can call a hemichroma; this makes the sequence 6/5 - 16/13 - 5/4 equidistant. This is considered a [[weak extension]], because the 13-limit intervals cannot be reached on the original chain of quills; specifically, the generator for this temperament is the rather complex 160/117, which splits the Porcupine perfect seventh (the octave complement of the quill) in half.

15 and 22edo also come to support this mapping when doubled to 30 and 44edo, adding 16/13 in the newly-acquired gap between the 5-limit thirds. Because of this, we may consider this mapping to define a canonical "hemi-Porcupine."

== Scales ==

=== MOS Scales ===

==== Equable Scale ====
Because the quill simultaneously represents each constituent element of 9:10:11:12, the equable [[tetrachord]] is equalized in Porcupine tunings. Using this tetrachord to form a full scale gives us a 7-note MOS with the signature sssLsss.

The mode names for this scale are given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[7]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Chinchillian
|Lssssss
|A2 - M3 - M4 - M5 - M6 - P7
|-
|2
|Badgerian
|sLsssss
|P2 - M3 - M4 - M5 - M6 - P7
|-
|1
|Zebrian
|ssLssss
|P2 - m3 - M4 - M5 - M6 - P7
|-
|0
|Dingoian
|sssLsss
|P2 - m3 - m4 - M5 - M6 - P7
|-
| -1
|Gazellian
|ssssLss
|P2 - m3 - m4 - m5 - M6 - P7
|-
| -2
|Lemurian
|sssssLs
|P2 - m3 - m4 - m5 - m6 - P7
|-
| -3
|Pandian
|ssssssL
|P2 - m3 - m4 - m5 - m6 - d7
|}

==== Octatonic Scale ====
The Porcupine Octatonic scale can be formed most simply by splitting the augmented quill step of the equable scale into a quill plus a Porcupine chroma, ensuring that the scale remains MV2. This one additional note actually creates a rather expressive addition to the harmony, creating double thirds over one mode and double fifths over another.

The modes of this scale are also given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[8]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|4
|Octopus
|LLLLLLLs
|P2 - m3 - m4 - m5 - m6 - d7 - d8
|-
|3
|Mantis
|LLLLLLsL
|P2 - m3 - m4 - m5 - m6 - d7 - P7
|-
|2
|Dolphin
|LLLLLsLL
|P2 - m3 - m4 - m5 - m6 - M6 - P7
|-
|1
|Crab
|LLLLsLLL
|P2 - m3 - m4 - m5 - M5 - M6 - P7
|-
| -1
|Tuna
|LLLsLLLL
|P2 - m3 - m4 - M4 - M5 - M6 - P7
|-
| -2
|Salmon
|LLsLLLLL
|P2 - m3 - M3 - M4 - M5 - M6 - P7
|-
| -3
|Starfish
|LsLLLLLL
|P2 - A2 - M3 - M4 - M5 - M6 - P7
|-
| -4
|Whale
|sLLLLLLL
|A1 - A2 - M3 - M4 - M5 - M6 - P7
|}

=== Non-MOS Scales ===

==== Zarlino ====
The [[Diatonic#Greek diatonic scales|Zarlino]] scale can be defined by an alternating stack of 5-limit thirds, 5/4 and 6/5. Porcupine temperament has several features which make it a notable system to support this scale: firstly, the intervals 81/80 and 25/24, which are the differences between steps of the Zarlino scale, are both represented by the Porcupine chroma; this means that there exists only one type of alteration which occurs on all intervals, rather than two types as in 5-limit JI. The steps are thus all consistently qualities of the quill, namely diminished, perfect, and augmented.

Secondly, the "wolf" fourth 27/20 is equated with 11/8 in Porcupine tuning, whose concordance makes it act less like a wolf interval and more like simply a different quality of fourth.

Because the Zarlino scale is chiral -- that is, its retroversion is not a rotation of the original -- we must distinguish left-hand and right-hand modes; in the generator sequence, left-hand modes have a chain which begins with 5/4 and ends with 6/5, whereas right-hand modes have the opposite.
{| class="wikitable"
|+Modes of LH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Lydian
|LMLsLMs
|A2 - M3 - A4 - M5 - A6 - A7
|-
|2
|Aeolian
|LsLMsLM
|A2 - m3 - M4 - M5 - m6 - P7
|-
|1
|Ionian
|LMsLMLs
|A2 - M3 - m4 - M5 - M6 - A7
|-
|0
|Phrygian
|sLMLsLM
|d2 - m3 - m4 - M5 - m6 - P7
|-
| -1
|Mixolydian
|MLsLMsL
|P2 - M3 - m4 - M5 - M6 - d7
|-
| -2
|Locrian
|sLMsLML
|d2 - m3 - m4 - d5 - m6 - d7
|-
| -3
|Dorian
|MsLMLsL
|P2 - d3 - m4 - m5 - M6 - d7
|}
{| class="wikitable"
|+Modes of RH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Dorian
|LsLMLsM
|A2 - m3 - M4 - M5 - A6 - P7
|-
|2
|Lydian
|LMLsMLs
|A2 - M3 - A4 - M5 - M6 - A7
|-
|1
|Aeolian
|LsMLsLM
|A2 - m3 - m4 - M5 - m6 - P7
|-
|0
|Ionian
|MLsLMLs
|P2 - M3 - m4 - M5 - M6 - A7
|-
| -1
|Phrygian
|sLMLsML
|d2 - m3 - m4 - M5 - m6 - d7
|-
| -2
|Mixolydian
|MLsMLsL
|P2 - M3 - m4 - m5 - M6 - d7
|-
| -3
|Locrian
|sMLsLML
|d2 - d3 - m4 - d5 - m6 - d7
|}

==== Blackdye ====
{{Main|10-form#Blackdye|l1=Blackdye}}
Just as we extended the equable scale into an octatonic form by turning the augmented quill into a perfect quill plus a Porcupine chroma, we can do the same to the augmented quills in the Zarlino scale to yield a 10-form called Blackdye. The scale remains MV3; the largest step represents a perfect quill, the middle step a diminished quill, and the smallest step a Porcupine chroma.

The Blackdye scale has several benefits over Zarlino, most notably uniting the two chiralities and creating longer unbroken fifth chains. More specifically, Blackdye is made up of two separate fifth chains, with their roots offset from one another by a quill; we can disambiguate similar modes based on which of the two chains contains the root. This gives us five "acute" modes and five "grave" modes, which can be organized further based on the brightness of their pentatonic chains.
{| class="wikitable"
|+Acute modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Aeolian
|LmLsLmLsLs
|P2 - d3 - d4 - m4 - m5 - d6 - dd7 - d7 - d8
|-
|1
|Dorian
|LmLsLsLmLs
|P2 - d3 - d4 - m4 - m5 - M5 - M6 - d7 - d8
|-
|0
|Mixolydian
|LsLmLsLmLs
|P2 - A2 - M3 - m4 - m5 - M5 - m6 - d7 - d8
|-
| -1
|Ionian
|LsLmLsLsLm
|P2 - A2 - M3 - m4 - m5 - M5 - M6 - A6 - A7
|-
| -2
|Lydian
|LsLsLmLsLm
|P2 - A2 - M3 - A3 - A4 - M5 - M6 - A6 - A7
|}
{| class="wikitable"
|+Grave modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Locrian
|mLsLmLsLsL
|d2 - d3 - m3 - m4 - d5 - d6 - m6 - d7 - P7
|-
|1
|Phrygian
|mLsLsLmLsL
|d2 - d3 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
|0
|Aeolian
|sLmLsLmLsL
|A1 - A2 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
| -1
|Dorian
|sLmLsLsLmL
|A1 - A2 - m3 - m4 - M4 - M5 - A5 - A6 - P7
|-
| -2
|Mixolydian
|sLsLmLsLmL
|A1 - A2 - AA2 - A3 - M4 - M5 - A5 - A6 - P7
|}
15edo is notable for [[Blackwood|collapsing the Blackdye scale into a MV2]] by merging the diminished quill with the Porcupine chroma, equivalent to tempering out (16/15) / (25/24) = 128/125; this means that its tuning of the scale only has two modes, a unified Acute Mode and a unified Grave Mode.

== Chords and harmony ==

=== Triad Qualities ===
Because equable and Zarlino, the basal scales of Porcupine, are both 7-form and are easy to take as alterations of one another, they can be used to define a system of triadic harmony not dissimilar to the kind seen in Western music.

==== Tertiary Triads ====
{| class="wikitable"
|+Triads of Porcupine
!Quality
!Intervals
!Formula
|-
|Super Augmented
|A3, A5
|A3 + M3
|-
|Augmented
|M3, A5
|M3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Minor
|m3, M5
|m3 + M3
|-
|Diminished
|m3, m5
|m3 + m3
|-
|Sub Diminished
|d3, d5
|d3 + m3
|}
Further qualities as appear in Zarlino will not be enumerated, and can simply be described as alterations of these triads.

==== Quartal Triads ====
In the octatonic scale, the two qualities of thirds do not occur on the same degree; however, the major third and the minor fourth do, which creates an inversion of the major or minor triad in the form of a quartal triad.
{| class="wikitable"
|+Quartal chords of Porcupine
!Quality
!Intervals
!Formula
|-
|Diminished Quartal
|m4, d7
|m4 + m4
|-
|Minor 6/4
|m4, M6
|m4 + M3
|-
|Major 6/3
|M3, M6
|M3 + m4
|-
|Augmented
|M3, A5
|M3 + M3
|}

==== Blackwood Triads ====
The Blackwood scale offers a different generalization of the triads, as the major and minor thirds occur alongside diminished fourths and double-augmented quills.
{| class="wikitable"
|+Triads of Blackwood
!Quality
!Intervals
!Formula
|-
|SusAA2
|AA2, M5
|AA2 + d4
|-
|Diminished
|m3, d6
|m3 + AA2
|-
|Minor
|m3, M5
|m3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Augmented
|M3, d6
|M3 + d4
|-
|Susd4
|d4, M5
|d4 + AA2
|}

=== Functional Harmony ===
Due to the structure of its scales, chord functions in Porcupine temperament are roughly midway between those of Meantone and Neutral temperaments. Because the equable and Zarlino scales both prioritize 5-limit major and minor triads, one can get a lot of functional leeway from simply translating familiar diatonic chord progressions into the temperament.

The quill is a useful leading interval, but its repeated occurrence throughout the equable and octatonic scales make its usage somewhat ambiguous and destabilize it as a directed step; comparatively, the diminished quill seen in the Zarlino scale is much narrower, and occurs much less frequently, allowing it to be used as a directed tension. Because of this property, major and minor triads on degree 5 both act equally as dominant functions, with the minor triad creating a soft and sweet pull while the major triad creates a powerful and confident pull back to the tonic triad.

User:Unque/Porcupine

2026-06-06T01:05:42Z

Unque:

'''Porcupine''' is the '''15 & 22''' [[temperament]] generated by an interval of approximately 160 - 165 cents known as a quill; a stack of two quills represents 6/5, and a stack of three represents 4/3, thus making the quill represent 10/9 in the 5-limit.

== Terminology, notation, and intervals ==
The most important property of the porcupine temperament is splitting 4/3 into three equal parts, and similarly splitting the [[apotome]] into three equal parts. Each third of 4/3 is the quill, representing 10/9~11/10~12/11, and each third of the apotome is the Porcupine chroma, representing 25/24~81/80.

The easiest way to notate Porcupine using a staff is to include a set of accidentals representing the Porcupine chroma. One can repurpose ♯/♭ for these purposes, but this may get confusing as these symbols traditionally represent the apotome rather than one-third of it, so the most common option is to use ^/v for the Porcupine chroma. The quill is thus a diatonic vM2 (read as "downmajor second" or "submajor second"), and 6/5 is a diatonic ^m3 (read as "upminor third" or "supraminor third").

The other possible paradigm is to redefine the ordinals to represent the equable scale (see scales) rather than the 3-limit diatonic scale, thus requiring fewer accidentals to effectively notate Porcupine structures. This requires composers and performers to unlearn familiar interval arithmetic, but it is ultimately more conducive to working with Porcupine as a system long-term. On this page, the equable labels will be used, however the conversion between equable labels and diatonic labels will be given in the interval table below.

The interval chain is shown below (for information on "Archy" and "Aberschismic," see extensions). Octave complements for diatonic and equable labels can be found using standard arithmetic (2-7, 3-6, 4-5), where major corresponds with minor, augmented corresponds with diminished, and ^ corresponds with v. Octave-reduced harmonics and subharmonics are shown in '''bold'''.
{| class="wikitable"
|+Interval Chain
!Quills up
!5-limit ratios
!11-limit ratios
!Archy ratios
!Aberschismic ratios
!Diatonic Label
!Equable Label
|-
|1
|10/9, 27/25
|11/10, 12/11
|
|
|vM2
|P2
|-
|2
|6/5
|11/9
|'''77/64'''
|
|^m3
|m3
|-
|3
|'''4/3'''
|
|'''21/16'''
|
|P4
|m4
|-
|4
|36/25, 40/27
|'''16/11''', 22/15
|35/24, 63/44
|
|vP5
|m5
|-
|5
|8/5
|44/27
|35/22
|
|^M6
|m6
|-
|6
|'''16/9'''
|
|'''7/4'''
|'''105/64'''
|m7
|d7
|-
|7
|48/25, 160/81
|'''64/33''', 88/45
|21/11, 35/18
|63/32
|vP8 / vP1
|d8
|-
|8
|'''16/15'''
|132/125
|21/20
|
|^m2
|d2
|-
|9
|'''32/27'''
|
|7/6
|'''77/64'''
|m3
|d3
|-
|10
|'''32/25'''
|'''128/99'''
|14/11
|'''21/16'''
|vP4
|d4
|-
|11
|'''64/45'''
|176/125
|7/5
|35/24, 63/44
|^d5
|d5
|-
|12
|'''128/81'''
|
|14/9
|35/22
|M6
|d6
|-
|13
|'''128/75'''
|
|56/33, 77/45
|'''7/4'''
|vm7
|dd7
|-
|14
|'''256/135'''
|
|28/15
|21/11, 36/35
|^d8 / ^d1
|dd8 / dd1
|-
|15
|'''256/243''', 156/125
|
|28/27
|21/20
|m2
|dd2
|-
|16
|104/75
|
|
|7/6
|vm3
|dd3
|}

== Supporting systems and extensions ==
Porcupine in the 5-limit is supported by 7, 8, 15, 22, 29, and 37edo; it also has a reasonable tuning in 36edo if we use the flat tuning for prime 5 rather than the sharp tuning borrowed from 12edo. All of these tunings additionally find prime 11 by equating the quill with 11/10~12/11, and equating the minor third with 11/9, making this extension quite ubiquitous. This also equalizes the harmonic segment 9:10:11:12 as a stack of quills, a feature which is relevant to scale construction.

=== Prime 7 ===
Prime 7 is slightly more contentious than 11 and 13 in this respect, as its mapping is not so ubiquitous across these supporting tunings.

==== Archy Porcupine (15 & 22) ====
The most commonly-accepted extension to add prime 7 is via Archy tuning, where 16/9 (found at six quills up) is equated with 7/4, a range which is reasonably supported by 15, 22, and 37edo. Note that all of these tunings have a quill which is on the narrower end of the range, around 160 cents; this provides a somewhat more accurate tuning of 6/5, but a very flat tuning of 4/3.

In this tuning, the Porcupine chroma is equated with 36/35, making the 7/6 - 6/5 - 5/4 - 9/7 segment equidistant.

==== Aberschismic Porcupine (29 & 36c) ====
Another reasonable extension to prime 7 is via Aberschismic tuning, where the Porcupine chroma is equated with 64/63, making the 7/6 - 32/27 - 6/5 segment equidistant; this range is supported primarily by 29edo and 36c. Both of these tunings have a quill around 165 cents, providing near-just 4/3 but a high-error 6/5; this property makes this extension act as something of a foil to the Archy extension, which has the opposite tendency.

Because 5/4 and 11/8 are both relatively high-error in this range, but their difference 11/10 is not, these tunings may be analyzed under the fractional subgroup 2.3.7.11/5 instead; however, this analysis has the detriment of providing no clear low-complexity interpretation for the thirds of the equable scale, which are 6/5~11/9 and 5/4~27/22 in the 2.3.5.7.11 subgroup.

=== Prime 13 ===
Because perfect fifth and the 5-limit thirds of porcupine tend to be tuned quite sharp, 16/13 tends to fall near the middle of 5/4 and 6/5 rather than leaning towards the sharp end; this is especially prevalent in 15edo, where 13/8 has almost precisely 50% error. This makes it rather difficult to convincingly map prime 13 to the porcupine generator chain.

==== (22 & 29) ====
Because 6/5 is equated with 11/9, making a third which is rather sharp of the former ratio and flat of the latter, we may map prime 13 by equating 16/13 with 5/4. Despite its higher error, this mapping for prime 13 is supported by 22, 29, and 36edo, but notably not 37edo.

This range has the property of making 13/8 and 5/4 into octave complements, equating the otonal chord 8:10:13:16 with its own utonal retroversion and making it symmetrical across the octave.

==== (30 & 37) ====
37edo does not support the above mapping, instead splitting the Porcupine chroma into two 65/64 intervals, which we can call a hemichroma; this makes the sequence 6/5 - 16/13 - 5/4 equidistant. This is considered a [[weak extension]], because the 13-limit intervals cannot be reached on the original chain of quills; specifically, the generator for this temperament is the rather complex 160/117, which splits the Porcupine perfect seventh (the octave complement of the quill) in half.

15 and 22edo also come to support this mapping when doubled to 30 and 44edo, adding 16/13 in the newly-acquired gap between the 5-limit thirds. Because of this, we may consider this mapping to define a canonical "hemi-Porcupine."

== Scales ==

=== MOS Scales ===

==== Equable Scale ====
Because the quill simultaneously represents each constituent element of 9:10:11:12, the equable [[tetrachord]] is equalized in Porcupine tunings. Using this tetrachord to form a full scale gives us a 7-note MOS with the signature sssLsss.

The mode names for this scale are given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[7]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Chinchillian
|Lssssss
|A2 - M3 - M4 - M5 - M6 - P7
|-
|2
|Badgerian
|sLsssss
|P2 - M3 - M4 - M5 - M6 - P7
|-
|1
|Zebrian
|ssLssss
|P2 - m3 - M4 - M5 - M6 - P7
|-
|0
|Dingoian
|sssLsss
|P2 - m3 - m4 - M5 - M6 - P7
|-
| -1
|Gazellian
|ssssLss
|P2 - m3 - m4 - m5 - M6 - P7
|-
| -2
|Lemurian
|sssssLs
|P2 - m3 - m4 - m5 - m6 - P7
|-
| -3
|Pandian
|ssssssL
|P2 - m3 - m4 - m5 - m6 - d7
|}

==== Octatonic Scale ====
The Porcupine Octatonic scale can be formed most simply by splitting the augmented quill step of the equable scale into a quill plus a Porcupine chroma, ensuring that the scale remains MV2. This one additional note actually creates a rather expressive addition to the harmony, creating double thirds over one mode and double fifths over another.

The modes of this scale are also given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[8]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|4
|Octopus
|LLLLLLLs
|P2 - m3 - m4 - m5 - m6 - d7 - d8
|-
|3
|Mantis
|LLLLLLsL
|P2 - m3 - m4 - m5 - m6 - d7 - P7
|-
|2
|Dolphin
|LLLLLsLL
|P2 - m3 - m4 - m5 - m6 - M6 - P7
|-
|1
|Crab
|LLLLsLLL
|P2 - m3 - m4 - m5 - M5 - M6 - P7
|-
| -1
|Tuna
|LLLsLLLL
|P2 - m3 - m4 - M4 - M5 - M6 - P7
|-
| -2
|Salmon
|LLsLLLLL
|P2 - m3 - M3 - M4 - M5 - M6 - P7
|-
| -3
|Starfish
|LsLLLLLL
|P2 - A2 - M3 - M4 - M5 - M6 - P7
|-
| -4
|Whale
|sLLLLLLL
|A1 - A2 - M3 - M4 - M5 - M6 - P7
|}

=== Non-MOS Scales ===

==== Zarlino ====
The [[Diatonic#Greek diatonic scales|Zarlino]] scale can be defined by an alternating stack of 5-limit thirds, 5/4 and 6/5. Porcupine temperament has several features which make it a notable system to support this scale: firstly, the intervals 81/80 and 25/24, which are the differences between steps of the Zarlino scale, are both represented by the Porcupine chroma; this means that there exists only one type of alteration which occurs on all intervals, rather than two types as in 5-limit JI. The steps are thus all consistently qualities of the quill, namely diminished, perfect, and augmented.

Secondly, the "wolf" fourth 27/20 is equated with 11/8 in Porcupine tuning, whose concordance makes it act less like a wolf interval and more like simply a different quality of fourth.

Because the Zarlino scale is chiral -- that is, its retroversion is not a rotation of the original -- we must distinguish left-hand and right-hand modes; in the generator sequence, left-hand modes have a chain which begins with 5/4 and ends with 6/5, whereas right-hand modes have the opposite.
{| class="wikitable"
|+Modes of LH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Lydian
|LMLsLMs
|A2 - M3 - A4 - M5 - A6 - A7
|-
|2
|Aeolian
|LsLMsLM
|A2 - m3 - M4 - M5 - m6 - P7
|-
|1
|Ionian
|LMsLMLs
|A2 - M3 - m4 - M5 - M6 - A7
|-
|0
|Phrygian
|sLMLsLM
|d2 - m3 - m4 - M5 - m6 - P7
|-
| -1
|Mixolydian
|MLsLMsL
|P2 - M3 - m4 - M5 - M6 - d7
|-
| -2
|Locrian
|sLMsLML
|d2 - m3 - m4 - d5 - m6 - d7
|-
| -3
|Dorian
|MsLMLsL
|P2 - d3 - m4 - m5 - M6 - d7
|}
{| class="wikitable"
|+Modes of RH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Dorian
|LsLMLsM
|A2 - m3 - M4 - M5 - A6 - P7
|-
|2
|Lydian
|LMLsMLs
|A2 - M3 - A4 - M5 - M6 - A7
|-
|1
|Aeolian
|LsMLsLM
|A2 - m3 - m4 - M5 - m6 - P7
|-
|0
|Ionian
|MLsLMLs
|P2 - M3 - m4 - M5 - M6 - A7
|-
| -1
|Phrygian
|sLMLsML
|d2 - m3 - m4 - M5 - m6 - d7
|-
| -2
|Mixolydian
|MLsMLsL
|P2 - M3 - m4 - m5 - M6 - d7
|-
| -3
|Locrian
|sMLsLML
|d2 - d3 - m4 - d5 - m6 - d7
|}

==== Blackdye ====
{{Main|10-form#Blackdye|l1=Blackdye}}
Just as we extended the equable scale into an octatonic form by turning the augmented quill into a perfect quill plus a Porcupine chroma, we can do the same to the augmented quills in the Zarlino scale to yield a 10-form called Blackdye. The scale remains MV3; the largest step represents a perfect quill, the middle step a diminished quill, and the smallest step a Porcupine chroma.

The Blackdye scale has several benefits over Zarlino, most notably uniting the two chiralities and creating longer unbroken fifth chains. More specifically, Blackdye is made up of two separate fifth chains, with their roots offset from one another by a quill; we can disambiguate similar modes based on which of the two chains contains the root. This gives us five "acute" modes and five "grave" modes, which can be organized further based on the brightness of their pentatonic chains.
{| class="wikitable"
|+Acute modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Aeolian
|LmLsLmLsLs
|P2 - d3 - d4 - m4 - m5 - d6 - dd7 - d7 - d8
|-
|1
|Dorian
|LmLsLsLmLs
|P2 - d3 - d4 - m4 - m5 - M5 - M6 - d7 - d8
|-
|0
|Mixolydian
|LsLmLsLmLs
|P2 - A2 - M3 - m4 - m5 - M5 - m6 - d7 - d8
|-
| -1
|Ionian
|LsLmLsLsLm
|P2 - A2 - M3 - m4 - m5 - M5 - M6 - A6 - A7
|-
| -2
|Lydian
|LsLsLmLsLm
|P2 - A2 - M3 - A3 - A4 - M5 - M6 - A6 - A7
|}
{| class="wikitable"
|+Grave modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Locrian
|mLsLmLsLsL
|d2 - d3 - m3 - m4 - d5 - d6 - m6 - d7 - P7
|-
|1
|Phrygian
|mLsLsLmLsL
|d2 - d3 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
|0
|Aeolian
|sLmLsLmLsL
|A1 - A2 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
| -1
|Dorian
|sLmLsLsLmL
|A1 - A2 - m3 - m4 - M4 - M5 - A5 - A6 - P7
|-
| -2
|Mixolydian
|sLsLmLsLmL
|A1 - A2 - AA2 - A3 - M4 - M5 - A5 - A6 - P7
|}
15edo is notable for [[Blackwood|collapsing the Blackdye scale into a MV2]] by merging the diminished quill with the Porcupine chroma, equivalent to tempering out (16/15) / (25/24) = 128/125; this means that its tuning of the scale only has two modes, a unified Acute Mode and a unified Grave Mode.

== Chords and harmony ==

=== Triad Qualities ===
Because equable and Zarlino, the basal scales of Porcupine, are both 7-form and are easy to take as alterations of one another, they can be used to define a system of triadic harmony not dissimilar to the kind seen in Western music.

==== Tertiary Triads ====
{| class="wikitable"
|+Triads of Porcupine
!Quality
!Intervals
!Formula
|-
|Super Augmented
|A3, A5
|A3 + M3
|-
|Augmented
|M3, A5
|M3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Minor
|m3, M5
|m3 + M3
|-
|Diminished
|m3, m5
|m3 + m3
|-
|Sub Diminished
|d3, d5
|d3 + m3
|}
Further qualities as appear in Zarlino will not be enumerated, and can simply be described as alterations of these triads.

==== Quartal Triads ====
In the octatonic scale, the two qualities of thirds do not occur on the same degree; however, the major third and the minor fourth do, which creates an inversion of the major or minor triad in the form of a quartal triad.
{| class="wikitable"
|+Quartal chords of Porcupine
!Quality
!Intervals
!Formula
|-
|Diminished Quartal
|m4, d7
|m4 + m4
|-
|Minor 6/4
|m4, M6
|m4 + M3
|-
|Major 6/3
|M3, M6
|M3 + m4
|-
|Augmented
|M3, A5
|M3 + M3
|}

==== Blackwood Triads ====
The Blackwood scale offers a different generalization of the triads, as the major and minor thirds occur alongside diminished fourths and double-augmented quills.
{| class="wikitable"
|+Triads of Blackwood
!Quality
!Intervals
!Formula
|-
|SusAA2
|AA2, M5
|AA2 + d4
|-
|Diminished
|m3, d6
|m3 + AA2
|-
|Minor
|m3, M5
|m3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Augmented
|M3, d6
|M3 + d4
|-
|Susd4
|d4, M5
|d4 + AA2
|}

=== Functional Harmony ===
Due to the structure of its scales, chord functions in Porcupine temperament are roughly midway between those of Meantone and Neutral temperaments. Because the equable and Zarlino scales both prioritize 5-limit major and minor triads, one can get a lot of functional leeway from simply translating familiar diatonic chord progressions into the temperament.

The quill is a useful leading interval, but its repeated occurrence throughout the equable and octatonic scales make its usage somewhat ambiguous and destabilize it as a directed step; comparatively, the diminished quill seen in the Zarlino scale is much narrower, and occurs much less frequently, allowing it to be used as a directed tension. Because of this property, major and minor triads on degree 5 both act equally as dominant functions, with the minor triad creating a soft and sweet pull while the major triad creates a powerful and confident pull back to the tonic triad.

User:Unque/Porcupine

2026-06-06T01:01:31Z

Unque:

'''Porcupine''' is the '''15 & 22''' [[temperament]] generated by an interval of approximately 160 - 165 cents known as a quill; a stack of two quills represents 6/5, and a stack of three represents 4/3, thus making the quill represent 10/9 in the 5-limit.

== Terminology, notation, and intervals ==
The most important property of the porcupine temperament is splitting 4/3 into three equal parts, and similarly splitting the [[apotome]] into three equal parts. Each third of 4/3 is the quill, representing 10/9~11/10~12/11, and each third of the apotome is the Porcupine chroma, representing 25/24~81/80.

The easiest way to notate Porcupine using a staff is to include a set of accidentals representing the Porcupine chroma. One can repurpose ♯/♭ for these purposes, but this may get confusing as these symbols traditionally represent the apotome rather than one-third of it, so the most common option is to use ^/v for the Porcupine chroma. The quill is thus a diatonic vM2 (read as "downmajor second" or "submajor second"), and 6/5 is a diatonic ^m3 (read as "upminor third" or "supraminor third").

The other possible paradigm is to redefine the ordinals to represent the equable scale (see scales) rather than the 3-limit diatonic scale, thus requiring fewer accidentals to effectively notate Porcupine structures. This requires composers and performers to unlearn familiar interval arithmetic, but it is ultimately more conducive to working with Porcupine as a system long-term. On this page, the equable labels will be used, however the conversion between equable labels and diatonic labels will be given in the interval table below.

The interval chain is shown below (for information on "Archy" and "Aberschismic," see extensions). Octave complements for diatonic and equable labels can be found using standard arithmetic (2-7, 3-6, 4-5), where major corresponds with minor, augmented corresponds with diminished, and ^ corresponds with v. Octave-reduced harmonics and subharmonics are shown in '''bold'''.
{| class="wikitable"
|+Interval Chain
!Quills up
!5-limit ratios
!11-limit ratios
!Archy ratios
!Aberschismic ratios
!Diatonic Label
!Equable Label
|-
|1
|10/9, 27/25
|11/10, 12/11
|
|
|vM2
|P2
|-
|2
|6/5
|11/9
|'''77/64'''
|
|^m3
|m3
|-
|3
|'''4/3'''
|
|'''21/16'''
|
|P4
|m4
|-
|4
|36/25, 40/27
|'''16/11''', 22/15
|35/24, 63/44
|
|vP5
|m5
|-
|5
|8/5
|44/27
|35/22
|
|^M6
|m6
|-
|6
|'''16/9'''
|
|'''7/4'''
|'''105/64'''
|m7
|d7
|-
|7
|48/25, 160/81
|'''64/33''', 88/45
|21/11, 35/18
|63/32
|vP8 / vP1
|d8
|-
|8
|'''16/15'''
|132/125
|21/20
|
|^m2
|d2
|-
|9
|'''32/27'''
|
|7/6
|'''77/64'''
|m3
|d3
|-
|10
|'''32/25'''
|'''128/99'''
|14/11
|'''21/16'''
|vP4
|d4
|-
|11
|'''64/45'''
|176/125
|7/5
|35/24, 63/44
|^d5
|d5
|-
|12
|'''128/81'''
|
|14/9
|35/22
|M6
|d6
|-
|13
|'''128/75'''
|
|56/33, 77/45
|'''7/4'''
|vm7
|dd7
|-
|14
|'''256/135'''
|
|28/15
|21/11, 36/35
|^d8 / ^d1
|dd8 / dd1
|-
|15
|'''256/243''', 156/125
|
|28/27
|21/20
|m2
|dd2
|-
|16
|104/75
|
|
|7/6
|vm3
|dd3
|}

== Supporting systems and extensions ==
Porcupine in the 5-limit is supported by 7, 8, 15, 22, 29, and 37edo; it also has a reasonable tuning in 36edo if we use the flat tuning for prime 5 rather than the sharp tuning borrowed from 12edo. All of these tunings additionally find prime 11 by equating the quill with 11/10~12/11, and equating the minor third with 11/9, making this extension quite ubiquitous. This also equalizes the harmonic segment 9:10:11:12 as a stack of quills, a feature which is relevant to scale construction.

=== Prime 7 ===
Prime 7 is slightly more contentious than 11 and 13 in this respect, as its mapping is not so ubiquitous across these supporting tunings.

==== Archy Porcupine (15 & 22) ====
The most commonly-accepted extension to add prime 7 is via Archy tuning, where 16/9 (found at six quills up) is equated with 7/4, a range which is reasonably supported by 15, 22, and 37edo. Note that all of these tunings have a quill which is on the narrower end of the range, around 160 cents; this provides a somewhat more accurate tuning of 6/5, but a very flat tuning of 4/3.

In this tuning, the Porcupine chroma is equated with 36/35, making the 7/6 - 6/5 - 5/4 - 9/7 segment equidistant.

==== Aberschismic Porcupine (29 & 36c) ====
Another reasonable extension to prime 7 is via Aberschismic tuning, where the Porcupine chroma is equated with 64/63, making the 7/6 - 32/27 - 6/5 segment equidistant; this range is supported primarily by 29edo and 36c. Both of these tunings have a quill around 165 cents, providing near-just 4/3 but a high-error 6/5; this property makes this extension act as something of a foil to the Archy extension, which has the opposite tendency.

Because 5/4 and 11/8 are both relatively high-error in this range, but their difference 11/10 is not, these tunings may be analyzed under the fractional subgroup 2.3.7.11/5 instead; however, this analysis has the detriment of providing no clear low-complexity interpretation for the thirds of the equable scale, which are 6/5~11/9 and 5/4~27/22 in the 2.3.5.7.11 subgroup.

=== Prime 13 ===
Because perfect fifth and the 5-limit thirds of porcupine tend to be tuned quite sharp, 16/13 tends to fall near the middle of 5/4 and 6/5 rather than leaning towards the sharp end; this is especially prevalent in 15edo, where 13/8 has almost precisely 50% error. This makes it rather difficult to convincingly map prime 13 to the porcupine generator chain.

==== (22 & 29) ====
Because 6/5 is equated with 11/9, making a third which is rather sharp of the former ratio and flat of the latter, we may map prime 13 by equating 16/13 with 5/4. Despite its higher error, this mapping for prime 13 is supported by 22, 29, and 36edo, but notably not 37edo.

This range has the property of making 13/8 and 5/4 into octave complements, equating the otonal chord 8:10:13:16 with its own utonal retroversion and making it symmetrical across the octave.

==== (30 & 37) ====
37edo does not support the above mapping, instead splitting the Porcupine chroma into two 65/64 intervals, which we can call a hemichroma; this makes the sequence 6/5 - 16/13 - 5/4 equidistant. This is considered a weak extension, because the 13-limit intervals cannot be reached on the original chain of quills; specifically, the generator for this temperament is the rather complex 160/117, which splits the quill's octave complement in half.

15 and 22edo also come to support this mapping when doubled to 30 and 44edo, adding 16/13 in the newly-acquired gap between the 5-limit thirds. Because of this, we may consider this mapping to be the canonical form of hemi-Porcupine.

== Scales ==

=== MOS Scales ===

==== Equable Scale ====
Because the quill simultaneously represents each constituent element of 9:10:11:12, the equable [[tetrachord]] is equalized in Porcupine tunings. Using this tetrachord to form a full scale gives us a 7-note MOS with the signature sssLsss.

The mode names for this scale are given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[7]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Chinchillian
|Lssssss
|A2 - M3 - M4 - M5 - M6 - P7
|-
|2
|Badgerian
|sLsssss
|P2 - M3 - M4 - M5 - M6 - P7
|-
|1
|Zebrian
|ssLssss
|P2 - m3 - M4 - M5 - M6 - P7
|-
|0
|Dingoian
|sssLsss
|P2 - m3 - m4 - M5 - M6 - P7
|-
| -1
|Gazellian
|ssssLss
|P2 - m3 - m4 - m5 - M6 - P7
|-
| -2
|Lemurian
|sssssLs
|P2 - m3 - m4 - m5 - m6 - P7
|-
| -3
|Pandian
|ssssssL
|P2 - m3 - m4 - m5 - m6 - d7
|}

==== Octatonic Scale ====
The Porcupine Octatonic scale can be formed most simply by splitting the augmented quill step of the equable scale into a quill plus a Porcupine chroma, ensuring that the scale remains MV2. This one additional note actually creates a rather expressive addition to the harmony, creating double thirds over one mode and double fifths over another.

The modes of this scale are also given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[8]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|4
|Octopus
|LLLLLLLs
|P2 - m3 - m4 - m5 - m6 - d7 - d8
|-
|3
|Mantis
|LLLLLLsL
|P2 - m3 - m4 - m5 - m6 - d7 - P7
|-
|2
|Dolphin
|LLLLLsLL
|P2 - m3 - m4 - m5 - m6 - M6 - P7
|-
|1
|Crab
|LLLLsLLL
|P2 - m3 - m4 - m5 - M5 - M6 - P7
|-
| -1
|Tuna
|LLLsLLLL
|P2 - m3 - m4 - M4 - M5 - M6 - P7
|-
| -2
|Salmon
|LLsLLLLL
|P2 - m3 - M3 - M4 - M5 - M6 - P7
|-
| -3
|Starfish
|LsLLLLLL
|P2 - A2 - M3 - M4 - M5 - M6 - P7
|-
| -4
|Whale
|sLLLLLLL
|A1 - A2 - M3 - M4 - M5 - M6 - P7
|}

=== Non-MOS Scales ===

==== Zarlino ====
The [[Diatonic#Greek diatonic scales|Zarlino]] scale can be defined by an alternating stack of 5-limit thirds, 5/4 and 6/5. Porcupine temperament has several features which make it a notable system to support this scale: firstly, the intervals 81/80 and 25/24, which are the differences between steps of the Zarlino scale, are both represented by the Porcupine chroma; this means that there exists only one type of alteration which occurs on all intervals, rather than two types as in 5-limit JI. The steps are thus all consistently qualities of the quill, namely diminished, perfect, and augmented.

Secondly, the "wolf" fourth 27/20 is equated with 11/8 in Porcupine tuning, whose concordance makes it act less like a wolf interval and more like simply a different quality of fourth.

Because the Zarlino scale is chiral -- that is, its retroversion is not a rotation of the original -- we must distinguish left-hand and right-hand modes; in the generator sequence, left-hand modes have a chain which begins with 5/4 and ends with 6/5, whereas right-hand modes have the opposite.
{| class="wikitable"
|+Modes of LH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Lydian
|LMLsLMs
|A2 - M3 - A4 - M5 - A6 - A7
|-
|2
|Aeolian
|LsLMsLM
|A2 - m3 - M4 - M5 - m6 - P7
|-
|1
|Ionian
|LMsLMLs
|A2 - M3 - m4 - M5 - M6 - A7
|-
|0
|Phrygian
|sLMLsLM
|d2 - m3 - m4 - M5 - m6 - P7
|-
| -1
|Mixolydian
|MLsLMsL
|P2 - M3 - m4 - M5 - M6 - d7
|-
| -2
|Locrian
|sLMsLML
|d2 - m3 - m4 - d5 - m6 - d7
|-
| -3
|Dorian
|MsLMLsL
|P2 - d3 - m4 - m5 - M6 - d7
|}
{| class="wikitable"
|+Modes of RH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Dorian
|LsLMLsM
|A2 - m3 - M4 - M5 - A6 - P7
|-
|2
|Lydian
|LMLsMLs
|A2 - M3 - A4 - M5 - M6 - A7
|-
|1
|Aeolian
|LsMLsLM
|A2 - m3 - m4 - M5 - m6 - P7
|-
|0
|Ionian
|MLsLMLs
|P2 - M3 - m4 - M5 - M6 - A7
|-
| -1
|Phrygian
|sLMLsML
|d2 - m3 - m4 - M5 - m6 - d7
|-
| -2
|Mixolydian
|MLsMLsL
|P2 - M3 - m4 - m5 - M6 - d7
|-
| -3
|Locrian
|sMLsLML
|d2 - d3 - m4 - d5 - m6 - d7
|}

==== Blackdye ====
{{Main|10-form#Blackdye|l1=Blackdye}}
Just as we extended the equable scale into an octatonic form by turning the augmented quill into a perfect quill plus a Porcupine chroma, we can do the same to the augmented quills in the Zarlino scale to yield a 10-form called Blackdye. The scale remains MV3; the largest step represents a perfect quill, the middle step a diminished quill, and the smallest step a Porcupine chroma.

The Blackdye scale has several benefits over Zarlino, most notably uniting the two chiralities and creating longer unbroken fifth chains. More specifically, Blackdye is made up of two separate fifth chains, with their roots offset from one another by a quill; we can disambiguate similar modes based on which of the two chains contains the root. This gives us five "acute" modes and five "grave" modes, which can be organized further based on the brightness of their pentatonic chains.
{| class="wikitable"
|+Acute modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Aeolian
|LmLsLmLsLs
|P2 - d3 - d4 - m4 - m5 - d6 - dd7 - d7 - d8
|-
|1
|Dorian
|LmLsLsLmLs
|P2 - d3 - d4 - m4 - m5 - M5 - M6 - d7 - d8
|-
|0
|Mixolydian
|LsLmLsLmLs
|P2 - A2 - M3 - m4 - m5 - M5 - m6 - d7 - d8
|-
| -1
|Ionian
|LsLmLsLsLm
|P2 - A2 - M3 - m4 - m5 - M5 - M6 - A6 - A7
|-
| -2
|Lydian
|LsLsLmLsLm
|P2 - A2 - M3 - A3 - A4 - M5 - M6 - A6 - A7
|}
{| class="wikitable"
|+Grave modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Locrian
|mLsLmLsLsL
|d2 - d3 - m3 - m4 - d5 - d6 - m6 - d7 - P7
|-
|1
|Phrygian
|mLsLsLmLsL
|d2 - d3 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
|0
|Aeolian
|sLmLsLmLsL
|A1 - A2 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
| -1
|Dorian
|sLmLsLsLmL
|A1 - A2 - m3 - m4 - M4 - M5 - A5 - A6 - P7
|-
| -2
|Mixolydian
|sLsLmLsLmL
|A1 - A2 - AA2 - A3 - M4 - M5 - A5 - A6 - P7
|}
15edo is notable for [[Blackwood|collapsing the Blackdye scale into a MV2]] by merging the diminished quill with the Porcupine chroma, equivalent to tempering out (16/15) / (25/24) = 128/125; this means that its tuning of the scale only has two modes, a unified Acute Mode and a unified Grave Mode.

== Chords and harmony ==

=== Triad Qualities ===
Because equable and Zarlino, the basal scales of Porcupine, are both 7-form and are easy to take as alterations of one another, they can be used to define a system of triadic harmony not dissimilar to the kind seen in Western music.

==== Tertiary Triads ====
{| class="wikitable"
|+Triads of Porcupine
!Quality
!Intervals
!Formula
|-
|Super Augmented
|A3, A5
|A3 + M3
|-
|Augmented
|M3, A5
|M3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Minor
|m3, M5
|m3 + M3
|-
|Diminished
|m3, m5
|m3 + m3
|-
|Sub Diminished
|d3, d5
|d3 + m3
|}
Further qualities as appear in Zarlino will not be enumerated, and can simply be described as alterations of these triads.

==== Quartal Triads ====
In the octatonic scale, the two qualities of thirds do not occur on the same degree; however, the major third and the minor fourth do, which creates an inversion of the major or minor triad in the form of a quartal triad.
{| class="wikitable"
|+Quartal chords of Porcupine
!Quality
!Intervals
!Formula
|-
|Diminished Quartal
|m4, d7
|m4 + m4
|-
|Minor 6/4
|m4, M6
|m4 + M3
|-
|Major 6/3
|M3, M6
|M3 + m4
|-
|Augmented
|M3, A5
|M3 + M3
|}

==== Blackwood Triads ====
The Blackwood scale offers a different generalization of the triads, as the major and minor thirds occur alongside diminished fourths and double-augmented quills.
{| class="wikitable"
|+Triads of Blackwood
!Quality
!Intervals
!Formula
|-
|SusAA2
|AA2, M5
|AA2 + d4
|-
|Diminished
|m3, d6
|m3 + AA2
|-
|Minor
|m3, M5
|m3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Augmented
|M3, d6
|M3 + d4
|-
|Susd4
|d4, M5
|d4 + AA2
|}

=== Functional Harmony ===
Due to the structure of its scales, chord functions in Porcupine temperament are roughly midway between those of Meantone and Neutral temperaments. Because the equable and Zarlino scales both prioritize 5-limit major and minor triads, one can get a lot of functional leeway from simply translating familiar diatonic chord progressions into the temperament.

The quill is a useful leading interval, but its repeated occurrence throughout the equable and octatonic scales make its usage somewhat ambiguous and destabilize it as a directed step; comparatively, the diminished quill seen in the Zarlino scale is much narrower, and occurs much less frequently, allowing it to be used as a directed tension. Because of this property, major and minor triads on degree 5 both act equally as dominant functions, with the minor triad creating a soft and sweet pull while the major triad creates a powerful and confident pull back to the tonic triad.

User:Unque/Porcupine

2026-06-06T00:44:04Z

Unque: Created page with "'''Porcupine''' is the '''15 & 22''' temperament generated by an interval of approximately 160 - 165 cents known as a quill; a stack of two quills represents 6/5, and a stack of three represents 4/3, thus making the quill represent 10/9 in the 5-limit. == Terminology, notation, and intervals == The most important property of the porcupine temperament is splitting 4/3 into three equal parts, and similarly splitting the apotome into three equal parts. Each third of 4/3 i..."

'''Porcupine''' is the '''15 & 22''' temperament generated by an interval of approximately 160 - 165 cents known as a quill; a stack of two quills represents 6/5, and a stack of three represents 4/3, thus making the quill represent 10/9 in the 5-limit.

== Terminology, notation, and intervals ==
The most important property of the porcupine temperament is splitting 4/3 into three equal parts, and similarly splitting the apotome into three equal parts. Each third of 4/3 is the quill, representing 10/9~11/10~12/11, and each third of the apotome is the Porcupine chroma, representing 25/24~81/80.

The easiest way to notate Porcupine using a staff is to include a set of accidentals representing the Porcupine chroma. One can repurpose ♯/♭ for these purposes, but this may get confusing as these symbols traditionally represent the apotome rather than one-third of it, so the most common option is to use ^/v for the Porcupine chroma. The quill is thus a diatonic vM2 (read as "downmajor second" or "submajor second"), and 6/5 is a diatonic ^m3 (read as "upminor third" or "supraminor third").

The other possible paradigm is to redefine the ordinals to represent the equable scale (see scales) rather than the 3-limit diatonic scale, thus requiring fewer accidentals to effectively notate Porcupine structures. This requires composers and performers to unlearn familiar interval arithmetic, but it is ultimately more conducive to working with Porcupine as a system long-term. On this page, the equable labels will be used, however the conversion between equable labels and diatonic labels will be given in the interval table below.

The interval chain is shown below (for information on "Archy" and "Aberschismic," see extensions). Octave complements for diatonic and equable labels can be found using standard arithmetic (2-7, 3-6, 4-5), where major corresponds with minor, augmented corresponds with diminished, and ^ corresponds with v. Octave-reduced harmonics and subharmonics are shown in '''bold'''.
{| class="wikitable"
|+Interval Chain
!Quills up
!5-limit ratios
!11-limit ratios
!Archy ratios
!Aberschismic ratios
!Diatonic Label
!Equable Label
|-
|1
|10/9, 27/25
|11/10, 12/11
|
|
|vM2
|P2
|-
|2
|6/5
|11/9
|'''77/64'''
|
|^m3
|m3
|-
|3
|'''4/3'''
|
|'''21/16'''
|
|P4
|m4
|-
|4
|36/25, 40/27
|'''16/11''', 22/15
|35/24, 63/44
|
|vP5
|m5
|-
|5
|8/5
|44/27
|35/22
|
|^M6
|m6
|-
|6
|'''16/9'''
|
|'''7/4'''
|'''105/64'''
|m7
|d7
|-
|7
|48/25, 160/81
|'''64/33''', 88/45
|21/11, 35/18
|63/32
|vP8 / vP1
|d8
|-
|8
|'''16/15'''
|132/125
|21/20
|
|^m2
|d2
|-
|9
|'''32/27'''
|
|7/6
|'''77/64'''
|m3
|d3
|-
|10
|'''32/25'''
|'''128/99'''
|14/11
|'''21/16'''
|vP4
|d4
|-
|11
|'''64/45'''
|176/125
|7/5
|35/24, 63/44
|^d5
|d5
|-
|12
|'''128/81'''
|
|14/9
|35/22
|M6
|d6
|-
|13
|'''128/75'''
|
|56/33, 77/45
|'''7/4'''
|vm7
|dd7
|-
|14
|'''256/135'''
|
|28/15
|21/11, 36/35
|^d8 / ^d1
|dd8 / dd1
|-
|15
|'''256/243''', 156/125
|
|28/27
|21/20
|m2
|dd2
|-
|16
|104/75
|
|
|7/6
|vm3
|dd3
|}

== Supporting systems and extensions ==
Porcupine in the 5-limit is supported by 7, 8, 15, 22, 29, and 37edo; it also has a reasonable tuning in 36edo if we use the flat tuning for prime 5 rather than the sharp tuning borrowed from 12edo. All of these tunings additionally find prime 11 by equating the quill with 11/10~12/11, and equating the minor third with 11/9, making this extension quite ubiquitous. This also equalizes the harmonic segment 9:10:11:12 as a stack of quills, a feature which is relevant to scale construction.

=== Prime 7 ===
Prime 7 is slightly more contentious than 11 and 13 in this respect, as its mapping is not so ubiquitous across these supporting tunings.

==== Archy Porcupine (15 & 22) ====
The most commonly-accepted extension to add prime 7 is via Archy tuning, where 16/9 (found at six quills up) is equated with 7/4, a range which is reasonably supported by 15, 22, and 37edo. Note that all of these tunings have a quill which is on the narrower end of the range, around 160 cents; this provides a somewhat more accurate tuning of 6/5, but a very flat tuning of 4/3.

In this tuning, the Porcupine chroma is equated with 36/35, making the 7/6 - 6/5 - 5/4 - 9/7 segment equidistant.

==== Aberschismic Porcupine (29 & 36c) ====
Another reasonable extension to prime 7 is via Aberschismic tuning, where the Porcupine chroma is equated with 64/63, making the 7/6 - 32/27 - 6/5 segment equidistant; this range is supported primarily by 29edo and 36c. Both of these tunings have a quill around 165 cents, providing near-just 4/3 but a high-error 6/5; this property makes this extension act as something of a foil to the Archy extension, which has the opposite tendency.

Because 5/4 and 11/8 are both relatively high-error in this range, but their difference 11/10 is not, these tunings may be analyzed under the fractional subgroup 2.3.7.11/5 instead; however, this analysis has the detriment of providing no clear low-complexity interpretation for the thirds of the equable scale, which are 6/5~11/9 and 5/4~27/22 in the 2.3.5.7.11 subgroup.

=== Prime 13 ===
Because perfect fifth and the 5-limit thirds of porcupine tend to be tuned quite sharp, 16/13 tends to fall near the middle of 5/4 and 6/5 rather than leaning towards the sharp end; this is especially prevalent in 15edo, where 13/8 has almost precisely 50% error. This makes it rather difficult to convincingly map prime 13 to the porcupine generator chain.

==== (22 & 29) ====
Because 6/5 is equated with 11/9, making a third which is rather sharp of the former ratio and flat of the latter, we may map prime 13 by equating 16/13 with 5/4. Despite its higher error, this mapping for prime 13 is supported by 22, 29, and 36edo, but notably not 37edo.

This range has the property of making 13/8 and 5/4 into octave complements, equating the otonal chord 8:10:13:16 with its own utonal retroversion and making it symmetrical across the octave.

==== (30 & 37) ====
37edo does not support the above mapping, instead splitting the Porcupine chroma into two 65/64 intervals, which we can call a hemichroma; this makes the sequence 6/5 - 16/13 - 5/4 equidistant. This is considered a weak extension, because the 13-limit intervals cannot be reached on the original chain of quills; specifically, the generator for this temperament is the rather complex 160/117, which splits the quill's octave complement in half.

15 and 22edo also come to support this mapping when doubled to 30 and 44edo, adding 16/13 in the newly-acquired gap between the 5-limit thirds. Because of this, we may consider this mapping to be the canonical form of hemi-Porcupine.

== Scales ==

=== MOS Scales ===

==== Equable Scale ====
Because the quill simultaneously represents each constituent element of 9:10:11:12, the equable tetrachord is equalized in Porcupine tunings. Using this tetrachord to form a full scale gives us a 7-note MOS with the signature sssLsss.

The mode names for this scale are given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[7]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|
|Lssssss
|A2 - M3 - M4 - M5 - M6 - P7
|-
|2
|
|sLsssss
|P2 - M3 - M4 - M5 - M6 - P7
|-
|1
|
|ssLssss
|P2 - m3 - M4 - M5 - M6 - P7
|-
|0
|
|sssLsss
|P2 - m3 - m4 - M5 - M6 - P7
|-
| -1
|
|ssssLss
|P2 - m3 - m4 - m5 - M6 - P7
|-
| -2
|
|sssssLs
|P2 - m3 - m4 - m5 - m6 - P7
|-
| -3
|
|ssssssL
|P2 - m3 - m4 - m5 - m6 - d7
|}

==== Octatonic Scale ====
The Porcupine Octatonic scale can be formed most simply by splitting the augmented quill step of the equable scale into a quill plus a Porcupine chroma, ensuring that the scale remains MV2. This one additional note actually creates a rather expressive addition to the harmony, creating double thirds over one mode and double fifths over another.

The modes of this scale are also given by William Lynch.
{| class="wikitable"
|+Modes of Porcupine[8]
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|4
|
|LLLLLLLs
|P2 - m3 - m4 - m5 - m6 - d7 - d8
|-
|3
|
|LLLLLLsL
|P2 - m3 - m4 - m5 - m6 - d7 - P7
|-
|2
|
|LLLLLsLL
|P2 - m3 - m4 - m5 - m6 - M6 - P7
|-
|1
|
|LLLLsLLL
|P2 - m3 - m4 - m5 - M5 - M6 - P7
|-
| -1
|
|LLLsLLLL
|P2 - m3 - m4 - M4 - M5 - M6 - P7
|-
| -2
|
|LLsLLLLL
|P2 - m3 - M3 - M4 - M5 - M6 - P7
|-
| -3
|
|LsLLLLLL
|P2 - A2 - M3 - M4 - M5 - M6 - P7
|-
| -4
|
|sLLLLLLL
|A1 - A2 - M3 - M4 - M5 - M6 - P7
|}

=== Non-MOS Scales ===

==== Zarlino ====
The Zarlino scale can be defined by an alternating stack of 5-limit thirds, 5/4 and 6/5. Porcupine temperament has several features which make it a notable system to support this scale: firstly, the intervals 81/80 and 25/24, which are the differences between steps of the Zarlino scale, are both represented by the Porcupine chroma; this means that there exists only one type of alteration which occurs on all intervals, rather than two types as in 5-limit JI. The steps are thus all consistently qualities of the quill, namely diminished, perfect, and augmented.

Secondly, the "wolf" fourth 27/20 is equated with 11/8 in Porcupine tuning, whose concordance makes it act less like a wolf interval and more like simply a different quality of fourth.

Because the Zarlino scale is chiral -- that is, its retroversion is not a rotation of the original -- we must distinguish left-hand and right-hand modes; in the generator sequence, left-hand modes have a chain which begins with 5/4 and ends with 6/5, whereas right-hand modes have the opposite.
{| class="wikitable"
|+Modes of LH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Lydian
|LMLsLMs
|A2 - M3 - A4 - M5 - A6 - A7
|-
|2
|Aeolian
|LsLMsLM
|A2 - m3 - M4 - M5 - m6 - P7
|-
|1
|Ionian
|LMsLMLs
|A2 - M3 - m4 - M5 - M6 - A7
|-
|0
|Phrygian
|sLMLsLM
|d2 - m3 - m4 - M5 - m6 - P7
|-
| -1
|Mixolydian
|MLsLMsL
|P2 - M3 - m4 - M5 - M6 - d7
|-
| -2
|Locrian
|sLMsLML
|d2 - m3 - m4 - d5 - m6 - d7
|-
| -3
|Dorian
|MsLMLsL
|P2 - d3 - m4 - m5 - M6 - d7
|}
{| class="wikitable"
|+Modes of RH Zarlino
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|3
|Dorian
|LsLMLsM
|A2 - m3 - M4 - M5 - A6 - P7
|-
|2
|Lydian
|LMLsMLs
|A2 - M3 - A4 - M5 - M6 - A7
|-
|1
|Aeolian
|LsMLsLM
|A2 - m3 - m4 - M5 - m6 - P7
|-
|0
|Ionian
|MLsLMLs
|P2 - M3 - m4 - M5 - M6 - A7
|-
| -1
|Phrygian
|sLMLsML
|d2 - m3 - m4 - M5 - m6 - d7
|-
| -2
|Mixolydian
|MLsMLsL
|P2 - M3 - m4 - m5 - M6 - d7
|-
| -3
|Locrian
|sMLsLML
|d2 - d3 - m4 - d5 - m6 - d7
|}

==== Blackdye ====
Just as we extended the equable scale into an octatonic form by turning the augmented quill into a perfect quill plus a Porcupine chroma, we can do the same to the augmented quills in the Zarlino scale to yield a 10-form called Blackdye. The scale remains MV3; the largest step represents a perfect quill, the middle step a diminished quill, and the smallest step a Porcupine chroma.

The Blackdye scale has several benefits over Zarlino, most notably uniting the two chiralities and creating longer unbroken fifth chains. More specifically, Blackdye is made up of two separate fifth chains, with their roots offset from one another by a quill; we can disambiguate similar modes based on which of the two chains contains the root. This gives us five "acute" modes and five "grave" modes, which can be organized further based on the brightness of their pentatonic chains.
{| class="wikitable"
|+Acute modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Aeolian
|LmLsLmLsLs
|P2 - d3 - d4 - m4 - m5 - d6 - dd7 - d7 - d8
|-
|1
|Dorian
|LmLsLsLmLs
|P2 - d3 - d4 - m4 - m5 - M5 - M6 - d7 - d8
|-
|0
|Mixolydian
|LsLmLsLmLs
|P2 - A2 - M3 - m4 - m5 - M5 - m6 - d7 - d8
|-
| -1
|Ionian
|LsLmLsLsLm
|P2 - A2 - M3 - m4 - m5 - M5 - M6 - A6 - A7
|-
| -2
|Lydian
|LsLsLmLsLm
|P2 - A2 - M3 - A3 - A4 - M5 - M6 - A6 - A7
|}
{| class="wikitable"
|+Grave modes of Blackdye
!Brightness
!Name
!Pattern
!Interval Qualities
|-
|2
|Locrian
|mLsLmLsLsL
|d2 - d3 - m3 - m4 - d5 - d6 - m6 - d7 - P7
|-
|1
|Phrygian
|mLsLsLmLsL
|d2 - d3 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
|0
|Aeolian
|sLmLsLmLsL
|A1 - A2 - m3 - m4 - M4 - M5 - m6 - d7 - P7
|-
| -1
|Dorian
|sLmLsLsLmL
|A1 - A2 - m3 - m4 - M4 - M5 - A5 - A6 - P7
|-
| -2
|Mixolydian
|sLsLmLsLmL
|A1 - A2 - AA2 - A3 - M4 - M5 - A5 - A6 - P7
|}
15edo is notable for collapsing the Blackdye scale into a MV2 by merging the diminished quill with the Porcupine chroma, equivalent to tempering out (16/15) / (25/24) = 128/125; this means that its tuning of the scale only has two modes, a unified Acute Mode and a unified Grave Mode.

== Chords and harmony ==

=== Triad Qualities ===
Because equable and Zarlino, the basal scales of Porcupine, are both 7-form and are easy to take as alterations of one another, they can be used to define a system of triadic harmony not dissimilar to the kind seen in Western music.

==== Tertiary Triads ====
{| class="wikitable"
|+Triads of Porcupine
!Quality
!Intervals
!Formula
|-
|Super Augmented
|A3, A5
|A3 + M3
|-
|Augmented
|M3, A5
|M3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Minor
|m3, M5
|m3 + M3
|-
|Diminished
|m3, m5
|m3 + m3
|-
|Sub Diminished
|d3, d5
|d3 + m3
|}
Further qualities as appear in Zarlino will not be enumerated, and can simply be described as alterations of these triads.

==== Quartal Triads ====
In the octatonic scale, the two qualities of thirds do not occur on the same degree; however, the major third and the minor fourth do, which creates an inversion of the major or minor triad in the form of a quartal triad.
{| class="wikitable"
|+Quartal chords of Porcupine
!Quality
!Intervals
!Formula
|-
|Diminished Quartal
|m4, d7
|m4 + m4
|-
|Minor 6/4
|m4, M6
|m4 + M3
|-
|Major 6/3
|M3, M6
|M3 + m4
|-
|Augmented
|M3, A5
|M3 + M3
|}

==== Blackwood Triads ====
The Blackwood scale offers a different generalization of the triads, as the major and minor thirds occur alongside diminished fourths and double-augmented quills.
{| class="wikitable"
|+Triads of Blackwood
!Quality
!Intervals
!Formula
|-
|SusAA2
|AA2, M5
|AA2 + d4
|-
|Diminished
|m3, d6
|m3 + AA2
|-
|Minor
|m3, M5
|m3 + M3
|-
|Major
|M3, M5
|M3 + m3
|-
|Augmented
|M3, d6
|M3 + d4
|-
|Susd4
|d4, M5
|d4 + AA2
|}

=== Functional Harmony ===
Due to the structure of its scales, chord functions in Porcupine temperament are roughly midway between those of Meantone and Neutral temperaments. Because the equable and Zarlino scales both prioritize 5-limit major and minor triads, one can get a lot of functional leeway from simply translating familiar diatonic chord progressions into the temperament.

The quill is a useful leading interval, but its repeated occurrence throughout the equable and octatonic scales make its usage somewhat ambiguous and destabilize it as a directed step; comparatively, the diminished quill seen in the Zarlino scale is much narrower, and occurs much less frequently, allowing it to be used as a directed tension. Because of this property, major and minor triads on the 5 degree both act equally as dominant functions, with the minor triad creating a soft and sweet pull while the major triad creates a powerful and confident pull back to the tonic triad.

Mosh

2026-05-02T23:10:17Z

Unque:

The '''Mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range.

== General scale theory ==
The Mosh scale has seven modes, whose names are given by Andrew Heathwaite.
{| class="wikitable"
|+Modes of Mosh
!Mode Name
!Pattern
!2nd
!3rd
!4th
!5th
!6th
!7th
|-
|Dril
|LsLsLss
|M
|P
|M
|M
|A
|M
|-
|Gil
|LsLssLs
|M
|P
|M
|M
|P
|M
|-
|Kleeth
|LssLsLs
|M
|P
|m
|M
|P
|M
|-
|Bish
|sLsLsLs
|m
|P
|m
|M
|P
|M
|-
|Fish
|sLsLssL
|m
|P
|m
|M
|P
|m
|-
|Jwl
|sLssLsL
|m
|P
|m
|m
|P
|m
|-
|Led
|ssLsLsL
|m
|d
|m
|m
|P
|m
|}
Like all [[MOS]] scales, tunings of Mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\4 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings explicitly do not; this creates a clear divide in how these two types of tunings are handled.

The qualities of scale degrees in soft and hard Mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:
{| class="wikitable"
|+Mosh vs. Diatonic
!Mosh Degree
!Mosh Quality
!Soft
!Hard
|-
| rowspan="2" |2
|minor
|dia n2
|m2
|-
|major
|dia M2
|A2
|-
| rowspan="2" |3
|diminished
|dia m3
|d3
|-
|perfect
|dia n3
|dia M3
|-
| rowspan="2" |4
|minor
|dia P4
|dia d4
|-
|major
|dia A4
|dia AA4
|-
| rowspan="2" |5
|minor
|dia d5
|dia dd5
|-
|major
|dia P5
|dia A5
|-
| rowspan="2" |6
|perfect
|dia n6
|dia m6
|-
|augmented
|dia M6
|dia A6
|-
| rowspan="2" |7
|minor
|dia m7
|dia d7
|-
|major
|dia n7
|dia M7
|}

== Chords and tendency tones ==
{{Wip}}

=== Soft Mosh ===
In the soft tuning of Mosh, we may notice that two generators create an approximation of 3/2, and thus the generator sequence creates useful tertian chords; on most degrees, these chords will be "neutral," splitting that fifth in half. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9). For clarity's sake, these chords will not be given new arcane names, and will simply be referred to with the names of their modes.

Unlike in diatonic, soft Mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this Mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of Mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.

Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the "diminished third" interval of the Mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.

From all this, we can see a presently emergent paradigm for functional harmony in Soft Mosh, a "neutral tonality" which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.
{| class="wikitable"
|+Chords of Kleeth
!Degrees
!Quality
!Functional description
|-
|1 - 3 - 5
|Neutral
|Tonic triad.
|-
|'''2''' - '''4''' - 6
|Led
|Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.
|-
|3 - 5 - '''7'''
|Neutral
|Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.
|-
|'''4''' - 6 - 1
|Neutral
|Expansion of subdominant area. Creates motion from tonic to elsewhere.
|-
|5 - '''7''' - '''2'''
|Neutral
|Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).
|-
|6 - 1 - 3
|Neutral
|Primary subdominant chord, "opposite" of III. Creates motion from tonic to elsewhere.
|-
|7 - '''2''' - '''4'''
|Jwl
|Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.
|}
Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.

The functional paradigm for tonal music in Mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.

Mosh

2026-05-02T20:26:51Z

Unque:

The '''Mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range.

== General scale theory ==
The Mosh scale has seven modes, whose names are given by Andrew Heathwaite.
{| class="wikitable"
|+Modes of Mosh
!Mode Name
!Pattern
!2nd
!3rd
!4th
!5th
!6th
!7th
|-
|Dril
|LsLsLss
|M
|P
|M
|M
|A
|M
|-
|Gil
|LsLssLs
|M
|P
|M
|M
|P
|M
|-
|Kleeth
|LssLsLs
|M
|P
|m
|M
|P
|M
|-
|Bish
|sLsLsLs
|m
|P
|m
|M
|P
|M
|-
|Fish
|sLsLssL
|m
|P
|m
|M
|P
|m
|-
|Jwl
|sLssLsL
|m
|P
|m
|m
|P
|m
|-
|Led
|ssLsLsL
|m
|d
|m
|m
|P
|m
|}
Like all [[MOS]] scales, tunings of Mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\4 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings explicitly do not; this creates a clear divide in how these two types of tunings are handled.

The qualities of scale degrees in soft and hard Mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:
{| class="wikitable"
|+Mosh vs. Diatonic
!Mosh Degree
!Mosh Quality
!Soft
!Hard
|-
| rowspan="2" |2
|minor
|dia n2
|m2
|-
|major
|dia M2
|A2
|-
| rowspan="2" |3
|diminished
|dia m3
|d3
|-
|perfect
|dia n3
|dia M3
|-
| rowspan="2" |4
|minor
|dia P4
|dia d4
|-
|major
|dia A4
|dia AA4
|-
| rowspan="2" |5
|minor
|dia d5
|dia dd5
|-
|major
|dia P5
|dia A5
|-
| rowspan="2" |6
|perfect
|dia n6
|dia m6
|-
|augmented
|dia M6
|dia A6
|-
| rowspan="2" |7
|minor
|dia m7
|dia d7
|-
|major
|dia n7
|dia M7
|}

== Chords and tendency tones ==
{{Wip}}

=== Soft Mosh ===
In the soft tuning of Mosh, we may notice that two generators create an approximation of 3/2, and thus the generator sequence creates useful tertian chords; on most degrees, these chords will be "neutral," splitting that fifth in half. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9). For clarity's sake, these chords will not be given new arcane names, and will simply be referred to with the names of their modes.

Unlike in diatonic, soft Mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this Mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of Mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.

Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the "diminished third" interval of the Mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.

From all this, we can see a presently emergent paradigm for functional harmony in Soft Mosh, a "neutral tonality" which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.
{| class="wikitable"
|+Chords of Kleeth
!Degrees
!Quality
!Functional description
|-
|1 - 3 - 5
|Neutral
|Tonic triad.
|-
|'''2''' - '''4''' - 6
|Led
|Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.
|-
|3 - 5 - '''7'''
|Neutral
|Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.
|-
|'''4''' - 6 - 1
|Neutral
|Expansion of subdominant area. Creates motion from tonic to elsewhere.
|-
|5 - '''7''' - '''2'''
|Neutral
|Expansion of dominant area, containing degree 2 (creating pull towards 3).
|-
|6 - 1 - 3
|Neutral
|Primary subdominant chord, "opposite" of III. Creates motion from tonic to elsewhere.
|-
|7 - '''2''' - '''4'''
|Jwl
|Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.
|}
Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.

The functional paradigm for tonal music in Mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.

User:Unque

2026-05-01T00:37:55Z

Unque:

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Diatonic#Greek diatonic scales|Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely half way between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for composers who want to skip the frivolities and pick up and play with a tuning, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot (temperament)|Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Chthonic harmony]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff I write that I don't believe deserves to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]]

L'Antica Musica

2026-04-30T11:57:48Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent discussing the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct character by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

=== Archicembalo Layout ===
The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size was likely variable throughout. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯.

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; so that if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Tuning ===

==== Vicentino's potential preferences ====
In the first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around [[Meantone#Quarter-comma Meantone|1/4-comma meantone]] would be preferred, providing a justly tuned [[5/4]] as the major third. In the 31-note MOS scale of this tuning, the wolf fifth between G♭-^C♯ is approximately the size of a Pythagorean fifth; this would ensure that the wolf forms of intervals would fall between the sizes of the Pythagorean tuning and the meantone tuning, making it a favorable tuning for providing consonant tunings for those wolves. Additionally, very little size adjustment from 1/4-comma is needed to provide a wolf fifth which is precisely the same size as the generating fifth, thus closing the circle by yielding precisely 31edo.

In the second paradigm, with a fraction of a comma between C-^C, the most likely preferred tuning would have been [[Meantone#1/3-comma Meantone|1/3-comma meantone]], providing a justly tuned 6/5 as the minor third. Since the two keyboards represent two 19-note circles that don't overlap, the wolf fifths are between B♯-G♭ and their comma-altered counterparts, and 1/3-comma meantone tunes this wolf fifth to a size almost precisely the same as the generating fifth (thus approximately [[19edo]] per keyboard). Additionally, Vicentino offers that a root on the lower keyboard may be paired with a fifth and major third from the upper keyboard, a suggestion that would only make sense if the major third were significantly flat of 5/4; in 1/3-comma tuning, we find a major third of approximately 379 cents. Sharpening this third by 1/3 of a syntonic comma in fact yields a tuning which is almost precisely justly tuned 5/4.

==== Modern tuning implications ====
By noticing that 1/3-comma meantone roughly closes at 19 steps along the circle of fifths, we can create a more rotationally-symmetrical version of the 38-note paradigm using the MOS scale 19L 19s, which is two instances of 19edo with their roots offset by a tiny amount. A near-pure 3/2 will yield a hardness of around 8:1, thus allowing us to cleanly approximate this paradigm by considering it a subset of 171edo. While the number is unwieldy, it provides an additional means of extending the paradigm beyond the two keyboards, such that nine keyboards (twenty-seven ranks) would yield a closed circle of 1/3-commas. This would be incredibly inefficient on an analog keyboard instrument of the 16th century, but could be quite simple and useful on isomorphic keyboards of the modern era, such as the Lumatone.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo's first paradigm:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated, can be taken as 7/6 based on relations to other intervals
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

Notation

2026-04-30T06:00:53Z

Unque:

There are many different forms of '''musical notation''' in xenharmony. This page will serve as an introduction to some of the concepts found in xenharmonic notation, and some of the more common types. Note that this does not cover [[Naming#Intervals|interval naming]] systems with no notational counterpart. Notation systems are usually designed for consistent tuning systems such as [[Regular temperament|regular temperaments]], or approximately consistent tuning systems.

In general, a ''notation system'' on this wiki allows for the transcription of musical information such as pitches, note durations, rhythms, etc. Most notation systems here act as extensions or modifications of standard staff notation; however, as this wiki focuses on xenharmony (that is, non-standard intervals), any way of naming and notating pitches (and optionally chords) is generally sufficient to be considered a notation system. A notation system may also include an interval naming system.

== Introduction ==
[[File:Minor.png|thumb|The C minor scale in standard diatonic notation. The notes are C, D, Eb, F, G, Ab, Bb, and C.]]

=== Staff notation ===

==== Staff ====
A ''staff'' is a set of lines (or other reference) that symbols representing notes are placed in relation to. In most notation systems, notes are placed on or in between the lines, resulting in (2n+1) staff positions for n staff lines (without needing additional lines, called ledger lines). Different staff positions represent different pitches. In standard diatonic notation, there are five lines per staff, and therefore 11 distinct staff positions without adding ledger lines.

==== Clef ====
A ''clef'' is a symbol placed on a staff (usually at the very start) to indicate what note a particular staff line represents. The other notes can then be determined from the clef and a characteristic scale.

==== Characteristic scale ====
The characteristic scale is the "default" set of notes that serve as the basis for a system being notated. The notes of the characteristic scale are represented in text by ''nominals'' (written symbols representing notes, usually alphabetic, numerical, or drawn from some other extramusical sequence) and on the staff by distinct staff positions. For standard notation, the characteristic scale is [[Diatonic#MOS diatonic|mosdiatonic]] in the mode LsLLsLL on A = 440Hz. The nominals are ABCDEFG, and then A an octave up.

Note that the actual order of the alphabet may be of secondary importance to some musicians. While retaining it (as favored in KISS and diamond-mos notation) may aid intuition for people getting into music theory through xenharmony, some experienced musicians may prefer to treat other parts of their relationships as more fundamental in nonstandard notation schemes. The letters are thus in those cases arranged in a non-alphabetical order based on pitch or chain position, more letters may be added or existing letters removed, and the series may be otherwise altered to support the intuition of the arranger.

==== Accidentals ====
[[File:C minir.png|thumb|The key signature for C minor in standard diatonic notation.]]
Accidentals are the term for any symbol representing an alteration to a nominal. In standard notation, the # accidental raises a note by a chromatic semitone, and the b accidental lowers by a chromatic semitone. Microtonal notation systems usually add new accidentals, rather than changing the set of nominals. For example, the ^ and v accidentals (arrows) raise and lower by an edostep. Note that accidentals other than the standard # and b (and derivatives) may instead be called ''inflections'' in some sources, wherein the category as a whole are ''alterations''. Accidentals generally continue to apply to the same staff position until the next barline is reached.

==== Key signatures ====
A key signature is a set of accidentals placed at the beginning of the sheet music that apply to the whole piece and serve as the "default" accidentals to apply to staff positions. These accidentals also apply to notes separated by octaves from their position, which is not true of other accidentals. In general, there are only about 15 key signatures in use (based on the circle of fifths) representing the various transpositions of the diatonic scale. However, key signatures are theoretically possible for any set of defined alterations from the base notes of the characteristic scale, so that (for instance) one could specify a non-MOS diatonic scale with alterations to the standard version. (Note that otherwise, accidentals work normally, so that if the key signature specifies C#, placing a b accidental before a C note will read as Cb, not C (which is what the natural accidental is used for).[[File:5555.png|thumb|201x201px|A key signature using FJS accidentals to specify the 5-limit zarlino scale in C major.]]

==== Note symbols ====
While not directly pitch- or interval-related, an important consideration in standard notation is that note duration is represented by differences in symbol shape, rather than by physical size, which allows the precise reading of durations.

=== Annotations ===
Figured bass is a prominent form of notation accompanying a standard score that indicates what notes and/or chords to play, most commonly used for analytical clarity, especially where performers are expected to improvise. It utilizes the numbers 1 through 7 to indicate the degrees of a scale. One notable constraint of figured bass is that the symbols for degrees must be singular digits or letters, disallowing constructions like "1/2" and "15"; this makes it rather painful to generalize to microtonal scales while keeping numeric indices for intuition. One system works around this by utilizing the letters of the Greek alphabet, which allow for 24 distinct degrees to be notated. Another option is to utilize alternative systems to figured bass, such as placing chord symbols above the score.

=== Scale degrees vs. intervals ===
A scale degree refers to a note defined by an offset from the tonal center, tonic, or otherwise a fixed reference point that does not change throughout a song or section. Scale degrees are conventionally notated by specifying alterations from the degrees of the characteristic scale. Intervals are the offsets themselves, which may be applied either to the tonal center (specifying a scale degree) or to any other pitch. Intervals are notated with qualities such as major or minor, so that "m3" ("minor third") is an interval while "b3" ("flat three") is a degree.

However, note that this is somewhat disputed:

* The established convention, with which Unque agrees, is to notate degrees with alterations.
* However, Inthar, HKM, and Vector prefer the use of qualities to notate degrees, as if they are intervals.
** Wad Wizard uses alterations as if they are qualities.

Additionally, conceptualization schemes vary massively as to the difference between scale degrees and intervals:

* Unque characterizes scale degrees as "points in space" and intervals as "types of motion within that space", characterizing pitch space relative to the tonal center as a form of "secondary absolute" space.
* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.
* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa.
* Inthar treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors.

== Microtonal notations ==

=== Diatonic notation ===
[[Diatonic notation|This page]] covers commonly accepted extensions to standard diatonic notation used in microtonality. The 7 nominals C, D, E, F, G, A, B form the C major scale, and # and b raise and lower notes by a chroma.

=== KISS notation ===
KISS notation generalizes the principles of diatonic notation to any arbitrary [[MOS]], involving alphabetic note naming based on the MOS in question, and accidentals based on the difference between that [[MOS]]' step sizes. KISS notation refers to a family of closely-related notation schemes.

==== Standard KISS notation ====
KISS notation, in its standard form, uses numbers to name notes, but the standard # and b symbols as accidentals, and places the treble and bass clefs around the middle note as they are in diatonic notation.

==== Quasi-diatonic MOS notation ====
Quasi-diatonic MOS notation uses letters as nominals, starting from A at 440Hz as the final note of the darkest mode of the scale (much as A is the final note of the Locrian scale before it repeats at B) uses standard accidentals, and has a custom system of clefs corresponding to each nominal.

==== Diamond-MOS notation ====
Diamond-MOS notation aims to disambiguate itself from diatonic notation by using different nominals (from J onwards), different accidentals (& = #, @ = b), and a custom clef system that makes it visually apparent where J is in different octaves on the staff and what MOS is being used.

=== Just notation systems ===
Conventionally, just intonation notation systems rely on a series of accidentals that move up or down a "formal comma" for each prime from a basic Pythagorean interval.

Common formal commas include 81/80 for 5 (the difference between the classical and diatonic major thirds), 64/63 for 7 (the difference between the diatonic and septimal major thirds), 33/32 (the difference between the perfect fourth and 11th harmonic) for 11, and 513/512 (the difference between the diatonic minor third and 19th harmonic) for 19.
{| class="wikitable"
|+
!
!FJS
!HEJI
!Neutral FJS
!Ben Johnston
|-
!Basic diatonic
| colspan="3" |Pythagorean
|Zarlino
|-
!7
| colspan="3" |64/63
|36/35
|-
!11
| colspan="2" |33/32
|sqrt(243/242)
|33/32
|-
!13
|1053/1024
|27/26
|sqrt(507/512)
|65/64
|-
!17
|4131/4096
|2187/2176
|4131/4096
|51/50
|-
!19
| colspan="3" |513/512
|96/95
|-
!Accidentals used
|Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction
|A selection of symbols
|Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction
|Numbers and inverted numbers, with otonal intervals all inflected in the same direction, except for the syntonic comma (+/-) and prime 11 (^/v),
|-
!Other notes
|All formal commas are derived systematically. Each prime harmonic is inflected from the simplest Pythagorean interval within 54.1 cents (or 57.8 cents).
|
|Similar to FJS, but includes "dicot" intervals for more intuitive notation of neutral intervals. Each prime harmonic is inflected from the simplest Pythagorean or neutral interval within 33.4 cents.
|Uses Zarlino diatonic as nominals, 36/35 as the accidental for 7, and 96/95 as the accidental for 19.
|}

==== Color notation ====
Color notation is a notation system that uses a series of syllables to notate just intervals, fully replacing the standard interval qualities. An interval's degree is determined from its mapping to 24edo.
{| class="wikitable"
|+
!Syllable
!Intervals
!Notes
|-
|la
|Raised by a chromatic semitone
|Not used, # is used instead
|-
|sa
|Lowered by a chromatic semitone
|Not used, b is used instead
|-
|wa
|3-limit, no alterations
|3-limit, no alterations
|-
|ru
|7 under, generally supermajor
| +64/63
|-
|yo
|5 over, generally ptolemaic major
| -81/80
|-
|gu
|5 under, generally ptolemaic minor
| +81/80
|-
|zo
|7 over, generally subminor
| -64/63
|-
|lu
|11 under
| -33/32, +729/704
|-
|lo
|11 over
| +33/32, -729/704
|}

== Ambiguous just intonation intervals ==
Systems of conceptualizing just intonation in relation to the diatonic scale disagree on the identities of certain intervals.

* 17/16 is a minor second in the Functional Just System and an augmented unison in the Helmholtz-Ellis system; all intervals of 17 are affected by this disparity. It is between the two in terms of size - the minor second is a simpler diatonic interval but it is closer in size to the augmented unison.
* 14/11 is often conceptualized as a major third (specifically, a neogothic major third), but most notation systems that use formal commas recognize it as an inflected perfect fourth. This is because it is the difference between 11/8 and 7/4 - 11/8 is about as sharp as a perfect fourth can reasonably be, and 7/4 is about as flat as a minor seventh can reasonably be. Therefore, their difference, another perfect fourth, is narrowed considerably. Systems like sagittal notation attempt to accurately capture 14/11's melodic quality, at the cost of sacrificing some stacking logic.

Magic

2026-04-13T18:22:54Z

Unque: Target tunings

{{Infobox regtemp
| Title = Magic
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[3125/3072]] (5-limit); [[225/224]], [[245/243]] (7-limit)
| Edo join 1 = 19 | Edo join 2 = 22
| Mapping = 1; 5 1 12
| Generators = 5/4
| Generators tuning = 380.5
| Optimization method = CWE
| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]
| Odd limit 1 = 5 | Mistuning 1 = 5.9 | Complexity 1 = 7
| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 13
}}

'''Magic''' (19 & 22) is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5. It is a 3-cluster temperament, as indicated by the edo join (22 - 19 = 3).

== Interval chain ==
In the following table, odd harmonics 1–15 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 380.5
| '''5/4'''
|-
| 2
| 760.9
| 14/9
|-
| 3
| 1141.4
| 27/14, 31/16
|-
| 4
| 321.8
| 6/5, 29/24
|-
| 5
| 702.3
| '''3/2'''
|-
| 6
| 1082.7
| '''15/8''', 28/15
|-
| 7
| 263.2
| 7/6
|-
| 8
| 643.7
| 36/25
|-
| 9
| 1024.1
| 9/5, 29/16
|-
| 10
| 204.6
| '''9/8'''
|-
| 11
| 585.0
| 7/5
|-
| 12
| 965.5
| '''7/4'''
|-
| 13
| 145.9
| 35/32
|}
<nowiki/>* In 7-limit CWE tuning

== Extensions ==
Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.

{{Adv|This can be seen from the following [[S-expression]] for the Magic comma:}}

3125/3072
= (25/24)^2/(16/15)
= (25/24)*(25/24)/(32/31*31/30)
= (25/24)S25*S26*S27*S28*S29*S30/(32/31)
= (S25*S26*S27*S28*S29*S30)^2*S31
= (S15*S28*S29*S30)^2*S31

{{Adv|[[canonical extension|structurally inducing]] the above 2.3.5.7.29.31 extension.}}

== Scales ==
{{Wip}}

=== MOS scales ===
Because three generators just barely fall short of an octave, the [[MOS]] scales generated by Magic are quite commatic. The Magic Diesis (a small step simultaneously representing 25/24, 28/27, 36/35, and 128/125) is the s step of all MOS scales up to the 22-form

==== 7-form ====
[[File:22edo Magic Cadence.png|thumb|An example of the Magic Cadence (III - I) in 22edo, written in SATB format with native fifths notation.]]
The 7-note Magic scale has the pattern 3L 4s, sometimes called [[Mosh]], with a large step of 6/5. The disparity in size between the two types of steps grants a quality to stepwise melodies that some find to sound awkward or lurching; while some composers may prefer or intend such a sound, those who do not are cautioned to avoid long stepwise runs for fear of creating an aimless and chromatic-sounding melodic core.

The 4:5:6 triad can be found on the tonic in two of the modes: LsLsLss and LsLssLs; these modes place the triad on degrees 1, 3, and 4. Generalizing from this, we can see that the other main type of triad that occurs on these degrees is 1/1 - 5/4 - 9/7, with an additional 1/1 - 16/15 - 9/7 triad in the ssLsLsL mode.

The LsLsLss mode has the clearest utility, with a clear cadence from the III chord to the I chord; we may consider this motion to be Magic temperament's analog to the Perfect Cadence from [[Meantone]][7]. This cadence creates contrary motion, with the 6 and 5 of the III chord resolving respectively to the 8 and 4 of the I chord, and the 3 sustained between both chords creates a strong sense of continuity.

The LsLssLs mode contains an inverse version of this cadence, which can be seen as a Magic analog to Meantone's Plagal Cadence.

=== Other scale forms ===
Due to the juxtaposition of the Magic Diesis against significantly larger step sizes, it is often desirable to use a non-MOS structure in Magic, such as [[Blackdye]] or the [[5-odd-limit|5-odd Diamond]].

== List of tunings ==

=== EDO patent vals ===
The following patent vals support 2.3.5 Magic. Vals that are contorted in 2.3.5 are not included.
{| class="wikitable sortable"
! Edo
!Extension to 7|| Generator || Fifth
|-
||3
| ||400.000||800.000
|-
||25
| ||384.000||720.000
|-
||22
|19 & 22||381.818||709.091
|-
||107
| ||381.308||706.542
|-
||85
|19 & 22||381.176||705.882
|-
||63
|19 & 22||380.952||704.762
|-
||104
| ||380.769||703.846
|-
||41
|19 & 22||380.488||702.439
|-
||60
|19 & 22||380.000||700.000
|-
||79
| ||379.747||698.734
|-
||19
|19 & 22||378.947||694.737
|-
||35
| ||377.143||685.714
|-
||16
| ||375.000||675.000
|}

=== Target-tunings ===
The following are some examples of useful target tunings available in Magic temperament.
{| class="wikitable"
!Just interval
!Comma fraction
!Generator
!Fifth
|-
|5/4
|0-comma
|386.314
|731.569
|-
|15/8
|1/6-comma
|381.378
|706.891
|-
|3/2
|1/5-comma
|380.391
|701.955
|-
|9/5
|2/9-comma
|379.733
|698.665
|-
|5/3
|1/4-comma
|378.910
|1694.552
|-
|25/24
|1/3-comma
|376.443
|682.213
|}
{{cat|Temperaments}}{{Navbox regtemp}}

Meantone

2026-04-13T17:21:00Z

Unque: Greatly expanded chords section

[[File:Meantone.png|thumb|Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.]]
'''Meantone''', or rarely '''Syntonic''' or '''Didymus,''' is a widespread historical [[temperament]] that forms the basis of Western music theory, where the [[Perfect fifth|fifths]] are flattened to about 696[[Cent|c]] to produce a [[diatonic major third]] tuned to roughly [[5/4]], enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a [[regular temperament]], where the period is the [[octave]], the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the '''syntonic comma''', which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.

As a monocot temperament, Meantone can be notated with standard [[diatonic notation]], and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. Meantone is a 7-form temperament, and is tuned well around the golden tuning of diatonic. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes. [7 & 12] is thus the ''edo join'' for meantone.

== Other primes ==
If an unmapped (not equated to a stack of anything else, as [[Classical major third|5/4]] is in blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7.

The edo join [7 & 12] results in an exotempered extension called ''dominant'' where 7/4 and 9/5 are equated, and is tuned best around Pythagorean tuning.

More accurate extensions of Meantone's diatonic structure to include other primes follow.

==== 7/4 as the augmented sixth (12 & 19) ====
This is the primary extension of Meantone to the 7-limit, where 7/4 is the augmented sixth (C-A#, +10 fifths). It is best tuned with the generator around 696 cents. 5/4 is 384 cents, and 7/4 is 960 cents. It is notable for being the most accurate extension, as well as containing the [[Golden sequences and tuning|golden tuning]] of the diatonic scale, and thus a melodically convenient chromatic and enharmonic scale. This means that the [[augmented diesis]] 128/125, is equated with the septimal quartertone 36/35, and the 5-limit supermajor and subminor intervals are equated with their septimal counterparts.

However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)

===== 11-limit[12 & 19] =====
The 11-limit form of 12 & 19 is an exotemperament called ''meanenneadecal'', which tunes 11/8 very sharp (because both 12edo and 19edo do so). More accurate extensions are below.

===== 11/8 as the double-augmented third (12 & 31) =====
This is best tuned around 697 cents, and places 11/9 as the double-augmented second (C-Dx, +16 fifths) and conflates 14/11 with [[Septimal supermajor third|9/7]] placed as the diminished fourth (C-Fb, -8 fifths). 13/8 is mapped to the double-diminished seventh (C-Bbb, -9 fifths).

===== 11/8 as the double-diminished fifth (19 & 31) =====
This is best tuned around 696 cents, and places 11/9 as the double-diminished fourth (C-Fbb, -15 fifths). 13/8 is mapped to the double-augmented fifth (C-Gxx, +15 fifths).

==== 7/4 as the diminished seventh (19 & 26) ====
This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo. It is a 7-cluster temperament, as indicated by the edo join (26 - 19 = 7).

== Chords ==
Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major triad P1-M3-P5, thus equating the [[Diatonic#MOS diatonic|MOS diatonic scale]] with the 5-limit tuning of [[Diatonic#Greek diatonic scales|Ancient Greek diatonic]] and allowing for 5-limit consonances to be easily accessed within a continuous circle of fifths. Modern Western music theory, which is derived in large part from meantone practice, treats triadic harmony (chords made by stacking two thirds over a root) as the basis of concordance, as the only way to fit three [[5-odd-limit]] intervals in one octave is via some permutation of 4/3, 5/4, and 6/5, which will always make some rotation or retroversion of 4:5:6.

The major and minor seventh chords in meantone diatonic can be enumerated as 8:10:12:15 and 10:12:15:18 respectively, and the half-diminished seventh chord as 25:30:36:45, or 35:42:50:63 in septimal meantone. Additionally, the 5:6:7:9 chord is available as P1-m3-A4-m7.

During the late Renaissance era, septimal meantone tunings were the basis of Augmented Sixth chords. The Italian Sixth chord can be enumerated as 4:5:7, with the intervals of a root, a major third, and an augmented sixth; the German Sixth chord adds an additional interval 3/2 above the root, providing a full 4:5:6:7, whereas the French Sixth chord adds the augmented fourth of 7/5, making a 20:25:28:35 chord.

The septimal triads, 6:7:9 and 14:18:21, can additionally be found at P1-A2-P5 and P1-d4-P5 respectively. These can be further extended to the septimal seventh chords, 12:14:18:21 and 14:18:21:27, which are respectively P1-A2-P5-A6 and P1-d4-P5-d1 in septimal meantone.

Meantone also contains an [[Collection of chords#Essentially tempered chords|essentially tempered chord]], where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 [[Wolf interval|wolf]] fifth).

== Tunings ==
As essentially the only temperament that is both [[regular]] and attested outside [[xenharmony]], Meantone has a number of historical tunings that today correspond to various extensions and approximate edos. Here, "comma" refers to the syntonic comma.

=== Quarter-comma Meantone ===
This tunes the fifth 1/4-comma flat, to a size of 696.57 cents. It has a just 5/4, and a step ratio of 1.65 (quasisoft). It approximates [[31edo]], and is often (rather insultingly to the rest of 31edo) seen as the latter's primary feature. It extends to 11-limit 19 & 31.

=== 1/3-comma Meantone ===
This is the tuning of the fifth to 694.78 cents, which has a just 6/5 and is extremely close to [[19edo]], having a step ratio of 1.503 (~monosoft). As a result, it does not cleanly extend to the 11-limit, although as it is slightly sharp of 19edo it does technically extend to 7-limit 12 & 19.

=== 2/5-comma Meantone ===
This is very close to the [[45edo]] tuning of Meantone, tuning the fifth 693.35 cents and having a just 27/25 (note that 27/25 is tempered together with 16/15 in this system, resulting in a sharp minor second). As a result of the flat tuning, this extends to Flattone, rather than to Septimal Meantone. Its step ratio is 1.401 (parasoft).

=== 1/2-comma Meantone ===
This is close to 33edo's diatonic tuning, which is not Meantone. As a result, it can be considered the lower bound of Meantone's tuning, where the tone is tuned to a just 10/9. It tunes the fifth to 691.2 cents. Its step ratio is 1.26 (ultrasoft).

=== 2/7-comma Meantone ===
This is the tuning of Meantone situated roughly between 50edo's and 69edo's tunings, with a fifth of 695.81 cents and a step ratio of 1.584 (quasisoft). Consequently, it extends to Septimal Meantone, but with a rather poor approximation of 7/4. However, it tunes other septimal intervals like 9/7 and 7/6 more accurately. It tunes 25/24 purely, and can thus be considered a compromise between 1/4-comma's perfect 5/4 and 1/3-comma's perfect 6/5.

=== 1/5-comma Meantone ===
This tuning of Meantone equalizes the error on 3/2 and 5/4, tuning the former to 697.65 cents and the latter to 390.61 cents. Equivalently, it tunes 16/15 purely. Its step ratio is 1.748 (minisoft), and consequently it is well approximated by 43edo.

=== 1/11-comma Meantone ===
This tuning of Meantone is almost perfectly approximated by 12edo, having a fifth tuning of nearly exactly 700 cents and a step ratio of nearly exactly 2 (~basic). 12edo by definition lowers the fifth by 1/12 of a Pythagorean comma; setting 1/12 of a Pythagorean comma to 1/11 of a syntonic comma is done in edos such as [[34edo#612edo|612edo]].

=== [[31edo]] ===

=== [[19edo]] ===

=== [[12edo]] ===

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!7-limit extensions
!11-limit extensions
!Generator tuning
|-
|5
|Dominant
|
|720c
|-
|12
|Dominant, Septimal Meantone
|[12 & 31], [12 & 19]
|700c
|-
|67
|
|
|698.5c
|-
|55
|
|
|698.2c
|-
|98
|
|
|698c
|-
|43
|Septimal Meantone
|[12 & 31]
|697.7c
|-
|117
|
|[12 & 31]
|697.4c
|-
|74
|Septimal Meantone
|[12 & 31]
|697.3c
|-
|105
|Septimal Meantone
|[12 & 31]
|697.1c
|-
|31
|Septimal Meantone
|[12 & 31], [19 & 31]
|696.8c
|-
|81
|Septimal Meantone
|[19 & 31]
|696.3c
|-
|50
|Septimal Meantone
|[19 & 31]
|696c
|-
|69
|
|[19 & 31]
|695.7c
|-
|88
|
|
|695.5c
|-
|19
|Septimal Meantone, Flattone
|[12 & 19], [19 & 31], (Flattone)
|694.7c
|-
|45
|Flattone
|(Flattone)
|693.3c
|-
|26
|Flattone
|(Flattone)
|692.3c
|-
|7
|Dominant, Flattone
|(Flattone)
|685.7c
|}

{{Navbox regtemp}}

{{Cat|Temperaments}}

Magic

2026-04-13T17:11:07Z

Unque: Started section on scales

{{Infobox regtemp
| Title = Magic
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[3125/3072]] (5-limit); [[225/224]], [[245/243]] (7-limit)
| Edo join 1 = 19 | Edo join 2 = 22
| Mapping = 1; 5 1 12
| Generators = 5/4
| Generators tuning = 380.5
| Optimization method = CWE
| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]
| Odd limit 1 = 5 | Mistuning 1 = 5.9 | Complexity 1 = 7
| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 13
}}

'''Magic''' (19 & 22) is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5. It is a 3-cluster temperament, as indicated by the edo join (22 - 19 = 3).

== Interval chain ==
In the following table, odd harmonics 1–15 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 380.5
| '''5/4'''
|-
| 2
| 760.9
| 14/9
|-
| 3
| 1141.4
| 27/14, 31/16
|-
| 4
| 321.8
| 6/5, 29/24
|-
| 5
| 702.3
| '''3/2'''
|-
| 6
| 1082.7
| '''15/8''', 28/15
|-
| 7
| 263.2
| 7/6
|-
| 8
| 643.7
| 36/25
|-
| 9
| 1024.1
| 9/5, 29/16
|-
| 10
| 204.6
| '''9/8'''
|-
| 11
| 585.0
| 7/5
|-
| 12
| 965.5
| '''7/4'''
|-
| 13
| 145.9
| 35/32
|}
<nowiki/>* In 7-limit CWE tuning

== Extensions ==
Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.

{{Adv|This can be seen from the following [[S-expression]] for the Magic comma:}}

3125/3072
= (25/24)^2/(16/15)
= (25/24)*(25/24)/(32/31*31/30)
= (25/24)S25*S26*S27*S28*S29*S30/(32/31)
= (S25*S26*S27*S28*S29*S30)^2*S31
= (S15*S28*S29*S30)^2*S31

{{Adv|[[canonical extension|structurally inducing]] the above 2.3.5.7.29.31 extension.}}

== List of patent vals ==
The following patent vals support 2.3.5 Magic. Vals that are contorted in 2.3.5 are not included.
{| class="wikitable sortable"
! Edo
!Extension to 7|| Generator || Fifth
|-
||3
| ||400.000||800.000
|-
||25
| ||384.000||720.000
|-
||22
|19 & 22||381.818||709.091
|-
||107
| ||381.308||706.542
|-
||85
|19 & 22||381.176||705.882
|-
||63
|19 & 22||380.952||704.762
|-
||104
| ||380.769||703.846
|-
||41
|19 & 22||380.488||702.439
|-
||60
|19 & 22||380.000||700.000
|-
||79
| ||379.747||698.734
|-
||19
|19 & 22||378.947||694.737
|-
||35
| ||377.143||685.714
|-
||16
| ||375.000||675.000
|}

== Scales ==
{{Wip}}

=== MOS scales ===
Because three generators just barely fall short of an octave, the [[MOS]] scales generated by Magic are quite commatic. The Magic Diesis (a small step simultaneously representing 25/24, 28/27, 36/35, and 128/125) is the s step of all MOS scales up to the 22-form

==== 7-form ====
[[File:22edo Magic Cadence.png|thumb|An example of the Magic Cadence (III - I) in 22edo, written in SATB format with native fifths notation.]]
The 7-note Magic scale has the pattern 3L 4s, sometimes called [[Mosh]], with a large step of 6/5. The disparity in size between the two types of steps grants a quality to stepwise melodies that some find to sound awkward or lurching; while some composers may prefer or intend such a sound, those who do not are cautioned to avoid long stepwise runs for fear of creating an aimless and chromatic-sounding melodic core.

The 4:5:6 triad can be found on the tonic in two of the modes: LsLsLss and LsLssLs; these modes place the triad on degrees 1, 3, and 4. Generalizing from this, we can see that the other main type of triad that occurs on these degrees is 1/1 - 5/4 - 9/7, with an additional 1/1 - 16/15 - 9/7 triad in the ssLsLsL mode.

The LsLsLss mode has the clearest utility, with a clear cadence from the III chord to the I chord; we may consider this motion to be Magic temperament's analog to the Perfect Cadence from [[Meantone]][7]. This cadence creates contrary motion, with the 6 and 5 of the III chord resolving respectively to the 8 and 4 of the I chord, and the 3 sustained between both chords creates a strong sense of continuity.

The LsLssLs mode contains an inverse version of this cadence, which can be seen as a Magic analog to Meantone's Plagal Cadence.

=== Other scale forms ===
Due to the juxtaposition of the Magic Diesis against significantly larger step sizes, it is often desirable to use a non-MOS structure in Magic, such as [[Blackdye]] or the [[5-odd-limit|5-odd Diamond]].

{{cat|Temperaments}}{{Navbox regtemp}}

File:22edo Magic Cadence.png

2026-04-13T16:39:57Z

Unque:

22edo Magic cadence, written in SATB format with native fifths notation.

Chthonic harmony

2026-04-07T22:02:05Z

Unque: The interval conversion between 7- and 14-form can be found two inches above the chord table if you just scroll up. Not to devalue the decimal notation scheme or anything, just that it takes up a lot of extra horizontal space here.

'''Chthonic harmony''', sometimes called '''semiquartal''' or '''latal harmony''' (''latal''/''latus'', pl. ''lati'', means "1-step of the 5-form", and is borrowed from Latin ''lātus'' (1st declension) "carried"), is a type of chord structure useful in many styles of xenharmonic music. It serves as an analog for tertian harmony that splits the [[perfect fourth]] rather than the [[Perfect fifth|fifth]].

== Chthonic intervals ==
The term "chthonic" is derived from the name of an [[interordinal]] interval which splits the perfect fourth into two roughly-equal parts. When this division is not equal, one may call the larger chthonic "major" and the smaller one "minor," by analog to diatonic ordinals; when it is equal or nearly so, such as the case with [[15/13]], we may call it a "neutral" or "perfect" chthonic. Where further types of chthonics exist, one may use [[ups and downs]] alteration (in an EDO) or [[ADIN]] (in any tuning) to describe the qualities.

=== In just intonation ===
In JI structures, chthonic intervals will usually have either a numerator or denominator divisible by 3, which creates a relatively concordant triad bounded by the perfect fourth.
{| class="wikitable"
|+Chthonic intervals in JI
!Ratio
!Cents
!ADIN
!Triad
!Complement
|-
|15/13
|247.7
|Neutral/Perfect
|39:45:52
|52/45
|-
|7/6
|266.9
|Submajor
|6:7:8
|8/7
|-
|32/27
|294.1
|Nearmajor
|27:32:36
|9/8
|-
|6/5
|315.6
|Farmajor
|15:18:20
|10/9
|-
|11/9
|347.4
|Ultramajor
|9:11:12
|12/11
|}

== On notation and stacking ==
Diatonic ordinals are typically notated by a symbol (P, M, m, etc.) to display its quality, plus a 1-indexed number (5, 3, 7, etc.) to display its place in the scale. The quality symbols can be maintained for describing interordinals. However, because these numbers are necessarily integers, it is difficult to slot interordinal intervals in between; while using decimal points has been suggested (and allows for intuitive placement of the intervals without much memorization), this comes with a number of issues, namely providing difficulty for reading and writing chords, figured bass, and scale degrees.

To fix this issue, Greek numerals will be used on this page to represent ordinals and interordinals together, based on the degrees (1-indexed) of the 14-form; odd-numbered degrees will therefore be equivalent to ordinals of the 7-form, and as such can be notated with either Greek or Arabic numerals. This has the advantage of being single characters; however, the ordering of the intervals is not immediately obvious to someone who does not know the Greek alphabet and may require memorization.

Just as there is a clear circle created by a stack of fifths or thirds, so too is there one created by a stack of chthonics; two chthonics will create some type of fourth, three will create a [[cocytic]] (fifth-inter-sixth), four will create a seventh, etc.
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+Interordinal notation
!Chthonics up
!Interval
!Arabic
!Greek
!14-form
|-
|0
|Unison
|1
|α
|0\14
|-
|1
|Chthonic
|2.5
|δ
|3\14
|-
|2
|Fourth
|4
|ζ
|6\14
|-
|3
|Cocytic
|5.5
|ι
|9\14
|-
|4
|Seventh
|7
|μ
|12\14
|-
|5
|Interizer†
|1.5
|β
|1\14
|-
|6
|Third
|3
|ε
|4\14
|-
|7
|Tritone
|4.5
|η
|7\14
|-
|8
|Sixth
|6
|κ
|10\14
|-
|9
|Antiïnterizer†
|7.5
|ν
|13\14
|-
|10
|Second
|2
|γ
|2\14
|-
|11
|Naiadic
|3.5
|ϛ
|5\14
|-
|12
|Fifth
|5
|θ
|8\14
|-
|13
|Ouranic
|6.5
|λ
|11\14
|}
†The term "interizer" is coined by [[User:Inthar|Inthar]], as it is the interval which separates a diatonic ordinal from an interordinal; thus it can be used to "inter-ize" those ordinals. Because it is also half of a diatonic minor second, it may also be considered a [[quartertone]], though this name may not be preferred, as four of them do not necessarily make a wholetone.

== Chthonic chords ==
Chords can be constructed most intuitively by alternating a stack of a chthonic with its fourth complement, just as tertian chords tend to follow an alternating pattern of a third with its fifth complement.

Where tertian harmony is most practical with chords of three and four notes, chthonic harmony is most practical with chords of four or five notes. Because a chthonic added to a fourth will always yield a cocytic, chords can be named by the type of chthonic and the type of cocytic, just as tertian chords can be named for the quality of their third and seventh. Unless otherwise specified, the fourth is always assumed to be perfect, and the seventh minor.

To display how these qualities propagate, this example will use only two types of chthonics.
{| class="wikitable"
|+Cocytic (four-note) Chords
!Quality
!Intervals
!Formula
|-
|Major Cocytic
|Mδ, Pζ, Mι
|Mδ + mδ + Mδ
|-
|Minor Cocytic
|mδ, Pζ, mι
|mδ + Mδ + mδ
|-
|Major Minor Cocytic
|Mδ, Pζ, mι
|Mδ + mδ + mδ
|-
|Minor Major Cocytic
|mδ, Pζ, Mι
|mδ + Mδ + Mδ
|-
|Half-Augmented Cocytic
|Mδ, Aζ, Mι
|Mδ + Mδ + mδ
|-
|Augmented Cocytic
|Mδ, Aζ, Aι
|Mδ + Mδ + Mδ
|-
|Half-Diminished Cocytic
|mδ, dζ, mι
|mδ + mδ + Mδ
|-
|Diminished Cocytic
|mδ, dζ, dι
|mδ + mδ + mδ
|}
== Chthonic scales ==
The scales which most prominently feature chthonic harmony are, as one might suspect, those which are generated by chthonic intervals, just as the diatonic scale can be generated by an alternating stack of major and minor thirds. Also just like the diatonic scale, many chthonic scales have alternative constructions as well, analogous to the diatonic circle of fifths construction.

=== Diasem ===
''See also: [[Aberrisma#Quasi-diatonic aberrismic scales|Quasi-diatonic aberrismic scales]]''

The most fundamental of chthonic scales is Diasem, a 9-note scale created by alternating a stack of major and minor chthonic intervals, such as 7/6 with 8/7. This construction ensures a maximum number of major and minor cocytic chords.

Diasem has two possible chiralities, based on whether 8/7 or 7/6 was stacked first; each chirality has nine unique modes.
{| class="wikitable sortable"
|+Left-Hand Diasem Modes
! rowspan="2" |Pattern
! colspan="8" |Degrees
! rowspan="2" |Cocytic Chord
|-
!1\9
!2\9
!3\9
!4\9
!5\9
!6\9
!7\9
!8\9
|-
|LsLmLsLmL
|9/8
|8/7
|9/7
|4/3
|3/2
|32/21
|12/7
|16/9
|Minor
|-
|sLmLsLmLL
|64/63
|8/7
|32/27
|4/3
|256/189
|32/21
|128/81
|16/9
|Minor
|-
|LmLsLmLLs
|9/8
|7/6
|21/16
|4/3
|3/2
|14/9
|7/4
|63/32
|Major
|-
|mLsLmLLsL
|28/27
|7/6
|32/27
|4/3
|112/81
|14/9
|7/4
|16/9
|Major
|-
|LsLmLLsLm
|9/8
|8/7
|9/7
|4/3
|3/2
|27/16
|12/7
|27/14
|Minor (wolf cocytic)
|-
|sLmLLsLmL
|64/63
|8/7
|32/27
|4/3
|3/2
|32/21
|12/7
|16/9
|Minor
|-
|LmLLsLmLs
|9/8
|7/6
|21/16
|189/128
|3/2
|27/16
|7/4
|63/32
|Half-Augmented (wolf cocytic)
|-
|mLLsLmLsL
|28/27
|7/6
|21/16
|4/3
|3/2
|14/9
|7/4
|16/9
|Major
|-
|LLsLmLsLm
|9/8
|81/64
|9/7
|81/56
|3/2
|27/16
|12/7
|27/14
|Wolf Major
|}
{| class="wikitable sortable"
|+Right-Hand Diasem Modes
! rowspan="2" |Pattern
! colspan="8" |Degrees
! rowspan="2" |Cocytic Chord
|-
!1\9
!2\9
!3\9
!4\9
!5\9
!6\9
!7\9
!8\9
|-
|LmLsLmLsL
|9/8
|7/6
|21/16
|4/3
|3/2
|14/9
|7/4
|16/9
|Major
|-
|mLsLmLsLL
|28/27
|7/6
|32/27
|4/3
|112/81
|14/9
|128/81
|16/9
|Major
|-
|LsLmLsLLm
|9/8
|8/7
|9/7
|4/3
|3/2
|32/21
|12/7
|27/14
|Minor
|-
|sLmLsLLmL
|64/63
|8/7
|32/27
|4/3
|256/189
|32/21
|12/7
|16/9
|Minor
|-
|LmLsLLmLs
|9/8
|7/6
|21/16
|4/3
|3/2
|27/16
|7/4
|63/32
|Major (wolf cocytic)
|-
|mLsLLmLsL
|28/27
|7/6
|32/27
|4/3
|3/2
|14/9
|7/4
|16/9
|Major
|-
|LsLLmLsLm
|9/8
|8/7
|9/7
|81/56
|3/2
|27/16
|12/7
|27/14
|Minor (wolf fourth)
|-
|sLLmLsLmL
|64/63
|8/7
|9/7
|4/3
|3/2
|32/21
|12/7
|16/9
|Minor
|-
|LLmLsLmLs
|9/8
|81/64
|21/16
|189/128
|3/2
|27/16
|7/4
|63/32
|Wolf Half-Augmented
|}

=== Superdiatonic ===
One may note that the Diasem scale pattern produces an imperfect "wolf" interval on each step, which creates an awkward and discordant quality in some of the chords. One potential way to fix this issue is by equating the large wolf chthonic with the regular major chthonic; this may be seen as analogous to how Meantone temperament equates the wolf intervals of the Zarlino pattern with the more frequent forms of the ordinals. This is at the cost of severely sharpening both the major chthonic and the fourth.

This scale is sometimes known as "superdiatonic," as it can be constructed by pentachords of LLLs in a similar manner to the diatonic scale's construction by tetrachords of LLs. It may also be called "armotonic", and is well-tuned in 16edo.
{| class="wikitable sortable"
|+Modes of Superdiatonic
! rowspan="2" |Pattern
! rowspan="2" |Brightness
! colspan="8" |Degrees
! rowspan="2" |Cocytic Chord
|-
!1\9
!2\9
!3\9
!4\9
!5\9
!6\9
!7\9
!8\9
|-
|LLLsLLLLs
| +3
|Maj.
|Maj.
|Maj.
|Perf.
|Perf.
|Maj.
|Maj.
|Maj.
|Major
|-
|LLsLLLLsL
| +1
|Maj.
|Maj.
|Min.
|Perf.
|Perf.
|Maj.
|Maj.
|Min.
|Major
|-
|LsLLLLsLL
| -1
|Maj.
|Min.
|Min.
|Perf.
|Perf.
|Maj.
|Min.
|Min.
|Minmaj
|-
|sLLLLsLLL
| -3
|Min.
|Min.
|Min.
|Perf.
|Perf.
|Min.
|Min.
|Min.
|Minor
|-
|LLLLsLLLs
| +4
|Maj.
|Maj.
|Maj.
|Aug.
|Perf.
|Maj.
|Maj.
|Maj.
|Half-Augmented
|-
|LLLsLLLsL
| +2
|Maj.
|Maj.
|Maj.
|Perf.
|Perf.
|Maj.
|Maj.
|Min.
|Major
|-
|LLsLLLsLL
|0
|Maj.
|Maj.
|Min.
|Perf.
|Perf.
|Maj.
|Min.
|Min.
|Major
|-
|LsLLLsLLL
| -2
|Maj.
|Min.
|Min.
|Perf.
|Perf.
|Min.
|Min.
|Min.
|Minor
|-
|sLLLsLLLL
| -4
|Min.
|Min.
|Min.
|Perf.
|Dim.
|Min.
|Min.
|Min.
|Minor
|}

=== Semiquartal ===
Another way to simplify the structure of Diasem is to use a single type of neutral/perfect chthonic rather than the two alternating major and minor chthonics. As such, the chord qualities need not be specified for this scale.

'''Todo: table'''

=== Chthonic chromatic ===
The Chthonic analog to the chromatic scale can be made by continuing the stack of neutral chthonics until you reach a scale of 14 notes. This scale has the pattern 5L 9s; because the step size is the interizer, the 14 notes of the scale can be considered to represent the seven diatonic ordinals plus the seven interordinal intervals. As a result, the degrees of this scale correspond exactly to the fourteen Greek numerals, and provide "canonical" representations for the major and minor qualities of each degree given a tuning system which contains this scale, just as diatonic provides such a representation for the major and minor qualities of the 7-form degrees.

For reference, the corresponding diatonic modes will also be noted here; every diatonic mode corresponds to two chthonic chromatic modes.
{| class="wikitable sortable"
|+Modes of Chthonic Chromatic
! rowspan="2" |Pattern
! rowspan="2" |Brightness
! colspan="13" |Degrees
! rowspan="2" |Diamode
|-
!β (1.5)
!γ (2)
!δ (2.5)
!ε (3)
!ϛ (3.5)
!ζ (4)
!η (4.5)
!θ (5)
!ι (5.5)
!κ (6)
!λ (6.5)
!μ (7)
!ν (7.5)
|-
|LssLssLsLssLss
| +5
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Ionian
|-
|ssLssLsLssLssL
| -5
|Min.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Phrygian
|-
|sLssLsLssLssLs
| +1
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Dorian
|-
|LssLsLssLssLss
| +6
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Aug.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Lydian
|-
|ssLsLssLssLssL
| -4
|Min.
|Min.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Phrygian
|-
|sLsLssLssLssLs
| +2
|Min.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Mixolydian
|-
|LsLssLssLssLss
| +7
|Maj.
|Maj.
|Aug.
|Maj.
|Maj.
|Aug.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Lydian
|-
|sLssLssLssLssL
| -3
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Aeolian
|-
|LssLssLssLssLs
| +3
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Mixolydian
|-
|ssLssLssLssLsL
| -7
|Min.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Dim.
|Min.
|Min.
|Dim.
|Min.
|Min.
|Locrian
|-
|sLssLssLssLsLs
| -2
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Maj.
|Aeolian
|-
|LssLssLssLsLss
| +4
|Maj.
|Maj.
|Perf.
|Maj.
|Maj.
|Perf.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Maj.
|Maj.
|Ionian
|-
|ssLssLssLsLssL
| -6
|Min.
|Min.
|Perf.
|Min.
|Min.
|Perf.
|Min.
|Dim.
|Min.
|Min.
|Perf.
|Min.
|Min.
|Locrian
|-
|sLssLssLsLssLs
| -1
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Perf.
|Min.
|Maj.
|Perf.
|Min.
|Maj.
|Dorian
|}
''todo: pajara''

Notation

2026-04-07T21:54:45Z

Unque: spellign

There are many different forms of '''musical notation''' in xenharmony. This page will serve as an introduction to some of the concepts found in xenharmonic notation, and some of the more common types. Note that this does not cover [[Naming#Intervals|interval naming]] systems with no notational counterpart. Notation systems are usually designed for consistent tuning systems such as [[Regular temperament|regular temperaments]], or approximately consistent tuning systems.

In general, a ''notation system'' on this wiki allows for the transcription of musical information such as pitches, note durations, rhythms, etc. Most notation systems here act as extensions or modifications of standard staff notation; however, as this wiki focuses on xenharmony (that is, non-standard intervals), any way of naming and notating pitches (and optionally chords) is generally sufficient to be considered a notation system. A notation system may also include an interval naming system.

== Introduction ==
[[File:Minor.png|thumb|The C minor scale in standard diatonic notation. The notes are C, D, Eb, F, G, Ab, Bb, and C.]]

=== Staff notation ===

==== Staff ====
A ''staff'' is a set of lines (or other reference) that symbols representing notes are placed in relation to. In most notation systems, notes are placed on or in between the lines, resulting in (2n+1) staff positions for n staff lines (without needing additional lines, called ledger lines). Different staff positions represent different pitches. In standard diatonic notation, there are five lines per staff, and therefore 11 distinct staff positions without adding ledger lines.

==== Clef ====
A ''clef'' is a symbol placed on a staff (usually at the very start) to indicate what note a particular staff line represents. The other notes can then be determined from the clef and a characteristic scale.

==== Characteristic scale ====
The characteristic scale is the "default" set of notes that serve as the basis for a system being notated. The notes of the characteristic scale are represented in text by ''nominals'' (written symbols representing notes, usually alphabetic, numerical, or drawn from some other extramusical sequence) and on the staff by distinct staff positions. For standard notation, the characteristic scale is [[Diatonic#MOS diatonic|mosdiatonic]] in the mode LsLLsLL on A = 440Hz. The nominals are ABCDEFG, and then A an octave up.

Note that the actual order of the alphabet may be of secondary importance to some musicians. While retaining it may aid intuition for people getting into music theory through xenharmony, some experienced musicians may prefer to treat other parts of their relationships as more fundamental in nonstandard notation schemes. The letters are thus often arranged in a non-alphabetical order based on pitch or chain position, more letters may be added or existing letters removed, and the series may be otherwise altered to support the intuition of the arranger.

==== Accidentals ====
[[File:C minir.png|thumb|The key signature for C minor in standard diatonic notation.]]
Accidentals are the term for any symbol representing an alteration to a nominal. In standard notation, the # accidental raises a note by a chromatic semitone, and the b accidental lowers by a chromatic semitone. Microtonal notation systems usually add new accidentals, rather than changing the set of nominals. For example, the ^ and v accidentals (arrows) raise and lower by an edostep. Note that accidentals other than the standard # and b (and derivatives) may instead be called ''inflections'' in some sources, wherein the category as a whole are ''alterations''. Accidentals generally continue to apply to the same staff position until the next barline is reached.

==== Key signatures ====
A key signature is a set of accidentals placed at the beginning of the sheet music that apply to the whole piece and serve as the "default" accidentals to apply to staff positions. These accidentals also apply to notes separated by octaves from their position, which is not true of other accidentals. In general, there are only about 15 key signatures in use (based on the circle of fifths) representing the various transpositions of the diatonic scale. However, key signatures are theoretically possible for any set of defined alterations from the base notes of the characteristic scale, so that (for instance) one could specify a non-MOS diatonic scale with alterations to the standard version. (Note that otherwise, accidentals work normally, so that if the key signature specifies C#, placing a b accidental before a C note will read as Cb, not C (which is what the natural accidental is used for).[[File:5555.png|thumb|201x201px|A key signature using FJS accidentals to specify the 5-limit zarlino scale in C major.]]

==== Note symbols ====
While not directly pitch- or interval-related, an important consideration in standard notation is that note duration is represented by differences in symbol shape, rather than by physical size, which allows the precise reading of durations.

== Microtonal notations ==

=== Diatonic notation ===
[[Diatonic notation|This page]] covers commonly accepted extensions to standard diatonic notation used in microtonality. The 7 nominals C, D, E, F, G, A, B form the C major scale, and # and b raise and lower notes by a chroma.

=== KISS notation ===
KISS notation generalizes the principles of diatonic notation to any arbitrary [[MOS]], involving alphabetic note naming based on the MOS in question, and accidentals based on the difference between that [[MOS]]' step sizes. KISS notation refers to a family of closely-related notation schemes.

==== Standard KISS notation ====
KISS notation, in its standard form, uses numbers to name notes, but the standard # and b symbols as accidentals, and places the treble and bass clefs around the middle note as they are in diatonic notation.

==== Quasi-diatonic MOS notation ====
Quasi-diatonic MOS notation uses letters as nominals, starting from A at 440Hz as the final note of the darkest mode of the scale (much as A is the final note of the Locrian scale before it repeats at B) uses standard accidentals, and has a custom system of clefs corresponding to each nominal.

==== Diamond-MOS notation ====
Diamond-MOS notation aims to disambiguate itself from diatonic notation by using different nominals (from J onwards), different accidentals (& = #, @ = b), and a custom clef system that makes it visually apparent where J is in different octaves on the staff and what MOS is being used.

=== Just notation systems ===
Conventionally, just intonation notation systems rely on a series of accidentals that move up or down a "formal comma" for each prime from a basic Pythagorean interval.

Common formal commas include 81/80 for 5 (the difference between the classical and diatonic major thirds), 64/63 for 7 (the difference between the diatonic and septimal major thirds), 33/32 (the difference between the perfect fourth and 11th harmonic) for 11, and 513/512 (the difference between the diatonic minor third and 19th harmonic) for 19.
{| class="wikitable"
|+
!
!FJS
!HEJI
!Neutral FJS
!Ben Johnston
|-
!Basic diatonic
| colspan="3" |Pythagorean
|Zarlino
|-
!7
| colspan="3" |64/63
|36/35
|-
!11
| colspan="2" |33/32
|sqrt(243/242)
|33/32
|-
!13
|1053/1024
|27/26
|sqrt(507/512)
|65/64
|-
!17
|4131/4096
|2187/2176
|4131/4096
|51/50
|-
!19
| colspan="3" |513/512
|96/95
|-
!Accidentals used
|Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction
|A selection of symbols
|Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction
|Numbers and inverted numbers, with otonal intervals all inflected in the same direction, except for the syntonic comma (+/-) and prime 11 (^/v),
|-
!Other notes
|All formal commas are derived systematically. Each prime harmonic is inflected from the simplest Pythagorean interval within 54.1 cents (or 57.8 cents).
|
|Similar to FJS, but includes "dicot" intervals for more intuitive notation of neutral intervals. Each prime harmonic is inflected from the simplest Pythagorean or neutral interval within 33.4 cents.
|Uses Zarlino diatonic as nominals, 36/35 as the accidental for 7, and 96/95 as the accidental for 19.
|}

==== Color notation ====
Color notation is a notation system that uses a series of syllables to notate just intervals, fully replacing the standard interval qualities. An interval's degree is determined from its mapping to 24edo.
{| class="wikitable"
|+
!Syllable
!Intervals
!Notes
|-
|la
|Raised by a chromatic semitone
|Not used, # is used instead
|-
|sa
|Lowered by a chromatic semitone
|Not used, b is used instead
|-
|wa
|3-limit, no alterations
|3-limit, no alterations
|-
|ru
|7 under, generally supermajor
| +64/63
|-
|yo
|5 over, generally ptolemaic major
| -81/80
|-
|gu
|5 under, generally ptolemaic minor
| +81/80
|-
|zo
|7 over, generally subminor
| -64/63
|-
|lu
|11 under
| -33/32, +729/704
|-
|lo
|11 over
| +33/32, -729/704
|}

Notation

2026-04-07T21:54:11Z

Unque: Fixed broken link and reworded some loaded language

There are many different forms of '''musical notation''' in xenharmony. This page will serve as an introduction to some of the concepts found in xenharmonic notation, and some of the more common types. Note that this does not cover [[Naming#Intervals|interval naming]] systems with no notational counterpart. Notation systems are usually designed for consistent tuning systems such as [[Regular temperament|regular temperaments]], or approximately consistent tuning systems.

In general, a ''notation system'' on this wiki allows for the transcription of musical information such as pitches, note durations, rhythms, etc. Most notation systems here act as extensions or modifications of standard staff notation; however, s this wiki focuses on xenharmony (that is, non-standard intervals), any way of naming and notating pitches (and optionally chords) is generally sufficient to be considered a notation system. A notation system may also include an interval naming system.

== Introduction ==
[[File:Minor.png|thumb|The C minor scale in standard diatonic notation. The notes are C, D, Eb, F, G, Ab, Bb, and C.]]

=== Staff notation ===

==== Staff ====
A ''staff'' is a set of lines (or other reference) that symbols representing notes are placed in relation to. In most notation systems, notes are placed on or in between the lines, resulting in (2n+1) staff positions for n staff lines (without needing additional lines, called ledger lines). Different staff positions represent different pitches. In standard diatonic notation, there are five lines per staff, and therefore 11 distinct staff positions without adding ledger lines.

==== Clef ====
A ''clef'' is a symbol placed on a staff (usually at the very start) to indicate what note a particular staff line represents. The other notes can then be determined from the clef and a characteristic scale.

==== Characteristic scale ====
The characteristic scale is the "default" set of notes that serve as the basis for a system being notated. The notes of the characteristic scale are represented in text by ''nominals'' (written symbols representing notes, usually alphabetic, numerical, or drawn from some other extramusical sequence) and on the staff by distinct staff positions. For standard notation, the characteristic scale is [[Diatonic#MOS diatonic|mosdiatonic]] in the mode LsLLsLL on A = 440Hz. The nominals are ABCDEFG, and then A an octave up.

Note that the actual order of the alphabet may be of secondary importance to some musicians. While retaining it may aid intuition for people getting into music theory through xenharmony, some experienced musicians may prefer to treat other parts of their relationships as more fundamental in nonstandard notation schemes. The letters are thus often arranged in a non-alphabetical order based on pitch or chain position, more letters may be added or existing letters removed, and the series may be otherwise altered to support the intuition of the arranger.

==== Accidentals ====
[[File:C minir.png|thumb|The key signature for C minor in standard diatonic notation.]]
Accidentals are the term for any symbol representing an alteration to a nominal. In standard notation, the # accidental raises a note by a chromatic semitone, and the b accidental lowers by a chromatic semitone. Microtonal notation systems usually add new accidentals, rather than changing the set of nominals. For example, the ^ and v accidentals (arrows) raise and lower by an edostep. Note that accidentals other than the standard # and b (and derivatives) may instead be called ''inflections'' in some sources, wherein the category as a whole are ''alterations''. Accidentals generally continue to apply to the same staff position until the next barline is reached.

==== Key signatures ====
A key signature is a set of accidentals placed at the beginning of the sheet music that apply to the whole piece and serve as the "default" accidentals to apply to staff positions. These accidentals also apply to notes separated by octaves from their position, which is not true of other accidentals. In general, there are only about 15 key signatures in use (based on the circle of fifths) representing the various transpositions of the diatonic scale. However, key signatures are theoretically possible for any set of defined alterations from the base notes of the characteristic scale, so that (for instance) one could specify a non-MOS diatonic scale with alterations to the standard version. (Note that otherwise, accidentals work normally, so that if the key signature specifies C#, placing a b accidental before a C note will read as Cb, not C (which is what the natural accidental is used for).[[File:5555.png|thumb|201x201px|A key signature using FJS accidentals to specify the 5-limit zarlino scale in C major.]]

==== Note symbols ====
While not directly pitch- or interval-related, an important consideration in standard notation is that note duration is represented by differences in symbol shape, rather than by physical size, which allows the precise reading of durations.

== Microtonal notations ==

=== Diatonic notation ===
[[Diatonic notation|This page]] covers commonly accepted extensions to standard diatonic notation used in microtonality. The 7 nominals C, D, E, F, G, A, B form the C major scale, and # and b raise and lower notes by a chroma.

=== KISS notation ===
KISS notation generalizes the principles of diatonic notation to any arbitrary [[MOS]], involving alphabetic note naming based on the MOS in question, and accidentals based on the difference between that [[MOS]]' step sizes. KISS notation refers to a family of closely-related notation schemes.

==== Standard KISS notation ====
KISS notation, in its standard form, uses numbers to name notes, but the standard # and b symbols as accidentals, and places the treble and bass clefs around the middle note as they are in diatonic notation.

==== Quasi-diatonic MOS notation ====
Quasi-diatonic MOS notation uses letters as nominals, starting from A at 440Hz as the final note of the darkest mode of the scale (much as A is the final note of the Locrian scale before it repeats at B) uses standard accidentals, and has a custom system of clefs corresponding to each nominal.

==== Diamond-MOS notation ====
Diamond-MOS notation aims to disambiguate itself from diatonic notation by using different nominals (from J onwards), different accidentals (& = #, @ = b), and a custom clef system that makes it visually apparent where J is in different octaves on the staff and what MOS is being used.

=== Just notation systems ===
Conventionally, just intonation notation systems rely on a series of accidentals that move up or down a "formal comma" for each prime from a basic Pythagorean interval.

Common formal commas include 81/80 for 5 (the difference between the classical and diatonic major thirds), 64/63 for 7 (the difference between the diatonic and septimal major thirds), 33/32 (the difference between the perfect fourth and 11th harmonic) for 11, and 513/512 (the difference between the diatonic minor third and 19th harmonic) for 19.
{| class="wikitable"
|+
!
!FJS
!HEJI
!Neutral FJS
!Ben Johnston
|-
!Basic diatonic
| colspan="3" |Pythagorean
|Zarlino
|-
!7
| colspan="3" |64/63
|36/35
|-
!11
| colspan="2" |33/32
|sqrt(243/242)
|33/32
|-
!13
|1053/1024
|27/26
|sqrt(507/512)
|65/64
|-
!17
|4131/4096
|2187/2176
|4131/4096
|51/50
|-
!19
| colspan="3" |513/512
|96/95
|-
!Accidentals used
|Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction
|A selection of symbols
|Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction
|Numbers and inverted numbers, with otonal intervals all inflected in the same direction, except for the syntonic comma (+/-) and prime 11 (^/v),
|-
!Other notes
|All formal commas are derived systematically. Each prime harmonic is inflected from the simplest Pythagorean interval within 54.1 cents (or 57.8 cents).
|
|Similar to FJS, but includes "dicot" intervals for more intuitive notation of neutral intervals. Each prime harmonic is inflected from the simplest Pythagorean or neutral interval within 33.4 cents.
|Uses Zarlino diatonic as nominals, 36/35 as the accidental for 7, and 96/95 as the accidental for 19.
|}

==== Color notation ====
Color notation is a notation system that uses a series of syllables to notate just intervals, fully replacing the standard interval qualities. An interval's degree is determined from its mapping to 24edo.
{| class="wikitable"
|+
!Syllable
!Intervals
!Notes
|-
|la
|Raised by a chromatic semitone
|Not used, # is used instead
|-
|sa
|Lowered by a chromatic semitone
|Not used, b is used instead
|-
|wa
|3-limit, no alterations
|3-limit, no alterations
|-
|ru
|7 under, generally supermajor
| +64/63
|-
|yo
|5 over, generally ptolemaic major
| -81/80
|-
|gu
|5 under, generally ptolemaic minor
| +81/80
|-
|zo
|7 over, generally subminor
| -64/63
|-
|lu
|11 under
| -33/32, +729/704
|-
|lo
|11 over
| +33/32, -729/704
|}

L'Antica Musica

2026-03-29T22:32:29Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

=== Archicembalo Layout ===
The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord. The fifth size was likely variable throughout. Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯.

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; so that if the lower keyboard were tuned to 2/7-comma meantone, then ^C in this tuning would be a 2/7 of a syntonic comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Tuning ===

==== Vicentino's potential preferences ====
In the first paradigm, with a fifth between B♯-^G♭, it is likely that a tuning around [[Meantone#Quarter-comma Meantone|1/4-comma meantone]] would be preferred, providing a justly tuned [[5/4]] as the major third. In the 31-note MOS scale of this tuning, the wolf fifth between G♭ ^C♯ is approximately the size of a Pythagorean fifth, which would ensure that the wolf forms of intervals would fall between the sizes of the Pythagorean tuning and the meantone tuning, making it a favorable tuning for providing consonant tunings for those wolves. Additionally, very little size adjustment from 1/4-comma is needed to provide a wolf fifth which is precisely the same size as the generating fifth, thus closing the circle by yielding precisely 31edo.

In the second paradigm, with a fraction of a comma between C-^C, the most likely preferred tuning would have been [[Meantone#1/3-comma Meantone|1/3-comma meantone]], providing a justly tuned 6/5 as the minor third. Since the two keyboards represent two 19-note circles that don't overlap, the wolf fifths are between B♯-G♭ and their comma-altered counterparts, and 1/3-comma meantone tunes this wolf fifth to a size almost precisely the same as the generating fifth (thus approximately [[19edo]] per keyboard). Additionally, Vicentino offers that a root on the lower keyboard may be paired with a fifth and major third from the upper keyboard, a suggestion that would only make sense if the major third were significantly flat of 5/4; in 1/3-comma tuning, we find a major third of approximately 379 cents. Sharpening this third by 1/3 of a syntonic comma in fact yields a tuning which is almost precisely justly tuned 5/4.

==== Modern tuning implications ====
By noticing that 1/3-comma meantone roughly closes at 19 steps along the circle of fifths, we can create a more rotationally-symmetrical version of the 38-note paradigm using the MOS scale 19L 19s, which is two instances of 19edo with their roots offset by a tiny amount. A purely-tuned 3/2 will yield a hardness of approximately 9:1, thus allowing us to cleanly approximate this paradigm by considering it a subset of 190edo. While the number is unwieldy, it provides an additional means of extending the paradigm beyond the two keyboards, such that ten keyboards (thirty ranks) would yield a closed circle of 1/3-commas. This would be incredibly inefficient on an analog keyboard instrument of the 16th century, but could be quite simple and useful on isomorphic keyboards of the modern era, such as the Lumatone.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo's first paradigm:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

User:Unque

2026-03-23T19:43:39Z

Unque:

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely half way between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for practical composers, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Chthonic harmony]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff I write that I don't believe deserves to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]]

User:Unque

2026-03-23T15:28:50Z

Unque: huge overhaul

I am '''Uncreative Name''', or '''Unque'''! I'm a student at a music school and linguistics school; I'm a music theorist, composer, performer, mathematician, worldbuilder, and linguist. Members of the community may know me as a vocal advocate for [[15edo|tunings of suboptimal popularity]]; an [[Dhembrwood|exotempering troll]]; a spewer of [[Regular temperament theory|recreational mathematics]]; or even the mythical [[User:Unque|guy who asked]].

After some chaos in the main Xenharmonic community, I've moved my base of operation to Xenreference. In my time here, I plan to take a music-first approach for sects of theory whose usefulness have been called into question due to the disproportionate amount of mathematical discussion and lack of musical composition. This primarily includes Regular Temperament Theory, a point of contention among xenharmonicists and a common punching bag for those musicians who dislike mathematically-heavy theory.

Some of my previous works can be found on the main xenharmonic wiki [https://en.xen.wiki/w/User:Unque here].

== Tuning Propaganda ==

=== [[15edo]] ===
This was the first tuning system that I ever truly fell in love with in my xenharmonic journey. Whereas most popular tunings such as 17, 19, and 31 felt like simple extensions to the familiar, 15edo allowed for a framework that was beautifully alien in ways that those other systems failed to be.

While 15edo does not provide accurate representations of the harmonic series, it does provide extremely useful melodic frameworks. The [[Pentawood|Blackwood Decatonic scale]], for instance, contains several copies of [[Nicetone]] over each degree, allowing for diatonic-like chord progressions to move smoothly between keys that may seem unrelated in systems with a more accurate chain of fifths.

Additionally, it can be noted that if one makes an [[Delta-rational chord|isodifferential]] triad with an interval of 400c (the familiar major third from 12edo) between the bottom two pitches, this chord will have a "fifth" which very closely resembles the "fifth" of 5edo. Thus, we can assume that a tuning which contains 3edo and 5edo as subsets has a close approximation of this chord. This can be seen as an alternative way to "fix" the lack of harmonic effect in the 12edo major triad, which detunes the fifth rather than the third.

=== [[30edt]] ===
There isn't too much to say about 30edt. Its most obvious application is as a stretched version of 19edo, which is desirable due to 19edo's consistent flat temperament of prime harmonics.

However, it can also be used on its own as a tritave-equivalent system. Because the tritave is significantly wider than the octave, using the semi-tritave as a period can be a good way to make scales more melodically coherent. The semi-tritave interval, unlike the semi-octave, is relatively consonant, and a very intuitive size for a period due to being precisely half way between the perfect fifth and the octave (the two most common spans for a scale).

=== [[29edo]] ===
I strongly believe that 29edo should be alongside systems such as 19edo and 31edo as introductions to xenharmonic tunings for beginners. Not only does it represent the third harmonic within less than two cents of error (making the diatonic scale roughly equivalent to its familiar representations in western music), but it additionally contains distinct, unambiguous representations for [[interordinal]] intervals. The ability to place these unfamiliar intervals onto the familiar circle of fifths is extremely beneficial, as it allows beginners to more clearly get a feel for how these intervals can be used to create sounds unavailable in 12edo. This provides a benefit over systems such as Meantone, in that the circle of fifths requires less relearning to account for the difference in intonation compared to 12edo, and in that the interordinals represented are distinct and unambiguous (compare to 31edo, where there is no clear representation for, say, the semifourth or the semisixth).

Additionally, 29edo finds the perfect fourth at 12 steps, a highly divisible size, supporting [[Porcupine]], [[Tesseract]], [[Negri]], [[Semaphore]], and other similar structures. This helps provide an introduction into systems that divide simple intervals into a certain number of steps, and how those divisions can apply to writing melodies and chord progressions.

Finally, for those who like microtemperaments, simple harmonics such as 5 and 7 are very easy to find in supersets such as [[87edo]], since these harmonics have a relative error very close to simple fractions. The perfect fifth of 29edo is optimal for [[Parapyth]] tuning, which makes supersets of 29edo extremely desirable if one seeks an extremely high accuracy equal temperament sequence; additionally, 87edo supports [[Rodan]] temperament, an extremely efficient system which extends the harmonies of the chain of fifths by adding a formal chroma S9~S8~S7 at (8/7) ^ 5. Rodan also contains [[Slendric]] and [[Hemifamity/Patent vals|Hemifamity]] as subset parts of its structure

=== [[36edo]] ===
I don't have too much to say regarding 36edo. It is a superset of 12edo, which provides a very accurate representation of prime 7. It is roughly the optimal tuning for [[Slendric]] temperament, as well as a good EDO representation for 2.3.7 JI scales such as Septimal Zarlino and [[Diasem]].

=== [[41edo]] ===
While a bit larger than the typical optimal size for practicality, physical instruments such as the Kite Guitar have made 41edo significantly more accessible than it may seem at first.

In terms of tone organization, 41edo is extremely accurate and efficient. The first fifteen harmonics are practically indistinguishable from JI (potentially excluding 13), and the edostep acts as an all-purpose formal comma, representing S10~S9~78/77~66/65~S8~S7~45/44. Additionally, 41edo is the unique intersection of [[Magic]], [[Sensamagic]], and [[Pentacircle]], all extremely intuitive relationships that make it a perfect choice for composers who want to access strong low-complexity JI-like sound while retaining all the benefits of an equal temperament sequence.

Additionally, and perhaps more convincingly for practical composers, 41edo maps the perfect fifth to 24 steps. Just like with the perfect fourth of 29edo, this highly divisible interval allows for many useful melodic structures, including (but not limited to) [[Neutral temperaments|Neutral]], [[Slendric]], [[Tetracot]], and [[Miracle]].

=== [[43edo]] ===
By the size of 43edo, even rank-2 thinking is rather difficult for conceptualizing how the system fits together. However, 43edo can very intuitively be taken as a Septimal Meantone system with 45/44~56/55~100/99 as a secondary chroma to provide access to 11-limit intervals. Because the chromatic semitone of the diatonic scale is three of these undecimal chromata, any interval size in the system can be notated with no more than one accidental of each kind.

This threefold division of the chroma additionally allows the wholetone to be altered into a down-wholetone such that three of them make a perfect fourth instead of an augmented one; the structure begotten by these down-wholetones resembles [[Porcupine]] in its form, but does not contain the classical minor third.

Finally, 43edo offers a rather accurate tuning of [[Bleu]], the temperament which cleaves the perfect fifth into five parts; two of these parts reaches 7/6, three reach 14/11, and four reach 11/8.

== Music ==
Here are some of the pieces I've written:

* Spring (26-EDO): [https://youtu.be/OjW8dgooG9Q?si=0YVI5V9rayxbavF_ https://youtu.be/OjW8dgooG9Q_]
* Isles of Despair (15-EDO): https://youtu.be/T49aXRutQxE
* Summer (17-EDO): https://youtu.be/UthDbDI31IY
* Nachtlandian Somersault (19-EDO): https://youtu.be/XZ3zB3EDKOM
* Bonetrousle Remix (22-EDO): https://youtu.be/wgyie6m7f5g
* Random Encounter Type Beat (2.3.7 JI): https://youtu.be/0OkCl2yfgBo
* Methane Lamentation (31-EDO): https://youtu.be/CBmYRoej2yQ
* Autumn (27-EDO): https://youtu.be/dcQe6ebpGFU
* North Star (15-EDT): [https://youtu.be/ftfnW9DsFVE?si=ho0-eiFfeOYSAo96&t=795 https://youtu.be/ftfnW9DsFVE&t=795]
* Winter (37-EDO): [https://youtu.be/rE9L56yZ1Kw?si=K9LGwj_VsbbAJn3H https://youtu.be/rE9L56yZ1Kw]
* September Sunset (18-EDO): https://youtu.be/HWUsCJUXOqg

== Contributions ==

=== Scales and tunings ===
One important part of my mission on xenreference is to provide more concrete composition theory, especially when it comes to concepts like scales and temperaments whose utility may not be obvious. My primary goal in so doing is to rectify the poor reputation of these systems and hopefully to beget greater interest and understanding of these systems, and utilization of their novelty for practical composition.

* [[29edo#MOS scales|29edo MOS scales]]
* [[Mosh#Chords and tendency tones|Mosh chords and tendency tones]]
* [[Orwell#General theory|Orwell general theory]]

=== Theory case studies ===
Another side goal of mine is to uncover writings of historical theorists that have application in tuning theory and further on practical xen composition. The Renaissance era in particular saw dozens of these writings, as musical tuning was a flourishing field of study at the time with many novel ideas yet to be explored.

* [[L'Antica Musica]]
* [[Lucidarium II]]

=== User space ===
Just like back in Xen Wiki, I will be using my user space to contain any idiosyncrasies, jokes, niche topics, subpar writing quality, and other stuff I write that I don't believe deserves to be put on the main wiki space. Some pages may be moved over to main space pages if they are deemed to be well-written pages about legitimately applicable ideas, but for now they remain here.

* [[User:Unque/On Solfege|On solfege]]

Lucidarium II

2026-03-20T17:09:19Z

Unque:

{{Wip}}

'''Lucidarium II''' was the second musical theory treatise in the ''Lucidarium'', published in the late 1310s by choral singer Marchettus of Padova. While Padova was not a composer by trade, his experience singing seemed to beget a great interest in understanding why music is what it is; as such, the ''Lucidarium'' treatises represent a very unique yet well-learnèd perspective on music theory of the Renaissance era.

Whereas the first book of the ''Lucidarium'' discussed the meaning of music and hypotheses on its origins, the second book begins to discuss the practice of how to make music, particularly as regards tuning systems.

== On the whole tone ==
The first several chapters cover the concept of the whole tone; according to Marchettus, this interval is the primary basis of musical tuning because it is the main type of step used in singing. He states that the 9/8 ratio for the whole tone is taught as common knowledge, but questions the origin of this knowledge, and thus sets out to determine why this ratio is used and not any other.

Marchetto begins with arithmetic on splitting a continuum, such as a string, into parts: 3, he declares, is the first nontrivial number of divisions, as 1 represents not splitting, and 2 represents a point which cannot be further divided, whereas a set of 3 divisions can be arranged as 2/3 + 1/3. Marchettus extends this by dividing the threefold division itself by three, thus producing nine arithmetic divisions of the continuum, and describes that this pattern can be continued indefinitely.

Marchettus then notes that by dividing divisions, one can make the trivial divisions become nontrivial; if the half is itself split into halves, then the full continuum is split into fourths, which can be arranged in patterns such as 1/4 + 3/4 or 2/4 + 2/4. Because relating two powers from the same series will always produce elements from that series (that is, 3^a : 3^b will reduce to 3^c for all a and b such that c = a - b), these two series must be related to one another to form nontrivial ratios.

What Marchettus describes here can be summarized in modern tuning theory term as the 3-limit: each ratio is some power of 3 compared to some power of 2. Because the 3-limit is here taken as the basis for tuning, the primary interval of music must be a 3-limit ratio, and 9/8 is the first such nontrivial ratio (3/2 is apparently trivial to consider in this case because it is made up solely of the original primes 3 and 2, rather than their powers).

While questionable, this long-winded explanation does follow a consistent and intriguing logic. The 9/8 basis is used throughout the rest of the treatise as the basis for all other notes, intervals, and scale forms.

== On the semitones ==
While the whole tone is the main type of step used in singing, and the basis of musical tuning according to Marchettus, it is not indivisible. In practice, the whole tone can be divided into several types of smaller parts known as semitones.

Marchettus notes that the 9th harmonic, being an odd number, cannot be split into two equal parts, and thus the reduced 9/8 ratio cannot be divided into equal semitones. However, 9 can be split up into five equidistant parts by way of the odd series: 1, 3, 5, 7, and 9 being its constituents. Thus, the whole tone, being the reduced ninth harmonic, can be divided into five equal parts, each of which Marchettus calls a diesis. Any number of these dieses fewer than five will constitute an interval smaller than a whole tone, and thus a semitone.

This explanation is noted by many later authors as rather questionable. It is unclear how the odd series as divisions of the number nine translate to actual division of a continuum, and it is further unclear how dividing the ninth harmonic into constituents will divide its octave-reduced counterpart into the same number.

Aside from the diesis, there are three types of intervals smaller than the whole tone, found respectively at two, three, and four times the size of the diesis. The two-diesis semitone is known as a limma, and is the small step of the diatonic scale, which Marchettus notes is used in plainchant. The three-diesis semitone is the apotome, the counterpart to the limma, and the difference between a large and small step in the diatonic scale; Marchettus notes that its larger size can create discordance when employed as a step, and hence it is not often used in plainchant. The final semitone is the counterpart of the diesis: Marchettus simply calls this interval a ''chroma'', from the Greek word for ''color''. This interval is used, as he claims, to "color" consonances for specific melodic effects.

Marchettus claims that the chroma is used in ascending melodies when two semitones occur in a row; for instance, a melodic pattern that travels F - F♯ - G will be sung as though F♯ were a chroma higher than F, and G a diesis higher than F♯. This claim may be representative of the tendency of vocal music to squish leading tones, especially at the end of a phrase.

== Implications for modern tuning theory ==
The ideas provided by Marchettus in the ''Lucidarium'' collection of treatises have many applications in modern tuning theory and xenharmonic composition, including construction of and composition in equal temperaments and regular temperaments of multiple kinds.

The only precise ratios outside the 3-limit cited in ''Lucidarium II'' are 18/17 for the limma and 17/16 for the apotome, which represent an otonal division of 9/8 as the trichord 16:17:18. If we assume that these semitones are tuned justly, then the diesis does not indicate an exact ratio, but rather a size range which is used to estimate the sizes of each part.

However, Marchettus does discuss the diesis as a consistent interval from which other intervals are formed, which would make this analysis quite strange. A likely alternative possibility is that Marchettus borrowed the ratios 18/17 and 17/16 from the works of Boethius and of his successors, without regard for the exact sizes of those intervals, and thus the diesis does indeed represent a consistent step size of which five produce a justly-tuned 9/8. This analysis would measure the size of the diesis as approximately 41 cents.

=== Equal temperaments ===
Because all intervals in conventional theory of the time are described by a formula of the whole tone and the limma, and because both such intervals are divisible by a diesis, one can determine the exact size of an interval measured in dieses based on these values. For instance, the perfect fourth is made up on twelve dieses, the perfect fifth is seventeen, and the octave is twenty-nine. Under this analysis, Marchettus' paradigm can be thought of in modern terms as [[29edo]] with a compressed octave to ensure that 9/8 is tuned exactly just.

Indeed, the diatonic scale in 29edo is made out of whole tones of 5\29, and semitones of 2\29, precisely consistent with the numbers provided by Marchettus. Knowing this, Marchettus' description of how to use other types of semitones to "color" the consonances may be applied to composition in 29edo.

=== Regular temperaments ===
One may construct a [[regular temperament]] based on Marchettus' description, with the diesis representing the difference between 18/17 and 17/16, and five of those dieses representing 9/8. The easiest way to do this is by equating a stack of three 18/17s with a stack of two 17/16s, as each of them create an interval of six dieses; the ratio for this comma is 1492992/1419857.

In the minimal [[subgroup]] of 2.3.17, this structure is supported by edos 29, 31, 33, and 35; if we restrict this to 2.9.17, then edos 30 and 32 additionally support it.

=== The diesis in other structures ===
Additionally, the fivefold division of the whole tone can be applied to meantone tunings as well, if we take the limma to be three dieses and the apotome to be two (the opposite of how Marchettus defined them). This is of note because quarter-comma meantone is practically indistinguishable from [[31edo]], and thus the diesis structure can be used as a means of understanding that system. Vicentino's [[L'Antica Musica|L'antica Musica]] explores a system not dissimilar from this one, even adopting the word diesis to describe his small step, though he does not define intervals by a number of dieses.

L'Antica Musica

2026-03-19T17:12:58Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music.

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord; it is unclear what tuning this entails, if any in particular, but the most popular frameworks of the time were meantone tunings between 1/4-comma (~697c fifth) and 1/3-comma (~694c fifth). Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).

Alternatively, one can tune it such that ^C is an untempered perfect fifth above F; if the lower keyboard were tuned to quarter-comma meantone, then ^C in this tuning would be a quarter comma higher in pitch than C. This tuning yields two wolf fifths, one between B♯ and G♭ in the lower keyboard, and one between ^B♯ and ^G♭ in the upper keyboard.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104, 385/384
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|65/63
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

L'Antica Musica

2026-03-19T17:01:42Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament. See [[L'Antica Musica#On Practice V: on the Archicembalo Instrument|Practice V]] for details on tuning.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord; it is unclear what tuning this entails, if any in particular, but the most popular frameworks of the time were meantone tunings between 1/4-comma (~697c fifth) and 1/3-comma (~694c fifth). Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).

Alternatively, one can tune it such that ^F is an untempered perfect fifth above F; if the lower keyboard were tuned to quarter-comma meantone, then ^F in this tuning would be a quarter comma higher in pitch than C. This tuning yields two wolf fifths, one on each keyboard between B♯ and G♭.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo. These ratio targets may inform how the instrument is tuned.
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|117/112
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

Lucidarium II

2026-03-19T16:21:15Z

Unque: Created page with "{{Wip}} '''Lucidarium II''' was the second musical theory treatise in the ''Lucidarium'', published in the late 1310s by choral singer Marchettus of Padova. While Padova was not a composer by trade, his experience singing seemed to beget a great interest in understanding why music is what it is; as such, the ''Lucidarium'' treatises represent a very unique yet well-learnèd perspective on music theory of the Renaissance era. Whereas the first book of the ''Lucidarium'..."

L'Antica Musica

2026-03-18T19:56:39Z

Unque:

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.
== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

=== Enharmonic modes ===
{{Wip}}

== On Practice V: on the Archicembalo Instrument ==
Because ''On Practice IV'' is spend describing notational paradigms such as clefs and note lengths, it will not be summarized here; any other source on musical notation will provide the same information.

Practice V is perhaps the most important book for understanding the rest of the treatise, as it describes in detail the mechanics and tuning of the Archicembalo. While Vicentino never gives outright values for any of the intervals, this book provides insight on how to play and understand the instrument, which in turn helps inform its tuning.

The Archicembalo's keys are split into six "ranks," with the lower three forming the lower keyboard and the upper three ranks forming the upper keyboard. The first rank outlines the diatonic scale, with the lowest note being the root of the Lydian mode; the next rank above it contains the notes which are a chromatic semitone above those ordinals, and the third rank contains notes which are a chromatic semitone below them. The fourth, fifth, and sixth rank are related to one another as the first, second, and third ranks are, but 1-4, 2-5, and 3-6 are all offset by some particular amount which varies based on the tuning.

Because the first two ranks form a familiar [http://tonalsoft.com/enc/h/halberstadt.aspx Halberstadt] layout, Vicentino instructs that they be tuned as if they were an organ or clavichord; it is unclear what tuning this entails, if any in particular, but the most popular frameworks of the time were meantone tunings between 1/4-comma (~697c fifth) and 1/3-comma (~694c fifth). Vicentino specifically says that from the lowest key (the root of the Lydian mode, or F on the Halberstadt layout), all intervals in the first two ranks should be with fifths upwards, which means the wolf fifth occurs between A♯ and F. Vicentino then instructs that the third rank be tuned by fifths downwards from the lowest key; ranks 1 and 3, without considering rank 2, will thus form a Halberstadt layout with its wolf between G♭ and B. There are two additional keys, B♯ and E♯, which are considered by Vicentino to be part of rank 3; however, they are tuned with respect to rank 2, such that E♯ is the tempered fifth above A♯.

Finally, Vicentino notes two possible tunings for the upper keyboard. The first is tuned such that ^G♭ is a tempered fifth above B♯ (therefore making it equivalently F𝄪), and the rest of the upper keyboard can be treated as a copy of the lower which has been transposed by the diesis (the amount by which ^G♭ exceeds G♭). This tuning yields a wolf fifth between Gb and ^C♯, though if tuned near quarter-comma, this wolf fifth can be nearly the same size as the regular tempered fifth (hence 31edo).

Alternatively, one can tune it such that ^F is an untempered perfect fifth above F; if the lower keyboard were tuned to quarter-comma meantone, then ^F in this tuning would be a quarter comma higher in pitch than C. This tuning yields two wolf fifths, one on each keyboard between B♯ and G♭.

=== Interval ratios ===
In addition to establishing an order of the circle of fifths, Vicentino provides a list of JI ratios represented by the 31 interval classes of the Archicembalo:
{| class="wikitable"
|+Ratios of the 31-form
!Step
!Name
!Ratios (5-limit)
!Ratios (7-limit)
!Ratios (higher)
|-
|0
|Comma
|81/80
|126/125
|105/104
|-
|1
|Diesis
|128/125, 648/625
|36/35, 50/49, 64/63
|40/39
|-
|2
|Chromatic Semitone
|25/24
|21/20
|117/112
|-
|3
|Diatonic Semitone
|16/15
|15/14
|14/13
|-
|4
|Minor Tone
| colspan="2" |N/A
|13/12
|-
|5
|Natural Tone
| colspan="3" |9/8, 10/9
|-
|6
|Major Tone
|N/A
| colspan="2" |8/7
|-
|7
|Minimal Third
| colspan="3" |Unstated
|-
|8
|Minor Third
| colspan="3" |6/5
|-
|9
|Proximate Third
| colspan="2" |N/A
|11/9
|-
|10
|Major Third
| colspan="3" |5/4
|-
|11
|Supramajor Third
|N/A
| colspan="2" |9/7
|-
|12
| colspan="4" |N/A (Vicentino skips this interval in his analysis)
|-
|13
|Perfect Fourth
| colspan="3" |4/3
|-
|14
|Superfourth
| colspan="3" |Unstated
|-
|15
|Augmented Fourth
|25/18
|7/5
|39/28
|-
|16+
|Octave Complements
| colspan="3" |Inverse ratios of their octave complements
|}

L'Antica Musica

2026-03-18T00:56:05Z

Unque:

{{Wip}}

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.

== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

=== Diatonic modes ===
Each diatonic mode has steps spanning 5\31 and 3\31, respectively the natural whole tone and the diatonic semitone; the pattern of the modes can thus be written using L and s to represent these two step sizes. The eight modes represent eight rotations of the Meantone[8] scale, with the final mode being a superfluous repetition of the first and the penultimate mode being noted as lacking distinct reason by proxy.
{| class="wikitable"
|+Diatonic Modes
!Number
!Name
!Pattern
!Modern Equivalent
!Notes
|-
|1
|Dorian
|LsLLLsL
|Dorian
|Said to sound devout and virtuous
|-
|2
|Hypodorian
|LsLLsLL
|Aeolian
|More "modest" version of Dorian
|-
|3
|Phrygian/
Trojan
|sLLLsLL
|Phrygian
|Naturally "lonely," but can be made cheerful when mixed with other modes
|-
|4
|Hypophrygian/
Hypotrojan
|sLLsLLL
|Locrian
|Said to sound sad and funereal
|-
|5
|Lydian
|LLLsLLs
|Lydian
|Said to sound haughty and cheerful
|-
|6
|Hypolydian
|LLsLLLs
|Ionian
|Somewhat both sad and cheerful
|-
|7
|Mixolydian
|LLsLLsL
|Mixolydian
|Said to be a simple mix of Dorian and Lydian, hence the name
|-
|8
|Hypomixolydian
|LsLLLsL
|Dorian
|Even Vicentino notes the superfluidity of this mode, as the modes represent 7 rotations of a 7-note scale
|}

=== Chromatic modes ===
The chromatic modes are more complex in construction than the diatonic, each having eight notes per octave compared to the seven notes of each diatonic mode. Each chromatic mode has three distinct types of steps instead of two (the incomposite third of 8\31, the diatonic semitone of 3\31, and the chromatic semitone of 2\31) which can be noted as L, m, and s; additionally, each mode has precisely 2 L steps, 3 m steps, and 3 s steps. These modes are not given unique names, instead simply being numbered.

The first mode is made by taking the fourth and fifth above the tonic degree, then adding a chromatic semitone below each of those three degrees. Finally, the supermajor third intervals which occur over the tonic and fifth are split into sequences of a diatonic semitone and an incomposite minor third, rounding out the scale to 8 notes. The second mode (or perhaps the hypofirst mode) can be found by rotating the scale to begin on the perfect fifth.

The third mode is formed in much the same way as the first, but the supermajor third is split into a minor third followed by a semitone, the inverse of the first mode. The fourth mode (hypothird) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the fifth mode, we begin with an upper tetrachord of P5 - m6 - M6 - 1, which terminates on the upper tonic, and then add a copy of the tetrachord that terminates on the fifth. A natural whole tone can be found above the root, which can be broken up into two unequal semitones to once again round out the scale to eight notes. The sixth mode (hypofifth) can once again be found by rotating the scale to begin on the perfect fifth.

To construct the seventh mode, take the root, fourth, and fifth as in the first two mode pairs; and then add a diatonic semitone above each degree. Just like in the first mode, this mode creates two large steps, though this time they are major thirds rather than supermajor; these can similarly be split into a semitone and minor third to round out the scale to eight notes, though the semitone is chromatic in this mode. The eighth (hyposeventh) and final mode, as with the previous, is found by rotating the scale to begin on the perfect fifth.
{| class="wikitable"
|+Chromatic Modes
!Number
!Pattern
!Comments
|-
|1
|mLsmsmLs
| rowspan="2" |Inverted pattern of modes 3 - 6
|-
|2
|mLsmLsms
|-
|3
|msLmsmsL
| rowspan="4" |All rotations of same pattern
|-
|4
|msLmsLms
|-
|5
|msmsLmsL
|-
|6
|msLmsmsL
|-
|7
|msLmsmLs
|
|-
|8
|mLsmsLms
|
|}

'''''Todo: finish Practice III, document rest of books in treatise'''''

L'Antica Musica

2026-03-16T16:59:29Z

Unque:

{{Wip}}

'''L'Antica Musica''' was a treatise published in 1555 by Nicola Vicentino, which explores how Renaissance-era advancements in musical tuning could be used to adapt the lost traditions of Ancient Greek musicians in such a way that blends them with the sensibilities of the 16th century. It is perhaps most revered in xenharmonic communities for its attestation and argument for [[31edo|31 equal divisions of the octave]]†, a tuning for which Vicentino had designed [[wikipedia:Archicembalo|his own custom instruments]].

†Vicentino's suggestion was likely not an equal temperament as we think of it today, but rather a [[well temperament]] or [[MOS|MOS scale]] based on [[meantone]] temperament.

== On Music Theory ==
The first book of L'Antica Musica, ''On Music Theory'', is intended as a sort of "recap" of the principles of Ancient Greek music theory, and the means by which those principles arose.

According to Greek musical tradition, [[Just Intonation|just intonation]] was discovered by the mathematician Pythagoras when he noticed that the hammers used by blacksmiths would produce more consonant sounds with one another if the sound that they displaced created certain frequency ratios; notable was the [[Perfect fifth|3/2]] ratio, which was considered the most consonant of all. Ancient Greek musicians considered the usage of low-complexity just intonation intervals as the primary way of tuning music

According to Vicentino, Pythagoras also went on to construct the first [[tetrachord]], by stacking two concordant whole tones within the span of a [[perfect fourth]]. This construction, Vicentino asserts, would pave the way for more such tetrachords to be devised, and they were split into three "genera" based on the step sizes. Vicentino provides diagrams to display the forms of these genera, with their steps arranged from the largest to the smallest.
{| class="wikitable"
|+Tetrachord Genera
! rowspan="2" |Genus
! colspan="3" |Step Sizes
! rowspan="2" |Comments
|-
!L
!M
!s
|-
|Diatonic
|Tone
|Tone
|Limma
|Vicentino seems to consider the two "tones" necessarily equal.
|-
|Chromatic
|Minor Third
|Apotome
|Limma
|
|-
|Enharmonic
|Major Third
|Diesis
|Diesis
|The diesis is said by Vicentino to divide the limma into two equal parts.
|}
Vicentino additionally sorts the genera by their "sweet" (Italian: ''dolce'') quality of sound. This appears to be an attempt to translate the Greek term ''μαλακός'' (literally meaning "gentle" or "soft"), which was used by Boethius to describe scales; however, most other Romance scholars used the word ''mollis'' (Italian ''molle'') for this. As Boethius used it, the term specifically describes melodies and scale forms that are said to create a sense of intimacy and resolution by the inclusion of exceedingly small intervals in specific places. Vicentino gives little elaboration on what precisely is meant by ''dolce'', but the term is always used alongside a note of a genus's smaller available step sizes. On the rest of this page, the term will be left untranslated to avoid the ambiguity with English senses of "sweet."

Vicentino goes on to discuss the various permutations of these tetrachords that can be constructed by putting the same step sizes in a different order; these permutations constitute the "species" of tetrachords. These permutations can be used to construct eight [[Historical modes|church modes]], which Vicentino enumerates as follows:
{| class="wikitable"
|+Eight Modes
!Mode
!Pattern
!Comments
|-
|Dorian
|<nowiki>| LMs sLM</nowiki>
|
|-
|Hypodorian
|<nowiki>sLM | LMs</nowiki>
|
|-
|Phrygian
|<nowiki>| MsL MsL</nowiki>
|In diatonic genus, resembles modern Mixolydian mode
|-
|Hypophrygian
|<nowiki>MsL | MsL</nowiki>
|In diatonic genus, resembles modern Dorian mode
|-
|Lydian
|<nowiki>| sLM LMs</nowiki>
|In diatonic genus, resembles modern Melodic Minor scale
|-
|Hypolydian
|<nowiki>LMs | sLM</nowiki>
|
|-
|Mixolydian
|<nowiki>sLM | sLM</nowiki>
|In diatonic genus, resembles modern Phrygian mode
|-
|Hypermixolydian
|<nowiki>sLM sLM |</nowiki>
|In diatonic genus, resembles modern Locrian mode
|}
The | symbol is here used to represent a step of precisely 9/8, which separates the two tetrachords from one another.

Note that these modes are constructed by Vicentino's enumeration of the tetrachords, and do not represent rotations of a single scale pattern (though the familiar "rotated" modes are indeed discussed in a later book). Some of the modes listed here are thus rather odd, such as Dorian having two consecutive semitones, or Lydian having four consecutive whole tones. Some translators suggest that this may represent multiple conflicting schemes of numbering the tetrachords.

== On Practice I ==
Following the book on ancient theory is the first book on contemporary practice of the Renaissance era when the treatise it was published. This first practice book is centered primarily around melodic composition, with the harmonic considerations not being featured until the following book.

The first several chapters summarize musical notation; to accommodate for the microtones of Vicentino's proposed tuning, he suggests a circular accidental which represents raising a note by the interval of a diesis (one step of the 31-form).

The majority of the book, however, is spent the intervals of the 31-form, gives each a name, and describes their unique nature and usage; Vicentino specifically uses properties such as consonance potential, subjective emotional content, and ''dolce'' qualities to describe the nature of the intervals used as steps or leaps in a melody.
{| class="wikitable"
|+Intervals of the 31-form
! rowspan="2" |Step
! rowspan="2" |Name
! colspan="2" |Melodic Quality
! rowspan="2" |Comments
|-
!Ascending
!Descending
|-
|0
|Comma
| colspan="2" |N/A (tempered out)
|Of note because the archicembalo was well-tempered
|-
|1
|Diesis
| colspan="2" |Concordant, lax, and ''dolce''
| rowspan="3" |To be used in alternating succession
|-
|2
|Chromatic Semitone
| rowspan="2" |Tense, yet cheerful
| rowspan="2" |Lax and sad
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| colspan="2" |Lax or tense
|Assimilates tension to nearby intervals
|-
|5
|Natural Tone
| rowspan="2" |Powerful and tense
| rowspan="2" |Lax
| rowspan="4" |[[64/63]] "is not discernible in singing, but in the tuning of instruments their difference is indispensable."
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Lax
| rowspan="3" |Somewhat tense and very sad
|-
|8
|Minor Third
|-
|9
|Supraminor Third
|Also known as "proximate" third; "neutral" had not yet been coined
|-
|10
|Major Third
|Tense and imperious
|Somewhat tense; sad if accidental
| rowspan="2" |Notably distinct despite being separated by 64/63
|-
|11
|Supermajor Third
|Extremely tense
|Extremely sad and lax
|-
|12
|N/A
| colspan="2" |N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
|Tense; points to the thirds
|Lax
|Level of tension when ascending depends on type of tetrachord used
|-
|14
|Superfourth
|Lively
|Sad and lax
|Also known as "proximate" fourth
|-
|15
|Augmented Fourth
|Vivacious and forceful
|Funereal and sad
|Vicentino advocates for liberal usage of tritones for "marvelous effect"
|-
|16+
|Octave Inversions
| colspan="2" |Inverse qualities of their inversions
|
|}

== On Practice II ==
In the second book on musical practice, Vicentino discusses the three methods of writing effective music. The first is the usage of melodic and harmonic intervals to mimic the character of sung lyrics; the second is the usage of melodic and harmonic intervals to create clear senses of tension and release; and the third is to balance different types of characteristics to create a clear sense of motion from start to finish. The bulk of the second book focuses on the principles of writing melodies and harmonies that play into these methods.

Vicentino applies the principles of counterpoint regardless of the method: all voices must begin and end a melodic line on the tonic pitch, contrary motion is preferred between voices, parallel perfect consonances are dispreferred between voices, and resolutions must be approached through stepwise motion.

In the following chapters, the natures of harmonic consonances are discussed, much the same as the melodic natures in the preceding book. Vicentino advocates specifically for the usage of syncopation, whereby two lines have the same rhythm offset by some consistent amount until the resolution; this syncopation will create dissonances between voices that would otherwise have been consonant, which allows for these dissonances to resolve. Vicentino notes that the ear can hear these dissonant intervals such as tritones as more concordant when approached via small steps such as dieses and semitones that create a more ''dolce'' melody.
{| class="wikitable"
|+Intervals of the 31-form
!Step
!Name
!Harmonic Function
!Comments
|-
|1
|Diesis
|N/A
|Does not occur between voices
|-
|2
|Chromatic Semitone
| rowspan="2" |Resolves to minor third or unison;
Bass voice moves down while top voice
moves up or remains fixed
| rowspan="5" |Vicentino prefers semitone suspensions over wholetones.
|-
|3
|Diatonic Semitone
|-
|4
|Minor Tone
| rowspan="3" |Resolves to third or unison;
Bass voice moves down while top voice
moves up or remains fixed
|-
|5
|Natural Tone
|-
|6
|Major Tone
|-
|7
|Minimal Third
| rowspan="3" |Imperfect consonance approached by seconds
|
|-
|8
|Minor Third
|
|-
|9
|Supraminor Third
|
|-
|10
|Major Third
| rowspan="2" |Imperfect consonance approached by fourths
Set up a more ''dulce'' sound to approach cadences by
|
|-
|11
|Supermajor Third
|
|-
|12
|N/A
|N/A
|Vicentino skips this interval in his analysis
|-
|13
|Perfect Fourth
| rowspan="2" |Resolves to major third;
Bass voice remains fixed while top voice moves down
|
|-
|14
|Superfourth
|
|-
|15
|Augmented Fourth
|Resolves by contrary motion to third or sixth
|
|-
|16
|Diminished Fifth
|Fudging of either an augmented fourth or imperfect fifth
|
|-
|17
|Imperfect Fifth
|Half-resolves to perfect fourth or superfourth;
or resolves to perfect fifth or minimal sixth
|
|-
|18
|Perfect Fifth
|Perfect consonance approached by fourths
|
|-
|19
|Superfifth
|N/A
|Does not occur between voices
|-
|20
|Minimal Sixth
|Imperfect consonance approached by diminished fifth
Leads to perfect fifth
| rowspan="3" |Concordant but directed;
Vicentino compares fifths and sixths to the sun and moon
|-
|21
|Minor Sixth
| rowspan="2" |Imperfect consonances leading to and from fifths
|-
|22
|Supraminor Sixth
|-
|23
|Major Sixth
| rowspan="2" |Imperfect consonances approached by sevenths
Can create strong directed motion anywhere if set up well
|
|-
|24
|Supermajor Sixth
|
|-
|25
|Minimal Seventh
| rowspan="3" |Resolves by oblique motion to sixth
|
|-
|26
|Minor Seventh
|
|-
|27
|Supraminor Seventh
|
|-
|28
|Major Seventh
| rowspan="2" |Resolves by oblique motion to sixth or octave
|
|-
|29
|Supermajor Seventh
|
|-
|30
|Imperfect Octave
|Used in place of octaves to evade parallel motion
|
|-
|31
|Octave
|Perfect consonance approached by ninths or sevenths
|
|}

== On Practice III ==
The third book on musical practice comprises demonstrations of the three tetrachord genera, with their associated octave species. The diatonic modes addressed in this book resemble the modern conception rather than the unconventional combination of tetrachord presented in the Theory book, further supporting the suggestion that the prior construction may have been an error.

'''''Todo: finish Practice III, document rest of books in treatise'''''

Mosh

2026-03-16T16:57:51Z

Unque: Began discussion on composition theory; also let's avoid using diatonic names for nondiatonic scales, especially in such a confusing fashion