kili 11:20 AM est jun 16: to start off, this is what the page on xenwkik loojks like:
i don't intend to copy it bt they are a lot closer to the source than i am
> Primodality (also informally called Zheanism after its originator Zhea Erose) is an approach to JI designed to emphasize the identity of the "tonic" as the pth harmonic and places importance on the particular timbre of chords with a given tonic. Scales and chords having the identity of the prime p as the tonic are collectively called a prime family, and can be denoted simply by /p. Zhea also uses various adjectives for specific primodalities, such as septimal, undecimal, tridecimal, septendecimal, novem(decimal) for /7, /11, /13, /17, and /19 respectively, which are not to be confused with the use of these adjectives to denote prime limits. (If disambiguation is needed, one can say over-7 and 7-limit respectively for the two meanings of septimal, for instance.) Zhea's ideas are new in that she not only treats higher JI as different from close irrational tunings (as some, like Johnny Reinhard, previously have done), but also claims that each prime comes with its own unique timbral "gestalt" which is in all chords built from small multiples of p (particularly 2p) as the tonic. The gestalt aspect is critical: while individual intervals in a primodal tuning may not be recognizable for what they are (using methods like harmonic entropy), when considered as a whole, their shared relationship to /p becomes apparent.

> Most importantly, primodality sees any overtone as valuable on its own, rather than relative to some fundamental. Taking a specific overtone as a tonic, even "non-xenharmonic" scales are said to gain the gestalt identity particular to the overtone, even when the corresponding fundamental is too low to be audible. In particular, primodality discards the concept of harmonic limit, which Zhea considers an artificial way to look at JI harmony. Zhea argues that prime families are a more natural way to categorize intervals; intervals from the same prime family (intervals with a common denominator for example, all /2, all /11 or all /13) tend to blend better together. For example, it is preferable to add 21/16 to 4:5:6:7 rather than 4/3, if one wants to retain the /2 gestalt (otherwise an /3 gestalt emerges from 12:15:16:18:21).

> To construct a primodal scale, we fix a prime p to be the denominator and take intervals of the form n/p, where n ≥ p. Zhea often takes n to range over a certain "lineal segment" (Mode mp of the harmonic series where m is a positive integer) or a subset thereof. For example, if we use p = 13 and take all n between 13 and 26 (inclusive), this would result in the scale 13:14:15:16:17:18:19:20:21:22:23:24:25:26. We may add a 3/2 to the scale root, which corresponds to adding 3p/p. (3/2 is a natural "halfway point" for harmonic scales, since if N is even, Mode N has a 3/2 as its N/2-th note.)

> Primodality, and Zhea's microtonal theory overall, emphasize subtle timbral effects, as opposed to lower-complexity JI identities such as 4:5:6:7:9 that are more common in composite modes. Mode p and Mode 2p (called respectively the first and second octaves of /p) are considered the most important for the identity of /p; those intervals are the most recognizable as distinct identities. For any prime p, the set of harmonics from p to 2p is unique in the sense that the sets {p/p, ..., 2p/p} and {n/n, ..., 2n/n} only intersect at {1/1, 2/1} for any positive integer n < p. Similarly, the second octaves of p and the second octave of any n Primodality could be understood as the use of prime modes of the harmonic series (hence "prime" + "mode" + "-ality") which is of musical interest because using a prime as the mode maximizes irreducible intervals; and as an additional step away from the exhibition of obvious low-limit JI intervals, primodality suggests the use of very large modes of the harmonic series (or subsets thereof), which as in higher harmonic tuning leverages JI instead for the "harmonic cloud" effect of a shared very low (sometimes infrasonic) fundamental.

> Neji
Primodality is often used in combination with another Zhea Erose technique: neji (or Near-Equal Just Intonation) tunings, which can be used to preserve the primodal aspects while producing tunings with the benefit of near-equal intervals.

Primodality

2026-06-16T15:20:19Z

Kili: cpmosianiot theory is i,m going to make

Talk:Mode

2026-06-16T15:05:54Z

Kili: merge into glolsarp

kili, jun 16 11:02 AM est:
this is my plan for merging the page MODE into the glossary so that it is not PRIVILIGED more so than SCALE.
fist: the head definition of mode will be slightly shortened to form the basis of the new paragraph
second: the example featured diatonic will be simplified, and the table will be left behind
third: the properties will be -> mostly reduced into the head definition and mention MOde of Limited Transposition since it contains that definition
fourth: inversion will be given its own glossary entry

Neutral

2026-06-15T18:52:30Z

Kili: fix

'''Neutral''' may refer to:
* [[Neutral interval]]s
* [[Neutral temperaments]], temperaments generated by 2/1 and a neutral third, including the rank-2 temperament 2.3.11[243/242], called [[Rastmatic]] (Dicot) or rarely Neutral
{{cat|Disambiguation pages}}

Meantone

2026-06-15T18:49:12Z

Kili: xlue on Nedo

[[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.
[[File:Meantone7.png|thumb|Meantone diatonic scale]]
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.

== Extensions ==
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.

If an unmapped (not equated to a stack of anything else, as [[5/4]] is in Blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7. It is supported by [[36edo]]'s patent val. (This is not technically an extension.)

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 and conflates 14/11 with 5/4 (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. The dominant seventh chord is 20:25:30:36, or in septimal meantone, the 1–5/4–3/2–9/5 [[collection of chords #Essentially tempered chords|essentially tempered chord]]; the half-diminished seventh chord is similarly 25:30:36:45, or in septimal meantone, the 1–6/5–10/7–9/5 essentially tempered chord. 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.

5-limit Meantone also contains an 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.

=== 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]].

=== 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]].

=== 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.

=== Golden meantone ===
''Main article: [[Golden generator]]''

Golden meantone is the tuning of meantone wherein the large and small steps of the diatonic scale are in the golden ratio. It is the only meantone tuning which produces exclusively soft scales, and meantone's general proximity to the golden tuning captures the difficulty of representing it (and the 5-limit as a whole) within a specific form.

=== 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.

=== 1/3-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.

=== Silver flattone ===
Silver flattone is the tuning of meantone such that the step size ratios of the diatonic and enharmonic (19-note) scale steps are the same, and that that ratio is the square root of 2. Alternatively, the step size ratio found in the chromatic scale is the silver ratio, sqrt(2)+1. It is somewhat sharp for flattone, tuning 7/4 flat of 960 cents. Silver flattone is the soft counterpart of [[argent]] tuning.

=== 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) and is thus close to silver flattone.

=== 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).

=== (Half Comma) Cleantone ===
Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 (in fact, any interval in the JI group (3/2).(5/4)) justly, but 2/1 complements of these intervals are detuned.

=== Lucy Tuning ([[88edo|88edo]]) ===
This is nearly indistinguishable from the 88edo tuning of Meantone, with a fifth (600 + 300/π cents) just 0.038 cents higher than [[88edo]]'s 695.55-ish fifth. Its major third (1200/π = 381.97 cents) is flat 4.3 cents, but closer than 1/3 Meantone's. The proper extension to the 7-limit (and possible 11-limit) is Mothra (8/7 = 200+100/π = 231.83 cents), which yields a 7/4 flat by only 0.659 cents.

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

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

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

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!7-limit strong extensions
!11-limit strong 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
|
|[26 & 31]
|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}}

Talk:Compositional theory

2026-06-15T18:41:26Z

Kili:

this is -kili 2026 2:36est jun 15, does anyone actually use Xenmodalism? it has been linked on this page for a while but no one has started it. i think i vaguely understand what it is supposed to be and maybe know some examples on youtube, but it would be really good if we could put a page onto xenmodalism to stop all the big red tomato text on what i consider an important page
-> ideally someone either uses "xenmodalism" enough to start the page on it, or they know about the music someone else makes and the method which qualifies as xenmodalilm...

Talk:Compositional theory

2026-06-15T18:39:12Z

Kili: Created page with "this is -kili 2026 2:36est jun 15, does anyone actually use Xenmodalism? it has been linked on this page for a while but no one has started it. i think i vaguely understand what it is supposed to be and maybe know some examples on youtube, but it would be really good if we could put a page onto xenmodalism to stop all the big red tomato text on what i consider an important page"

Tcherepnin

2026-06-15T18:16:01Z

Kili:

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 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.

Glossary

2026-06-15T18:12:23Z

Kili: /* Monzo */ mode of limetod tretansoposition

This page lists various terms conventionally used in xenharmony (or in some cases, general music theory as it applies to xen) that can be briefly described.

'''Don't put idiosyncratic terms here.''' When using personal terminology in an article, either explain it there or link to an article about your theory that explains the term.

== Basis ==
A '''basis''' (pl. ''bases'') for a [[#JI group|JI group]], or similar group, is a list of intervals called ''generators'' such that:
# anything in the group can be written as a stack of intervals of the basis or their inverses (possibly with repetition).
# the list is non-redundant in the sense that there is only one way to write any particular interval in the group as a stack of generators.

Examples:
* [2, 3/2, 5/4] is a basis for the [[5-limit]]; so is [2, 3, 5].
* [2, 5/3] and [2, 9, 5] are not bases for the 5-limit, on account of not satisfying condition 1.
* [2, 3, 5, 15] is not a basis for the 5-limit, on account of not satisfying condition 2.

JI groups are denoted using basis elements separated by full stops, for example 2.5.11/3.

Categories: RTT, JI, Math terms

== Binary ==
A '''binary''' scale is a scale with exactly two step sizes (usually denoted L and s). [[MOS]] scales are binary, but binary scales need not be MOS scales (e.g. melodic minor, LsLLLLs, is binary but not a MOS).

Categories: Scales

== Cent ==
A '''cent''' (abbreviated to c or ¢) is the conventional measurement unit of the logarithmic (perceptual) distance between [[Frequency|frequencies]]; in other words, the size of the [[interval]] between them. A cent is defined as a frequency ratio of 2^(1/1200), or a factor of about 1.0005778, such that the octave ([[2/1]]) spans exactly 1200 cents, and therefore that each step of [[12edo]] spans exactly 100.

Categories: Core knowledge

== Chiral ==
When a '''chiral''' scale has its step pattern reversed, it is no longer a mode of the original scale. A scale is '''achiral''' when this does not hold.

MOS scales and certain ternary scales such as [[blackdye]] are achiral, but many scales of interest such as [[zarlino]], [[diasem]] and [[Zil|Zil[14]]] are chiral.

The chiral pair of a chiral scale is conventionally denoted "right-hand", "RH", or "R", and "left-hand", "LH", or "L". An algorithm is usually used to determine the choice of L or R, but it's musically inconsequential. All you have to know is that one is L and one is R.

Categories: Scales

== Chord ==
A '''chord''' is a finite set of (usually three or more) pitches, often implying a context when the pitches are played together. Two chords are usually considered the same chord if they only differ by transposition.

Categories: Core knowledge

== Comma ==
'''Comma''' may refer to:
# a small JI interval. A fundamental fact about JI is that no stack of one prime is a stack of any other set of primes. Commas hence occur frequently in stacking-based JI.
# The commas of a regular temperament are the intervals it tempers out, {{adv|which can all be written as stacks of a certain number of commas known as the ''comma basis'' which suffice to determine every comma that is tempered out or every pair of intervals that is equated.}} "Tempering out" means that all JI ratios/stacks that are separated by that comma are equated, e.g. tempering out 81/80 not only equates 81/64 and 5/4 but also equates 40/27 and 3/2. This follows from the principles of regular temperament.
# An interval region of intervals around 20 cents, less than about 30 cents.

Categories: JI, RTT

== Complexity ==
The '''complexity''' of a rank-2 temperament is fairly easy to intuit: it is how many stacked generators are needed to reach simple JI ratios. There is often a tradeoff between simplicity and accuracy in temperaments. For example, 5-limit Schismic is a more accurate but more complex temperament than 5-limit Meantone, since more generators are needed to reach 5/4 in the former.

Categories: RTT

== Consistency ==
An approximation of an interval in an EDO (or otherwise an equal-step tuning) is '''consistent''' when it is both the closest direct approximation of the just interval available in the tuning, and the approximation regularly dictated by the [[#Val|val]] being used (whether [[#Patent val|patent]] or otherwise). Approximations where these two deviate from each other are correspondingly '''inconsistent'''.

For example, the interval [[7/6]] is inconsistent in [[34edo|34et]], since while [[7/4]] is defined as 27 steps and [[3/2]] as 20 steps (implying 7/6 to be 7 steps), 7/6 itself is slightly closer to 8 steps than to 7 steps of 34edo. {{Adv|In the alternative val 34d, where 7/4 is mapped to 28 steps instead, 7/6 becomes consistent but 7/4 itself is now inconsistent.}}

When discussing equal tunings, it is common to speak of the '''consistency limit''', the largest [[odd-limit]] in which every interval is mapped consistently, as well as the '''distinct consistency limit''', which adds the criterion that all intervals of the consistent odd-limit must be mapped to distinct intervals in the tuning. Such odd-limits can further be restricted to a particular [[#JI group|subgroup]] which the ET is considered as an approximation to.

Categories: EDO, JI

== Constant structure ==
A '''constant structure''' (CS; Erv Wilson's term) is a scale such that no two of its interval classes share a common interval.

Pythagorean diatonic is a constant structure:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
|-
!1/1
|9/8
|81/64
|4/3
|3/2
|27/16
|243/128
|-
!9/8
|9/8
|32/27
|4/3
|3/2
|27/16
|16/9
|-
!81/64
|256/243
|32/27
|4/3
|3/2
|128/81
|16/9
|-
!4/3
|9/8
|81/64
|729/512
|3/2
|27/16
|243/128
|-
!3/2
|9/8
|81/64
|4/3
|3/2
|27/16
|16/9
|-
!27/16
|9/8
|32/27
|4/3
|3/2
|128/81
|16/9
|-
!243/128
|256/243
|32/27
|4/3
|1024/729
|128/81
|16/9
|}
But 12edo diatonic is not, because 600c is both a 3-step interval and a 4-step one:
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
|-
!0\12
|200.0
|400.0
|500.0
|700.0
|900.0
|1100.0
|-
!2\12
|200.0
|300.0
|500.0
|700.0
|900.0
|1000.0
|-
!4\12
|100.0
|300.0
|500.0
|700.0
|800.0
|1000.0
|-
!5\12
|200.0
|400.0
|class="thl"|600.0
|700.0
|900.0
|1100.0
|-
!7\12
|200.0
|400.0
|500.0
|700.0
|900.0
|1000.0
|-
!9\12
|200.0
|300.0
|500.0
|700.0
|800.0
|1000.0
|-
!11\12
|100.0
|300.0
|500.0
|class="thl|600.0
|800.0
|1000.0
|}

Some find CS a desirable property for JI scales, and some people find constant structure scales easier to navigate on keyboards.

A JI scale being a CS is ''not'' equivalent to it being a detempering of an equal temperament. The latter implies the former, but not vice versa.

Categories: Scales

== Detempering ==
'''Detempering''' a tempered scale results in a scale that has pitches in JI (or a temperament that tempers less). Each tempered pitch corresponds to one or more pitches in the detempered scale, which map to the tempered pitch under the temperament.

The Zarlino scale in 5-limit JI is a detempering of Meantone diatonic. Pental blackdye is another detempering of Meantone diatonic, but with some cases of multiple detempered pitches corresponding to a tempered pitch.

Categories: RTT, JI

== Enharmonic ==

=== Sense 1 ===
Two notes or intervals are enharmonic, or enharmonically equivalent, if they map to the same degree of the chromatic scale (the 12-note [[MOS]] scale generated by a [[perfect fifth]]). This can be generalized to pairs of notes separated by the difference between a chroma and a small step in a given scale, where enharmonic intervals are separated by a diesis, and can be equated by tempering out said diesis.

=== Sense 2 ===
A 17- or 19-note MOS scale generated by a perfect fifth, which assigns enharmonically equivalent diatonic intervals their own scale degrees by making the diatonic diesis a small scale step. Schismic[17] is usable as a scale for [[Schismic]] temperament.

=== Sense 3 ===
A Greek scale in which the lower two of the three intervals of a [[tetrachord]] are less than a semitone each.

=== Sense 4 (proscribed) ===
In [[12edo]], enharmonic notes in sense 1 are equated, which has led to a secondary use of "enharmonic" to refer to other equations between notes of a scaleform in some tuning system (such as B# = Cb in [[19edo]]). This particular use is discouraged due to the potential for confusion with other meanings of this already overloaded term.

== Equave ==
An '''equave''' or '''interval of equivalence''' is an interval that separates notes that are considered equivalent. Most commonly the octave (2/1), but 3/1, 3/2, and other intervals are sometimes used. See also [[Glossary#Non-Octave|Non-Octave]].

Categories: Core knowledge
== Extension, contorsion ==
An '''extension''' of a temperament is a temperament that interprets the tempered intervals of the original temperament within a larger [[Glossary#JI group|JI group]]. A '''weak extension''' introduces new tempered intervals in addition to those of the original temperament, whereas a '''strong extension''' uses the same set of intervals as the original temperament. The opposite of an extension is a '''restriction''', which interprets a temperament as a subset of the original JI group, and strong and weak restrictions are defined similarly.

For instance, [[Meantone]] introduces [[5-limit]] interpretations of intervals on a [[Chain of fifths|chain]] of tempered fifths by making the equivalence ([[3/2]])4 = [[Octave|2]]2 × 5/4 (tempering out the comma [[81/80]] and finding 5 at 4 fifths up). But if the chain of fifths is continued further, [[7-limit]] harmonies can be introduced: (3/2)2 × (5/4)2 = 2 × [[7/4]], which can be worked out to place 7 at 10 fifths up, a mapping of 7 known as ''septimal Meantone'', which is a strong extension of 5-limit Meantone.

Weak extensions are created by dividing the original period or (a choice of) generator into equal parts and then interpreting the split parts. As an example, Mothra is a temperament where the 3/2 Meantone generator is split into 3 parts, and then (3/2)^(1/3) is interpreted as [[8/7]]. It is a weak extension of pental Meantone, as Meantone natively doesn't have something that is one-third of a 3/2, to the 7-limit. Sometimes a weak extension may split the period instead of the generator; for example, Pajara (2.3.5.7[10 & 22]) is a weak extension of Archy (2.3.7[5 & 22]) that splits 2/1 into two 7/5's.

If you don't interpret the new intervals of a weak extension, the result is called '''contorsion'''. For example, if one were to take 2.3.5 Meantone and split the fifth into three equal parts without interpreting ~(3/2)1/3 as JI, the resulting temperament is a contorted 2.3.5 Meantone. If a rank-1 temperament is said to be contorted, it's still an instance of the concept of not fully interpreted weak extensions, but implies that the equal tuning is just a multiple of, and has the same cent value mappings as a subset equal tuning. For example, 36edo is contorted in the 5-limit because it uses the same cent value mappings as 12edo but adds uninterpreted intervals outside of 12edo. Note that the new generator 1\36 has no 5-limit interpretation. However, interpreting 1\36 as 64/63 interprets 36edo as a weak extension of 5-limit 12edo that adds prime 7.

Note that temperaments of different ranks are ''not'' considered extensions or restrictions of one another.

''For this wiki's guidelines on what extensions a given temperament name refers to, see [[Xenharmonic Reference:Guidelines]].''

Categories: RTT, Somewhat technical

== Generator ==
In xen theory, a '''generator''' is an interval that is stacked to obtain various intervals. Technically, ''generator'' has a number of slightly different senses:
* An element of a (chosen) [[glossary#Basis|basis]] for a [[glossary#JI_group|group]].
* For a rank-2 structure (temperament or [[MOS]]), a non-period generator (the choice of which is not unique). Examples:
** In [[Meantone]], which has period 1\1, the generator is (assuming pure 2/1) a flattened ~3/2 or a sharpened ~4/3. This corresponds to a generator of the MOS 5L2s (MOS diatonic).
** In [[Pajara]], which has period 1\2, there are 4 choices of octave-reduced generator: ~16/15, ~4/3, ~3/2, ~15/8. This is a generator of the MOS 2L8s.
* An element of a [[generator sequence]].

Categories: RTT, JI

== Hardness ==
The '''hardness''' of a [[binary scale]] (a scale with two distinct step sizes, ''L'' > ''s''), principally a [[MOS]], is the ratio ''L'':''s''. For instance, the diatonic ([[5L 2s]]) scale in [[17edo]] comprises the steps 3-3-1-3-3-3-1; its hardness is therefore 3:1. In [[34edo]], the diatonic is inherited from 17edo, and its step sizes are 6-6-2-6-6-6-2. 6:2 reduces down to 3:1.

Scales with a hardness greater than 2:1 (i.e. ''L'' > 2''s'') are called "hard", while scales with a hardness less than 2:1 (i.e. ''L'' < 2''s'') are called "soft". Binary scales with a hardness of exactly 2:1 are called "basic". A finer gradation of terms for hardnesses is provided by [[TAMNAMS]].

The concept of hardness can also be extended to [[ternary scale]]s and higher - for instance, [[22edo]] [[zarlino]], 3-4-2-4-3-4-2, has hardness 4:3:2.

Categories: Scales

== Harmonic mode ==
A harmonic segment of the form ''n''::2''n'', considered as an octave-equivalent scale. For example, mode 7 of the harmonic series is 7:8:9:10:11:12:13:14.

Categories: JI
== Harmonic segment ==
Any finite set of consecutive harmonics in the harmonic series. Can be denoted ''m''::''n''. For example, 5:6:7:8:9:10 is written 5::10.

Categories: JI
== Harmonic series ==
The infinite sequence of whole-number frequency multiples, called ''harmonics'', above a fundamental frequency. The harmonics of 110 Hz are:
* 1st harmonic (fundamental): 110 Hz
* 2nd harmonic: 220 Hz
* 3rd harmonic: 330 Hz
* 4th harmonic: 440 Hz
* 5th harmonic: 550 Hz
* 6th harmonic: 660 Hz
* ...
Every JI interval occurs in the harmonic series as the pitch difference between some pair of harmonics.

Differences in relative loudnesses of various harmonics above a note, as well as deviations from mathematically exact harmonics (called ''inharmonicity''), are perceived as different timbres of the same note.

Categories: JI, Core knowledge

== Interval class ==
An '''interval class''' or '''generic interval''' is the set of all intervals that occur as a given number of steps in a given scale. For example, the interval class of fifths (4-step intervals) in 12edo diatonic is {700c, 600c}. Sometimes called an '''ordinal''', because these are called ordinal numbers in conventional diatonic theory: "seconds", "thirds", etc. Other schemes such as [https://en.xen.wiki/w/TAMNAMS TAMNAMS] use a 0-indexing scheme: "1-step" for "seconds", "2-step" for "thirds", etc. See also [[#k-step]].

Categories: Scales

== JI group ==
A '''JI group''' is the set of all intervals that are formed by stacking a given set of JI ratios or their inverses finitely many times. JI groups are often called '''subgroups''', as they can be seen as subgroups (subsets of a group that are also groups) of infinite-limit just intonation. Additionally, "subgroup" may be used in older materials to refer to JI groups that are not [[Glossary#Limit|prime-limit]]s, because older RTT theorists thought of non-full-prime-limit groups as subgroups of full prime-limits. A JI group (or the interpretation-agnostic tuning of intervals to a JI group) may also be called a '''JI lattice''', though "lattice" can also mean a diagram of how the pitches of a particular JI or tempered scale look in such a JI group.

JI groups are denoted by generators, called ''basis elements'' (the standard mathematical term) or ''formal primes'' in this context, separated by full stops: for example, 2.3.5.7 denotes the [[7-limit|7-prime-limit]]. Usually, the first basis element is assumed to represent the [[equave]]: "3.2.5" would be a version of 2.3.5 that repeats on the [[3/1|tritave]], though note that mathematically speaking, 2.3.5, 3.2.5, 3/2.3.5, and so on are the same group.

Prime-limits are JI groups. Non-prime-limit JI groups include groups of primes (such as [[2.3.7 subgroup|2.3.7]]), as well as groups including composites (like 2.3.25.13 or 2.9.15.7) or fractions (like 2.5.7/3.11/3). By convention, composite and fractional basis elements are sorted by the prime-limit that they belong to. On XR, 2.b/a.c/a.d/a may be written {{nowrap|2.(a:b:c:d)}} for brevity, for example {{nowrap|2.(5:7:11:13)}} = 2.7/5.11/5.13/5 (note that this is the JI group generated by 2/1 and the intervals of 5:7:11:13).

Groups can be generalized to non-JI generators, for example 2.√6 (representing a chain of perfect hemififths), or 2.φ.

A regular temperament starts with a JI group and maps the group to a tempered group. For example, Meantone maps 2.3.5 to the group generated by tempered 2 and tempered 3/2.

{{adv|Mathematically, a '''group''' is a set with}}
* {{adv|a binary operation * (for all group elements ''g'' and ''h'', ''g'' * ''h'' is also an element of the group)}}
* {{adv|the binary operation * is associative (thus no parentheses are needed when writing the group operation on more than two elements)}}
* {{adv|an identity element: a unique element ''e'' such that {{nowrap|''g'' * ''e'' {{=}} ''e'' * ''g'' {{=}} ''g''}} for all ''g'' in the group}}
* {{adv|an inverse element for every element: every ''g'' corresponds to a unique element ''g''-1 such that {{nowrap|''g'' * ''g''-1 {{=}} ''g''-1 * ''g'' {{=}} ''e''}}}}
{{adv|A subgroup ''generated by'' a subset of a group is the group formed by iterating the binary operation on elements in the subset. Equivalently, it is the smallest subgroup of the larger group containing that subset.}}

{{Adv|Groups in xen theory are typically a much more specific type of groups, namely [[wikipedia:Free abelian group|free abelian groups]].}}

Categories: JI, RTT, Math terms

== Limit ==
In just intonation, '''limit''' most commonly has two distinct senses:
* The ''p''-'''[[prime-limit]]''' is the set of all JI ratios with primes up to ''p'' in their prime factorization. 3/2, 5/3, 7/4, and 49/36 are all in the 7-prime-limit, but 11/7 is not.
* The ''n''-'''[[odd-limit]]''' is a set of JI intervals with both numerator and denominator at most ''n'' after all factors of 2 are removed. Equivalently, it is the set of all intervals that appear in the harmonic series scale ''k'':(''k''+1):...:2''k'' (and all their octave equivalents), where ''k'' = ''n''/2 + 1/2. For example, the 15-odd-limit is the set of intervals that occur in the harmonic series scale 8:9:10:11:12:13:14:15:16; 21/16 is not in the 15-odd-limit.
The term "limit" without qualification today more commonly means prime-limit, though Harry Partch who coined the term ''limit'' originally meant odd-limit.

=== Proper limit ===
While a prime-limit encompasses all ratios up to a given prime, '''proper prime-limit''' classifies JI ratios based only based on the ''highest'' prime they contain in either the numerator or denominator. Equivalently, it is all of the intervals of a prime limit that are not found in a lower prime limit. This has been called '''harmonic class''', but this is discouraged because (a) it's a vague term and there are potentially many situations where intervals could be classified into "classes", and (b) people e.g. often informally use "7-limit" to denote the proper 7-limit.

The proper ''p''-prime limit contains ''only'' ratios for which ''p'' is the highest prime number found in their factorizations. For example:
* [[7/4]] is in the proper 7-limit because 7 is the highest prime in its factorization.
* [[5/4]] is in the proper 5-limit, not proper 7-limit, even though it's within the 7-limit.
* [[9/7]] is in the proper 7-limit because the highest prime is 7 (since {{nowrap| 9 {{=}} 32 }}).

Similarly, the '''proper ''n''-odd-limit''' is the set of all ''n''-odd-limit intervals that are in no lower odd-limits. For example, 7/4 is in the proper 7-odd-limit, and 9/7 is in the proper 9-odd-limit.

This distinction helps differentiate between intervals that merely fall within a limit versus those that specifically use a particular prime or odd. Unlike regular harmonic limits, proper harmonic limits are mutually exclusive categories.

Categories: JI

== Linearly independent ==
A set of vectors (such as a set of [[monzo]]s or a set of [[val]]s) is '''linearly independent''' if no vector in the set is redundant: no nonzero multiple of a vector can be written as a sum of multiples of other vectors. In the Xenharmonic Reference we will often shorten this to '''independent'''. In other sources the term ''co-unique'' may be used. {{Adv|This is technically <math>\mathbb{Z}</math>-linear independence; <math>\mathbb{Z}</math>-modules and abelian groups are the same concept.}}

Examples (for vals):
* {{val|12 19 28}} and {{val|19 30 44}} ([[12edo]] and [[19edo]] [[Glossary#Val|patent val]]s in the [[5-limit]]) are independent.
* {{val|12 19 28}}, {{val|19 30 44}}, and {{val|31 49 72}} are not independent, since the [[31edo]] val is a sum of the 12edo and 19edo patent vals. {{adv|We say that three vectors are ''collinear'' if they taken together are not linearly independent though any two of them are.}}
* {{val|24 38 96}} and {{val|36 57 84}} are not independent, since they share a common multiple.

Examples of where this concept shows up in RTT:
* Basis elements for any applicable group must be independent.
* Two ''independent'' vals (equal temperaments) determine a rank-2 temperament, three ''independent'' vals determine a rank-3 one, ...

Categories: RTT, Math terms, Somewhat technical

== Mode of Limited Transposition ==
A scale is considered to be Mode of Limited Transposition should it fulfill this criterion:

* it can be transposed by any pitch in its superset while at least 2 of those transpositions result in no change to the roster of pitches available.

The Diatonic Major of 12edo is '''not''' of Limited Transposition, because each possible disposition changes at least one of the pitches, until you return to the scale's period. However, the Diatonic Major of 7edo '''is''' of Limited transposition, because multiple transpositions of n\7 will result the tenure of the original pitches.

== Monzo ==
A '''monzo''' is a vector (list of coordinates) representing a JI ratio, whose coordinates are (usually) prime exponents. Also called an '''interval vector''' or a '''prime count vector'''.

Example: 81/80 = 3^4/(2^4 * 5^1) = 2^-4 * 3^4 * 5^-1 can be written in monzo form as {{monzo|-4 4 -1}}.

Categories: RTT
== Neji ==
A '''neji''' ("near-equal/equivalent JI") is a (possibly somewhat loose) JI approximation to a non-JI scale (often an edo), usually a subset of a chosen harmonic mode. The term was introduced by Zhea Erose.

Nejis are usually written as enumerated chords (i.e. written in the form a:b:...:z in ascending order): for example, the 12edo neji used in Zhea Erose's Eurybia is 22:23:25:26:28:30:31:33:35:37:39:42:44.

Categories: JI

== Non-octave ==
A tuning or temperament which does not have 2/1 is called non-octave. This includes JI subgroups that do not include 2, such as 3.5.7, as well as equal temperaments such as [[Bohlen-Pierce|13edt]] or Bohlen-Pierce. Tunings and temperaments that map a multiple of 2/1 (but not 2/1 itself), such as 41ed4 or 4.5.7, are also included.

== Period ==
'''Period''' has the following related but different senses:
* The smallest unit at which a given scale repeats — a fraction of the equave but not necessarily the equave itself.
** Example: Pentawood (5L5s, LsLsLsLsLs) has period 1\5 (240c).
* One of the generators of a regular temperament, specifically chosen to be a fraction of the equave (usually 2/1). (We make this choice for musical reasons, though a group mathematically doesn't have a distinguished element called the "period".)
** Example: The temperament Blackwood has period 1\5.
The two senses are related in that a multiperiod scale or equal division often supports a multiperiod temperament interpretation, and a multiperiod temperament requires an equal division that supports it to be divisible by some number (namely, the number of temperament-periods in the equave).

Categories: Scales, RTT

== Pitch class ==
Assuming an equave, two pitches or two intervals belong to the same '''pitch class''' if they are separated by a multiple of the equave. Pitch class space is a circle, whereas pitch space is a line.

Lattice diagrams of JI or tempered scales show the pitches in a pitch-class lattice, a lattice one dimension lower than the original JI group, where equave differences are ignored.

Categories: Core knowledge
== Rank ==
The term '''rank''' just means "dimensionality". The rank of a temperament is the dimension of the group of tempered JI ratios under that temperament. A temperament like [[Meantone]] has rank (dimension) 2 because any interval in Meantone can be written as a stack of some number of tempered octaves and some number of tempered fifths. Any [[equal tuning]] is rank 1 because all intervals in an equal tuning are a stack of that tuning's step size.

Rank-2 temperaments deserve special mention as they can be described as stacking a single generator against a [[Glossary#Period|period]]. As a result, a very clear method for constructing scales from rank-2 temperaments exists, that being forming a [[MOS]] from the temperament's generator and period, which is quite nontrivial to generalize to systems of higher rank.

Categories: RTT, Math terms

== Regular temperament ==
:''Main article: [[Regular temperament]]''

A '''regular temperament''' (often just '''temperament''') is a way of assigning JI interpretations (from a chosen JI group) to intervals in a non-JI tuning. We assign the interpretations so that the stack of two JI ratios gets assigned to the stack of the corresponding tempered versions of the two ratios. We also assume that each JI ratio is assigned to one and only one cent value, unlike in irregular/well temperaments.

If you know what notes of a tempered tuning the ''basis generators'' of a chosen JI group get assigned to, that suffices to determine the interpretations assigned to any particular interval {{adv|(provided that every interval is indeed interpreted, as in the overwhelming majority of practical cases).}} This is how vals and mappings for regular temperaments work — they specify what tempered notes correspond to the basis elements of the JI group.

The study of regular temperaments is called [[regular temperament theory]] (RTT).

Categories: RTT

== Scale ==
A '''scale''' is a collection of pitches; two scales are considered the same scale if they only differ by transposition. Unlike chords, scales are usually ''periodic'', i.e. the same pattern of intervals repeats at some interval called the ''equave''. On the Xenharmonic Reference, ''scales are periodic or repeating unless stated otherwise.'' (Though of course, finite but nonrepeating pitch material may be useful to consider in some contexts like voicing and register, especially in harmonic series or spectralist music.) A scale can be visualized as a set of points in the circle of equave-equivalent pitch classes.

Categories: Core knowledge, Scale

== Signature ==
A '''signature''' is a list of numbers giving useful but incomplete information about an object. Usually refers to one of:
* a ''step signature'', a list of how many of each step size a scale has; e.g. 4L3m2s.
* a ''[[delta signature]]'', a list of frequency increases between adjacent notes measured relative to a reference frequency increase, e.g. +1+1+2 for the chord 6.465:7.465:8.465:10.465.

Categories: Somewhat technical
== ''k''-step ==
An abbreviation for "''k''-step interval". For example, the fifth in the diatonic scale is a 4-step. See also [[#Interval class]].

Categories: Scales

== Square-superparticular ==
A '''square-superparticular''' or '''square-particular''' is a superparticular of the form

<math>\frac{k^2}{k^2-1} = \frac{k}{k-1}\frac{k}{k+1} = \frac{\frac{k}{k-1}}{\frac{k+1}{k}},</math>

denoted S''k'' or S(''k'') in xen math.

A square-superparticular is the difference between consecutive suparparticulars. When a square-superparticular S''k'' is tempered out, it makes harmonics {{nowrap|''k'' - 1}}, ''k'', and {{nowrap|''k'' + 1}} equally spaced. For example, tempering out S9 = 81/80 makes harmonics 8, 9, and 10 equally spaced. Factoring a comma into a product of square-particulars, called an '''S-expression''', is often helpful for understanding it.

{{Adv|The ratio between two consecutive square-superparticulars is called an ''ultraparticular'', which has the form S''k''/S(''k'' + 1). Tempering out an ultraparticular equates the differences between three consecutive superparticulars.}}

Categories: RTT

== Superparticular ==
A '''superparticular''' or Delta-1 ratio is a ratio between two whole numbers which differ by 1: e.g. [[2/1]], [[3/2]], [[4/3]], [[5/4]], etc, representing intervals between consecutive members of the [[#Harmonic series|harmonic series]]. These are distinguished from '''superpartient''' ratios (all other rational ratios), which can be classified as Delta-2, Delta-3, etc. by the difference between their numerator and denominator. Note that the [[Glossary#Square-superparticular|ratio between consecutive superparticulars]] is itself superparticular.

Categories: JI, Math terms

== Ternary ==
A '''ternary''' scale is a scale with exactly three step sizes (usually denoted L, m, and s).

Categories: Scales

== Tertian ==
In standard music theory, '''tertian''' harmony refers to harmony where thirds are privileged as the main component of chords. The most basic tertian chords are root-third-fifth triads and their inversions, but larger chords such as dom7 (stacked M3-m3-m3) and major 11th (stacked M3-m3-M3-m3-M3) are often also considered tertian.

Categories: Chords

== Val ==
A '''val''' (short for "valuation") is a vector whose coordinates are step mappings of primes in an [[equal temperament]]. {{Adv|It can mathematically be called a "covector", since it is a kind of a vector "dual" (complementary) to interval vectors.}}

Example: 12et maps 2/1 to 12 steps, 3/1 to 19 steps (reduced: 7 steps), and 5/1 to 28 steps (reduced: 4 steps). We write this in val form as {{val|12 19 28}}. Vals can be ''evaluated'' at monzos (showing how the equal temperament maps the JI ratio) by multiplying each pair of corresponding entries and summing the results together. This can be seen as, for a monzo with entries m and a val with entries v, "stepping" by each v m times for its corresponding m. {{Adv|In linear algebra, this operation is called the dot product.}} This is denoted by {{val|val}}{{monzo|monzo}}. Evaluating this val at {{monzo|-4 4 -1}} (the monzo for 81/80) shows that 12et tempers out 81/80:

{{val|12 19 28}}{{monzo|-4 4 -1}} = 12 * -4 + 19 * 4 + 28 * -1 = -48 + 76 - 28 = 0.

Categories: RTT

=== Patent val ===
'''Patent vals''' are the most common kinds of vals to consider. The "patent" means that the closest approximations in the edo tuning in question are used for the step mappings. The above val is the 12edo patent val in the 5-limit. An example of a non-patent val is {{val|12 19 27}}, since the closest approximation to 5/1 in 12edo is not 27 steps, but 28 steps.

{{Adv|The concept of a patent val can be extended into the notion of a '''generalized patent val''' (GPV), or a "uniform map" in some sources. A GPV is essentially a patent val corresponding to an equal-step tuning that might not necessarily divide an exact 2/1. For instance, the val {{val|17 27 40}} is a GPV, as this consists of the closest approximations of primes 2, 3, and 5 in 17.1edo, where 5 differs from the approximation in 17edo proper (which is 39 steps).}}

Categories: RTT

== Variety ==
'''Variety''' (or '''interval variety''') refers to how many interval sizes an [[Glossary#Interval class|interval class]] comes in. We often refer to
* '''maximum variety''' (MV) if all varieties satisfy a certain bound and there is some variety equal to the bound (thus MV2 scales are ''not'' MV3)
* '''strict variety''' (SV) if all varieties (except equave multiples) are equal to some value.
For example, [[MOS]] scales can be defined as scales that are MV2. Mosdiatonic is SV2.

Categories: Scales

== Virtual fundamental ==
The '''virtual fundamental''' or '''missing fundamental''' associated with the pitches of a JI chord is the perception of a common frequency such that all of the notes of the chord are overtones of it. Humans perceive a virtual fundamental when they hear a (complete enough) set of overtones with the fundamental missing (though it's not as simple as just matching harmonics to a template; see [[Delta-rational chord]]). For example, the virtual fundamental of {220 Hz, 330 Hz, 440 Hz, 550 Hz, 660 Hz} is 110 Hz, so when a 2:3:4:5:6 chord is played on 220 Hz, you may hear a pitch at 110 Hz.

Categories: JI, Acoustics

5-odd-limit

2026-06-01T18:57:18Z

Kili: /* Perfect fifth (3/2) */ lijnkao

{{Odd-limit navigation}}
The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are considered equivalent. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino, constructed from the numbers 1, 2, 3, 4, 5, 6, and 8. (Note the absence of 7.)

== Table of 5-odd-limit intervals ==
Reduced to an octave, the intervals of the 5-odd-limit are:
{| class="wikitable"
!Interval
!Cents
!Name
!Type
|-
|1/1
|0.0
|Unison
|Equivalence
|-
|6/5
|315.6
|Classical minor 3rd
|Imperfect consonance
|-
|5/4
|386.4
|Classical major 3rd
|Imperfect consonance
|-
|4/3
|498.0
|Perfect 4th
|Perfect consonance
|-
|3/2
|702.0
|Perfect 5th
|Perfect consonance
|-
|8/5
|813.6
|Classical minor 6th
|Imperfect consonance
|-
|5/3
|884.4
|Classical major 6th
|Imperfect consonance
|-
|2/1
|1200.0
|Octave
|Equivalence
|}

== Approximation by edos ==
The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[Dicot]], does not lead to categorical conflicts the same way 3edo's [[Father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, [[19edo]], is a [[Meantone]] tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as [[Magic|five major thirds stacking to a fifth]]. [[22edo]], a non-Meantone tuning, has the opposite tuning tendencies to 12edo.

== Intervals of the 5-odd-limit ==

=== Perfect consonances ===

==== Perfect fourth (4/3) ====
''Main article: [[Perfect fourth]]''

The perfect fourth is a perfect consonance and the bounding interval for [[chthonic]] harmony. It also exists between the fifth over the root and the root an octave up. It is the dark generator of the diatonic scale; stacking it produces the Locrian [[mode]].

In certain triadic musical traditions that use 4:5:6 as a consonant chord, the perfect fourth over the root can be considered dissonant, as it resolves downwards to the major third.

==== Perfect fifth (3/2) ====
''Main article: [[Perfect fifth]]''

The perfect fifth is an unambiguous perfect consonance. It appears in musical systems worldwide, and can be easily tuned by ear. This is the reason behind the prevalence of the [[Pythagorean tuning|Pythagorean]] system of tuning.

In the context of 5-limit consonances, the perfect fifth serves as the bounding interval of the triads 10:12:15 (minor) and 4:5:6 (major), which utilize the other 5-odd-limit consonances 5/4 and 6/5.

=== Imperfect consonances ===

==== Classical major third (5/4) ====
The major third 5/4 serves primarily as a component in tertian chords like 4:5:6. It is the most consonant "third" interval. Building a scale by stacking 4:5:6 triads produces the Zarlino [[diatonic]] major scale. The 4:5:6 triad is well-represented in 15edo (which has a stretched triad), 19edo, 22edo, 31edo, 34edo, 41edo, 46edo, and 53edo.

5/4 is also the octave-reduced generator of the [[5-limit|prime 5]] axis in [[lattice-based just intonation]].

==== Classical minor third (6/5) ====
The classical minor third 6/5 is the fifth complement of 5/4. The distinction between the two leads to the paradigm of major vs. minor in interval classification and in triadic harmony; it is why the "third" category of intervals exists at all, and additionally why thirds are often considered the "default" example of interval qualities.

One quality of 6/5 worth noting is that chords with 6/5 as a lower interval are, as a rule, not "rooted" (in that their root note is not a power of 2 in the harmonic series). The significance of this is debated by xenharmonic theorists; Lamplight uses it as a model for the different "feels" of the chords 4:5:6 and 10:12:15.

==== Classical major sixth (5/3) ====
The classical major sixth is the octave complement of 6/5. It is according to some the next most consonant interval within the octave after 4/3; Leriendil sees it as an important target interval on the level of 4/3 and it is also the bounding interval of the chord 3:4:5, which may be seen as an inversion of 4:5:6 or as the primary focus of 5-limit "/3" harmony (such as in [[Kleismic]]).

==== Classical minor sixth (8/5) ====
The classical minor sixth is, while consonant on its own, unusually dissonant for a 5-odd-limit consonance in certain contexts; the result is likely a combination of factors. First is its complex ratio - it is the only 5-odd-limit interval in the octave that uses 8 in the numerator. Second is its proximity to the golden ratio, which serves as a distinctly dissonant target (similar to the [[semioctave]]'s influence on 7/5). Third is its proximity to 3/2, which produces a 'zone' of dissonance around it. Also of relevance to the discussion is the 12edo augmented triad, which contains a note tuned the same way as the classical minor sixth (and which may be voiced as it in JI depending on interpretation) yet is considered a dissonant chord.

This and the major sixth mainly show up in chords in Western harmony as the bounding intervals of triads in certain inversions.

Mabilic

2026-06-01T18:55:47Z

Kili: linkoe

'''Mabilic''' is a rank-2 [[regular temperament]] based around the [[antidiatonic]] and [[armotonic]] scale structures. [[5/4|5/2]] is split into three [[Generator|generators]] which are somewhat sharper than a fourth (~520-530 cents), five of which stack (octave-reduced) to make [[8/7]].
Mabilic in its basic form is a [[2.5.7 subgroup]] temperament, because 3 cannot be added without a high complexity or a significant loss of accuracy. Mabilic[7] or [9], Semabila[16], and Trismegistus[25] are reasonable forms.

{{Cat|Temperaments}}

Extensions of Mabilic include

* Trismegistus (best tuned around 527 cents) which finds [[3/2]] at 15 generators up, equating it to both three 8/7s ([[Slendric]] temperament) and five 5/4s ([[Magic]] temperament).
** Mnemonic: tris- is Greek for "thrice"; three 8/7's are equated to 3/2; -megistus refers to the Magic mapping of 2.3.5
* Semabila (best tuned around 530 cents) which finds [[4/3]] at 10 generators up, equating it to two 8/7s ([[Alpha-dicot|Semaphore]] temperament).
** Semabila also tunes the fifth to 25/17, finding 17 at 7 generators; then 8 generators may be assigned to 23/16. This is possible but less accurate in Trismegistus.
*** In fact, Semabila easily extends to the full 23-limit by finding 16/13 at 12 generators and 16/11 at 8 generators, which is not accurate at all in Trismegistus.
* Mavila (best tuned around 522-528 cents), an exotemperament which sets the generator itself equal to 4/3, and somewhat functions as an opposite to [[Meantone]].
** Mavila and trismegistus can be considered co-structural temperaments.
In Meantone, 4 fifths make a 5/4; in Mavila they make a [[6/5]].

In any tuning, the flat fifth generator may be identified wth 28/19. This produces a 2.5.7.19 temperament with a flat tendency for 19, and justifies the generator as an imperfect fifth of 95/64 created by stacking 5/4 and 19/16.

== Intervals ==
Trismegistus/Mavila:
{| class="wikitable"
! rowspan="2" |Generators
! rowspan="2" |Tuning
! colspan="4" |Interpretation
|-
!2.5.7.19
!Trismegistus
!Mavila
!+17.23 interpretation
|-
|0
|0
|1/1
|
|
|
|-
|1
|527
|19/14
|
|4/3
|23/17, 34/25
|-
|2
|1054
|64/35
|
|16/9, 15/8
|
|-
|3
|381
|'''5/4'''
|
|9/7, 24/19
|
|-
|4
|908
|'''32/19'''
|
|5/3
|
|-
|5
|235
|'''8/7'''
|
|10/9
|
|-
|6
|762
|25/16
|
|32/21
|
|-
|7
|89
|
|21/20
|15/14
|17/16
|-
|8
|616
|10/7
|
|
|23/16
|-
|9
|1143
|
|
|40/21
|
|-
|10
|470
|
|21/16
|
|
|-
|11
|997
|
|
|
|34/19
|-
|12
|324
|
|6/5
|
|
|-
|13
|851
|
|
|
|
|-
|14
|178
|
|
|
|17/15
|-
|15
|705
|
|3/2
|
|
|}
Semabila:
{| class="wikitable"
!Generators
!Tuning
!Interpretation (2.3.5.7.19)
!Interpretation (2...29)
|-
|0
|0
|1/1
|
|-
|1
|530
|19/14
|23/17, 34/25, 15/11
|-
|2
|1060
|28/15, 64/35
|
|-
|3
|390
|5/4, 19/15
|14/11
|-
|4
|920
|32/19
|
|-
|5
|250
|7/6, 8/7
|15/13
|-
|6
|780
|25/16, 19/12
|
|-
|7
|110
|16/15
|17/16
|-
|8
|640
|10/7
|23/16, 16/11
|-
|9
|1170
|
|
|-
|10
|500
|4/3
|
|-
|11
|1030
|
|34/19, 29/16, 20/11
|-
|12
|360
|
|16/13
|-
|13
|890
|5/3
|
|-
|14
|220
|
|17/15
|-
|15
|750
|32/21
|20/13
|}

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

Talk:Tertian structure

2026-06-01T18:51:47Z

Kili:

"In just intonation, there is no rational relationship in size between 25/24 and 36/35. The ratio of their sizes is log36/35(25/24) = 1.44908497." if you can create a rational relationship between 25/24 ani 36/35 isn1t that really important for tertian structure? idrk -KILI

"Common rank-3 temperaments can be expressed as tertian structures, and in fact, a tertian structure can always be thought of as a rank-3 temperament. These include Mint, Dicot.7, Keemic, and Mandos.
Mint
Ratio (36/35:25/24): 0:1" what does this mean, i can't read this strange fraction of a bajillion fractions, so the article should explain this notation and what it means - KILI

2026-05-26T18:26:51Z

Kili:

WHat:
"The two sizes of second in a 7-form rank-2 temperament provide a set of generators for the entire system in question." Which Temparament? Which seconds? -kili

ALSOO!
is there a page that is just for secondy intervals, like chromatic semitone goes to the ratio, and second seems like a diatonic category, can this be a page that Chroma and Semitone and all the tones that are not unisons go - also kili

Talk:Second

Kili: /* Aberration Nicepent Eod13 */

User:Kili/Scales

2026-05-14T19:29:31Z

Kili: /* Aberration Nicepent Eod13 */

User:Kili/Scales

2026-05-14T18:53:55Z

Kili: