This page serves as a summary of the differences in terminology used by different composers. '''It is not an invitation to add fuel to these debates,''' rather it is a tool to ease communication.

== Accidentals across notation systems ==

=== Formal commas ===
{| class="wikitable"
|+
! colspan="2" rowspan="2" |Accidental
!Vector
!Ben Johnston
!FJS
!HEJI
! colspan="2" |Color
! colspan="3" |Sagittal
|-
!Name
!Symbol
!Symbol
!Symbol
!Phonetic coding
!Symbol
!Name
!Phonetic coding
!Symbol
|-
| rowspan="2" |81/80
|Raise
|5-sharp
| +
|/5
|(An up arrow on a natural sign)
|gu
|g
| rowspan="2" |5-comma
|pai
|<nowiki>/|</nowiki>
|-
|Lower
|5-flat
| -
|5
|(A down arrow on a natural sign)
|yo
|y
|pao
|\!
|-
| rowspan="2" |64/63
|Raise
|7-sharp
| rowspan="2" |(not used in BJ)
|/7
|Γ
|ru
|r
| rowspan="2" |7-comma
|tai
|<nowiki>|)</nowiki>
|-
|Lower
|7-flat
|7
|L
|zo
|z
|tao
|!)
|}

=== General purpose ===
{| class="wikitable"
|+
! colspan="2" rowspan="2" |Accidental
!KISS
! colspan="2" |Diamond-MOS
!
!Kite's ups and downs
|-
!Symbol
!Name
!Symbol
!
!Symbol
|-
| rowspan="2" |MOSchroma
|Raise
|#
|am
|&
|
| rowspan="2" |(Not used in Kite)
|-
|Lower
|b
|at
|@
|
|-
| rowspan="2" |Subchroma
|Raise
| rowspan="4" |(Not used in KISS)
| colspan="2" rowspan="2" |(Not used in Diamond-MOS)
|
|
|-
|Lower
|
|
|-
| rowspan="2" |Edostep
|Raise
|
|
|
|
|-
|Lower
|
|
|
|
|}

== Section heading ==
What counts as a minor third, how do people categorize the things that aren't, and how would they refer to other senses of 'minor third'?

TODO: make this into a table with different people's thoughts
{| class="wikitable"
|+
!Interval
!{{UserTag|SS|Vector|cebaff|Interval size}}
!{{UserTag|SS|Vector|cebaff|Interval category}}
!Quasi-diatonic
!Just notation
|-
|32/27
|class="prime19"|Pythagorean minor third
|class="prime19"|minor third
|class="prime19"|minor third
|class="prime19"|minor third
|-
|An interval inflected by formal commas from 32/27
| -
|class="prime19"|minor third
|''(out of scope)''
|class="prime19"|minor third
|-
|3\12, representing 32/27
|class="prime19"|MOSdiatonic minor third
|class="prime19"|minor third
|class="prime19"|minor third
|class="prime19"|minor third
|-
|3\12, representing an interval inflected by formal commas from 32/27
|class="prime19"|minor third
|class="prime19"|minor third
|''(out of scope)''
|class="prime19"|minor third
|-
|3\12, in general
|class="prime19"|minor third
|3/12
|''(out of scope)''
| ''(out of scope)''
|-
|Any interval in the vicinity of 3\12
|class="prime19"|minor third
|~3/12
|''(out of scope)''
| ''(out of scope)''
|-
|Any interval in the vicinity of 3\12, provided it is mapped as the third degree of a heptatonic scale
|class="prime19"|minor third
|~3/12
|''(out of scope)''
| ''(out of scope)''
|-
|Any interval in the vicinity of 3\12, excluding Pythagorean intervals other than 32/27
|class="prime19"|minor third
|~3/12
|''(out of scope)''
| ''(out of scope)''
|-
|Any interval in the vicinity of 3\12, provided it is inflected by formal commas from 32/27
|class="prime19"|minor third
|class="prime19"|minor third
|''(out of scope)''
|class="prime19"|minor third
|-
|Any interval in the vicinity of 3\12, provided it is a tempered version of an interval inflected by formal commas from 32/27
|class="prime19"|minor third
|class="prime19"|minor third
|''(out of scope)''
|class="prime19"|minor third
|-
|The smaller version of the third degree of a JI-agnostic diatonic scale
| -
|class="prime19"|minor third
|class="prime19"|minor third
|''(out of scope)''
|-
|The smaller version of the third degree of a JI-agnostic diatonic MOS
|class="prime19"|MOSdiatonic minor third
|class="prime19"|minor third
|class="prime19"|minor third
|''(out of scope)''
|-
|The smaller version of the third degree of a ternary (or greater) diatonic scale, even if it is not mapped to something inflected by formal commas from 32/27
| -
|class="prime19"|minor third
|class="prime19"|minor third
| ''(out of scope)''
|-
|The smaller version of the third degree of any heptatonic scale
|minor 2-step
|minor 2-step
|class="prime19"|minor third
| ''(out of scope)''
|-
|The smaller version of the third degree of any heptatonic MOS
|minor 2-step
|minor 2-step
|class="prime19"|minor third
| ''(out of scope)''
|-
|The larger version of the third degree of antidiatonic, when mapped as 32/27 and tuned near 400c
|major third
|major third
|major third
|class="prime19"|minor third
|-
|The larger version of the third degree of antidiatonic when tuned near 400c, in general
|major third
|major third
|major third
| ''(out of scope)''
|-
|The smaller version of the fourth degree of a decatonic scale
|class="prime19"|decatonic minor third
|class="prime19"|minor third
|minor fourth
|''(out of scope)''
|-
|The fourth degree of a dodecatonic scale
| -
| class="prime19" |minor third
|fourth
|''(out of scope)''
|-
|The smaller version of the third degree of any scale
|minor 2-step
|minor 2-step
| class="prime19" |minor third
|''(out of scope)''
|-
|The b3 degree
|''(out of scope)''
|''(out of scope)''
|''(out of scope)''
|''(out of scope)''
|}

== Miscellaneous disputes ==

Overview of terminology differences

2026-10-26T05:09:22Z

Vector:

2026-10-24T19:45:06Z

Vector:

'''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.

XenReference:To-do list

2026-10-22T06:53:10Z

Vector:

5edo

2026-10-22T06:43:01Z

Vector:

[[File:5edo comparison.png|thumb|5edo compared to other equipentatonic scales. Relative to just intonation, it narrows 7/4, 7/6, and 4/3, and widens 12/7, 8/7, and 3/2.]]
'''5edo''', or 5 equal divisions of the octave (sometimes called '''5-TET''' or '''5-tone equal temperament'''), is the [[Equal temperament|equal tuning]] featuring steps of (1200/5) = 240 [[Cent|cents]] exactly, 5 of which stack to the perfect octave [[2/1]]. It is the basic equipentatonic, where all five steps are tuned to be exactly equal.

== General theory ==

==== Derivation ====
5edo is derived by equalizing an equipentatonic scale, which is interpreted by Leriendil as one of the two primary basic forms of scale produced by human cultures, alongside equiheptatonic (or Pythagorean heptatonic) scales; this makes 5edo rather natural as a simplified model for such scales.

==== JI approximation ====
5edo is most obviously a 2.3.7 system (and this property carries to 5-form systems as a whole). although its 3 is audibly sharper than a just 3/2. There is no [[diatonic]] scale in 5edo; there are too few notes (although for theoretical purposes, 5edo can be taken as having a MOS diatonic where the small step is tuned to 0 cents). However, 5edo interestingly is simple enough, and at least vaguely accurate enough to just intonation, that essentially any combination of notes taken from it sounds consonant to some extent.

Some interpretations treat 5edo as a 2.3.5.7 (5-limit or full 7-limit) system, in which the 5/4 interval, detuned by almost 100 cents, is equated with 4/3, equalizing the triad 3:4:5. Due to the inaccuracy of this temperament, this is mostly only a theoretical curiosity.

In higher limits, 37/32 is the next prime represented reasonably accurately, although as for composite intervals 17/13 is rather accurately represented; so is 55/32.

{{Harmonics in ED|5|37|edo=5|EDO=5|ET=5}}

==== Edostep interpretations ====
5edo's edostep has the following interpretations in the 2.3.7 subgroup:

* 7/6
* 8/7
* 9/8

=== Intervals and notation ===
5edo may be notated with any diatonic notation, as long as you keep in mind that E and F are the same note, and B and C are also the same note; the note sequence is usually C-D-E-G-A like a major pentatonic scale, and it is preferable to avoid accidentals. (However, it may be useful to some to treat the third step as E and F simultaneously; Kite calls this "eef".)
{| class="wikitable"
|+
! rowspan="2" |Edostep
! rowspan="2" |Cents
!Intervals
! colspan="2" |Notation
! rowspan="2" |Name
|-
!2.3.7
!Diatonic
!Eef
|-
|0
|0
|
| colspan="2" |D
|Unison
|-
|1
|240
|9/8, 8/7, 7/6
|E, F
|ꘙ
|Major second
|-
|2
|480
|21/16, 4/3
| colspan="2" |G
|Perfect fourth
|-
|3
|720
|3/2, 32/21
| colspan="2" |A
|Perfect fifth
|-
|4
|960
|16/9, 7/4, 12/7
|B, C
|C
|Minor seventh
|-
|5
|1200
|2/1
| colspan="2" |D
|Octave
|}
Solfege functions according to standard diatonic solfege: do, re, mi, so, la.

== Tempering properties ==

=== Tempered commas ===
Important commas tempered out by the 2.3.7 subgroup interpretation of 5edo include:

* 49/48 (S7), equating 7/6 with 8/7 and 9/7 with 21/16
* 64/63 (S8), equating 8/7 with 9/8 and 21/16 with 4/3

=== Arithmetic progressions ===
Equivalently, 5et in the 2.3.7 subgroup may be defined by equalizing 6:7:8:9. In the inaccurate 5-limit, this extends further to 5:6:7:8:9:10.

=== Notable structural chains ===
5edo, due to its size, only has two edo-distinct rank-2 temperaments: [[Slendric]] (or alternatively [[Semaphore]]) and [[Archy|Archy.]] Semaphore or Slendric is generated by the 240c 8/7 interval, two of which reach 4/3 (semaphore) and three of which reach 3/2 (slendric). Archy is 5edo's version of the standard chain of fifths, where two 4/3s reach 7/4. 5edo is on the sharper end of the tuning range of slendric and archy (which are optimally tuned around 232c and 710c respectively) and on the flatter end of the semaphore tuning range (which is usually around 245c).

== Compositional theory ==

=== Tertian structure ===
As the smallest edo in which diatonic interval categories make much sense, 5edo only has thirds by the broadest possible definition of the interval region (which places thirds in the middle third of 3/2) or by the definition based on the degenerate MOS diatonic scale, where the primary major and minor third are 480c and 240c.

Diatonic interval qualities are, however, not particularly useful when composing in 5edo.
{| class="wikitable"
|+Thirds in 5edo
!Quality
|'''Minor'''
|'''Major'''
|-
!Cents
|'''240'''
|'''480'''
|-
!Just interpretation
|'''7/6''' (-27{{c}}), '''8/7''' (+9c), '''9/8''' (+36c)
|'''9/7''' (+45{{c}}), '''4/3''' (-18c)
|}
Diatonic thirds are bolded, though note that all thirds shown here appear in the MOS diatonic scale.

Technically, 5edo is a [[Tertian structure#Mint|mint]] temperament, tempering together 6/5 and 7/6 to produce 2 qualities from the 4 main 7-limit thirds, although its 6/5 is inaccurate enough as to be basically useless.

=== Scales ===
5edo is simple enough that the entire edo is usually used as a scale (although composer Hideya Amano avoids this by using subsets such as [0 2 3 5] in order to establish a clear tonal center). The construction of scales from 5edo is usually not by taking subsets but by detuning the notes to reach one of the equipentatonic scales discussed in [[Equipentatonic]]. In fact, in real world musical cultures that use equipentatonic scales, they are almost never perfect 5edo.

Every possible octave-equivalent subset of 5edo is a MOS; 5edo is the largest edo with this property. There are seven such scales, up to rotation: 5edo, 1edo, 4-1, 3-2, 3-1-1, 2-2-1, and 2-1-1-1. Each scale is generated by at least one of 5edo's temperaments:
{| class="wikitable"
|+
!Scale
!Chart
!Notes
!Temperament
|-
|4-1
|{{Interval ruler|5|0, 240, 1200}}
|
|Slendric
|-
|3-2
|{{Interval ruler|5|0, 720, 1200}}
|
|Archy
|-
|3-1-1
|{{Interval ruler|5|0, 240, 480, 1200}}
|
|Slendric
|-
|2-2-1
|{{Interval ruler|5|0, 480, 720, 1200}}
|Used by Amano
|Archy
|-
|2-1-1-1
|{{Interval ruler|5|0, 240, 480, 720, 1200}}
|Used by Amano
|Slendric, Archy
|}
(Archy generates Lsss by coincidence; this is not a general property of archy.)

=== Harmony ===
Essentially any combination of notes in 5edo is a chord of some kind. Up to inversion and modulation, there are a total of 5 chords in 5edo, not counting a single note or the whole scale. Predictably, triadic harmony is rare in 5edo, and most harmonic systems in this tuning rely on drones or modulation.

The full set of chords or scales including modes is as follows:
{| class="wikitable"
!Chord
!Chart
!Info
|-
|0 1
|{{Interval ruler|5|0, 240, 1200}}
|
|-
|0 2
|{{Interval ruler|5|0, 480, 1200}}
|Perfect fourth trine
|-
|0 3
|{{Interval ruler|5|0, 720, 1200}}
|Perfect fifth trine
|-
|0 4
|{{Interval ruler|5|0, 960, 1200}}
|
|-
|0 1 2
|{{Interval ruler|5|0, 240, 480, 1200}}
|Chthonic triad, see [[Chthonic harmony]]
|-
|0 1 3
|{{Interval ruler|5|0, 240, 720, 1200}}
|"Minor triad", see [[Extraclassical tonality]]
|-
|0 1 4
|{{Interval ruler|5|0, 240, 960, 1200}}
|
|-
|0 2 3
|{{Interval ruler|5|0, 480, 720, 1200}}
|"Major triad", see [[Extraclassical tonality]]
|-
|0 2 4
|{{Interval ruler|5|0, 480, 960, 1200}}
|
|-
|0 3 4
|{{Interval ruler|5|0, 720, 960, 1200}}
|Representation of the chord 4:6:7(:8).
|-
|0 1 2 3
|{{Interval ruler|5|0, 240, 480, 720, 1200}}
|"Major/minor triad", see [[Extraclassical tonality]]. Also the slendric tetrad.
|-
|0 1 2 4
|{{Interval ruler|5|0, 240, 480, 960, 1200}}
|
|-
|0 1 3 4
|{{Interval ruler|5|0, 240, 720, 960, 1200}}
|"Minor seventh" chord
|-
|0 2 3 4
|{{Interval ruler|5|0, 480, 720, 960, 1200}}
|"Dominant seventh" chord
|}
However, it is often better to avoid relying on chord progressions, especially as most useful chrods are inversions of each other and the step of the edo is found as the step in many chords, making voice leading difficult.{{UserTag|SS|Vector|cebaff|It's best in my opinion to use a monophonic drone and modulate between different keys, that or use a perfect fifth drone in harmony.}}

== Instruments ==

=== On a Lumatone ===
As a diatonic edo, 5edo can be cleanly represented on a Lumatone:{{Lumatone edo mapping|n=5|start=0|xstep=1|ystep=-1}}

=== On a guitar ===
If a guitar is tuned according to its standard tuning in 5edo, it becomes isomorphic, because the diatonic major third is also the perfect fourth. It's not necessary to re-fret the guitar for most notes (meaning that a guitar tuned like this actually plays in a subset of [[60edo]]).

== Supersets and subsets ==
Owing to 5edo's accuracy in 2.3.7 but failure to represent 5, a perfectly tuned 5edo can be taken as the 2.3.7 structure in a rank-2 temperament, with 5 (or any other prime) added as a [[generator]]. This is called '''blackwood''' temperament. Blackwood tempers out the [[diatonic semitone]], meaning that the 3-limit diatonic MOS scale is degenerate, collapsing to a 5-note scale. However, intervals of 5 are distinct, and may be used to generate the Zarlino [[diatonic]].

Supersets of 5edo that support blackwood include [[10edo]], [[15edo]], [[20edo]], [[25edo]], and [[30edo]]. All of these share 5edo's 2.3.7 subgroup, and its slendric, semaphore, and archy temperaments.

== Comparisons to other tuning systems ==
5edo's temperaments are shared by other edos separated by 5:

* [[16edo]], [[21edo]], [[26edo]], [[31edo]], [[36edo]], [[41edo]], and [[46edo]] all support [[Slendric]].
* [[9edo]], [[14edo]], [[19edo]], [[24edo]], [[29edo]], and [[34edo]] all support [[Semaphore]].
* [[12edo]], [[17edo]], [[22edo]], [[27edo]], [[32edo]], and [[37edo]] all support [[Archy]].
* As previously mentioned, the multiples of 5edo itself support Blackwood.

{{Cat|Edos}}{{Navbox EDO}}

5edo

2026-10-21T11:30:24Z

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