2026-03-09T00:13:23Z

Hotcrystal0: 120edo

[[File:Circle of fifths.png|thumb|The circle of fifths in 12edo. Source: Wikipedia]]
'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave|2/1]]. As the dominant tuning system in the world, it is as such covered by the Xenharmonic Reference for completeness as it is definitionally not "xenharmonic".

12edo's most important feature is its approximation of [[3/2]] at 7 steps, which is less than 2{{c}} flat of just, and which, when stacked against the octave, produces the basic [[diatonic]] scale, 2-2-1-2-2-2-1. As the fifth is still slightly flat, this has allowed 12edo to be conventionally interpreted as a tuning of [[Meantone]] and hence a [[5-limit]] system, where the minor third of 3 steps and the major third of 4 steps represent [[6/5]] and [[5/4]] respectively. Also important to 12edo's structure is the fact that these thirds derive from [[4edo]] and [[3edo]], allowing for the construction of scales that repeat at fractions of the octave.

However, as the 12edo thirds are 16{{c}} and 14{{c}} out of tune with the 5-limit, another important interpretation is the 2.3.17.19 subgroup, which is tuned more accurately than prime 5. In particular, the minor third closely approximates [[19/16]], and the 100{{c}} semitone can be thought of as [[17/16]]~[[18/17]].

== Theory ==
=== Edostep interpretations ===
12edo's edostep has the following interpretations in the 2.3.5 subgroup:
* 16/15 (the difference between 5/4 and 4/3)
* 25/24 (the difference between 6/5 and 5/4)
* 27/25 (the difference between 10/9 and 6/5)

When primes 17 and 19 are included, it also serves as 17/16, 18/17, [[19/18]], and [[20/19]].

12edo tempers out the following important commas in its 5-limit:
* 81/80 (equating 9/8 with 10/9, and four 3/2s to 5/1)
* 128/125 (causing three 5/4s to reach an octave exactly)
* 648/625 (causing four 6/5s to reach an octave exactly)
* 2048/2025 (splitting 9/8 into two 16/15s).

It can be defined in the subgroup 2.3.5.17.19 by equalizing the arithmetic progression 15::20, and therefore tempering out [[square superparticular]]s S16 through S19, and combinations thereof.

=== JI approximation ===
12edo is conventionally seen as a [[5-limit|2.3.5]] edo, though perhaps more salient to conventional Western musical practice is the fact that it contains the basic [[diatonic]] scale, 2-2-1-2-2-2-1.
{{Harmonics in ED|12|31|0}}
{| class="wikitable"
|+Thirds in 12edo
!Quality
|'''Minor'''
|'''Major'''
|-
!Cents
|'''300'''
|'''400'''
|-
!Just interpretation
|'''6/5'''
|'''5/4'''
|}
Diatonic thirds are bolded.

=== Notation ===
12edo has a standard notation system, consistent with classical theory. As a result, [[ups and downs]] notation, [[KISS notation]] for diatonic, [[Pythagorean notation]], and [[sagittal]] notation all converge on 12edo.

== Compositional theory ==
=== Regular temperaments ===
Notable 5-limit temperaments supported by 12edo are Augmented (12 & 15), Diminished (12 & 16), and Meantone (12 & 19). These temperaments lead to the 3L 3s (or 3L 6s) MOS, the 4L 4s MOS, and the 5L 2s (or 3L 2s) MOS, respectively. Of these, it is a particularly good tuning of Diminished.

=== Chords ===
12edo is notable for its tritone of exactly 600c, major third of exactly 400c, and minor third of exactly 300c. This makes available a fully symmetrical [[Diminished seventh tetrad|diminished seventh chord]] and also a fully symmetrical [[augmented triad]], and enables tritone substitution of [[Dominant tetrad|dominant tetrads]].

Due to 12edo's accuracy in the 2.3.17.19 subgroup, the minor triad of [0 3 7] can be analyzed as 16:19:24, which some theorists believe to contribute to its stable sound.

=== Scales ===
12edo, due to its large number of factors, contains many MOS scales with periods that are some fraction of the octave. One well-known example is [[4L 4s]], which is known as the octatonic scale. Additionally, it contains [[6edo]] as a subset, which is the whole tone scale.

12edo is small enough that the EDO itself functions as a chromatic scale.

== Multiples ==

===24edo===
''Main article: [[24edo]]''

24edo corrects 12edo's approximate 11th and 13th harmonics (which fall about halfway between the steps in 12edo) and it can be used in the 2.3.(5).11.13.17.19 subgroup, however its 5 now accumulates error too quickly and is less usable than 12edo's.

{{Harmonics in ED|24|31|0}}

===36edo===

36edo is another way to extend 12edo while maintaining a manageable amount of notes, correcting the 7th and 13th harmonics (in a way distinct from 24edo). Its 5 is still inherited from 12edo, although it is now about halfway between the steps. It is a good tuning for the 2.3.7.13.17.19 subgroup, and its 8/7 makes it a good tuning for [[Slendric]] temperament.

{{Harmonics in ED|36|31|0}}

===72edo===
''Main article: [[72edo]]''

===120edo===

''See [[40edo#120edo]].''

===612edo===

''See [[34edo#612edo]].''

=== Compton temperament ===
If 12edo is taken as a temperament of [[Pythagorean tuning]] instead of a 2.3.5 temperament, an independent dimension for any other prime (conventionally 5) may be added. This temperament, called Compton, takes advantage of the fact that due to 12edo's extraordinary accuracy in the 3-limit, higher-limit intervals tend to be off by roughly consistent offsets from 12edo ones. Compton temperament tempers out the Pythagorean comma (531441/524288, the ~24c difference between the Pythagorean chroma and minor second), and equates any mosdiatonic interval with its enharmonic counterpart. However, intervals of 5 are distinct, with 5/4 usually being tuned around 384 cents. As such, this is well-tuned in [[72edo]], and is supported by any multiple of 12 up to and including 300edo.

{{Cat|edos}}{{Navbox EDO}}

3/1

2026-03-06T14:00:19Z

Hotcrystal0: adding infobox

{{Infobox interval|3/1|Name = 3rd harmonic, tritave, triple, perfect twelfth}}
3/1, the '''tritave''' or '''perfect twelfth''', is the second most common equave after [[2/1]]. In octave-equivalent systems, it is a fifth plus an octave, and can thus be seen as one of the two generators of Pythagorean tuning (see [[Perfect fifth]] for more info).

It can be seen as the most consonant interval after the octave, which is the reason for its usage as an equave in systems such as [[Bohlen-Pierce]] tuning. Tritave-equivalent systems tend to avoid prime 2, only involving ratios between odd numbers (such as [[9/7]] and [[5/3]]). As such, timbres chosen for tritave-equivalent music tend to include mostly odd harmonics, such as the clarinet.

{{Cat|Intervals}}{{Interval regions}}

26edo

2026-03-05T17:37:55Z

Hotcrystal0:

26edo

2026-03-05T17:35:21Z

Hotcrystal0:

User:Hotcrystal0/26edo

2026-03-03T01:12:33Z

Hotcrystal0: edostep interpretations

34edo

2026-03-02T18:58:12Z

Hotcrystal0: fix in 306edo

[[File:34edo chart.png|thumb|377x377px|The structure of the 5-limit in 34et, visualized.]]
'''34edo''' is the equal tuning system which splits the octave into 34 equal steps, of about (1200/34) ~= 35.3 cents each. It is a 5-limit and 2.3.5.13 system with a number of melodically intuitive structures.

== Derivation ==

=== From doubling 17edo ===
One can observe that 17edo's step is a nearly perfectly tuned 25/24, and also that 5/4 and 6/5 are almost exactly halfway in-between notes of 17edo. Thus, 17edo can be doubled to improve the tunings of 5-limit intervals.

===== From the usage of Pythagorean diatonic semitones as classical chromatic semitones =====
Forcing 17edo's near-just 25/24, which is a Pythagorean diatonic semitone, to surround a neutral third and function as a chromatic semitone, requires offsetting the chain of fifths by a perfect semioctave, effectively allowing one to 'swap' the tunings of diatonic and chromatic semitones. This results in the 34edo tuning of Diaschismic.

=== From the DKW step sizes ===
[https://en.xen.wiki/w/DKW_theory DKW theory] suggests that the core step sizes of the 5-limit are 9/8, 16/15, and 25/24. It can be observed that 16/15 stacks twice to approximate 9/8, and that 25/24 stacks 3 times to approximate 9/8. Tempering these equivalences together results in 34edo. Because the latter (kleismic) equalizes 24:25:26:27, and the former (diaschismic) equalizes 15:16:17:18, 34edo can be seen as a 2.3.5.13.17 system. 34edo can, thus, be broken up as 6-3-2-3-6-3-2-3-6, with 6 representing 9/8, 3 representing 16/15, and 2 representing 25/24.

== Theory ==

==== Edostep interpretations ====
34edo's edostep, the ''sextula'', has the following interpretations in the 2.3.5.13.17 subgroup:

* [[81/80]], the difference between the fifth-generated major third and the classical major third
* [[128/125]], the difference between the 5-limit enharmonic intervals
* [[40/39]], the difference between [[13/10]] and [[4/3]] and between [[15/13]] and [[9/8]]; also between [[6/5]] and [[16/13]], among others
* [[65/64]], the difference between [[8/5]] and [[13/8]]

''TODO: add to list''

==== JI approximation ====
34edo is straddle-7, straddle-11, and straddle-19, but has good accuracy on the 2.3.5.13.17.23 subgroup. 34edo inherits 17edo's mosdiatonic scale, 6-6-2-6-6-6-2, with the "optimally tuned" leading tone approximating 25/24. It also supports the zarlino scale, but because it does not support [[Porcupine]], the zarlino scale requires 2 sets of accidentals to notate, making it awkward to use as the basis of notation. (The best option is to use 5-sharp and 5-flat accidentals from 17edo's diatonic, as if you are notating 5-limit JI, which may in 34edo be represented as ups and downs.)
{{Harmonics in ED|34|31}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Inframinor
|'''Farminor'''
|Nearminor
|Neutral
|Nearmajor
|'''Farmajor'''
|Ultramajor
|-
!Cents
|247
|'''282'''
|318
|353
|388
|'''424'''
|459
|-
!Just interpretation
|15/13
|'''20/17'''
|6/5
|16/13
|5/4
|'''32/25'''
|13/10
|}
MOS diatonic thirds are bolded.

=== Chords ===
34edo supports [[arto and tendo theory]] with its inframinor and ultramajor thirds. Being a Diaschismic edo, it has a series of tetrads wherein the third and seventh are separated by 600 cents, but due to not supporting [[Pajara]], these do not approximate simple 7-limit chords.

=== Scales ===
34edo contains 17edo's diatonic scale and alongside it a zarlino scale. Other scales it includes are:

* the blackdye scale, with steps 1-5-3-5-1-5-3-5-1-5 (sLmLsLmLsL)
* Diaschismic[12], with steps 3-3-3-3-3-2-3-3-3-3-3-2 (LLLLLsLLLLLs)
** the Delkian scale, a MODMOS of diaschismic[12] with steps 3-3-3-2-3-3-3-3-3-3-2-3 (LLLsLLLLLLsL)
* the hemipythagorean decatonic, with steps 3-4-3-4-3-3-4-3-4-3 (sLsLssLsLs)
** the Sixanian scale, a MODMOS of the above with steps 3-4-3-4-3-3-3-4-3-4 (sLsLsssLsL)
* the Roklotian scale, 2-2-2-3-2-3-2-2-2-3-2-3-2-2-2 (sssLsLsssLsLsss)
* the MOS pentatonic, pythagorean[5], 6-8-6-8-6 (sLsLs)
* the equable pentatonic, Semaphore[5], 7-7-6-7-7 (LLsLL)
* the vertical pentatonic, 5-9-6-5-9 (sLmsL)

=== Additional regular temperaments ===
Alongside [[Kleismic]] (shared with [[15edo]] and [[19edo]]), and [[Diaschismic]] (shared with [[12edo]]), 34edo supports the following temperaments:

* [[Tetracot]] (splitting 3/2 into four 10/9s), shared with [[Equiheptatonic|7edo]] and [[27edo]])
* [[Gammic]] (setting 25/24 to a tenth of 3/2), shared with 103edo

== Multiples ==
=== 68edo ===
68edo is the double of 34edo, and improves its mapping of 7 much as 34edo improves 17edo's mapping of 5. This improves the mappings of 11 and 19 as well, making 68edo function as a general 19-limit system.

The new 7/4 supports [[Sensamagic]], doubling 9/7 to reach 5/3, and 2.5.7 [[Didacus]], splitting 5/4 into two wholetones that stack 5 times to reach 7/4. Additionally, the new 11/8 makes 14/11 equal to 81/64, supporting [[Pentacircle]] (and various gentle/neogothic temperaments).

{{Harmonics in ED|68|31|0}}

=== 306edo ===
306 is the decominator of a continued fraction convergent to log2(3/2), and as such 306edo has a nearly perfectly accurate 3/2 representation. Its step is the difference between 34edo's 3/2 and the near-just one. It also has a 7/4 accurate to within 0.2 cents.

{{Harmonics in ED|306|31|0}}

=== 612edo ===
612edo doubles 306edo, adding the perfect fifth from [[12edo]] and a nearly perfect [[5/4]]. Its main utility is as a fine-grained interval size measurement system for the 11-limit, wherein 3/2 is 358 steps and 5/4 is 197 steps, as its step size is almost exactly a (consistently represented) [[schisma]]. The 12edo perfect fifth is 357 steps, 34edo's is 360 steps. Thus, 34edo is about twice as inaccurate as 12edo in its tuning of 3/2.

{{Harmonics in ED|612|31|0}}

{{Navbox EDO}}

{{Cat|Edos}}

26edo

2026-03-02T18:40:49Z

Hotcrystal0: reorganize