Xenharmonic Reference - User contributions [en]

EDO

2026-07-13T05:17:54Z

Tristanbay: /* List of edos */ 2edo has a good harmonic 11 and 23 for its size

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps. It is a type of equal temperament.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties, as will one with the same number of notes as the MOS has L steps. These two edos form the boundaries of how the MOS can be tuned.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== Uses ==

Edos are the most common type of tuning system in contemporary xenharmony. Unlike other types such as rank-2 temperaments and just intonation scales, equal temperaments allow for free modulation and transposition due to their uniform step size. That is, every n-step interval is the same as every other n-step interval. This comes at the expense of less freedom in approximating target intervals. It also encourages a less structured approach to composition where pitch shifts and interval quality changes can happen without much deeper meaning.

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |23/16
|2.23
|-
|2.<3.>>5.>>7.>11.<17.<23
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo. Good approximation of 2.5.7 for its size.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5
|-
|2.9.5.>7
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting Porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.21.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.>3.<5.>>7.<11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports Semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an Archy 3, as in Porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of Mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate [[Aberschismic]] edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma Archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|Huygens]]". Like all Meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of Flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate [[Aberschismic]] edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports Magic and Archy with its sharp fifth and Deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of Schismic analogue of Didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with [[Buzzard]] temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of Archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 Meantone (or, more conventionally, Didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden Meantone, and serves as a definitive tuning of Meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp Archy fifth and a more accurate 5/4 which support Porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the Pajara tuning range. Can alternatively be used as a very flat Deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp Meantone tuning, which is so sharp that it does not even support Septimal Meantone, and is best interpreted as Mohajira as it pertains to Meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports Porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting Magic temperament and having 10edo's 7 and 13, also supporting 7-limit Compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of Flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-Garibaldi Schismic system (in fact, it supports Sensi), and a straddle-7 and -13 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma Meantone and Slendric edo, which also supports [[Orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting Sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma Meantone, somewhat between standard Septimal Meantone and 19edo. As a result, it is a Mohajira system (setting 7/4 to the semiflat minor seventh) but not a Septimal Meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports Archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is an [[Aberschismic]] system, as is typical with tunings with slightly sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the Superpyth tuning range (and produces the same mapping for 5 as Superpyth), despite not supporting Archy. The flat fifth, analogously, produces Flattone's mapping for 7 and is well-tuned for Flattone, but does not support Flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|[[72edo|72]]
|A multiple of 12edo and a very good Miracle and Compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support Meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a Meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|[[94edo|94]]
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|[[99edo|99]]
|Arguably the most accurate 7-limit edo below 100. Supports [[Aberschismic]], [[Didacus]], and [[Ennealimmal]].
|{{First 12 edo intervals|edo=99}}
|703.0
|126/125, 225/224, kleisma
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|{{First 12 edo intervals|edo=140}}
|702.9
|
| -
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|171
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.
|{{First 12 edo intervals|edo=171}}
|701.8
| 225/224, 5120/5103, [[kleisma]]
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|270
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.
|{{First 12 edo intervals|edo=270}}
|702.2
|385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-04-02T06:14:26Z

Tristanbay: /* List of edos */ added edostep interpretations for 270edo and 171edo

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5
|-
|2.9.5.>7
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.21.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.>3.<5.>>7.<11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|{{First 12 edo intervals|edo=140}}
|702.9
|
| -
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|171
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.
|{{First 12 edo intervals|edo=171}}
|701.8
| 225/224, 5120/5103, [[kleisma]]
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|270
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.
|{{First 12 edo intervals|edo=270}}
|702.2
|385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-04-02T05:53:51Z

Tristanbay: /* List of edos */ Added edostep info for 140edo

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5
|-
|2.9.5.>7
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.21.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.>3.<5.>>7.<11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|{{First 12 edo intervals|edo=140}}
|702.9
|
| -
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|171
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.
|{{First 12 edo intervals|edo=171}}
|701.8
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|270
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.
|{{First 12 edo intervals|edo=270}}
|702.2
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

EDO

2026-04-02T05:51:56Z

Tristanbay: /* List of edos */ Added 171edo and 270edo, tweaked some other entries

An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps.

The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.

An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties.

The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').

== List of edos ==
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''
{| class="wikitable"
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].
|-
!Edo
!Description
!First twelve steps (¢)
!Fifth (¢)
!Edostep interpretation
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups
|-
| rowspan="2" |1
| rowspan="2" |Equivalent to the 2-limit.
| rowspan="2" |1200
| rowspan="2" |1200
| rowspan="2" |2/1
|2
|-
|2
|-
| rowspan="2" |2
| rowspan="2" |Just a 12edo tritone.
| rowspan="2" |600, 1200
| rowspan="2" |600
| rowspan="2" |''none available in basic subgroup''
|2
|-
|2.<3.>>5.>>7.<17
|-
| rowspan="2" |3
| rowspan="2" |An augmented triad.
| rowspan="2" |400, 800, 1200
| rowspan="2" |800
| rowspan="2" |5/4
|2.5
|-
|2.>3.5.>19?
|-
| rowspan="2" |4
| rowspan="2" |A diminished tetrad.
| rowspan="2" |300, 600, 900, 1200
| rowspan="2" |600
| rowspan="2" |19/16
|2.19
|-
|2.<3.<5.<7.<17
|-
| rowspan="2" class="thl" |[[5edo|5]]
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.
| rowspan="2" |240, 480, 720, 960, 1200
| rowspan="2" |720
| rowspan="2" |9/8, 8/7, 7/6
|2.3.7
|-
|2.>>3.<7
|-
| rowspan="2" |6
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.
| rowspan="2" |200, 400, 600, 800, 1000, 1200
| rowspan="2" |600, 800
| rowspan="2" |9/8, 10/9, 28/25, 8/7
|2.9.5
|-
|2.9.5.>7
|-
| rowspan="2" class="thl" |[[7edo|7]]
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200
| rowspan="2" |685.7
| rowspan="2" |9/8, 10/9, 16/15
|2.3.5.11.13
|-
|2.<3.<<5.>13
|-
| rowspan="2" |[[8edo|8]]
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200
| rowspan="2" |750
| rowspan="2" |''none available in basic subgroup''
|2.19
|-
|2.x3.x5.x7.x11.x13
|-
| rowspan="2" |[[9edo|9]]
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group).
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200
| rowspan="2" |666.7
| rowspan="2" |9/8, 16/15, 25/24
|2.5.11
|-
|2.<<3.>5.<<7
|-
| rowspan="2" |[[10edo|10]]
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200
| rowspan="2" |720
| rowspan="2" |16/15, 10/9, 81/80, 36/35
|2.3.5.7.13
|-
|2.>>3.<7.13
|-
| rowspan="2" |[[11edo|11]]
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9
| rowspan="2" |654.5, 763.6
| rowspan="2" |128/119, 17/16, 18/17
|2.9.7.11
|-
|2.x3.x5.7.11
|-
| rowspan="2" class="thl" |[[12edo|12]]
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.
| rowspan="2" |{{First 12 edo intervals|edo=12}}
| rowspan="2" |700
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24
|2.3.5.17.19
|-
|2.3.>5.>>7.17.19
|-
|[[13edo|13]]
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].
|{{First 12 edo intervals|edo=13}}
|646.2, 738.5
|17/16, 18/17, 19/18, 20/19
|2.5.11.13.17.19.23
|-
|[[14edo|14]]
|Basic [[semiquartal]].
|{{First 12 edo intervals|edo=14}}
|685.7
|28/27, 21/20, 15/14
|2.3.7.13
|-
| class="thl" |[[15edo|15]]
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.
|{{First 12 edo intervals|edo=15}}
|720
|81/80, 25/24, 16/15, 33/32, 36/35
|2.3.5.7.11.23
|-
|[[16edo|16]]
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].
|{{First 12 edo intervals|edo=16}}
|675, 750
|20/19, 133/128, 26/25
|2.5.7.13.19
|-
| class="thl" |[[17edo|17]]
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.
|{{First 12 edo intervals|edo=17}}
|705.9
|256/243, 24/23, 27/26, 33/32
|2.3.13.23
|-
| rowspan="2" |[[18edo|18]]
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric.
| rowspan="2" |{{First 12 edo intervals|edo=18}}
| rowspan="2" |666.6, 733.3
| rowspan="2" |
|2.9.5.21.13
|-
|2.xx3.>5.>>7.<11.<13
|-
| class="thl" |[[19edo|19]]
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.
|{{First 12 edo intervals|edo=19}}
|694.7
|25/24, [[diaschisma]], 36/35, 28/27
|2.3.5.23
|-
| |20
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.
|{{First 12 edo intervals|edo=20}}
|660, 720
|
|2.7.11.13.19
|-
| rowspan="2" |[[21edo|21]]
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.
| rowspan="2" |{{First 12 edo intervals|edo=21}}
| rowspan="2" |685.7, 742.9
| rowspan="2" |
|2.3.5.7.23
|-
|2.x>3.x<5.7.x<11.x<13.23
|-
| rowspan="2" class="thl" |[[22edo|22]]
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.
| rowspan="2" |{{First 12 edo intervals|edo=22}}
| rowspan="2" |709.1
| rowspan="2" |
|2.3.5.7.11.17
|-
|2.>3.<5.>>7.<11
|-
|23
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.
|{{First 12 edo intervals|edo=23}}
|678.3
|
|2.x3.x5.x7.x11.13.17.23
|-
| div class="thl"|[[24edo|24]]
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.
|{{First 12 edo intervals|edo=24}}
|700
|
|2.3.11.13.17.19
|-
|25
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=25}}
|720
|
|2.5.7.19
|-
|[[26edo|26]]
|A simple tuning of Flattone. Has an absurdly accurate 7/4.
|{{First 12 edo intervals|edo=26}}
|692.7
|
|2.3.7.11.13
|-
|27
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.
|{{First 12 edo intervals|edo=27}}
|711.1
|
|2.3.5.7.13.23
|-
|28
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.
|{{First 12 edo intervals|edo=28}}
|685.7, 728.6
|
|2.3.5.7.11
|-
|[[29edo|29]]
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.
|{{First 12 edo intervals|edo=29}}
|703.4
|
|2.3.7/5.11/5.13/5.19.23
|-
|30
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.
|{{First 12 edo intervals|edo=30}}
|680, 720
|
|2.x3.x5.7.11.13
|-
| class="thl" |[[31edo|31]]
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=31}}
|696.8
|
|2.3.5.7.11.23
|-
|32
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.
|{{First 12 edo intervals|edo=32}}
|712.5
|
|2.3.7.11.17.19.23
|-
|33
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.
|{{First 12 edo intervals|edo=33}}
|690.9
|
|2.3.11.13.17.19.23
|-
| rowspan="2" class="thl"|[[34edo|34]]
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.
It is the double of 17edo, which it takes its circle of fifths from.
| rowspan="2" |{{First 12 edo intervals|edo=34}}
| rowspan="2" |705.9
| rowspan="2" |
|2.3.5.13.23
|-
|2.3.5.x7.13.x19.23
|-
|35
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.
|{{First 12 edo intervals|edo=35}}
|685.7, 720.0
|
|2.5.7.11.17
|-
|36
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.
|{{First 12 edo intervals|edo=36}}
|700
|
|2.3.7.13.17.19.23
|-
| rowspan="2" class="thl" |[[37edo|37]]
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.
| rowspan="2" |{{First 12 edo intervals|edo=37}}
| rowspan="2" |681.1, 713.5
| rowspan="2" |
|2.9.5.7.11.13.17.19
|-
|2.<x3.5.7.11.13.17.19
|-
|38
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.
|{{First 12 edo intervals|edo=38}}
|694.7
|
|2.3.5.7.13.17.23
|-
|39
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.
|{{First 12 edo intervals|39|edo=39}}
|707.7
|
|2.3.11
|-
|[[40edo|40]]
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.
|{{First 12 edo intervals|40|edo=40}}
|690
|
|
|-
| class="thl" |[[41edo|41]]
|The first reasonably accurate Hemifamity edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=41}}
|702.4
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44
|2.3.5.7.11.13.19
|-
|42
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.
|{{First 12 edo intervals|edo=42}}
|685.7, 714.3
|
|2.7.11.17.23
|-
|43
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.
|{{First 12 edo intervals|edo=43}}
|697.7
|
|2.3.5.7.11.13.17
|-
|44
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo).
|{{First 12 edo intervals|edo=44}}
|709.1
|
|2.3.5.11.13.17.19.23
|-
|45
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.
|{{First 12 edo intervals|edo=45}}
|693.3, 720
|
|2.3.7.11.17.19
|-
| class="thl"|46
|The second reasonably accurate Hemifamity edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}
|{{First 12 edo intervals|edo=46}}
|704.3
|
|2.3.5.7.11.13.17.23
|-
|47
|The first edo with two distinct mosdiatonic scales. Supports magic and archy with its sharp fifth and deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of schismic analogue of didacus.
|{{First 12 edo intervals|edo=47}}
|689.4, 714.9
|
|2.5.7.13.17
|-
|48
|Four times 12edo, associated with buzzard temperament.
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
|700
|
|2.3.7.11.17.19.23
|-
|49
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).
|{{First 12 edo intervals|edo=49}}
|710.2
|
|2.5.17.19
|-
|50
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.
|{{First 12 edo intervals|edo=50}}
|696
|
|2.3.5.11.13.23
|-
|51
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.
|{{First 12 edo intervals|edo=51}}
|705.9
|
|2.3.7.13
|-
|52
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.
|{{First 12 edo intervals|edo=52}}
|692.3, 715.4
|
|2.5.7.11.19.23
|-
|class="thl"|[[53edo|53]]
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).
|{{First 12 edo intervals|edo=53}}
|701.9
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90
|2.3.5.7.13.19
|-
|54
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.
|
|688.9, 711.1
|
|2.11.13.17.23
|-
|55
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.
|
|698.2
|
|2.3.5.11.17.23
|-
|56
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.
|
|
|
|
|-
|57
|Has 19edo's 5-limit combined with better interpretations of higher limits.
|
|
|
|
|-
|58
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals.
|{{First 12 edo intervals|edo=58}}
|703.4
|
|2.3.7.17
|-
|59
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.
|
|
|
|
|-
|60
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.
|
|
|
|
|-
|61
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.
|
|
|
|
|-
|62
|Doubled 31edo, which shares its mappings through the 11-limit.
|
|
|
|
|-
|63
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.
|{{First 12 edo intervals|edo=63}}
|704.8
|
|2.3.5.7.11.13.23
|-
|64
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.
|{{First 12 edo intervals|edo=64}}
|693.8, 712.5
|
|2.13.19
|-
|65
|A non-garibaldi schismic system (in fact, it supports Sensi), or a straddle-7 system.
|{{First 12 edo intervals|edo=65}}
|701.5
|
|2.3.5.11.17.19
|-
|66
|Tripled 22edo, with an improved approximation to 7 that supports Slendric.
|{{First 12 edo intervals|edo=66}}
|709.1, 690.9
|
|2.5.7.11.13.17
|-
|67
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].
|{{First 12 edo intervals|edo=67}}
|698.5
|
|2.3.7.11.13.17.23
|-
|68
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.
|{{First 12 edo intervals|edo=68}}
|705
|
|2.3.5.7.11.17.19
|-
|69
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.
|{{First 12 edo intervals|edo=69}}
|695.7, 713
|
|2.5.7.11.13.17.19.23
|-
|70
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.
|{{First 12 edo intervals|edo=70}}
|702.9
|
|2.3.11.13.17
|-
|71
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.
|{{First 12 edo intervals|edo=71}}
|693. 709.9
|
|2.5.7.13.17.23
|-
|72
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.
|{{First 12 edo intervals|edo=72}}
|700
|
|2.3.5.7.11.17.19.23
|-
| colspan="6" |...
|-
|80
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.
|{{First 12 edo intervals|edo=80}}
|705
|
|2.3.5.11.13.17.19.23
|-
|81
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.
|{{First 12 edo intervals|edo=81}}
|696.3
|
|2.9.5.11.13.17.19
|-
| colspan="6" |...
|-
|84
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].
|{{First 12 edo intervals|edo=84}}
|700
|
|2.3.5.7.13.19.23
|-
| colspan="6" |...
|-
|87
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.
|{{First 12 edo intervals|edo=87}}
|703.4
|
|2.3.5.7.11.13.17
|-
| colspan="6" |...
|-
|93
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.
|{{First 12 edo intervals|edo=93}}
|696.8, 709.7
|
|2.5.7.11.13.17.19.23
|-
|94
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi.
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.
|{{First 12 edo intervals|edo=94}}
|702.1
|
| -
|-
| colspan="6" |...
|-
|140
|A significant edo for interval categorization, as the next resolution level up from 58edo.
|702.9
|
| -
|-
| colspan="6" |...
|-
|[[159edo|159]]
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.
|{{First 12 edo intervals|edo=159}}
|701.9
|
| -
|-
| colspan="6" |...
|-
|171
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.
|{{First 12 edo intervals|edo=171}}
|701.8
|
| -
|-
| colspan="6" |...
|-
|200
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.
|{{First 12 edo intervals|edo=200}}
|702
|
| -
|-
| colspan="6" |...
|-
|270
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.
|{{First 12 edo intervals|edo=270}}
|702.2
|
| -
|-
| colspan="6" |...
|-
|306
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.
|{{First 12 edo intervals|edo=306}}
|702
|
| -
|-
| colspan="6" |...
|-
|311
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.
|{{First 12 edo intervals|edo=311}}
|702.3
|
| -
|-
| colspan="6" |...
|-
|612
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).
|{{First 12 edo intervals|edo=612}}
|702
|
| -
|-
| colspan="6" |...
|-
|665
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.
|{{First 12 edo intervals|edo=665}}
|702
|
| -
|}
{{Cat|Core knowledge}}

User:Tristanbay/Gallery of just intonation scales

2026-01-04T05:07:01Z

Tristanbay: Gave each scale their own section

All of the scales below are constant structure and repeat at 2/1.

== Boutique scales (less than 12 notes) ==
===24:26:28:30:32:33:36:39:42:45:48===
<pre>13/12
7/6
5/4
4/3
11/8
3/2
13/8
7/4
15/8
2/1</pre>

===24:27:28:30:33:36:39:42:45:48===
<pre>9/8
7/6
5/4
11/8
3/2
13/8
7/4
15/8
2/1</pre>

===60:63:64:70:80:84:96:105:120===
<pre>35/32
5/4
21/16
4/3
35/24
5/3
7/4
2/1</pre>

===56:60:64:70:84:90:96:105:112===
<pre>35/32
7/6
5/4
4/3
35/24
7/4
15/8
2/1</pre>

===18:20:22:24:27:30:33:36===
<pre>10/9
11/9
4/3
3/2
5/3
11/6
2/1</pre>

===14:16:18:20:21:24:27:28===
<pre>8/7
9/7
10/7
3/2
12/7
27/14
2/1</pre>

===27:28:33:36:39:42:45:54===
<pre>28/27
11/9
4/3
13/9
14/9
5/3
2/1</pre>

===7:8:9:10:11:12:13:14===
<pre>8/7
9/7
10/7
11/7
12/7
13/7
2/1</pre>

===56:60:64:70:72:80:84:90:96:105:112===
<pre>15/14
8/7
5/4
9/7
10/7
3/2
45/28
12/7
15/8
2/1</pre>

===52:58:64:71:78:86:95:104===
<pre>71/64
39/32
43/32
95/64
13/8
29/16
2/1</pre>

===48:52:58:64:72:78:87:96===
<pre>13/12
29/24
4/3
3/2
13/8
29/16
2/1</pre>

===33:36:40:42:44:48:54:60:63:66===
<pre>12/11
40/33
14/11
4/3
16/11
18/11
20/11
21/11
2/1</pre>

===720:780:840:910:945:1008:1092:1170:1260:1365:1440===
<pre>13/12
7/6
91/72
21/16
7/5
91/60
13/8
7/4
91/48
2/1</pre>

== 12-note scales ==
===Antitonic 17-limit===
<pre>17/16
39/32
5/4
21/16
11/8
3/2
51/32
13/8
5/3
7/4
11/6
2/1</pre>

===Bicycle===
<pre>13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
11/6
2/1</pre>

===Duodene===
<pre>16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1</pre>

===Septimal Catgirl===
<pre>28/27
9/8
7/6
5/4
4/3
45/32
3/2
14/9
5/3
7/4
15/8
2/1</pre>

===North Dakota===
<pre>33/32
10/9
7/6
5/4
4/3
11/8
3/2
14/9
5/3
7/4
11/6
2/1</pre>

===Johnston Piano Scale (Ringer 12)===
<pre>15/14
8/7
17/14
9/7
19/14
10/7
3/2
11/7
12/7
13/7
27/14
2/1</pre>

===Ephemeral Soup Scale===
<pre>77/72
9/8
77/64
21/16
693/512
189/128
3/2
77/48
7/4
231/128
63/32
2/1</pre>

===Once Upon A Time Scale===
<pre>10/9
9/8
5/4
9/7
21/16
35/24
3/2
5/3
12/7
7/4
35/18
2/1</pre>

===Centaur===
<pre>21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1</pre>

===Pental Diachrome===
<pre>10/9
9/8
5/4
81/64
4/3
40/27
3/2
5/3
27/16
15/8
160/81
2/1</pre>

===Margo Scale===
<pre>104/99
9/8
13/11
14/11
4/3
63/44
3/2
52/33
56/33
16/9
21/11
2/1</pre>

===Meta-Slendro===
<pre>50/49
8/7
57/49
64/49
65/49
72/49
74/49
151/98
12/7
86/49
96/49
2/1</pre>

===Succulent Undecimal Scale===
<pre>10/9
7/6
5/4
21/16
4/3
11/8
3/2
5/3
7/4
16/9
11/6
2/1</pre>

== 14-note scales ==
===No-13s ultrajust===
17/16
9/8
19/16
5/4
21/16
11/8
23/16
3/2
19/12
5/3
7/4
11/6
23/12
2/1

== 17-note scales ==
===Walkie===
65/64
13/12
9/8
585/512
39/32
5/4
4/3
65/48
45/32
3/2
195/128
13/8
5/3
16/9
117/64
15/8
2/1
===Mohahadene===
55/54
88/81
9/8
55/48
11/9
5/4
4/3
11/8
45/32
3/2
55/36
44/27
5/3
55/32
11/6
15/8
2/1
===Dichotidene===
28/27
35/32
9/8
7/6
5/4
35/27
4/3
45/32
35/24
3/2
14/9
105/64
27/16
7/4
15/8
35/18
2/1
===Just Spoogalaxy===
28/27
29/27
9/8
7/6
29/24
81/64
21/16
87/64
116/81
3/2
14/9
29/18
27/16
7/4
29/16
243/128
2/1
===Cartwheel===
28/27
13/12
9/8
7/6
11/9
5/4
4/3
11/8
13/9
3/2
14/9
13/8
5/3
7/4
11/6
15/8
2/1
===Ringer 17-quad===
33/32
17/16
9/8
19/16
77/64
5/4
21/16
11/8
23/16
3/2
49/32
25/16
13/8
7/4
29/16
15/8
2/1
===Ringer 17-tri===
49/48
17/16
9/8
19/16
29/24
5/4
21/16
11/8
23/16
3/2
74/48
25/16
13/8
7/4
11/6
15/8
2/1
===Schismadene===
135/128
10/9
9/8
32/27
5/4
81/64
4/3
45/32
40/27
3/2
128/81
5/3
27/16
16/9
15/8
160/81
2/1
===Traditional Baglama Fret Scale===
18/17
12/11
9/8
81/68
27/22
81/64
4/3
24/17
16/11
3/2
27/17
18/11
27/16
16/9
32/17
64/33
2/1
===23-limit Pseudoringer===
33/32
13/12
9/8
7/6
39/32
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1
===Zothogothic===
28/27
13/12
9/8
7/6
39/32
91/72
4/3
112/81
13/9
3/2
14/9
13/8
91/54
7/4
117/64
91/48
2/1
===Zolugothic===
28/27
12/11
9/8
7/6
27/22
14/11
4/3
112/81
16/11
3/2
14/9
18/11
56/33
7/4
81/44
21/11
2/1
===Persian Johnston===
33/32
17/16
9/8
19/16
5/4
21/16
4/3
11/8
23/16
3/2
13/8
5/3
7/4
57/32
15/8
63/32
2/1
===Succulent Novemdecimal scale===
13/12
10/9
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
19/12
5/3
7/4
16/9
11/6
17/9
2/1
===Aberrismic Neopersian===
64/63
13/12
9/8
8/7
39/32
26/21
4/3
256/189
13/9
3/2
32/21
13/8
104/63
12/7
117/64
13/7
2/1
===Latwethagothic===
33/32
69/64
9/8
32/27
11/9
23/18
4/3
11/8
23/16
3/2
99/64
207/128
27/16
16/9
11/6
23/12
2/1
===Zalatwethagothic===
33/32
69/64
9/8
7/6
11/9
23/18
21/16
11/8
23/16
3/2
14/9
44/27
46/27
7/4
11/6
23/12
2/1
===Chariot===
19/18
13/12
10/9
7/6
11/9
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1

== 19-note scales ==
===Not-quite-ringer 19p===
33/32
17/16
9/8
7/6
19/16
5/4
21/16
4/3
11/8
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1
===Quasi-hemipyth detemper===
28/27
13/12
9/8
7/6
32/27
11/9
21/16
4/3
11/8
13/9
3/2
14/9
13/8
27/16
7/4
16/9
11/6
63/32
2/1
===19-limit dual-harmonic-segment-based===
25/24
13/12
9/8
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
25/16
13/8
27/16
7/4
57/32
15/8
63/32
2/1
===Pseudo-fives scale===
33/32
77/72
9/8
7/6
77/64
11/9
21/16
4/3
11/8
77/54
3/2
14/9
77/48
44/27
7/4
16/9
11/6
154/81
2/1
===Chipped Tetromino===
21/20
35/32
9/8
7/6
6/5
5/4
21/16
4/3
7/5
35/24
3/2
63/40
8/5
5/3
7/4
9/5
15/8
63/32
2/1
===Not-quite-ringer 19egh===
17/16
35/32
9/8
19/16
39/32
5/4
21/16
11/8
45/32
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1

== 20-note scales ==
===Yatha Double Blackdye===
65/64
13/12
10/9
9/8
65/54
39/32
5/4
4/3
65/48
13/9
40/27
3/2
130/81
13/8
5/3
27/16
65/36
117/64
15/8
2/1
===Yala Double Blackdye===
55/54
33/32
10/9
9/8
55/48
11/9
5/4
4/3
110/81
11/8
40/27
3/2
55/36
99/64
5/3
27/16
55/32
11/6
15/8
2/1

== 22-note scales ==
===Yala Dart===
33/32
88/81
10/9
9/8
32/27
11/9
5/4
81/64
4/3
11/8
45/32
40/27
3/2
99/64
44/27
5/3
27/16
16/9
11/6
15/8
160/81
2/1
===Close-to-even Scale===
33/32
17/16
35/32
9/8
7/6
29/24
5/4
31/24
4/3
11/8
17/12
35/24
3/2
14/9
29/18
5/3
31/18
16/9
11/6
17/9
35/18
2/1
===Cheendene===
1053/1024
13/12
10/9
9/8
32/27
39/32
5/4
81/64
4/3
351/256
45/32
40/27
3/2
3159/2048
13/8
5/3
27/16
16/9
117/64
15/8
160/81
2/1
===Compact 11-limit detemper===
33/32
77/72
35/32
9/8
7/6
77/64
5/4
165/128
4/3
11/8
45/32
35/24
3/2
14/9
77/48
5/3
55/32
7/4
11/6
15/8
35/18
2/1

== 24-note scales ==
===Car===
33/32
17/16
13/12
9/8
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
57/32
11/6
15/8
23/12
2/1
===Sedan===
33/32
17/16
13/12
10/9
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
11/6
15/8
23/12
2/1
===Modstraw===
28/27
35/32
10/9
9/8
7/6
32/27
5/4
35/27
21/16
4/3
45/32
35/24
40/27
3/2
14/9
105/64
5/3
27/16
7/4
16/9
15/8
35/18
63/32
2/1
===17-limit scale===
33/32
17/16
13/12
10/9
7/6
32/27
11/9
5/4
21/16
4/3
11/8
17/12
13/9
3/2
14/9
51/32
13/8
5/3
7/4
16/9
11/6
15/8
52/27
2/1
===31-limit scale===
33/32
17/16
13/12
9/8
7/6
19/16
29/24
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
29/16
15/8
31/16
2/1
===No-fives undecimal scale===
33/32
77/72
88/81
9/8
7/6
77/64
11/9
81/64
21/16
4/3
11/8
77/54
352/243
3/2
14/9
77/48
44/27
27/16
7/4
16/9
11/6
154/81
63/32
2/1
===37-limit scale===
31/30
16/15
11/10
17/15
7/6
6/5
37/30
19/15
13/10
4/3
11/8
17/12
35/24
3/2
31/20
8/5
33/20
17/10
7/4
9/5
37/20
19/10
39/20
2/1
===Hexquad===
28/27
77/72
35/32
9/8
7/6
77/64
11/9
5/4
35/27
4/3
11/8
45/32
35/24
3/2
14/9
77/48
44/27
5/3
7/4
231/128
11/6
15/8
35/18
2/1
===bus-like 11-limit detemper===
33/32
77/72
35/32
9/8
7/6
77/64
11/9
5/4
21/16
4/3
11/8
45/32
35/24
3/2
14/9
77/48
105/64
5/3
7/4
16/9
11/6
15/8
35/18
2/1

== 26-note scales ==
===Mothra Detemper===
135/128
15/14
35/32
9/8
8/7
7/6
315/256
5/4
9/7
21/16
4/3
45/32
10/7
35/24
3/2
32/21
45/28
105/64
5/3
12/7
7/4
945/512
15/8
40/21
63/32
2/1

== 29-note scales ==
===andromeda[29] detemper===
33/32
135/128
13/12
10/9
9/8
7/6
32/27
11/9
5/4
81/64
21/16
4/3
11/8
45/32
13/9
189/128
3/2
14/9
405/256
13/8
5/3
27/16
7/4
16/9
11/6
15/8
52/27
63/32
2/1

== 31-note scales ==
===Otonal Sevenice===
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
128/105
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1
===Utonal Sevenice===
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
315/256
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1
===Ringer 31p===
39/38
20/19
41/38
21/19
43/38
22/19
45/38
23/19
47/38
24/19
49/38
25/19
51/38
26/19
27/19
55/38
28/19
3/2
29/19
30/19
61/38
31/19
32/19
33/19
67/38
34/19
35/19
36/19
37/19
75/38
2/1
===Symmetrizine===
64/63
28/27
16/15
12/11
9/8
8/7
7/6
6/5
11/9
5/4
9/7
21/16
4/3
11/8
45/32
64/45
16/11
3/2
32/21
14/9
8/5
18/11
5/3
12/7
7/4
16/9
11/6
15/8
27/14
63/32
2/1

== Large scales (greater than 31 notes) ==
===41-tone 5-limit JI===
81/80
25/24
135/128
16/15
27/25
10/9
9/8
256/225
75/64
32/27
6/5
100/81
5/4
81/64
32/25
320/243
4/3
27/20
25/18
45/32
64/45
36/25
40/27
3/2
243/160
25/16
128/81
8/5
81/50
5/3
27/16
128/75
225/128
16/9
9/5
50/27
15/8
256/135
48/25
160/81
2/1
===46-tone septimal JI===
64/63
36/35
21/20
16/15
15/14
35/32
10/9
9/8
8/7
7/6
189/160
6/5
128/105
315/256
5/4
80/63
9/7
21/16
4/3
27/20
48/35
7/5
45/32
10/7
35/24
189/128
3/2
32/21
14/9
63/40
8/5
45/28
105/64
5/3
27/16
12/7
7/4
16/9
9/5
64/35
28/15
15/8
40/21
27/14
63/32
2/1
===53-tone Semantic Scale===
81/80
128/125
25/24
135/128
16/15
27/25
800/729
10/9
9/8
256/225
144/125
75/64
32/27
6/5
243/200
100/81
5/4
81/64
32/25
125/96
320/243
4/3
27/20
512/375
25/18
45/32
64/45
36/25
375/256
40/27
3/2
243/160
192/125
25/16
128/81
8/5
81/50
400/243
5/3
27/16
128/75
125/72
225/128
16/9
9/5
729/400
50/27
15/8
256/135
48/25
125/64
160/81
2/1
===Boston===
81/80
45/44
33/32
25/24
21/20
35/33
15/14
27/25
12/11
11/10
10/9
9/8
25/22
55/48
231/200
7/6
33/28
25/21
6/5
40/33
11/9
99/80
5/4
63/50
14/11
9/7
100/77
55/42
33/25
4/3
27/20
15/11
11/8
25/18
7/5
140/99
10/7
36/25
16/11
22/15
40/27
3/2
50/33
55/36
77/50
14/9
11/7
100/63
8/5
160/99
18/11
33/20
5/3
27/16
300/176
55/32
693/400
7/4
99/56
25/14
9/5
20/11
11/6
297/160
15/8
189/100
21/11
27/14
150/77
55/28
99/50
2/1
===130-tone 13-limit JI===
225/224
105/104
65/64
45/44
36/35
33/32
28/27
25/24
21/20
135/128
35/33
16/15
15/14
14/13
13/12
12/11
35/32
11/10
72/65
10/9
28/25
9/8
112/99
25/22
8/7
55/48
15/13
65/56
7/6
168/143
13/11
32/27
25/21
6/5
135/112
40/33
128/105
11/9
16/13
26/21
56/45
5/4
63/50
81/64
14/11
32/25
9/7
84/65
13/10
72/55
21/16
33/25
143/108
4/3
75/56
27/20
65/48
15/11
48/35
11/8
18/13
25/18
7/5
45/32
99/70
64/45
10/7
36/25
13/9
16/11
35/24
22/15
96/65
40/27
112/75
3/2
448/297
50/33
32/21
55/36
20/13
65/42
14/9
25/16
11/7
128/81
100/63
8/5
45/28
21/13
13/8
18/11
105/64
33/20
224/135
5/3
42/25
27/16
22/13
143/84
12/7
112/65
26/15
96/55
7/4
44/25
99/56
16/9
25/14
9/5
65/36
20/11
64/35
11/6
24/13
13/7
28/15
15/8
66/35
256/135
40/21
48/25
27/14
64/33
35/18
49/25
63/32
99/50
143/72
2/1

User:Tristanbay/Gallery of just intonation scales

2026-01-04T04:59:01Z

Tristanbay: /* Large scales (greater than 31 notes) */ Added Boston scale

All of the scales below are constant structure and repeat at 2/1.

== Boutique scales (less than 12 notes) ==
'''24:26:28:30:32:33:36:39:42:45:48'''
<pre>13/12
7/6
5/4
4/3
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''24:27:28:30:33:36:39:42:45:48'''
<pre>9/8
7/6
5/4
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''60:63:64:70:80:84:96:105:120'''
<pre>35/32
5/4
21/16
4/3
35/24
5/3
7/4
2/1</pre>

'''56:60:64:70:84:90:96:105:112'''
<pre>35/32
7/6
5/4
4/3
35/24
7/4
15/8
2/1</pre>

'''18:20:22:24:27:30:33:36'''
<pre>10/9
11/9
4/3
3/2
5/3
11/6
2/1</pre>

'''14:16:18:20:21:24:27:28'''
<pre>8/7
9/7
10/7
3/2
12/7
27/14
2/1</pre>

'''27:28:33:36:39:42:45:54'''
<pre>28/27
11/9
4/3
13/9
14/9
5/3
2/1</pre>

'''7:8:9:10:11:12:13:14'''
<pre>8/7
9/7
10/7
11/7
12/7
13/7
2/1</pre>

'''56:60:64:70:72:80:84:90:96:105:112'''
<pre>15/14
8/7
5/4
9/7
10/7
3/2
45/28
12/7
15/8
2/1</pre>

'''52:58:64:71:78:86:95:104'''
<pre>71/64
39/32
43/32
95/64
13/8
29/16
2/1</pre>

'''48:52:58:64:72:78:87:96'''
<pre>13/12
29/24
4/3
3/2
13/8
29/16
2/1</pre>

'''33:36:40:42:44:48:54:60:63:66'''
<pre>12/11
40/33
14/11
4/3
16/11
18/11
20/11
21/11
2/1</pre>

'''720:780:840:910:945:1008:1092:1170:1260:1365:1440'''
<pre>13/12
7/6
91/72
21/16
7/5
91/60
13/8
7/4
91/48
2/1</pre>

== 12-note scales ==
'''Antitonic 17-limit'''
<pre>17/16
39/32
5/4
21/16
11/8
3/2
51/32
13/8
5/3
7/4
11/6
2/1</pre>

'''Bicycle'''
<pre>13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
11/6
2/1</pre>

'''Duodene'''
<pre>16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1</pre>

'''Septimal Catgirl'''
<pre>28/27
9/8
7/6
5/4
4/3
45/32
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''North Dakota'''
<pre>33/32
10/9
7/6
5/4
4/3
11/8
3/2
14/9
5/3
7/4
11/6
2/1</pre>

'''Johnston Piano Scale (Ringer 12)'''
<pre>15/14
8/7
17/14
9/7
19/14
10/7
3/2
11/7
12/7
13/7
27/14
2/1</pre>

'''Ephemeral Soup Scale'''
<pre>77/72
9/8
77/64
21/16
693/512
189/128
3/2
77/48
7/4
231/128
63/32
2/1</pre>

'''Once Upon A Time Scale'''
<pre>10/9
9/8
5/4
9/7
21/16
35/24
3/2
5/3
12/7
7/4
35/18
2/1</pre>

'''Centaur'''
<pre>21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''Pental Diachrome'''
<pre>10/9
9/8
5/4
81/64
4/3
40/27
3/2
5/3
27/16
15/8
160/81
2/1</pre>

'''Margo Scale'''
<pre>104/99
9/8
13/11
14/11
4/3
63/44
3/2
52/33
56/33
16/9
21/11
2/1</pre>

'''Meta-Slendro'''
<pre>50/49
8/7
57/49
64/49
65/49
72/49
74/49
151/98
12/7
86/49
96/49
2/1</pre>

'''Succulent Undecimal Scale'''
<pre>10/9
7/6
5/4
21/16
4/3
11/8
3/2
5/3
7/4
16/9
11/6
2/1</pre>

== 14-note scales ==
'''No-13s ultrajust'''
17/16
9/8
19/16
5/4
21/16
11/8
23/16
3/2
19/12
5/3
7/4
11/6
23/12
2/1

== 17-note scales ==
'''Walkie'''
65/64
13/12
9/8
585/512
39/32
5/4
4/3
65/48
45/32
3/2
195/128
13/8
5/3
16/9
117/64
15/8
2/1
'''Mohahadene'''
55/54
88/81
9/8
55/48
11/9
5/4
4/3
11/8
45/32
3/2
55/36
44/27
5/3
55/32
11/6
15/8
2/1
'''Dichotidene'''
28/27
35/32
9/8
7/6
5/4
35/27
4/3
45/32
35/24
3/2
14/9
105/64
27/16
7/4
15/8
35/18
2/1
'''Just Spoogalaxy'''
28/27
29/27
9/8
7/6
29/24
81/64
21/16
87/64
116/81
3/2
14/9
29/18
27/16
7/4
29/16
243/128
2/1
'''Cartwheel'''
28/27
13/12
9/8
7/6
11/9
5/4
4/3
11/8
13/9
3/2
14/9
13/8
5/3
7/4
11/6
15/8
2/1
'''Ringer 17-quad'''
33/32
17/16
9/8
19/16
77/64
5/4
21/16
11/8
23/16
3/2
49/32
25/16
13/8
7/4
29/16
15/8
2/1
'''Ringer 17-tri'''
49/48
17/16
9/8
19/16
29/24
5/4
21/16
11/8
23/16
3/2
74/48
25/16
13/8
7/4
11/6
15/8
2/1
'''Schismadene'''
135/128
10/9
9/8
32/27
5/4
81/64
4/3
45/32
40/27
3/2
128/81
5/3
27/16
16/9
15/8
160/81
2/1
'''Traditional Baglama Fret Scale'''
18/17
12/11
9/8
81/68
27/22
81/64
4/3
24/17
16/11
3/2
27/17
18/11
27/16
16/9
32/17
64/33
2/1
'''23-limit Pseudoringer'''
33/32
13/12
9/8
7/6
39/32
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1
'''Zothogothic'''
28/27
13/12
9/8
7/6
39/32
91/72
4/3
112/81
13/9
3/2
14/9
13/8
91/54
7/4
117/64
91/48
2/1
'''Zolugothic'''
28/27
12/11
9/8
7/6
27/22
14/11
4/3
112/81
16/11
3/2
14/9
18/11
56/33
7/4
81/44
21/11
2/1
'''Persian Johnston'''
33/32
17/16
9/8
19/16
5/4
21/16
4/3
11/8
23/16
3/2
13/8
5/3
7/4
57/32
15/8
63/32
2/1
'''Succulent Novemdecimal scale'''
13/12
10/9
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
19/12
5/3
7/4
16/9
11/6
17/9
2/1
'''Aberrismic Neopersian'''
64/63
13/12
9/8
8/7
39/32
26/21
4/3
256/189
13/9
3/2
32/21
13/8
104/63
12/7
117/64
13/7
2/1
'''Latwethagothic'''
33/32
69/64
9/8
32/27
11/9
23/18
4/3
11/8
23/16
3/2
99/64
207/128
27/16
16/9
11/6
23/12
2/1
'''Zalatwethagothic'''
33/32
69/64
9/8
7/6
11/9
23/18
21/16
11/8
23/16
3/2
14/9
44/27
46/27
7/4
11/6
23/12
2/1
'''Chariot'''
19/18
13/12
10/9
7/6
11/9
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1

== 19-note scales ==
'''Not-quite-ringer 19p'''
33/32
17/16
9/8
7/6
19/16
5/4
21/16
4/3
11/8
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1
'''Quasi-hemipyth detemper'''
28/27
13/12
9/8
7/6
32/27
11/9
21/16
4/3
11/8
13/9
3/2
14/9
13/8
27/16
7/4
16/9
11/6
63/32
2/1
'''19-limit dual-harmonic-segment-based'''
25/24
13/12
9/8
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
25/16
13/8
27/16
7/4
57/32
15/8
63/32
2/1
'''Pseudo-fives scale'''
33/32
77/72
9/8
7/6
77/64
11/9
21/16
4/3
11/8
77/54
3/2
14/9
77/48
44/27
7/4
16/9
11/6
154/81
2/1
'''Chipped Tetromino'''
21/20
35/32
9/8
7/6
6/5
5/4
21/16
4/3
7/5
35/24
3/2
63/40
8/5
5/3
7/4
9/5
15/8
63/32
2/1
'''Not-quite-ringer 19egh'''
17/16
35/32
9/8
19/16
39/32
5/4
21/16
11/8
45/32
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1

== 20-note scales ==
'''Yatha Double Blackdye'''
65/64
13/12
10/9
9/8
65/54
39/32
5/4
4/3
65/48
13/9
40/27
3/2
130/81
13/8
5/3
27/16
65/36
117/64
15/8
2/1
'''Yala Double Blackdye'''
55/54
33/32
10/9
9/8
55/48
11/9
5/4
4/3
110/81
11/8
40/27
3/2
55/36
99/64
5/3
27/16
55/32
11/6
15/8
2/1

== 22-note scales ==
'''Yala Dart'''
33/32
88/81
10/9
9/8
32/27
11/9
5/4
81/64
4/3
11/8
45/32
40/27
3/2
99/64
44/27
5/3
27/16
16/9
11/6
15/8
160/81
2/1
'''Close-to-even Scale'''
33/32
17/16
35/32
9/8
7/6
29/24
5/4
31/24
4/3
11/8
17/12
35/24
3/2
14/9
29/18
5/3
31/18
16/9
11/6
17/9
35/18
2/1
'''Cheendene'''
1053/1024
13/12
10/9
9/8
32/27
39/32
5/4
81/64
4/3
351/256
45/32
40/27
3/2
3159/2048
13/8
5/3
27/16
16/9
117/64
15/8
160/81
2/1
'''Compact 11-limit detemper'''
33/32
77/72
35/32
9/8
7/6
77/64
5/4
165/128
4/3
11/8
45/32
35/24
3/2
14/9
77/48
5/3
55/32
7/4
11/6
15/8
35/18
2/1

== 24-note scales ==
'''Car'''
33/32
17/16
13/12
9/8
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
57/32
11/6
15/8
23/12
2/1
'''Sedan'''
33/32
17/16
13/12
10/9
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
11/6
15/8
23/12
2/1
'''Modstraw'''
28/27
35/32
10/9
9/8
7/6
32/27
5/4
35/27
21/16
4/3
45/32
35/24
40/27
3/2
14/9
105/64
5/3
27/16
7/4
16/9
15/8
35/18
63/32
2/1
'''17-limit scale'''
33/32
17/16
13/12
10/9
7/6
32/27
11/9
5/4
21/16
4/3
11/8
17/12
13/9
3/2
14/9
51/32
13/8
5/3
7/4
16/9
11/6
15/8
52/27
2/1
'''31-limit scale'''
33/32
17/16
13/12
9/8
7/6
19/16
29/24
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
29/16
15/8
31/16
2/1
'''No-fives undecimal scale'''
33/32
77/72
88/81
9/8
7/6
77/64
11/9
81/64
21/16
4/3
11/8
77/54
352/243
3/2
14/9
77/48
44/27
27/16
7/4
16/9
11/6
154/81
63/32
2/1
'''37-limit scale'''
31/30
16/15
11/10
17/15
7/6
6/5
37/30
19/15
13/10
4/3
11/8
17/12
35/24
3/2
31/20
8/5
33/20
17/10
7/4
9/5
37/20
19/10
39/20
2/1
'''Hexquad'''
28/27
77/72
35/32
9/8
7/6
77/64
11/9
5/4
35/27
4/3
11/8
45/32
35/24
3/2
14/9
77/48
44/27
5/3
7/4
231/128
11/6
15/8
35/18
2/1
'''bus-like 11-limit detemper'''
33/32
77/72
35/32
9/8
7/6
77/64
11/9
5/4
21/16
4/3
11/8
45/32
35/24
3/2
14/9
77/48
105/64
5/3
7/4
16/9
11/6
15/8
35/18
2/1

== 26-note scales ==
'''Mothra Detemper'''
135/128
15/14
35/32
9/8
8/7
7/6
315/256
5/4
9/7
21/16
4/3
45/32
10/7
35/24
3/2
32/21
45/28
105/64
5/3
12/7
7/4
945/512
15/8
40/21
63/32
2/1

== 29-note scales ==
'''andromeda[29] detemper'''
33/32
135/128
13/12
10/9
9/8
7/6
32/27
11/9
5/4
81/64
21/16
4/3
11/8
45/32
13/9
189/128
3/2
14/9
405/256
13/8
5/3
27/16
7/4
16/9
11/6
15/8
52/27
63/32
2/1

== 31-note scales ==
'''Otonal Sevenice'''
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
128/105
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1
'''Utonal Sevenice'''
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
315/256
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1
'''Ringer 31p'''
39/38
20/19
41/38
21/19
43/38
22/19
45/38
23/19
47/38
24/19
49/38
25/19
51/38
26/19
27/19
55/38
28/19
3/2
29/19
30/19
61/38
31/19
32/19
33/19
67/38
34/19
35/19
36/19
37/19
75/38
2/1
'''Symmetrizine'''
64/63
28/27
16/15
12/11
9/8
8/7
7/6
6/5
11/9
5/4
9/7
21/16
4/3
11/8
45/32
64/45
16/11
3/2
32/21
14/9
8/5
18/11
5/3
12/7
7/4
16/9
11/6
15/8
27/14
63/32
2/1

== Large scales (greater than 31 notes) ==
'''41-tone 5-limit JI'''
81/80
25/24
135/128
16/15
27/25
10/9
9/8
256/225
75/64
32/27
6/5
100/81
5/4
81/64
32/25
320/243
4/3
27/20
25/18
45/32
64/45
36/25
40/27
3/2
243/160
25/16
128/81
8/5
81/50
5/3
27/16
128/75
225/128
16/9
9/5
50/27
15/8
256/135
48/25
160/81
2/1
'''46-tone septimal JI'''
64/63
36/35
21/20
16/15
15/14
35/32
10/9
9/8
8/7
7/6
189/160
6/5
128/105
315/256
5/4
80/63
9/7
21/16
4/3
27/20
48/35
7/5
45/32
10/7
35/24
189/128
3/2
32/21
14/9
63/40
8/5
45/28
105/64
5/3
27/16
12/7
7/4
16/9
9/5
64/35
28/15
15/8
40/21
27/14
63/32
2/1
'''53-tone Semantic Scale'''
81/80
128/125
25/24
135/128
16/15
27/25
800/729
10/9
9/8
256/225
144/125
75/64
32/27
6/5
243/200
100/81
5/4
81/64
32/25
125/96
320/243
4/3
27/20
512/375
25/18
45/32
64/45
36/25
375/256
40/27
3/2
243/160
192/125
25/16
128/81
8/5
81/50
400/243
5/3
27/16
128/75
125/72
225/128
16/9
9/5
729/400
50/27
15/8
256/135
48/25
125/64
160/81
2/1
'''Boston'''
81/80
45/44
33/32
25/24
21/20
35/33
15/14
27/25
12/11
11/10
10/9
9/8
25/22
55/48
231/200
7/6
33/28
25/21
6/5
40/33
11/9
99/80
5/4
63/50
14/11
9/7
100/77
55/42
33/25
4/3
27/20
15/11
11/8
25/18
7/5
140/99
10/7
36/25
16/11
22/15
40/27
3/2
50/33
55/36
77/50
14/9
11/7
100/63
8/5
160/99
18/11
33/20
5/3
27/16
300/176
55/32
693/400
7/4
99/56
25/14
9/5
20/11
11/6
297/160
15/8
189/100
21/11
27/14
150/77
55/28
99/50
2/1
'''130-tone 13-limit JI'''
225/224
105/104
65/64
45/44
36/35
33/32
28/27
25/24
21/20
135/128
35/33
16/15
15/14
14/13
13/12
12/11
35/32
11/10
72/65
10/9
28/25
9/8
112/99
25/22
8/7
55/48
15/13
65/56
7/6
168/143
13/11
32/27
25/21
6/5
135/112
40/33
128/105
11/9
16/13
26/21
56/45
5/4
63/50
81/64
14/11
32/25
9/7
84/65
13/10
72/55
21/16
33/25
143/108
4/3
75/56
27/20
65/48
15/11
48/35
11/8
18/13
25/18
7/5
45/32
99/70
64/45
10/7
36/25
13/9
16/11
35/24
22/15
96/65
40/27
112/75
3/2
448/297
50/33
32/21
55/36
20/13
65/42
14/9
25/16
11/7
128/81
100/63
8/5
45/28
21/13
13/8
18/11
105/64
33/20
224/135
5/3
42/25
27/16
22/13
143/84
12/7
112/65
26/15
96/55
7/4
44/25
99/56
16/9
25/14
9/5
65/36
20/11
64/35
11/6
24/13
13/7
28/15
15/8
66/35
256/135
40/21
48/25
27/14
64/33
35/18
49/25
63/32
99/50
143/72
2/1

User:Tristanbay/Gallery of just intonation scales

2026-01-04T04:52:26Z

Tristanbay: Formatting and named a scale

All of the scales below are constant structure and repeat at 2/1.

== Boutique scales (less than 12 notes) ==
'''24:26:28:30:32:33:36:39:42:45:48'''
<pre>13/12
7/6
5/4
4/3
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''24:27:28:30:33:36:39:42:45:48'''
<pre>9/8
7/6
5/4
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''60:63:64:70:80:84:96:105:120'''
<pre>35/32
5/4
21/16
4/3
35/24
5/3
7/4
2/1</pre>

'''56:60:64:70:84:90:96:105:112'''
<pre>35/32
7/6
5/4
4/3
35/24
7/4
15/8
2/1</pre>

'''18:20:22:24:27:30:33:36'''
<pre>10/9
11/9
4/3
3/2
5/3
11/6
2/1</pre>

'''14:16:18:20:21:24:27:28'''
<pre>8/7
9/7
10/7
3/2
12/7
27/14
2/1</pre>

'''27:28:33:36:39:42:45:54'''
<pre>28/27
11/9
4/3
13/9
14/9
5/3
2/1</pre>

'''7:8:9:10:11:12:13:14'''
<pre>8/7
9/7
10/7
11/7
12/7
13/7
2/1</pre>

'''56:60:64:70:72:80:84:90:96:105:112'''
<pre>15/14
8/7
5/4
9/7
10/7
3/2
45/28
12/7
15/8
2/1</pre>

'''52:58:64:71:78:86:95:104'''
<pre>71/64
39/32
43/32
95/64
13/8
29/16
2/1</pre>

'''48:52:58:64:72:78:87:96'''
<pre>13/12
29/24
4/3
3/2
13/8
29/16
2/1</pre>

'''33:36:40:42:44:48:54:60:63:66'''
<pre>12/11
40/33
14/11
4/3
16/11
18/11
20/11
21/11
2/1</pre>

'''720:780:840:910:945:1008:1092:1170:1260:1365:1440'''
<pre>13/12
7/6
91/72
21/16
7/5
91/60
13/8
7/4
91/48
2/1</pre>

== 12-note scales ==
'''Antitonic 17-limit'''
<pre>17/16
39/32
5/4
21/16
11/8
3/2
51/32
13/8
5/3
7/4
11/6
2/1</pre>

'''Bicycle'''
<pre>13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
11/6
2/1</pre>

'''Duodene'''
<pre>16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1</pre>

'''Septimal Catgirl'''
<pre>28/27
9/8
7/6
5/4
4/3
45/32
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''North Dakota'''
<pre>33/32
10/9
7/6
5/4
4/3
11/8
3/2
14/9
5/3
7/4
11/6
2/1</pre>

'''Johnston Piano Scale (Ringer 12)'''
<pre>15/14
8/7
17/14
9/7
19/14
10/7
3/2
11/7
12/7
13/7
27/14
2/1</pre>

'''Ephemeral Soup Scale'''
<pre>77/72
9/8
77/64
21/16
693/512
189/128
3/2
77/48
7/4
231/128
63/32
2/1</pre>

'''Once Upon A Time Scale'''
<pre>10/9
9/8
5/4
9/7
21/16
35/24
3/2
5/3
12/7
7/4
35/18
2/1</pre>

'''Centaur'''
<pre>21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''Pental Diachrome'''
<pre>10/9
9/8
5/4
81/64
4/3
40/27
3/2
5/3
27/16
15/8
160/81
2/1</pre>

'''Margo Scale'''
<pre>104/99
9/8
13/11
14/11
4/3
63/44
3/2
52/33
56/33
16/9
21/11
2/1</pre>

'''Meta-Slendro'''
<pre>50/49
8/7
57/49
64/49
65/49
72/49
74/49
151/98
12/7
86/49
96/49
2/1</pre>

'''Succulent Undecimal Scale'''
<pre>10/9
7/6
5/4
21/16
4/3
11/8
3/2
5/3
7/4
16/9
11/6
2/1</pre>

== 14-note scales ==
'''No-13s ultrajust'''
17/16
9/8
19/16
5/4
21/16
11/8
23/16
3/2
19/12
5/3
7/4
11/6
23/12
2/1

== 17-note scales ==
'''Walkie'''
65/64
13/12
9/8
585/512
39/32
5/4
4/3
65/48
45/32
3/2
195/128
13/8
5/3
16/9
117/64
15/8
2/1
'''Mohahadene'''
55/54
88/81
9/8
55/48
11/9
5/4
4/3
11/8
45/32
3/2
55/36
44/27
5/3
55/32
11/6
15/8
2/1
'''Dichotidene'''
28/27
35/32
9/8
7/6
5/4
35/27
4/3
45/32
35/24
3/2
14/9
105/64
27/16
7/4
15/8
35/18
2/1
'''Just Spoogalaxy'''
28/27
29/27
9/8
7/6
29/24
81/64
21/16
87/64
116/81
3/2
14/9
29/18
27/16
7/4
29/16
243/128
2/1
'''Cartwheel'''
28/27
13/12
9/8
7/6
11/9
5/4
4/3
11/8
13/9
3/2
14/9
13/8
5/3
7/4
11/6
15/8
2/1
'''Ringer 17-quad'''
33/32
17/16
9/8
19/16
77/64
5/4
21/16
11/8
23/16
3/2
49/32
25/16
13/8
7/4
29/16
15/8
2/1
'''Ringer 17-tri'''
49/48
17/16
9/8
19/16
29/24
5/4
21/16
11/8
23/16
3/2
74/48
25/16
13/8
7/4
11/6
15/8
2/1
'''Schismadene'''
135/128
10/9
9/8
32/27
5/4
81/64
4/3
45/32
40/27
3/2
128/81
5/3
27/16
16/9
15/8
160/81
2/1
'''Traditional Baglama Fret Scale'''
18/17
12/11
9/8
81/68
27/22
81/64
4/3
24/17
16/11
3/2
27/17
18/11
27/16
16/9
32/17
64/33
2/1
'''23-limit Pseudoringer'''
33/32
13/12
9/8
7/6
39/32
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1
'''Zothogothic'''
28/27
13/12
9/8
7/6
39/32
91/72
4/3
112/81
13/9
3/2
14/9
13/8
91/54
7/4
117/64
91/48
2/1
'''Zolugothic'''
28/27
12/11
9/8
7/6
27/22
14/11
4/3
112/81
16/11
3/2
14/9
18/11
56/33
7/4
81/44
21/11
2/1
'''Persian Johnston'''
33/32
17/16
9/8
19/16
5/4
21/16
4/3
11/8
23/16
3/2
13/8
5/3
7/4
57/32
15/8
63/32
2/1
'''Succulent Novemdecimal scale'''
13/12
10/9
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
19/12
5/3
7/4
16/9
11/6
17/9
2/1
'''Aberrismic Neopersian'''
64/63
13/12
9/8
8/7
39/32
26/21
4/3
256/189
13/9
3/2
32/21
13/8
104/63
12/7
117/64
13/7
2/1
'''Latwethagothic'''
33/32
69/64
9/8
32/27
11/9
23/18
4/3
11/8
23/16
3/2
99/64
207/128
27/16
16/9
11/6
23/12
2/1
'''Zalatwethagothic'''
33/32
69/64
9/8
7/6
11/9
23/18
21/16
11/8
23/16
3/2
14/9
44/27
46/27
7/4
11/6
23/12
2/1
'''Chariot'''
19/18
13/12
10/9
7/6
11/9
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1

== 19-note scales ==
'''Not-quite-ringer 19p'''
33/32
17/16
9/8
7/6
19/16
5/4
21/16
4/3
11/8
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1
'''Quasi-hemipyth detemper'''
28/27
13/12
9/8
7/6
32/27
11/9
21/16
4/3
11/8
13/9
3/2
14/9
13/8
27/16
7/4
16/9
11/6
63/32
2/1
'''19-limit dual-harmonic-segment-based'''
25/24
13/12
9/8
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
25/16
13/8
27/16
7/4
57/32
15/8
63/32
2/1
'''Pseudo-fives scale'''
33/32
77/72
9/8
7/6
77/64
11/9
21/16
4/3
11/8
77/54
3/2
14/9
77/48
44/27
7/4
16/9
11/6
154/81
2/1
'''Chipped Tetromino'''
21/20
35/32
9/8
7/6
6/5
5/4
21/16
4/3
7/5
35/24
3/2
63/40
8/5
5/3
7/4
9/5
15/8
63/32
2/1
'''Not-quite-ringer 19egh'''
17/16
35/32
9/8
19/16
39/32
5/4
21/16
11/8
45/32
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1

== 20-note scales ==
'''Yatha Double Blackdye'''
65/64
13/12
10/9
9/8
65/54
39/32
5/4
4/3
65/48
13/9
40/27
3/2
130/81
13/8
5/3
27/16
65/36
117/64
15/8
2/1
'''Yala Double Blackdye'''
55/54
33/32
10/9
9/8
55/48
11/9
5/4
4/3
110/81
11/8
40/27
3/2
55/36
99/64
5/3
27/16
55/32
11/6
15/8
2/1

== 22-note scales ==
'''Yala Dart'''
33/32
88/81
10/9
9/8
32/27
11/9
5/4
81/64
4/3
11/8
45/32
40/27
3/2
99/64
44/27
5/3
27/16
16/9
11/6
15/8
160/81
2/1
'''Close-to-even Scale'''
33/32
17/16
35/32
9/8
7/6
29/24
5/4
31/24
4/3
11/8
17/12
35/24
3/2
14/9
29/18
5/3
31/18
16/9
11/6
17/9
35/18
2/1
'''Cheendene'''
1053/1024
13/12
10/9
9/8
32/27
39/32
5/4
81/64
4/3
351/256
45/32
40/27
3/2
3159/2048
13/8
5/3
27/16
16/9
117/64
15/8
160/81
2/1
'''Compact 11-limit detemper'''
33/32
77/72
35/32
9/8
7/6
77/64
5/4
165/128
4/3
11/8
45/32
35/24
3/2
14/9
77/48
5/3
55/32
7/4
11/6
15/8
35/18
2/1

== 24-note scales ==
'''Car'''
33/32
17/16
13/12
9/8
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
57/32
11/6
15/8
23/12
2/1
'''Sedan'''
33/32
17/16
13/12
10/9
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
11/6
15/8
23/12
2/1
'''Modstraw'''
28/27
35/32
10/9
9/8
7/6
32/27
5/4
35/27
21/16
4/3
45/32
35/24
40/27
3/2
14/9
105/64
5/3
27/16
7/4
16/9
15/8
35/18
63/32
2/1
'''17-limit scale'''
33/32
17/16
13/12
10/9
7/6
32/27
11/9
5/4
21/16
4/3
11/8
17/12
13/9
3/2
14/9
51/32
13/8
5/3
7/4
16/9
11/6
15/8
52/27
2/1
'''31-limit scale'''
33/32
17/16
13/12
9/8
7/6
19/16
29/24
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
29/16
15/8
31/16
2/1
'''No-fives undecimal scale'''
33/32
77/72
88/81
9/8
7/6
77/64
11/9
81/64
21/16
4/3
11/8
77/54
352/243
3/2
14/9
77/48
44/27
27/16
7/4
16/9
11/6
154/81
63/32
2/1
'''37-limit scale'''
31/30
16/15
11/10
17/15
7/6
6/5
37/30
19/15
13/10
4/3
11/8
17/12
35/24
3/2
31/20
8/5
33/20
17/10
7/4
9/5
37/20
19/10
39/20
2/1
'''Hexquad'''
28/27
77/72
35/32
9/8
7/6
77/64
11/9
5/4
35/27
4/3
11/8
45/32
35/24
3/2
14/9
77/48
44/27
5/3
7/4
231/128
11/6
15/8
35/18
2/1
'''bus-like 11-limit detemper'''
33/32
77/72
35/32
9/8
7/6
77/64
11/9
5/4
21/16
4/3
11/8
45/32
35/24
3/2
14/9
77/48
105/64
5/3
7/4
16/9
11/6
15/8
35/18
2/1

== 26-note scales ==
'''Mothra Detemper'''
135/128
15/14
35/32
9/8
8/7
7/6
315/256
5/4
9/7
21/16
4/3
45/32
10/7
35/24
3/2
32/21
45/28
105/64
5/3
12/7
7/4
945/512
15/8
40/21
63/32
2/1

== 29-note scales ==
'''andromeda[29] detemper'''
33/32
135/128
13/12
10/9
9/8
7/6
32/27
11/9
5/4
81/64
21/16
4/3
11/8
45/32
13/9
189/128
3/2
14/9
405/256
13/8
5/3
27/16
7/4
16/9
11/6
15/8
52/27
63/32
2/1

== 31-note scales ==
'''Otonal Sevenice'''
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
128/105
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1
'''Utonal Sevenice'''
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
315/256
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1
'''Ringer 31p'''
39/38
20/19
41/38
21/19
43/38
22/19
45/38
23/19
47/38
24/19
49/38
25/19
51/38
26/19
27/19
55/38
28/19
3/2
29/19
30/19
61/38
31/19
32/19
33/19
67/38
34/19
35/19
36/19
37/19
75/38
2/1
'''Symmetrizine'''
64/63
28/27
16/15
12/11
9/8
8/7
7/6
6/5
11/9
5/4
9/7
21/16
4/3
11/8
45/32
64/45
16/11
3/2
32/21
14/9
8/5
18/11
5/3
12/7
7/4
16/9
11/6
15/8
27/14
63/32
2/1

== Large scales (greater than 31 notes) ==
'''41-tone 5-limit JI'''
81/80
25/24
135/128
16/15
27/25
10/9
9/8
256/225
75/64
32/27
6/5
100/81
5/4
81/64
32/25
320/243
4/3
27/20
25/18
45/32
64/45
36/25
40/27
3/2
243/160
25/16
128/81
8/5
81/50
5/3
27/16
128/75
225/128
16/9
9/5
50/27
15/8
256/135
48/25
160/81
2/1
'''46-tone septimal JI'''
64/63
36/35
21/20
16/15
15/14
35/32
10/9
9/8
8/7
7/6
189/160
6/5
128/105
315/256
5/4
80/63
9/7
21/16
4/3
27/20
48/35
7/5
45/32
10/7
35/24
189/128
3/2
32/21
14/9
63/40
8/5
45/28
105/64
5/3
27/16
12/7
7/4
16/9
9/5
64/35
28/15
15/8
40/21
27/14
63/32
2/1
'''53-tone Semantic Scale'''
81/80
128/125
25/24
135/128
16/15
27/25
800/729
10/9
9/8
256/225
144/125
75/64
32/27
6/5
243/200
100/81
5/4
81/64
32/25
125/96
320/243
4/3
27/20
512/375
25/18
45/32
64/45
36/25
375/256
40/27
3/2
243/160
192/125
25/16
128/81
8/5
81/50
400/243
5/3
27/16
128/75
125/72
225/128
16/9
9/5
729/400
50/27
15/8
256/135
48/25
125/64
160/81
2/1
'''130-tone 13-limit JI'''
225/224
105/104
65/64
45/44
36/35
33/32
28/27
25/24
21/20
135/128
35/33
16/15
15/14
14/13
13/12
12/11
35/32
11/10
72/65
10/9
28/25
9/8
112/99
25/22
8/7
55/48
15/13
65/56
7/6
168/143
13/11
32/27
25/21
6/5
135/112
40/33
128/105
11/9
16/13
26/21
56/45
5/4
63/50
81/64
14/11
32/25
9/7
84/65
13/10
72/55
21/16
33/25
143/108
4/3
75/56
27/20
65/48
15/11
48/35
11/8
18/13
25/18
7/5
45/32
99/70
64/45
10/7
36/25
13/9
16/11
35/24
22/15
96/65
40/27
112/75
3/2
448/297
50/33
32/21
55/36
20/13
65/42
14/9
25/16
11/7
128/81
100/63
8/5
45/28
21/13
13/8
18/11
105/64
33/20
224/135
5/3
42/25
27/16
22/13
143/84
12/7
112/65
26/15
96/55
7/4
44/25
99/56
16/9
25/14
9/5
65/36
20/11
64/35
11/6
24/13
13/7
28/15
15/8
66/35
256/135
40/21
48/25
27/14
64/33
35/18
49/25
63/32
99/50
143/72
2/1

User:Tristanbay/Gallery of just intonation scales

2026-01-04T04:43:56Z

Tristanbay: Created gallery of JI scales (expanded from my user page)

All of the scales below are constant structure and repeat at 2/1.

== Boutique scales (less than 12 notes) ==
'''24:26:28:30:32:33:36:39:42:45:48'''
<pre>13/12
7/6
5/4
4/3
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''24:27:28:30:33:36:39:42:45:48'''
<pre>9/8
7/6
5/4
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''60:63:64:70:80:84:96:105:120'''
<pre>35/32
5/4
21/16
4/3
35/24
5/3
7/4
2/1</pre>

'''56:60:64:70:84:90:96:105:112'''
<pre>35/32
7/6
5/4
4/3
35/24
7/4
15/8
2/1</pre>

'''18:20:22:24:27:30:33:36'''
<pre>10/9
11/9
4/3
3/2
5/3
11/6
2/1</pre>

'''14:16:18:20:21:24:27:28'''
<pre>8/7
9/7
10/7
3/2
12/7
27/14
2/1</pre>

'''27:28:33:36:39:42:45:54'''
<pre>28/27
11/9
4/3
13/9
14/9
5/3
2/1</pre>

'''7:8:9:10:11:12:13:14'''
<pre>8/7
9/7
10/7
11/7
12/7
13/7
2/1</pre>

'''56:60:64:70:72:80:84:90:96:105:112'''
<pre>15/14
8/7
5/4
9/7
10/7
3/2
45/28
12/7
15/8
2/1</pre>

'''52:58:64:71:78:86:95:104'''
<pre>71/64
39/32
43/32
95/64
13/8
29/16
2/1</pre>

'''48:52:58:64:72:78:87:96'''
<pre>13/12
29/24
4/3
3/2
13/8
29/16
2/1</pre>

'''33:36:40:42:44:48:54:60:63:66'''
<pre>12/11
40/33
14/11
4/3
16/11
18/11
20/11
21/11
2/1</pre>

'''720:780:840:910:945:1008:1092:1170:1260:1365:1440'''
<pre>13/12
7/6
91/72
21/16
7/5
91/60
13/8
7/4
91/48
2/1</pre>

== 12-note scales ==
'''Antitonic 17-limit'''
<pre>17/16
39/32
5/4
21/16
11/8
3/2
51/32
13/8
5/3
7/4
11/6
2/1</pre>

'''Bicycle'''
<pre>13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
11/6
2/1</pre>

'''Duodene'''
<pre>16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1</pre>

'''Septimal Catgirl'''
<pre>28/27
9/8
7/6
5/4
4/3
45/32
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''North Dakota'''
<pre>33/32
10/9
7/6
5/4
4/3
11/8
3/2
14/9
5/3
7/4
11/6
2/1</pre>

'''Johnston Piano Scale (Ringer 12)'''
<pre>15/14
8/7
17/14
9/7
19/14
10/7
3/2
11/7
12/7
13/7
27/14
2/1</pre>

'''Ephemeral Soup Scale'''
<pre>77/72
9/8
77/64
21/16
693/512
189/128
3/2
77/48
7/4
231/128
63/32
2/1</pre>

'''Once Upon A Time Scale'''
<pre>10/9
9/8
5/4
9/7
21/16
35/24
3/2
5/3
12/7
7/4
35/18
2/1</pre>

'''Centaur'''
<pre>21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''Pental Diachrome'''
<pre>10/9
9/8
5/4
81/64
4/3
40/27
3/2
5/3
27/16
15/8
160/81
2/1</pre>

'''Margo Scale'''
<pre>104/99
9/8
13/11
14/11
4/3
63/44
3/2
52/33
56/33
16/9
21/11
2/1</pre>

'''Meta-Slendro'''
<pre>50/49
8/7
57/49
64/49
65/49
72/49
74/49
151/98
12/7
86/49
96/49
2/1</pre>

'''Succulent Undecimal Scale'''
<pre>10/9
7/6
5/4
21/16
4/3
11/8
3/2
5/3
7/4
16/9
11/6
2/1</pre>

== 14-note scales ==
'''No-13s ultrajust'''
17/16
9/8
19/16
5/4
21/16
11/8
23/16
3/2
19/12
5/3
7/4
11/6
23/12
2/1

== 17-note scales ==
'''Walkie'''
65/64
13/12
9/8
585/512
39/32
5/4
4/3
65/48
45/32
3/2
195/128
13/8
5/3
16/9
117/64
15/8
2/1

'''Mohahadene'''
55/54
88/81
9/8
55/48
11/9
5/4
4/3
11/8
45/32
3/2
55/36
44/27
5/3
55/32
11/6
15/8
2/1

'''Dichotidene'''
28/27
35/32
9/8
7/6
5/4
35/27
4/3
45/32
35/24
3/2
14/9
105/64
27/16
7/4
15/8
35/18
2/1

'''Just Spoogalaxy'''
28/27
29/27
9/8
7/6
29/24
81/64
21/16
87/64
116/81
3/2
14/9
29/18
27/16
7/4
29/16
243/128
2/1

'''Cartwheel'''
28/27
13/12
9/8
7/6
11/9
5/4
4/3
11/8
13/9
3/2
14/9
13/8
5/3
7/4
11/6
15/8
2/1

'''Ringer 17-quad'''
33/32
17/16
9/8
19/16
77/64
5/4
21/16
11/8
23/16
3/2
49/32
25/16
13/8
7/4
29/16
15/8
2/1

'''Ringer 17-tri'''
49/48
17/16
9/8
19/16
29/24
5/4
21/16
11/8
23/16
3/2
74/48
25/16
13/8
7/4
11/6
15/8
2/1

'''Schismadene'''
135/128
10/9
9/8
32/27
5/4
81/64
4/3
45/32
40/27
3/2
128/81
5/3
27/16
16/9
15/8
160/81
2/1

'''Traditional Baglama Fret Scale'''
18/17
12/11
9/8
81/68
27/22
81/64
4/3
24/17
16/11
3/2
27/17
18/11
27/16
16/9
32/17
64/33
2/1

'''23-limit Pseudoringer'''
33/32
13/12
9/8
7/6
39/32
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1

'''Zothogothic'''
28/27
13/12
9/8
7/6
39/32
91/72
4/3
112/81
13/9
3/2
14/9
13/8
91/54
7/4
117/64
91/48
2/1

'''Zolugothic'''
28/27
12/11
9/8
7/6
27/22
14/11
4/3
112/81
16/11
3/2
14/9
18/11
56/33
7/4
81/44
21/11
2/1

'''Persian Johnston'''
33/32
17/16
9/8
19/16
5/4
21/16
4/3
11/8
23/16
3/2
13/8
5/3
7/4
57/32
15/8
63/32
2/1

'''Succulent Novemdecimal scale'''
13/12
10/9
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
19/12
5/3
7/4
16/9
11/6
17/9
2/1

'''Aberrismic Neopersian'''
64/63
13/12
9/8
8/7
39/32
26/21
4/3
256/189
13/9
3/2
32/21
13/8
104/63
12/7
117/64
13/7
2/1

'''Latwethagothic'''
33/32
69/64
9/8
32/27
11/9
23/18
4/3
11/8
23/16
3/2
99/64
207/128
27/16
16/9
11/6
23/12
2/1

'''Zalatwethagothic'''
33/32
69/64
9/8
7/6
11/9
23/18
21/16
11/8
23/16
3/2
14/9
44/27
46/27
7/4
11/6
23/12
2/1

'''Chariot'''
19/18
13/12
10/9
7/6
11/9
5/4
4/3
11/8
17/12
3/2
19/12
13/8
5/3
7/4
11/6
23/12
2/1

== 19-note scales ==
'''Not-quite-ringer 19p'''
33/32
17/16
9/8
7/6
19/16
5/4
21/16
4/3
11/8
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1

'''Quasi-hemipyth detemper'''
28/27
13/12
9/8
7/6
32/27
11/9
21/16
4/3
11/8
13/9
3/2
14/9
13/8
27/16
7/4
16/9
11/6
63/32
2/1

'''19-limit dual-harmonic-segment-based'''
25/24
13/12
9/8
7/6
19/16
5/4
21/16
4/3
11/8
17/12
3/2
25/16
13/8
27/16
7/4
57/32
15/8
63/32
2/1

'''Pseudo-fives scale'''
33/32
77/72
9/8
7/6
77/64
11/9
21/16
4/3
11/8
77/54
3/2
14/9
77/48
44/27
7/4
16/9
11/6
154/81
2/1

'''Chipped Tetromino'''
21/20
35/32
9/8
7/6
6/5
5/4
21/16
4/3
7/5
35/24
3/2
63/40
8/5
5/3
7/4
9/5
15/8
63/32
2/1

'''Not-quite-ringer 19egh'''
17/16
35/32
9/8
19/16
39/32
5/4
21/16
11/8
45/32
23/16
3/2
25/16
13/8
27/16
7/4
29/16
15/8
31/16
2/1

== 20-note scales ==
'''Yatha Double Blackdye'''
65/64
13/12
10/9
9/8
65/54
39/32
5/4
4/3
65/48
13/9
40/27
3/2
130/81
13/8
5/3
27/16
65/36
117/64
15/8
2/1

'''Yala Double Blackdye'''
55/54
33/32
10/9
9/8
55/48
11/9
5/4
4/3
110/81
11/8
40/27
3/2
55/36
99/64
5/3
27/16
55/32
11/6
15/8
2/1

== 22-note scales ==
'''Yala Dart'''
33/32
88/81
10/9
9/8
32/27
11/9
5/4
81/64
4/3
11/8
45/32
40/27
3/2
99/64
44/27
5/3
27/16
16/9
11/6
15/8
160/81
2/1

'''Close-to-even Scale'''
33/32
17/16
35/32
9/8
7/6
29/24
5/4
31/24
4/3
11/8
17/12
35/24
3/2
14/9
29/18
5/3
31/18
16/9
11/6
17/9
35/18
2/1

'''cheen detempering'''
1053/1024
13/12
10/9
9/8
32/27
39/32
5/4
81/64
4/3
351/256
45/32
40/27
3/2
3159/2048
13/8
5/3
27/16
16/9
117/64
15/8
160/81
2/1

'''Compact 11-limit detemper'''
33/32
77/72
35/32
9/8
7/6
77/64
5/4
165/128
4/3
11/8
45/32
35/24
3/2
14/9
77/48
5/3
55/32
7/4
11/6
15/8
35/18
2/1

== 24-note scales ==
'''Car'''
33/32
17/16
13/12
9/8
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
57/32
11/6
15/8
23/12
2/1

'''Sedan'''
33/32
17/16
13/12
10/9
7/6
19/16
39/32
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
11/6
15/8
23/12
2/1

'''Modstraw'''
28/27
35/32
10/9
9/8
7/6
32/27
5/4
35/27
21/16
4/3
45/32
35/24
40/27
3/2
14/9
105/64
5/3
27/16
7/4
16/9
15/8
35/18
63/32
2/1

'''17-limit scale'''
33/32
17/16
13/12
10/9
7/6
32/27
11/9
5/4
21/16
4/3
11/8
17/12
13/9
3/2
14/9
51/32
13/8
5/3
7/4
16/9
11/6
15/8
52/27
2/1

'''31-limit scale'''
33/32
17/16
13/12
9/8
7/6
19/16
29/24
5/4
21/16
4/3
11/8
17/12
23/16
3/2
14/9
19/12
13/8
5/3
7/4
16/9
29/16
15/8
31/16
2/1

'''No-fives undecimal scale'''
33/32
77/72
88/81
9/8
7/6
77/64
11/9
81/64
21/16
4/3
11/8
77/54
352/243
3/2
14/9
77/48
44/27
27/16
7/4
16/9
11/6
154/81
63/32
2/1

'''37-limit scale'''
31/30
16/15
11/10
17/15
7/6
6/5
37/30
19/15
13/10
4/3
11/8
17/12
35/24
3/2
31/20
8/5
33/20
17/10
7/4
9/5
37/20
19/10
39/20
2/1

'''Hexquad'''
28/27
77/72
35/32
9/8
7/6
77/64
11/9
5/4
35/27
4/3
11/8
45/32
35/24
3/2
14/9
77/48
44/27
5/3
7/4
231/128
11/6
15/8
35/18
2/1

'''bus-like 11-limit detemper'''
33/32
77/72
35/32
9/8
7/6
77/64
11/9
5/4
21/16
4/3
11/8
45/32
35/24
3/2
14/9
77/48
105/64
5/3
7/4
16/9
11/6
15/8
35/18
2/1

== 26-note scales ==
'''Mothra Detemper'''
135/128
15/14
35/32
9/8
8/7
7/6
315/256
5/4
9/7
21/16
4/3
45/32
10/7
35/24
3/2
32/21
45/28
105/64
5/3
12/7
7/4
945/512
15/8
40/21
63/32
2/1

== 29-note scales ==
'''andromeda[29] detemper'''
33/32
135/128
13/12
10/9
9/8
7/6
32/27
11/9
5/4
81/64
21/16
4/3
11/8
45/32
13/9
189/128
3/2
14/9
405/256
13/8
5/3
27/16
7/4
16/9
11/6
15/8
52/27
63/32
2/1

== 31-note scales ==
'''Otonal Sevenice'''
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
128/105
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1

'''Utonal Sevenice'''
36/35
21/20
15/14
35/32
9/8
8/7
7/6
6/5
315/256
5/4
9/7
21/16
4/3
48/35
7/5
10/7
35/24
3/2
32/21
63/40
8/5
105/64
5/3
12/7
7/4
9/5
64/35
15/8
40/21
63/32
2/1

'''Ringer 31p'''
39/38
20/19
41/38
21/19
43/38
22/19
45/38
23/19
47/38
24/19
49/38
25/19
51/38
26/19
27/19
55/38
28/19
3/2
29/19
30/19
61/38
31/19
32/19
33/19
67/38
34/19
35/19
36/19
37/19
75/38
2/1

'''Symmetrizine'''
64/63
28/27
16/15
12/11
9/8
8/7
7/6
6/5
11/9
5/4
9/7
21/16
4/3
11/8
45/32
64/45
16/11
3/2
32/21
14/9
8/5
18/11
5/3
12/7
7/4
16/9
11/6
15/8
27/14
63/32
2/1

== Large scales (greater than 31 notes) ==
'''41-tone 5-limit JI'''
81/80
25/24
135/128
16/15
27/25
10/9
9/8
256/225
75/64
32/27
6/5
100/81
5/4
81/64
32/25
320/243
4/3
27/20
25/18
45/32
64/45
36/25
40/27
3/2
243/160
25/16
128/81
8/5
81/50
5/3
27/16
128/75
225/128
16/9
9/5
50/27
15/8
256/135
48/25
160/81
2/1

'''46-tone septimal JI'''
64/63
36/35
21/20
16/15
15/14
35/32
10/9
9/8
8/7
7/6
189/160
6/5
128/105
315/256
5/4
80/63
9/7
21/16
4/3
27/20
48/35
7/5
45/32
10/7
35/24
189/128
3/2
32/21
14/9
63/40
8/5
45/28
105/64
5/3
27/16
12/7
7/4
16/9
9/5
64/35
28/15
15/8
40/21
27/14
63/32
2/1

'''53-tone Semantic Scale'''
81/80
128/125
25/24
135/128
16/15
27/25
800/729
10/9
9/8
256/225
144/125
75/64
32/27
6/5
243/200
100/81
5/4
81/64
32/25
125/96
320/243
4/3
27/20
512/375
25/18
45/32
64/45
36/25
375/256
40/27
3/2
243/160
192/125
25/16
128/81
8/5
81/50
400/243
5/3
27/16
128/75
125/72
225/128
16/9
9/5
729/400
50/27
15/8
256/135
48/25
125/64
160/81
2/1

'''130-tone 13-limit JI'''
225/224
105/104
65/64
45/44
36/35
33/32
28/27
25/24
21/20
135/128
35/33
16/15
15/14
14/13
13/12
12/11
35/32
11/10
72/65
10/9
28/25
9/8
112/99
25/22
8/7
55/48
15/13
65/56
7/6
168/143
13/11
32/27
25/21
6/5
135/112
40/33
128/105
11/9
16/13
26/21
56/45
5/4
63/50
81/64
14/11
32/25
9/7
84/65
13/10
72/55
21/16
33/25
143/108
4/3
75/56
27/20
65/48
15/11
48/35
11/8
18/13
25/18
7/5
45/32
99/70
64/45
10/7
36/25
13/9
16/11
35/24
22/15
96/65
40/27
112/75
3/2
448/297
50/33
32/21
55/36
20/13
65/42
14/9
25/16
11/7
128/81
100/63
8/5
45/28
21/13
13/8
18/11
105/64
33/20
224/135
5/3
42/25
27/16
22/13
143/84
12/7
112/65
26/15
96/55
7/4
44/25
99/56
16/9
25/14
9/5
65/36
20/11
64/35
11/6
24/13
13/7
28/15
15/8
66/35
256/135
40/21
48/25
27/14
64/33
35/18
49/25
63/32
99/50
143/72
2/1

Xenharmonic Reference:Color schemes

2026-01-02T04:29:45Z

Tristanbay: /* Color scheme E */ Removed E.b2, added E.b1 cent-value palette

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b1 ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color scheme ''E.b1'' is similar to that of the original color scheme E. In this scheme, the base colors are all full-saturation except for the first and last ones, and all colors for primes up to 31 were picked manually. It also has a color palette for all 1200 cents in all 3 color modes.

[[File:E_b1.png|thumb|400x300px|Color Scheme E.b1 cent-value palette]]

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|990031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#222" |Error in steps
|-
! style="background-color:#222" |Error range
| style="background-color:#444" |0
| style="background-color:#444" |(0, 1/24]
| style="background-color:#444" |(1/24, 1/12]
| style="background-color:#444" |(1/12, 1/8]
| style="background-color:#444" |(1/8, 1/6]
| style="background-color:#444" |(1/6, 1/4]
| style="background-color:#444" |(1/4, 1/3]
| style="background-color:#444" |(1/3, 5/12]
| style="background-color:#444" |>5/12
|-
! style="background-color:#222" |Dark mode
| style="background-color:#664488" |664488
| style="background-color:#336565" |336565
| style="background-color:#337550" |337550
| style="background-color:#397733" |397733
| style="background-color:#547733" |547733
| style="background-color:#6d7733" |6D7733
| style="background-color:#776733" |776733
| style="background-color:#774c33" |774C33
| style="background-color:#773333" |773333
|-
! style="background-color:#222" |Light mode
| style="background-color:#bd80eb;color:black" |BD80EB
| style="background-color:#6ed7df;color:black" |6ED7DF
| style="background-color:#6ce3a9;color:black" |6CE3A9
| style="background-color:#75e675;color:black" |75E675
| style="background-color:#abe56f;color:black" |ABE56F
| style="background-color:#cfe36d;color:black" |CFE36D
| style="background-color:#e5cd72;color:black" |E5CD72
| style="background-color:#ecaa7a;color:black" |ECAA7A
| style="background-color:#f18385;color:black" |F18385
|}

File:E b1.png

2026-01-02T04:15:44Z

Tristanbay: Gradient of color scheme E.b1; Light mode colors on top, base colors in the middle, and dark mode colors on the bottom. The x-coordinate of the pixel corresponds to the number of cents in the represented interval, octave-reduced and rounded down to the next whole cent.

== Summary ==
Gradient of color scheme E.b1; Light mode colors on top, base colors in the middle, and dark mode colors on the bottom. The x-coordinate of the pixel corresponds to the number of cents in the represented interval, octave-reduced and rounded down to the next whole cent.

User:Tristanbay

2026-01-01T05:38:30Z

Tristanbay: Formatting

Hi, I'm Tristan Bay, a microtonal electronic musician living in Portland, Oregon, USA. My stuff can be found at https://tristanbay.com.

== JI scale dump ==
Here are some just intonation scales I've been collecting. All of them are constant structure (i.e. the same just interval is always represented by the same number of scale degrees) and repeat at the octave.

=== Boutique scales (less than 12 notes) ===
'''24:26:28:30:32:33:36:39:42:45:48'''
<pre>13/12
7/6
5/4
4/3
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''24:27:28:30:33:36:39:42:45:48'''
<pre>9/8
7/6
5/4
11/8
3/2
13/8
7/4
15/8
2/1</pre>

'''60:63:64:70:80:84:96:105:120'''
<pre>35/32
5/4
21/16
4/3
35/24
5/3
7/4
2/1</pre>

'''56:60:64:70:84:90:96:105:112'''
<pre>35/32
7/6
5/4
4/3
35/24
7/4
15/8
2/1</pre>

'''18:20:22:24:27:30:33:36'''
<pre>10/9
11/9
4/3
3/2
5/3
11/6
2/1</pre>

'''14:16:18:20:21:24:27:28'''
<pre>8/7
9/7
10/7
3/2
12/7
27/14
2/1</pre>

'''27:28:33:36:39:42:45:54'''
<pre>28/27
11/9
4/3
13/9
14/9
5/3
2/1</pre>

'''7:8:9:10:11:12:13:14'''
<pre>8/7
9/7
10/7
11/7
12/7
13/7
2/1</pre>

'''56:60:64:70:72:80:84:90:96:105:112'''
<pre>15/14
8/7
5/4
9/7
10/7
3/2
45/28
12/7
15/8
2/1</pre>

'''52:58:64:71:78:86:95:104'''
<pre>71/64
39/32
43/32
95/64
13/8
29/16
2/1</pre>

'''48:52:58:64:72:78:87:96'''
<pre>13/12
29/24
4/3
3/2
13/8
29/16
2/1</pre>

'''33:36:40:42:44:48:54:60:63:66'''
<pre>12/11
40/33
14/11
4/3
16/11
18/11
20/11
21/11
2/1</pre>

'''720:780:840:910:945:1008:1092:1170:1260:1365:1440'''
<pre>13/12
7/6
91/72
21/16
7/5
91/60
13/8
7/4
91/48
2/1</pre>

=== 12-note scales ===
'''Antitonic 17-limit'''
<pre>17/16
39/32
5/4
21/16
11/8
3/2
51/32
13/8
5/3
7/4
11/6
2/1</pre>

'''Bicycle'''
<pre>13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
11/6
2/1</pre>

'''Duodene'''
<pre>16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1</pre>

'''Septimal Canright'''
<pre>28/27
9/8
7/6
5/4
4/3
45/32
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''North Dakota'''
<pre>33/32
10/9
7/6
5/4
4/3
11/8
3/2
14/9
5/3
7/4
11/6
2/1</pre>

'''Johnston Piano Scale (Ringer 12)'''
<pre>15/14
8/7
17/14
9/7
19/14
10/7
3/2
11/7
12/7
13/7
27/14
2/1</pre>

'''Ephemeral Soup Scale'''
<pre>77/72
9/8
77/64
21/16
693/512
189/128
3/2
77/48
7/4
231/128
63/32
2/1</pre>

'''Once Upon A Time Scale'''
<pre>10/9
9/8
5/4
9/7
21/16
35/24
3/2
5/3
12/7
7/4
35/18
2/1</pre>

'''Centaur'''
<pre>21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1</pre>

'''Pental Diachrome'''
<pre>10/9
9/8
5/4
81/64
4/3
40/27
3/2
5/3
27/16
15/8
160/81
2/1</pre>

'''Margo Scale'''
<pre>104/99
9/8
13/11
14/11
4/3
63/44
3/2
52/33
56/33
16/9
21/11
2/1</pre>

'''Meta-Slendro'''
<pre>50/49
8/7
57/49
64/49
65/49
72/49
74/49
151/98
12/7
86/49
96/49
2/1</pre>

'''Succulent Undecimal Scale'''
<pre>10/9
7/6
5/4
21/16
4/3
11/8
3/2
5/3
7/4
16/9
11/6
2/1</pre>

=== 14-note scales ===
=== 17-note scales ===
=== 19-note scales ===
=== 20-note scales ===
=== 22-note scales ===
=== 24-note scales ===
=== 26-note scales ===
=== 29-note scales ===
=== Large scales (greater than 29 notes) ===

User:Tristanbay

2026-01-01T05:34:31Z

Tristanbay: Created my user page

Hi, I'm Tristan Bay, a microtonal electronic musician living in Portland, Oregon, USA. My stuff can be found at https://tristanbay.com.

== JI scale dump ==
Here are some just intonation scales I've been collecting. All of them are constant structure (i.e. the same just interval is always represented by the same number of scale degrees) and repeat at the octave.

=== Boutique scales (less than 12 notes) ===
'''24:26:28:30:32:33:36:39:42:45:48'''
13/12
7/6
5/4
4/3
11/8
3/2
13/8
7/4
15/8
2/1

'''24:27:28:30:33:36:39:42:45:48'''
9/8
7/6
5/4
11/8
3/2
13/8
7/4
15/8
2/1

'''60:63:64:70:80:84:96:105:120'''
35/32
5/4
21/16
4/3
35/24
5/3
7/4
2/1

'''56:60:64:70:84:90:96:105:112'''
35/32
7/6
5/4
4/3
35/24
7/4
15/8
2/1

'''18:20:22:24:27:30:33:36'''
10/9
11/9
4/3
3/2
5/3
11/6
2/1

'''14:16:18:20:21:24:27:28'''
8/7
9/7
10/7
3/2
12/7
27/14
2/1

'''27:28:33:36:39:42:45:54'''
28/27
11/9
4/3
13/9
14/9
5/3
2/1

'''7:8:9:10:11:12:13:14'''
8/7
9/7
10/7
11/7
12/7
13/7
2/1

'''56:60:64:70:72:80:84:90:96:105:112'''
15/14
8/7
5/4
9/7
10/7
3/2
45/28
12/7
15/8
2/1

'''52:58:64:71:78:86:95:104'''
71/64
39/32
43/32
95/64
13/8
29/16
2/1

'''48:52:58:64:72:78:87:96'''
13/12
29/24
4/3
3/2
13/8
29/16
2/1

'''33:36:40:42:44:48:54:60:63:66'''
12/11
40/33
14/11
4/3
16/11
18/11
20/11
21/11
2/1

'''720:780:840:910:945:1008:1092:1170:1260:1365:1440'''
13/12
7/6
91/72
21/16
7/5
91/60
13/8
7/4
91/48
2/1

=== 12-note scales ===
'''Antitonic 17-limit'''
17/16
39/32
5/4
21/16
11/8
3/2
51/32
13/8
5/3
7/4
11/6
2/1

'''Bicycle'''
13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
11/6
2/1

'''Duodene'''
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
9/5
15/8
2/1

'''Septimal Canright'''
28/27
9/8
7/6
5/4
4/3
45/32
3/2
14/9
5/3
7/4
15/8
2/1

'''North Dakota'''
33/32
10/9
7/6
5/4
4/3
11/8
3/2
14/9
5/3
7/4
11/6
2/1

'''Johnston Piano Scale (Ringer 12)'''
15/14
8/7
17/14
9/7
19/14
10/7
3/2
11/7
12/7
13/7
27/14
2/1

'''Ephemeral Soup Scale'''
77/72
9/8
77/64
21/16
693/512
189/128
3/2
77/48
7/4
231/128
63/32
2/1

'''Once Upon A Time Scale'''
10/9
9/8
5/4
9/7
21/16
35/24
3/2
5/3
12/7
7/4
35/18
2/1

'''Centaur'''
21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1

'''Pental Diachrome'''
10/9
9/8
5/4
81/64
4/3
40/27
3/2
5/3
27/16
15/8
160/81
2/1

'''Margo Scale'''
104/99
9/8
13/11
14/11
4/3
63/44
3/2
52/33
56/33
16/9
21/11
2/1

'''Meta-Slendro'''
50/49
8/7
57/49
64/49
65/49
72/49
74/49
151/98
12/7
86/49
96/49
2/1

'''Succulent Undecimal Scale'''
10/9
7/6
5/4
21/16
4/3
11/8
3/2
5/3
7/4
16/9
11/6
2/1

=== 14-note scales ===
=== 17-note scales ===
=== 19-note scales ===
=== 20-note scales ===
=== 22-note scales ===
=== 24-note scales ===
=== 26-note scales ===
=== 29-note scales ===
=== Large scales (greater than 29 notes) ===

72edo

2026-01-01T05:14:02Z

Tristanbay: More link text formatting

'''72edo''', or 72 equal divisions of the octave, is an equal tuning system with a step size of exactly ⅙ of a semitone, or 16⅔ cents. It is a remarkably accurate model of 11-limit just intonation.

== Theory ==
72edo is a superset of [[12edo]], sharing the same perfect fifth, major second, and other 3-limit intervals, as well as the same approximations of prime harmonics 17 and 19. It also inherits [[24edo]]'s prime 11 and [[36edo]]'s primes 7 and 13. However, its approximation of prime harmonic 5 is unique compared to all [[EDO|edo]]<nowiki/>s lower than it.
{{Harmonics in ED|72|31}}

Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three ~[[8/7]]<nowiki/>s, leading to 36edo's Slendric scales. Combining these divisions yields Miracle scales, dividing the fifth into six ~[[16/15]]<nowiki/>s.

72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a 2.3.7-subgroup (septal) interval. This makes extending [[Diatonic notation|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.

=== Octave stretch ===
72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

== Use in software ==
72edo can be achieved (at least to the nearest cent or so) with most 12edo software instruments by using six instances of it that are all detuned ⅙ of a semitone from each other. In some DAWs, all six instances can be controlled from a single MIDI track on the same piano roll view by routing one MIDI channel to each instance.

72edo

2026-01-01T05:13:10Z

Tristanbay: Added ~s and reformatted text in and around links

'''72edo''', or 72 equal divisions of the octave, is an equal tuning system with a step size of exactly ⅙ of a semitone, or 16⅔ cents. It is a remarkably accurate model of 11-limit just intonation.

== Theory ==
72edo is a superset of [[12edo]], sharing the same perfect fifth, major second, and other 3-limit intervals, as well as the same approximations of prime harmonics 17 and 19. It also inherits [[24edo]]'s prime 11 and [[36edo]]'s primes 7 and 13. However, its approximation of prime harmonic 5 is unique compared to all [[EDO|edo]]s lower than it.
{{Harmonics in ED|72|31}}

Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three ~[[8/7]]<nowiki/>s, leading to 36edo's Slendric scales. Combining these divisions yields Miracle scales, dividing the fifth into six ~[[16/15]]<nowiki/>s.

72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a 2.3.7-subgroup (septal) interval. This makes extending [[Diatonic notation|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.

=== Octave stretch ===
72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

== Use in software ==
72edo can be achieved (at least to the nearest cent or so) with most 12edo software instruments by using six instances of it that are all detuned ⅙ of a semitone from each other. In some DAWs, all six instances can be controlled from a single MIDI track on the same piano roll view by routing one MIDI channel to each instance.

72edo

2026-01-01T05:11:21Z

Tristanbay: /* Theory */ Removed extra newline

'''72edo''', or 72 equal divisions of the octave, is an equal tuning system with a step size of exactly ⅙ of a semitone, or 16⅔ cents. It is a remarkably accurate model of 11-limit just intonation.

== Theory ==
72edo is a superset of [[12edo]], sharing the same perfect fifth, major second, and other 3-limit intervals, as well as the same approximations of prime harmonics 17 and 19. It also inherits [[24edo]]'s prime 11 and [[36edo]]'s primes 7 and 13. However, its approximation of prime harmonic 5 is unique compared to all [[EDO|edo]]s lower than it.
{{Harmonics in ED|72|31}}

Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three [[8/7]]s, leading to 36edo's Slendric scales. Combining these divisions yields Miracle scales, dividing the fifth into six [[16/15]]s.

72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a 2.3.7-subgroup (septal) interval. This makes extending [[Diatonic notation|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.

=== Octave stretch ===
72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

== Use in software ==
72edo can be achieved (at least to the nearest cent or so) with most 12edo software instruments by using six instances of it that are all detuned ⅙ of a semitone from each other. In some DAWs, all six instances can be controlled from a single MIDI track on the same piano roll view by routing one MIDI channel to each instance.

72edo

2026-01-01T05:10:13Z

Tristanbay: Added harmonics table

'''72edo''', or 72 equal divisions of the octave, is an equal tuning system with a step size of exactly ⅙ of a semitone, or 16⅔ cents. It is a remarkably accurate model of 11-limit just intonation.

== Theory ==
72edo is a superset of [[12edo]], sharing the same perfect fifth, major second, and other 3-limit intervals, as well as the same approximations of prime harmonics 17 and 19. It also inherits [[24edo]]'s prime 11 and [[36edo]]'s primes 7 and 13. However, its approximation of prime harmonic 5 is unique compared to all [[EDO|edo]]s lower than it.

{{Harmonics in ED|72|31}}

Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three [[8/7]]s, leading to 36edo's Slendric scales. Combining these divisions yields Miracle scales, dividing the fifth into six [[16/15]]s.

72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a 2.3.7-subgroup (septal) interval. This makes extending [[Diatonic notation|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.

=== Octave stretch ===
72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

== Use in software ==
72edo can be achieved (at least to the nearest cent or so) with most 12edo software instruments by using six instances of it that are all detuned ⅙ of a semitone from each other. In some DAWs, all six instances can be controlled from a single MIDI track on the same piano roll view by routing one MIDI channel to each instance.

Ploidacot

2026-01-01T05:07:45Z

Tristanbay: Created page for ploidacot

Ploidacot is a naming scheme for rank-2 tuning/temperament structures based on the number of periods per octave and the number of periods and independent generators per fifth, created by [[Praveen Venkataramana]].

The general form is '''x'''-ploid '''y'''-sheared '''z'''-cot, where '''x''' is the number of periods in the octave, and where '''y''' periods ''down'' and '''z''' generators ''up'' are needed to reach ~3/2. '''x''' and '''z''' typically expressed as numerical prefixes (e.g. mono, di, tri, tetra) instead of regular numbers, and '''y'''-sheared is usually replaced with the number '''y''' in Greek gematria (e.g. 1 = alpha, 2 = beta, 3 = gamma, 10 = iota). A temperament with a ~3/2 of an exact number of periods is ''acot'' or ''0-cot''. In this situation, the shear amount tends to be left out of the ploidacot name, but it can be seen as -1 times however many periods there are in the fifth.

== Examples ==
Blackwood has a ⅕-octave period with the perfect fifth at exactly 3\5, making it a ''pentaploid (-3-sheared) acot'' temperament.

Meantone and Schismic have a full-octave period and a perfect fifth generator, making them ''(monoploid) monocot'' temperaments.

Diaschismic can be interpreted as having a half-octave period and a perfect fifth generator, so it is a ''diploid monocot'' temperament.

Harry has a half-octave period and splits ~6/1 into 6 equal generator steps, so ~3/2 can be reached by going 4 periods down from there. This makes it a ''diploid delta-hexacot'' temperament.

== Relationship to pergens ==
[[Pergen]]s are an alternative naming system to ploidacot which notates 3-limit intervals in terms of diatonic scale degrees, created by [[Kite Giedraitis]]. While ploidacot describes ''how'' to get to ~3/2, pergens describe what 3-limit interval is divided into how many parts to get the generator. To use an above example, Harry, which is a ''diploid delta-hexacot'' temperament, would have the pergen ''(P8/2, P19/6)'', which simplifies to ''(P8/2, P4/6)''. Note the use of the perfect fourth. This does not appear to be in the "official" ploidacot standard, although [[Osmium]] has proposed the use of ''omega'' to refer to -1-shearing, corresponding to ~4/3 for single-period-octave (monoploid) temperaments.

72edo

2025-12-31T21:08:34Z

Tristanbay: Created page fro 72edo

'''72edo''', or 72 equal divisions of the octave, is an equal tuning system with a step size of exactly ⅙ of a semitone, or 16⅔ cents. It is a remarkably accurate model of 11-limit just intonation.

== Theory ==
72edo is a superset of [[12edo]], sharing the same perfect fifth, major second, and other 3-limit intervals, as well as the same approximations of prime harmonics 17 and 19. It also inherits [[24edo]]'s prime 11 and [[36edo]]'s primes 7 and 13. However, its approximation of prime harmonic 5 is unique compared to all [[EDO|edo]]s lower than it.

Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three [[8/7]]s, leading to 36edo's slendric scales. Combining these divisions yields miracle scales, dividing the fifth into six [[16/15]]s.

72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a 2.3.7-subgroup (septal) interval. This makes extending [[Diatonic notation|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.

=== Octave stretch ==
72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

== Use in software ==
72edo can be achieved (at least to the nearest cent or so) with most 12edo software instruments by using six instances of it that are all detuned ⅙ of a semitone from each other. In some DAWs, all six instances can be controlled from a single MIDI track on the same piano roll view by routing one MIDI channel to each instance.

31edo

2025-12-27T23:41:37Z

Tristanbay: 41edo is a better example of miracle imo

'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 cents. Aside from [[12edo]], it is a popular tuning of [[Meantone]] and has accurate approximations of harmonics 5 and 7.

== Theory ==

==== JI approximation ====
31edo is best understood as a 2.3.5.7.11.23 system, although it has a sharp but functional prime 13. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval. It has a rather functional diatonic scale, with the whole tone split into 2 and 3, with 2 steps making a chromatic semitone (or "chromatone") and 3 steps making a diatonic semitone (or "diatone").

{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Subminor
|'''Pentaminor'''
|Neutral
|'''Pentamajor'''
|Supermajor
|-
!Cents
|271
|'''310'''
|348
|'''387'''
|426
|-
!Just interpretation
|7/6
|'''6/5'''
|11/9
|'''5/4'''
|9/7
|}
Diatonic thirds are bolded.

=== Chords ===
Along with its diatonic major and minor chords which approximate 5-limit harmony, 31edo also has a narrow but functional supermajor triad, and a well-tuned subminor triad. It also supports arto and tendo chords, with its slendric chords of [0 6 18] and [0 12 18], and has a neutral triad [0 9 18] which represents both artoneutral and tendoneutral triads in the 11- and 13-limit.

=== Scales ===
31edo does not temper out 64/63, meaning that it can be used to tune [[Diasem]] while representing some simpler 5-limit intervals. 31edo's step is called a [[diesis]], and can function as an [[aberrisma]]. Due to being a prime number, 31edo has a large number of full-period MOS scales that exist in the edo. Orwell[9] ([[gramitonic]]) is one example, so is Mohajira[7] ([[mosh]]).

31edo also has a usable 12-note chromatic scale, approximating [[Golden sequences and tuning|golden]] Meantone/monocot.

=== Regular temperaments ===
Besides [[Meantone]] (for which it provides an excellent tuning and which is shared with 19edo), 31edo also supports variations of Rastmic temperament (like 24edo), Slendric (like 36edo), Miracle (like 41edo), and Orwell (like 22edo).

== Notation ==
31edo, as one of the more popular edos, has a somewhat agreed-upon notation system. This notation is simply neutral [[diatonic notation]] applied to the edo, where a half-# or half-b represents an alteration by one diesis. In this manner, all notes can be spelled in a way that does not require multiple sharps or flats.

31edo

2025-12-25T23:44:18Z

Tristanbay: Fixed edostep cent value

'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 cents. Aside from [[12edo]], it is a popular tuning of [[meantone]] and has accurate approximations of harmonics 5 and 7.

== Theory ==

==== JI approximation ====
31edo has a somewhat flat prime 3, a slightly sharp prime 5, a slightly flat prime 7, and a flat prime 11. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval.

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

=== Scales ===
31edo does not also temper out 64/63, meaning that it can be used to tune [[Diasem]] while representing some simpler 5-limit intervals.

=== Regular temperaments ===
Besides meantone, 31edo also supports variations of neutral temperament, slendric, miracle, and orwell.

31edo

2025-12-25T23:43:18Z

Tristanbay: Began page for 31edo

'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. Aside from [[12edo]], it is a popular tuning of [[meantone]] and has accurate approximations of harmonics 5 and 7.

== Theory ==

==== JI approximation ====
31edo has a somewhat flat prime 3, a slightly sharp prime 5, a slightly flat prime 7, and a flat prime 11. The flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral interval.

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

=== Scales ===
31edo does not also temper out 64/63, meaning that it can be used to tune [[Diasem]] while representing some simpler 5-limit intervals.

=== Regular temperaments ===
Besides meantone, 31edo also supports variations of neutral temperament, slendric, miracle, and orwell.

Xenharmonic Reference:Color schemes

2025-12-18T22:29:13Z

Tristanbay: /* Color scheme E.b */ Added E.b2 primes 2-31 + highPrime colors

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b1 ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color schemes ''E.b'' are similar to that of color scheme E. In E.b1, the base colors are all full-saturation except for the first and last ones, and all colors were picked manually.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|990031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

=== Color scheme E.b2 ===

E.b2 is the successor to E.b1. E.b2 ditches E.b1's full-saturation base colors for the property of being within the sRGB gamut for the 31-limit primes. Furthermore, every 1-cent increment in this scheme has been assigned a color by interpolating linearly (or roughly linearly) through Oklch color space.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#676767"|676767
|style="background-color:#e6e6e6;color:black"|E6E6E6
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#83152e"|83152E
|style="background-color:#f81d58"|F81D58
|style="background-color:#fd8994;color:black"|FD8994
|-
|5
|style="background-color:#597f05"|597F05
|style="background-color:#b7fa39;color:black"|B7FA39
|style="background-color:#9ec860;color:black"|9EC860
|-
|7
|style="background-color:#4b069c"|4B069C
|style="background-color:#853afe"|853AFE
|style="background-color:#aa90f9;color:black"|AA90F9
|-
|11
|style="background-color:#836411"|836411
|style="background-color:#f9c02e;color:black"|F9C02E
|style="background-color:#dbb155;color:black"|DBB155
|-
|13
|style="background-color:#6d088a"|6D088A
|style="background-color:#c51ff7"|C51FF7
|style="background-color:#d87efb;color:black"|D87EFB
|-
|17
|style="background-color:#127177"|127177
|style="background-color:#05d7e2;color:black"|05D7E2
|style="background-color:#71c4ca;color:black"|71C4CA
|-
|19
|style="background-color:#058541"|058541
|style="background-color:#31fb87;color:black"|31FB87
|style="background-color:#69cd86;color:black"|69CD86
|-
|23
|style="background-color:#983d15"|983D15
|style="background-color:#ff7236"|FF7236
|style="background-color:#f89d79;color:black"|F89D79
|-
|29
|style="background-color:#1311ba"|1311BA
|style="background-color:#375ffe"|375FFE
|style="background-color:#83a4f8;color:black"|83A4F8
|-
|31
|style="background-color:#085190"|085190
|style="background-color:#2d92f3"|2D92F3
|style="background-color:#73b3f8;color:black"|73B3F8
|-
|Higher primes
|style="background-color:#484848"|484848
|style="background-color:#838383"|838383
|style="background-color:#8f8f8f;color:black"|8f8f8f
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#222" |Error in steps
|-
! style="background-color:#222" |Error range
| style="background-color:#444" |0
| style="background-color:#444" |(0, 1/24]
| style="background-color:#444" |(1/24, 1/12]
| style="background-color:#444" |(1/12, 1/8]
| style="background-color:#444" |(1/8, 1/6]
| style="background-color:#444" |(1/6, 1/4]
| style="background-color:#444" |(1/4, 1/3]
| style="background-color:#444" |(1/3, 5/12]
| style="background-color:#444" |>5/12
|-
! style="background-color:#222" |Dark mode
| style="background-color:#664488" |664488
| style="background-color:#336565" |336565
| style="background-color:#337550" |337550
| style="background-color:#397733" |397733
| style="background-color:#547733" |547733
| style="background-color:#6d7733" |6D7733
| style="background-color:#776733" |776733
| style="background-color:#774c33" |774C33
| style="background-color:#773333" |773333
|-
! style="background-color:#222" |Light mode
| style="background-color:#bd80eb;color:black" |BD80EB
| style="background-color:#6ed7df;color:black" |6ED7DF
| style="background-color:#6ce3a9;color:black" |6CE3A9
| style="background-color:#75e675;color:black" |75E675
| style="background-color:#abe56f;color:black" |ABE56F
| style="background-color:#cfe36d;color:black" |CFE36D
| style="background-color:#e5cd72;color:black" |E5CD72
| style="background-color:#ecaa7a;color:black" |ECAA7A
| style="background-color:#f18385;color:black" |F18385
|}

Xenharmonic Reference:Color schemes

2025-12-18T01:12:42Z

Tristanbay: /* Color scheme E.b */ fixed typo

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color scheme is similar to that of color scheme E. However, the base colors are all full-saturation except for the first and last ones, and all colors were picked manually.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|990031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#222" |Error in steps
|-
! style="background-color:#222" |Error range
| style="background-color:#444" |0
| style="background-color:#444" |(0, 1/24]
| style="background-color:#444" |(1/24, 1/12]
| style="background-color:#444" |(1/12, 1/8]
| style="background-color:#444" |(1/8, 1/6]
| style="background-color:#444" |(1/6, 1/4]
| style="background-color:#444" |(1/4, 1/3]
| style="background-color:#444" |(1/3, 5/12]
| style="background-color:#444" |>5/12
|-
! style="background-color:#222" |Dark mode
| style="background-color:#664488" |664488
| style="background-color:#336565" |336565
| style="background-color:#337550" |337550
| style="background-color:#397733" |397733
| style="background-color:#547733" |547733
| style="background-color:#6d7733" |6D7733
| style="background-color:#776733" |776733
| style="background-color:#774c33" |774C33
| style="background-color:#773333" |773333
|-
! style="background-color:#222" |Light mode
| style="background-color:#bd80eb;color:black" |BD80EB
| style="background-color:#6ed7df;color:black" |6ED7DF
| style="background-color:#6ce3a9;color:black" |6CE3A9
| style="background-color:#75e675;color:black" |75E675
| style="background-color:#abe56f;color:black" |ABE56F
| style="background-color:#cfe36d;color:black" |CFE36D
| style="background-color:#e5cd72;color:black" |E5CD72
| style="background-color:#ecaa7a;color:black" |ECAA7A
| style="background-color:#f18385;color:black" |F18385
|}

Xenharmonic Reference:Color schemes

2025-12-18T00:35:18Z

Tristanbay: Made light mode colors brighter for my accuracy color proposal

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color scheme is similar to that of color scheme E. However, the base colors are all full-saturation except for the first and last ones, and all colors were picked manually.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|99031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#222" |Error in steps
|-
! style="background-color:#222" |Error range
| style="background-color:#444" |0
| style="background-color:#444" |(0, 1/24]
| style="background-color:#444" |(1/24, 1/12]
| style="background-color:#444" |(1/12, 1/8]
| style="background-color:#444" |(1/8, 1/6]
| style="background-color:#444" |(1/6, 1/4]
| style="background-color:#444" |(1/4, 1/3]
| style="background-color:#444" |(1/3, 5/12]
| style="background-color:#444" |>5/12
|-
! style="background-color:#222" |Dark mode
| style="background-color:#664488" |664488
| style="background-color:#336565" |336565
| style="background-color:#337550" |337550
| style="background-color:#397733" |397733
| style="background-color:#547733" |547733
| style="background-color:#6d7733" |6D7733
| style="background-color:#776733" |776733
| style="background-color:#774c33" |774C33
| style="background-color:#773333" |773333
|-
! style="background-color:#222" |Light mode
| style="background-color:#bd80eb;color:black" |BD80EB
| style="background-color:#6ed7df;color:black" |6ED7DF
| style="background-color:#6ce3a9;color:black" |6CE3A9
| style="background-color:#75e675;color:black" |75E675
| style="background-color:#abe56f;color:black" |ABE56F
| style="background-color:#cfe36d;color:black" |CFE36D
| style="background-color:#e5cd72;color:black" |E5CD72
| style="background-color:#ecaa7a;color:black" |ECAA7A
| style="background-color:#f18385;color:black" |F18385
|}

Xenharmonic Reference:Color schemes

2025-12-18T00:14:08Z

Tristanbay: /* Tristan's Proposal */ Uncapitalized word in heading

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color scheme is similar to that of color scheme E. However, the base colors are all full-saturation except for the first and last ones, and all colors were picked manually.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|99031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#222" |Error in steps
|-
! style="background-color:#222" |Error range
| style="background-color:#444" |0
| style="background-color:#444" |(0, 1/24]
| style="background-color:#444" |(1/24, 1/12]
| style="background-color:#444" |(1/12, 1/8]
| style="background-color:#444" |(1/8, 1/6]
| style="background-color:#444" |(1/6, 1/4]
| style="background-color:#444" |(1/4, 1/3]
| style="background-color:#444" |(1/3, 5/12]
| style="background-color:#444" |>5/12
|-
! style="background-color:#222" |Dark mode
| style="background-color:#664488" |664488
| style="background-color:#336565" |336565
| style="background-color:#337550" |337550
| style="background-color:#397733" |397733
| style="background-color:#547733" |547733
| style="background-color:#6d7733" |6D7733
| style="background-color:#776733" |776733
| style="background-color:#774c33" |774C33
| style="background-color:#773333" |773333
|-
! style="background-color:#222" |Light mode
| style="background-color:#9d70cb;color:black" |9D70CB
| style="background-color:#63aeae;color:black" |63AEAE
| style="background-color:#5cc389;color:black" |5CC389
| style="background-color:#61c661;color:black" |61C661
| style="background-color:#8bc55f;color:black" |8BC55F
| style="background-color:#afc35d;color:black" |AFC35D
| style="background-color:#c5ad62;color:black" |C5AD62
| style="background-color:#cc8a6a;color:black" |CC8A6A
| style="background-color:#d16d6f;color:black" |D16D6F
|}

Xenharmonic Reference:Color schemes

2025-12-18T00:13:10Z

Tristanbay: /* Tristan's Proposal */ Added light mode colors, added hex labels, and reorganized table

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color scheme is similar to that of color scheme E. However, the base colors are all full-saturation except for the first and last ones, and all colors were picked manually.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|99031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's Proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#222" |Error in steps
|-
! style="background-color:#222" |Error range
| style="background-color:#444" |0
| style="background-color:#444" |(0, 1/24]
| style="background-color:#444" |(1/24, 1/12]
| style="background-color:#444" |(1/12, 1/8]
| style="background-color:#444" |(1/8, 1/6]
| style="background-color:#444" |(1/6, 1/4]
| style="background-color:#444" |(1/4, 1/3]
| style="background-color:#444" |(1/3, 5/12]
| style="background-color:#444" |>5/12
|-
! style="background-color:#222" |Dark mode
| style="background-color:#664488" |664488
| style="background-color:#336565" |336565
| style="background-color:#337550" |337550
| style="background-color:#397733" |397733
| style="background-color:#547733" |547733
| style="background-color:#6d7733" |6D7733
| style="background-color:#776733" |776733
| style="background-color:#774c33" |774C33
| style="background-color:#773333" |773333
|-
! style="background-color:#222" |Light mode
| style="background-color:#9d70cb;color:black" |9D70CB
| style="background-color:#63aeae;color:black" |63AEAE
| style="background-color:#5cc389;color:black" |5CC389
| style="background-color:#61c661;color:black" |61C661
| style="background-color:#8bc55f;color:black" |8BC55F
| style="background-color:#afc35d;color:black" |AFC35D
| style="background-color:#c5ad62;color:black" |C5AD62
| style="background-color:#cc8a6a;color:black" |CC8A6A
| style="background-color:#d16d6f;color:black" |D16D6F
|}

Xenharmonic Reference:Color schemes

2025-12-17T23:58:56Z

Tristanbay: /* Absolute error */ Added my own proposal for absolute error colors

[[File:Color Scheme E, neutral and standard version.png|thumb|342x342px|Color Scheme E]]

== Color scheme E ==
'''Color scheme E''' is a tentative color scheme to use for [[prime harmonics]] and potentially other intervals on the wiki. It aligns with various strong consensus opinions about the colors associated with primes (most notably, "7 is blue"). It is based on Hojo Minori's color scheme, in that it defines a gradient to be used throughout the octave and pulls colors from that.

The "neutral" color is to be used for background colors of e.g. text boxes, tables, etc.

The hex codes for Color Scheme E for the first 9 primes are the following. Note the extremely similar shades associated with 7, 29, and 31.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#EA4335"|EA4335
|style="background-color:#BB4E45"|BB4E45
|-
|5
|style="background-color:#59CD1F"|59CD1F
|style="background-color:#5B963D"|5B963D
|-
|7
|style="background-color:#3C4AD8"|3C4AD8
|style="background-color:#4C55AB"|4C55AB
|-
|11
|style="background-color:#E0CB1D"|E0CB1D
|style="background-color:#A3983F"|A3983F
|-
|13
|style="background-color:#B33FD0"|B33FD0
|style="background-color:#924FA3"|924FA3
|-
|17
|style="background-color:#1BA6EA"|1BA6EA
|style="background-color:#3D88AC"|3D88AC
|-
|19
|style="background-color:#1CCF9D"|1CCF9D
|style="background-color:#3B977D"|3B977D
|-
|23
|style="background-color:#E69138"|E69138
|style="background-color:#B98147"|B98147
|-
|29
|style="background-color:#375ADB"|375ADB
|style="background-color:#495EAB"|495EAB
|-
|31
|style="background-color:#2B7AE1"|2B7AE1
|style="background-color:#4961AB"|4961AB
|}

=== Color scheme E.a3 ===

This variation of color scheme E intend to give the primes in the upper region of the octave more distinct colors. It is the current color scheme used on the wiki.

{| class="wikitable"
|+
!Prime
!Color
!Color (Neutral)
|-
|2
|style="background-color:#EEEEEE;color:black"|EEEEEE
|style="background-color:#888888"|888888
|-
|3
|style="background-color:#BA2C00"|BA2C00
|style="background-color:#C7634F"|C7634F
|-
|5
|style="background-color:#83FE35"|83FE35
|style="background-color:#8DCF56"|8DCF56
|-
|7
|style="background-color:#2D00AE"|2D00AE
|style="background-color:#654FC4"|654FC4
|-
|11
|style="background-color:#EEF400"|EEF400
|style="background-color:#CEC94F"|CEC94F
|-
|13
|style="background-color:#7700A0"|7700A0
|style="background-color:#9D4FC3"|9D4FC3
|-
|17
|style="background-color:#58E3FE"|58E3FE
|style="background-color:#59C3CF"|59C3CF
|-
|19
|style="background-color:#50FFAC"|50FFAC
|style="background-color:#59CE8F"|59CE8F
|-
|23
|style="background-color:#D59100"|D59100
|style="background-color:#CA9B4F"|CA9B4F
|-
|29
|style="background-color:#0007C8"|0007C8
|style="background-color:#4F5BC7"|4F5BC7
|-
|31
|style="background-color:#0C7BFE"|0C7BFE
|style="background-color:#4F92CF"|4F92CF
|}

=== Color scheme E.b ===

[[User:Tristanbay|Tristan Bay]]'s prime harmonic color scheme is similar to that of color scheme E. However, the base colors are all full-saturation except for the first and last ones, and all colors were picked manually.

{| class="wikitable"
|+
! rowspan="2" |Prime
! colspan="3" |Color
|-
!Dark mode
!Base
!Light mode
|-
|2
|style="background-color:#888888"|888888
|style="background-color:#eeeeee;color:black"|EEEEEE
|style="background-color:#f7f7f7;color:black"|F7F7F7
|-
|3
|style="background-color:#990031"|99031
|style="background-color:#ff0052"|FF0052
|style="background-color:#ff6697;color:black"|FF6697
|-
|5
|style="background-color:#6b9900"|6B9900
|style="background-color:#b3ff00;color:black"|B3FF00
|style="background-color:#d1ff66;color:black"|D1FF66
|-
|7
|style="background-color:#330099"|330099
|style="background-color:#5500ff"|5500FF
|style="background-color:#9966ff;color:black"|9966FF
|-
|11
|style="background-color:#997500"|997500
|style="background-color:#ffc300;color:black"|FFC300
|style="background-color:#ffdb66;color:black"|FFDB66
|-
|13
|style="background-color:#660099"|660099
|style="background-color:#aa00ff"|AA00FF
|style="background-color:#cc66ff;color:black"|CC66FF
|-
|17
|style="background-color:#008699"|008699
|style="background-color:#00e0ff;color:black"|00E0FF
|style="background-color:#66ecff;color:black"|66ECFF
|-
|19
|style="background-color:#00994d"|00994D
|style="background-color:#00ff80;color:black"|00FF80
|style="background-color:#66ffb3;color:black"|66FFB3
|-
|23
|style="background-color:#993b00"|993B00
|style="background-color:#ff6300"|FF6300
|style="background-color:#ffa166;color:black"|FFA166
|-
|29
|style="background-color:#000599"|000599
|style="background-color:#0008ff"|0008FF
|style="background-color:#666bff;color:black"|666BFF
|-
|31
|style="background-color:#004799"|004799
|style="background-color:#0077ff"|0077FF
|style="background-color:#66adff;color:black"|66ADFF
|-
|Higher primes
|style="background-color:#444444"|444444
|style="background-color:#777777"|777777
|style="background-color:#aaaaaa;color:black"|AAAAAA
|}

== Tuning error color scheme ==
A color scheme for tuning error in relative cents.

=== Vector's proposal ===
{| class="wikitable"
| style="background-color:#737" |0.0
| style="background-color:#353" |<4
| style="background-color:#373" |<9
| style="background-color:#573" |<16
| style="background-color:#773" |<25
| style="background-color:#753" |<36
| style="background-color:#733" |>36
|}

The magenta color is used for the perfectly in-tune interval, usually the equave.

=== Vector's 2nd proposal ===

==== Relative error ====
{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#336565" |<3
| style="background-color:#33774B" |<7
| style="background-color:#447733" |<11
| style="background-color:#607733" |<16
| style="background-color:#777033" |<25
| style="background-color:#775033" |<36
| style="background-color:#773333" |>36
|}
==== Absolute error ====

{| class="wikitable"
| style="background-color:#664488" |0.0
| style="background-color:#733" |>300
| style="background-color:#633" |<300
| style="background-color:#533" |<150
| style="background-color:#433" |<75
| style="background-color:#773" |<38
| style="background-color:#663" |<18.8
| style="background-color:#553" |<9.4
| style="background-color:#443" |<4.7
| style="background-color:#373" |<2.34
| style="background-color:#363" |<1.17
| style="background-color:#353" |<0.59
| style="background-color:#343" |<0.293
|-
|
| style="background-color:#377" |<0.146
| style="background-color:#366" |<0.073
| style="background-color:#355" |<0.037
| style="background-color:#344" |<0.0183
| style="background-color:#337" |<0.0092
| style="background-color:#336" |<0.0046
| style="background-color:#335" |<0.00229
| style="background-color:#334" |<0.00114
| style="background-color:#737" |<0.00057
| style="background-color:#636" |<0.00029
| style="background-color:#535" |<0.00014
| style="background-color:#434" |<0.00007
|}

=== Tristan's Proposal ===

The relative error table colors are based off the ones from Vector's 2nd proposal.

{| class="wikitable"
! colspan="10" ! style="background-color:#444" |Error in steps
|-
! style="background-color:#444" |Dark mode
| style="background-color:#664488" |0
| style="background-color:#336565" |(0, 1/24]
| style="background-color:#337550" |(1/24, 1/12]
| style="background-color:#397733" |(1/12, 1/8]
| style="background-color:#547733" |(1/8, 1/6]
| style="background-color:#6d7733" |(1/6, 1/4]
| style="background-color:#776733" |(1/4, 1/3]
| style="background-color:#774c33" |(1/3, 5/12]
| style="background-color:#773333" |>5/12
|-
! style="background-color:#444" |Light mode
| style="background-color:#9d70cb;color=black" |0
| style="background-color:#63aeae;color=black" |(0, 1/24]
| style="background-color:#5cc389;color=black" |(1/24, 1/12]
| style="background-color:#61c661;color=black" |(1/12, 1/8]
| style="background-color:#8bc55f;color=black" |(1/8, 1/6]
| style="background-color:#afc35d;color=black" |(1/6, 1/4]
| style="background-color:#c5ad62;color=black" |(1/4, 1/3]
| style="background-color:#cc8a6a;color=black" |(1/3, 5/12]
| style="background-color:#d16d6f;color=black" |>5/12
|}

Xenharmonic Reference:Color schemes

2025-12-17T22:47:50Z

Tristanbay: /* Color scheme E.b */ Added label for the cell I forgot to add the label for

Xenharmonic Reference:Color schemes

2025-12-17T22:47:10Z

Tristanbay: /* Color scheme E.b */ Added dark and light mode color variations

41edo

2025-12-17T22:26:33Z

Tristanbay: just call it the freaking kite guitar please

'''41edo''', or 41 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. It is known for its relatively good approximation of 11-limit just intonation.

== Theory ==

==== JI approximation ====
41edo is most accurately a 2.3.5.7.11 tuning, though it also has an acceptable if sharp 13th harmonic, notably widening the difference between the [[Collection of chords#Arto triad|arto]] and [[Collection of chords#Tendo triad|tendo]] triads such that they become simple [[Slendric]] divisions of the fifth. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 7-6-4-7-6-7-4. However, it also features a MOS diatonic of 7-7-3-7-7-7-3.

{{Harmonics in ED|41|31|0}}

=== Chords ===
41edo has three different flavors of minor and major intervals as well as neutral intervals. Its subminor and supermajor intervals approximate simpler septal ratios such as 7/4 and 9/7, while its supraminor ("pentaminor") and submajor ("pentamajor") intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its plain major and plain minor intervals approximate classic 3-limit ratios. As a result, 41edo has nine qualities of tertian, fifth-bounded triad: tendo, supermajor, novamajor, pentamajor, neutral, pentaminor, novaminor, subminor. However, 41edo lacks true interseptimal intervals (to reach a tuning with both neutrals and interseptimals, 58edo must be used), so as for latal fourth-bounded triads, there are only four qualities.

=== Scales ===
41edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being an augmented second and 5/4 a diminished fourth. Notably, 41edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-1-3-3-1-3-3-3-1-3-3-1-3-3-1, which features a mosdegree representing both the 5-limit major and minor thirds, and thus has familiar chord qualities.

=== Regular temperaments ===
41edo shares Schismic (and its extension Garibaldi, and thus Marvel and Hemifamity) with 29edo, Slendric (and its extension Miracle) with 31edo, and Tetracot with 34edo, and Magic with 22edo. Magic is especially important here as it forms the fret layout and main string tuning for the Kite guitar.

It also contains a slightly-stretched version of equal Bohlen-Pierce tuning (where the perfect twelfth of 3/1 is split into 13 equal parts) via every fifth step. If used in a linear temperament as the generator, this temperament is called Bohpier.

== Notation ==
Since 41edo has a perfect fifth which is split exactly in half, semisharps and semiflats (as in [[Diatonic notation|neutral diatonic notation]]) can be used to notate it. A useful addition is ups and downs, which naturally reflect 41edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub".)

== Practice ==
41edo is used by the musician and conlanger Lamplight as a standard tuning for their ''Shasavic'' theory of music.

Xenharmonic Reference:Color schemes

2025-12-17T09:13:24Z

Tristanbay: /* Color scheme E */ Began work on a prime harmonic color scheme of my own

15edo

2025-12-17T03:40:14Z

Tristanbay: Changed template to 15edo from 41edo

'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|15|31|0}}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

=== Scales ===
15edo contains a large number of useful scales. Among them are [[onyx]] tuned to 2-2-2-2-2-2-3, the aforementioned Zarlino diatonic, and [[pentawood]] tuned to 2-1-2-1-2-1-2-1-2-1, which splits each 5edo-step into alternating large and small steps and contains the Zarlino diatonic as a subset. Pentawood is notable in that there is a perfect fifth on every note, which is distinct from even mosdiatonic where there is one diminished fifth, and the triads alternate between major and minor, with a harmonic seventh available on every root. This is distinct from 12edo's diminished scale (which follows a similar pattern, splitting 4edo) in which half of the notes lack a perfect fifth above them entirely.

The perfect fourth halves to 8/7 and doubles to 7/4.

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 24edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

22edo

2025-12-17T03:39:43Z

Tristanbay: Replaced old table with template

'''22edo''', or 22 equal divisions of the octave, is the equal tuning featuring steps of (1200/22) ~= 54.5 cents, 22 of which stack to the perfect octave [[2/1]]. It is not a meantone system, but it is a functional 11-limit system, with 3 at ~709 cents, 5 at ~382 cents, 7 at ~982 cents, and 11 at ~545 cents.

== Theory ==

==== JI approximation ====
22edo is comparable in accuracy in the 7-limit to 12edo in the 5-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 22edo as 4-3-2-4-3-4-2. However, it also features a MOS diatonic of 4-4-1-4-4-4-1, characteristic of superpyth systems. 22edo can also be seen as an 11-limit system, however due to equating 11/9 with 6/5, it lacks much of the function of prime 11.
{{Harmonics in ED|22|31|0}}

==== Chords ====
Because it approximates the 7-limit, 22edo supports the [[harmonic tetrad]] 4:5:6:7, tuned as [0 7 13 18], and because 5/4 and 7/4 are separated by a [[perfect semioctave]], it also supports an alteration shared with any [[jubilic]] temperament in which the 5 and 7 are both flattened by a chroma, resulting in the "minor harmonic tetrad" [0 6 13 17], approximating [1/1 6/5 3/2 12/7]. As a consequence, the distance between 5/4 and 6/5 is narrowed, and the distance between 7/4 and 12/7 is widened.

==== Scales ====
A scale in 22edo with similar properties to 12edo's diatonic that takes advantage of the important structural role of the semioctave in the aforementioned tetrads is [[jaric]] (2L 8s), with the tuning 2-2-2-2-3-2-2-2-2-3. This means that 22edo can be usefully thought of as not just adding more qualities to existing ordinals, but adding three new ordinals with their own qualities, roughly surrounding 8/7 (the "unilatus"), the semioctave, and 7/4 (the "antilatus"). This has the function of giving the simplest 7-limit intervals their own category separate from sixths and sevenths, much as the simplest 5-limit intervals have their own diatonic category in the form of thirds.

From a diatonic perspective, 22edo has four varieties of third: subminor (7/6, 5\22), pentaminor (6/5, 6\22), pentamajor (5/4, 7\22), and supermajor (9/7, 8\22).

=== Regular temperaments ===

22edo shares [[superpyth]] temperament with 27edo, [[pajara]] temperament with 12edo, and [[porcupine]] temperament with 15edo. It shares [[keemic]] temperament with 15edo and 19edo.

== Notation ==

22edo may be notated with [[diamond-mos notation]] (or another [[KISS notation]]) for [[jaric]], or with [[ups and downs]] notation for diatonic. Simple Pythagorean notation may also be used, as in 12edo, although it does not support commatic alterations of interval categories (as is common in microtonal music in order to notate, for example, supermajor triads, or in 22edo's case pentamajor triads).

15edo

2025-12-17T03:33:07Z

Tristanbay: Replaced old table with template

'''15edo''', or 15 equal divisions of the octave, is the equal tuning featuring steps of (1200/15) = 80 cents, 15 of which stack to the perfect octave [[2/1]]. It is notable for its acceptable but rather distant approximation of the 11-limit featuring a near-isoharmonic 4:5:6, and for its contorted mappings.

== Theory ==

==== JI approximation ====
15edo has roughly 10-20% error on prime harmonics 3 through 11, which is a deviation from just intonation significant enough to severely affect its structure, without fully compromising the function of the prime harmonics. It is best seen as a crude approximation of the 11-limit. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 15edo as 3-2-1-3-2-3-1. Note that 15edo lacks a standard MOS diatonic scale due to its [[Perfect fifth|fifth]] being 720 cents.
Significantly, 15edo is 5 x 3, and inherits its tunings of 3 and 7 from 5edo, and 5 from 3edo. This requires either a chain of 11/8s or 23/16 or a 2-dimensional lattice be used to visualize 15edo's structure in a similar manner to the circle of fifths in 12edo.
{{Harmonics in ED|41|31|0}}

==== Chords ====
15edo contains 5edo's suspended triads, now functioning as a kind of "tendo and arto" triads. However, it adds to 5edo standard major and minor triads. Its major triad is especially notable for being close to an isoharmonic 50:63:76 triad, a property not shared by either other 5n-edos like 25 or 12edo. Additionally, the wolf chords coming with the Zarlino diatonic have a wolf fifth of 640 cents, which is also the tuning for 16/11 and thus significantly more functional than the wolf fifth in diatonic is in general. Additionally, 15edo approximates the harmonic tetrad 4:5:6:7 as [0 5 9 12]. 9:10:11:12 is equidistant (spanning a perfect fourth), and so is 6:7:8:9 (spanning a perfect fifth).

=== Scales ===
15edo contains a large number of useful scales. Among them are [[onyx]] tuned to 2-2-2-2-2-2-3, the aforementioned Zarlino diatonic, and [[pentawood]] tuned to 2-1-2-1-2-1-2-1-2-1, which splits each 5edo-step into alternating large and small steps and contains the Zarlino diatonic as a subset. Pentawood is notable in that there is a perfect fifth on every note, which is distinct from even mosdiatonic where there is one diminished fifth, and the triads alternate between major and minor, with a harmonic seventh available on every root. This is distinct from 12edo's diminished scale (which follows a similar pattern, splitting 4edo) in which half of the notes lack a perfect fifth above them entirely.

The perfect fourth halves to 8/7 and doubles to 7/4.

=== Regular temperaments ===
15edo shares Porcupine with 22edo, Augmented with 12edo, Semaphore with 24edo, and Blackwood with 10edo.

== Notation ==
Due to MOS-diatonic-based notations being nonfunctional with edos that have multiple chains of fifths (except for [[Diatonic notation#Ups and downs notation|ups and downs notation]], and even that requires E and F be treated as enharmonic), they are somewhat inconvenient for working with 15edo. Notation is often [[Notation#KISS notation|KISS notation]] based on onyx or pentawood, or notation based on the Zarlino diatonic scale.

12edo

2025-12-17T03:32:35Z

Tristanbay: Replaced old table with template

'''12edo''' is the equal tuning featuring steps of (1200/12) = 100 cents, by definition, as 12 steps stack to the octave [[Octave|2/1]]. It is the dominant tuning system in the world, and as such is covered by Xenbase for completeness as it is not 'xenharmonic'. Its fifth is at 7 steps, and its major third is at 4 steps.

== Theory ==

==== 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}}

==== 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 functions as the diminished 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.

=== Regular temperaments ===
12edo shares [[augmented]] with [[15edo]], [[diminished]] with [[16edo]], [[meantone]] with [[19edo]], boethian (equating the diatonic minor third with [[19/16]]) with 41edo, and [[archy]] with [[5edo]]. Of these, it is a particularly good tuning of diminished.

== 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.

== 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.

41edo

2025-12-17T03:30:01Z

Tristanbay: replaced old table with template

'''41edo''', or 41 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. It is known for its relatively good approximation of 11-limit just intonation.

== Theory ==
41edo has three different flavors of minor and major intervals as well as neutral intervals. Its subminor and supermajor intervals approximate simpler septal ratios such as 7/4 and 9/7, while its superminor and submajor intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its plain major and plain minor intervals approximate classic 3-limit ratios as well as ones in the 2.3.7/5, 2.3.19, and 2.3.19/5 just intonation subgroups (such as 63/40, 19/12, and 19/15, respectively). Its neutral intervals approximate 2.3.11 and, less accurately, 2.3.13.

41edo is used by the musician and conlanger Lamplight as a standard tuning for their ''Shasavic'' theory of music.

{{Harmonics in ED|41|31|0}}

=== Notation ===
Since 41edo has a perfect fifth which is split exactly in half, Stein-Zimmerman accidentals can be used alongside standard accidentals to notate it. These accidentals represent half-sharps, half-flats, sesquisharps, and sesquiflats.

Ups and downs notation is a great fit for 41edo. An up arrow represents 1 step up in pitch, and a down arrow represents 1 step down in pitch. These arrows represent comma-sized deviations from the central chain of fifths so that, for example, C to ↓E represents a tempered 5/4 ratio, and C to ↓B♭ represents a tempered 7/4.

=== Regular temperaments ===
41edo supports a number of notable linear temperaments including schismic, slendric, tetracot, miracle, and magic. Magic is especially important here as it forms the fret layout and main string tuning for the Kite Guitar.

It also contains a slightly-stretched version of equal Bohlen-Pierce tuning (where the perfect twelfth of 3/1 is split into 13 equal parts) via every fifth step. If used in a linear temperament as the generator, this temperament is called bohpier.

Planar temperaments it supports include marvel and aberschismic (also known as hemifamity).

Module:Harmonics in ED

2025-12-17T03:25:54Z

Tristanbay: Tweaked accuracy color thresholds

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
local p_cols = {"prime2", "prime3", "prime5", "prime7", "prime11", "prime13", "prime17", "prime19", "prime23", "prime29", "prime31", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
if(abs_err == 0) then
return "acc0"
end
if(abs_err < 0.03125) then
return "acc1"
end
return "acc" .. math.floor(abs_err * 12) + 2
end

function p.table(frame) -- making the table itself
local p_len = #primes
local et = math.floor(frame.args["et"])
local nowiki = frame.args["nowiki"]
local p_lim = math.floor(frame.args["p_lim"])

local tab = "{| class=\"wikitable\"\n"
if (tonumber(nowiki) == 1) then
tab = "<nowiki>" .. tab end
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! class=\""
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, primes[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| class=\"" .. rel_col(er) .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|}"
if (tonumber(nowiki) == 1) then
tab = tab .. "</nowiki>" end
return tab
end

return p

Module:Harmonics in ED

2025-12-17T03:01:06Z

Tristanbay: Cleaned up code a little more

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
local p_cols = {"prime2", "prime3", "prime5", "prime7", "prime11", "prime13", "prime17", "prime19", "prime23", "prime29", "prime31", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
if(abs_err == 0) then
return "acc0"
end
if(abs_err < 0.03) then
return "acc1"
end
if(abs_err < 0.07) then
return "acc2"
end
if(abs_err < 0.11) then
return "acc3"
end
if(abs_err < 0.16) then
return "acc4"
end
if(abs_err < 0.25) then
return "acc5"
end
if(abs_err < 0.36) then
return "acc6"
end
return "acc7"
end

function p.table(frame) -- making the table itself
local p_len = #primes
local et = math.floor(frame.args["et"])
local nowiki = frame.args["nowiki"]
local p_lim = math.floor(frame.args["p_lim"])

local tab = "{| class=\"wikitable\"\n"
if (tonumber(nowiki) == 1) then
tab = "<nowiki>" .. tab end
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! class=\""
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, primes[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| class=\"" .. rel_col(er) .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|}"
if (tonumber(nowiki) == 1) then
tab = tab .. "</nowiki>" end
return tab
end

return p

Module:Harmonics in ED

2025-12-17T02:59:47Z

Tristanbay: Cleaned up code

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
-- TODO: switch out these colors for proper classes
local p_cols = {"prime2", "prime3", "prime5", "prime7", "prime11", "prime13", "prime17", "prime19", "prime23", "prime29", "prime31", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
if(abs_err == 0) then
return "acc0"
end
if(abs_err < 0.03) then
return "acc1"
end
if(abs_err < 0.07) then
return "acc2"
end
if(abs_err < 0.11) then
return "acc3"
end
if(abs_err < 0.16) then
return "acc4"
end
if(abs_err < 0.25) then
return "acc5"
end
if(abs_err < 0.36) then
return "acc6"
end
return "acc7"
end

function p.table(frame) -- making the table itself
-- TODO: switch out these colors for proper classes
local p_len = #primes
local et = math.floor(frame.args["et"])
local nowiki = frame.args["nowiki"]
local p_lim = math.floor(frame.args["p_lim"])

local tab = "{| class=\"wikitable\"\n"
if (tonumber(nowiki) == 1) then
tab = "<nowiki>" .. tab end
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! class=\""
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, primes[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| class=\"" .. rel_col(er) .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|}"
if (tonumber(nowiki) == 1) then
tab = tab .. "</nowiki>" end
return tab
end

return p

Template:Harmonics in ED

2025-12-17T02:29:09Z

Tristanbay: fix 1st argument point

<includeonly>{{#invoke: Harmonics_in_ED | table
| et = {{{1}}}
| p_lim = {{{2}}}
| nowiki = {{{3}}}
}}</includeonly><noinclude>
Takes an equal tuning and returns a table with approximations of its prime harmonics up to a given number.
==== Arguments ====
* '''et:''' The edo to display the table for. At the moment, only edos are supported, but more equal tunings should be supported in the future.
* '''p_lim:''' The maximum harmonic to display if prime; otherwise, display the largest prime smaller than the value of this argument.
* '''nowiki:''' Whether or not to ignore wiki formatting; 0 for ''false'', 1 for ''true''.
=== Usage example ===
<pre>{{Harmonics in ED|41|23|0}}</pre>
{{Harmonics in ED|41|23|0}}
</noinclude>

Template:Harmonics in ED

2025-12-17T02:27:09Z

Tristanbay: Updated documentation

<includeonly>{{#invoke: Harmonics_in_ED | table
| et = {{{1}}}
| p_lim = {{{2}}}
| nowiki = {{{3}}}
}}</includeonly><noinclude>
Takes an equal tuning and returns a table with approximations of its prime harmonics up to a given number.
==== Arguments ====
* '''edo:''' At the moment, only edos are supported, but more equal tunings should be supported in the future.
* '''p_lim:''' The maximum harmonic to display if prime; otherwise, display the largest prime smaller than the value of this argument.
* '''nowiki:''' Whether or not to ignore wiki formatting; 0 for ''false'', 1 for ''true''.
=== Usage example ===
<pre>{{Harmonics in ED|41|23|0}}</pre>
{{Harmonics in ED|41|23|0}}
</noinclude>

Template:Harmonics in ED

2025-12-16T21:30:38Z

Tristanbay: Explained template arguments

<includeonly>{{#invoke: Harmonics_in_ED | table
| et = {{{1}}}
| p_lim = {{{2}}}
}}</includeonly><noinclude>
Takes an equal tuning and returns a table with approximations of its prime harmonics up to a given number.<br>
The first argument is the edo (at the moment, only edos are supported, but more equal tunings should be supported in the future), and the second argument is the maximum harmonic to display (if prime; otherwise, display the largest prime smaller than the value of this argument).
=== Usage example ===
<pre>{{Harmonics in ED|41|23}}</pre>
{{Harmonics in ED|41|23}}
</noinclude>

Template:Harmonics in ED

2025-12-16T21:25:03Z

Tristanbay: fixed preformatted text

Template:Harmonics in ED

2025-12-16T21:24:24Z

Tristanbay: switched include tags around and added placeholders back in

Module:Harmonics in ED

2025-12-16T21:19:29Z

Tristanbay: Fixed prime limit logic

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
-- TODO: switch out these colors for proper classes
local p_cols = {"888888", "BB4E45", "5B963D", "4C55AB", "A3983F", "924FA3", "AF7E3D", "614B8D", "3A8DA3", "829C3A", "AD6679", "885B2E", "743F34", "25573F", "40445F", "2C617A", "6A752A", "674F58", "604220", "5F2D23", "11493A", "273450", "55203A", "454C19"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
-- TODO: switch out these colors for proper classes
abs_err = math.abs(err)
red = math.pow(abs_err * 2, 1 / 3.0) * 150
green = math.pow(1 - (abs_err * 2), 0.5) * 150
blue = (1 - (abs_err * 10)) * 150
if blue < 10 then blue = 10 end
return string.format("%02x%02x%02x", math.floor(red), math.floor(green), math.floor(blue))
end

function p.table(frame) -- making the table itself
-- TODO: switch out these colors for proper classes
local p_len = #primes
local et = math.floor(frame.args["et"])
local p_lim = math.floor(frame.args["p_lim"])
local tab = "{| class=\"wikitable\"\n"
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! style=\"background-color:#"
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, primes[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| style=\"background-color:#" .. rel_col(er)
tab = tab .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] >= p_lim then break end
end
tab = tab .. "|}"
return tab
end

return p

Module:Harmonics in ED

2025-12-16T21:17:44Z

Tristanbay: Fixed colors hopefully

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
-- TODO: switch out these colors for proper classes
local p_cols = {"888888", "BB4E45", "5B963D", "4C55AB", "A3983F", "924FA3", "AF7E3D", "614B8D", "3A8DA3", "829C3A", "AD6679", "885B2E", "743F34", "25573F", "40445F", "2C617A", "6A752A", "674F58", "604220", "5F2D23", "11493A", "273450", "55203A", "454C19"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
-- TODO: switch out these colors for proper classes
abs_err = math.abs(err)
red = math.pow(abs_err * 2, 1 / 3.0) * 150
green = math.pow(1 - (abs_err * 2), 0.5) * 150
blue = (1 - (abs_err * 10)) * 150
if blue < 10 then blue = 10 end
return string.format("%02x%02x%02x", math.floor(red), math.floor(green), math.floor(blue))
end

function p.table(frame) -- making the table itself
-- TODO: switch out these colors for proper classes
local p_len = #primes
local et = math.floor(frame.args["et"])
local p_lim = math.floor(frame.args["p_lim"])
local tab = "{| class=\"wikitable\"\n"
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! style=\"background-color:#"
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, primes[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| style=\"background-color:#" .. rel_col(er)
tab = tab .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|}"
return tab
end

return p

Module:Harmonics in ED

2025-12-16T21:10:28Z

Tristanbay: Fixed some logic I guess

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
-- TODO: switch out these colors for proper classes
local p_cols = {"888888", "BB4E45", "5B963D", "4C55AB", "A3983F", "924FA3", "AF7E3D", "614B8D", "3A8DA3", "829C3A", "AD6679", "885B2E", "743F34", "25573F", "40445F", "2C617A", "6A752A", "674F58", "604220", "5F2D23", "11493A", "273450", "55203A", "454C19"}

local function steps(et, harm) -- steps of harmonic
return math.log(harm) / math.log(2) * et
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = steps(et, harm)
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
-- TODO: switch out these colors for proper classes
abs_err = math.abs(err)
red = err * 270
green = 135 - (err * 270)
blue = 135 - (err * 1350)
if blue < 0 then blue = 0 end
return string.format("%02x%02x%02x", red, green, blue)
end

function p.table(frame) -- making the table itself
-- TODO: switch out these colors for proper classes
local p_len = #primes
local et = math.floor(frame.args["et"])
local p_lim = math.floor(frame.args["p_lim"])
local tab = "{| class=\"wikitable\"\n"
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! style=\"background-color:#"
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, primes[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| style=\"background-color:#" .. rel_col(er)
tab = tab .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|}"
return tab
end

return p

Template:Harmonics in ED

2025-12-16T21:07:51Z

Tristanbay: testing input

<noinclude>{{#invoke: Harmonics_in_ED | table
| et = 41
| p_lim = 23
}}</noinclude><includeonly>
Takes an equal tuning and returns a table with approximations of its prime harmonics up to a given number.
== Usage example ==
`{{Harmonics in ED|41|23}}`
{{Harmonics in ED|41|23}}
</includeonly>

Module:Harmonics in ED

2025-12-16T04:54:17Z

Tristanbay: fix even more errors hopefully

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
local p_cols = {"888888", "BB4E45", "5B963D", "4C55AB", "A3983F", "924FA3", "AF7E3D", "614B8D", "3A8DA3", "829C3A", "AD6679", "885B2E", "743F34", "25573F", "40445F", "2C617A", "6A752A", "674F58", "604220", "5F2D23", "11493A", "273450", "55203A", "454C19"}

local function steps(et, harm) -- steps of harmonic
return math.log(harm) / math.log(2) * et
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = steps(et, harm)
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
red = err * 270
green = 135 - (err * 270)
blue = 135 - (err * 1350)
if blue < 0 then blue = 0 end
return string.format("%02x%02x%02x", red, green, blue)
end

function p.table(frame) -- making the table itself
local p_len = #primes
local et = math.floor(frame.args["et"])
local p_lim = math.floor(frame.args["p_lim"])
local tab = "{| class=\"wikitable\"\n"
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! style=\"background-color:#"
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| " .. (rel_err(et, primes[i]) > 0 and "+")
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| style=\"background-color:#" .. rel_col(er)
tab = tab .. "\" | " .. (er > 0 and "+")
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|}"
return tab
end

return p

Module:Harmonics in ED

2025-12-16T04:42:54Z

Tristanbay: Fixed more errors

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
local p_cols = {"888888", "BB4E45", "5B963D", "4C55AB", "A3983F", "924FA3", "AF7E3D", "614B8D", "3A8DA3", "829C3A", "AD6679", "885B2E", "743F34", "25573F", "40445F", "2C617A", "6A752A", "674F58", "604220", "5F2D23", "11493A", "273450", "55203A", "454C19"}

local function steps(et, harm) -- steps of harmonic
return math.log(harm) / math.log(2) * et
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = steps(et, harm)
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
red = err * 270
green = 135 - (err * 270)
blue = 135 - (err * 1350)
if blue < 0 then blue = 0 end
return string.format("%02x%02x%02x", red, green, blue)
end

function p.table(frame) -- making the table itself
local p_len = #primes
local et = tonumber(frame.args["et"])
local p_lim = tonumber(frame.args["p_lim"])
local tab = "{| class=\"wikitable\"\n"
tab = tab .. "|+Approximation of prime harmonics in "
tab = tab .. et .. " edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! style=\"background-color:#"
tab = tab .. p_cols[i] .. "\" |" .. primes[i] .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| " .. (rel_err(et, primes[i]) > 0 and "+")
tab = tab .. string.format("%.1f", abs_err(et, primes[i])) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, primes[i])
tab = tab .. "| style=\"background-color:#" .. rel_col(er)
tab = tab .. "\" | " .. (er > 0 and "+")
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, primes[i]) .. "\n(" .. steps_re(et, primes[i]) .. ")\n"
if primes[i] > p_lim then break end
end
tab = tab .. "|}"
return tab
end

return p