User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

This is a user page, not one of the main wiki pages.

This page is only opinion, not fact.

This user page details how I personally assign each EDO to a subgroup of just intonation.

Types of subgroups

Prime

• lim = Prime limit
• no-n = No-n subgroup
• dual = Dual-n subgroup
• cen = Directional subgroup
  • Example: -40cen+1 includes all primes where -40¢<absolute error<+1¢
  • Example: -1cen+35 includes all primes where -1¢<absolute error<+35¢
  • Example: -25cen+10 includes all primes where -25¢<relative error<+10¢
  • Note that lim, no-n and most dual subgroups obey -15cen+15 or stricter

Composite

• comp = Other composite subgroup

Fractional

• nth-b = Nth-basis subgroup
• frac = Other fractional subgroup

(Technically any fractional subgroup can be said to be nth-basis, so an arbitrary cutoff must be drawn somewhere. This page considers 200th-basis or higher to not be nth-basis, while 199th or lower is accepted.)

Procedure for choosing a subgroup

Remember: All of these rules are made to be broken. Bend the rules to fit the EDO. Don't bend the EDO to fit the rules.

Aims

The aim of these procedures is to make visibly available all of the simple consonant intervals an EDO has to offer, without falsely including too many ones it doesn’t have, and without allowing less-important intervals to create unnecessary clutter.

The aim is also to make subgroups of similar-sized EDOs look fairly similar so that it’s easy to cross-compare between them at a glance.

Why different sized EDOs have different procedures

As EDOs get bigger and their step size gets smaller, their step size gets closer and closer to the just-noticeable difference.

This means that if a smaller EDO has high relative error on a prime, it will sound like the prime is not there at all (no-no), but if a larger EDO has high relative error on a prime, especially a small prime, it will sound like there are two versions of the prime (dual).

Different approaches are needed for different EDO sizes to reflect this.

Also, as EDOs get bigger, more notes per octave need to be labelled with a JI approximation, so more basis elements are needed to produce those labels. Whereas, as EDOs get smaller, too many basis elements just make it needlessly complicated to navigate them, and fewer basis elements are better. So this is another reason for differing approaches at different EDO sizes.

EDOs with 1 to 27 tones/octave

1. The subgroup should have:
   1. 3 basis elements if the EDO has 1-6 tones
   2. 5 basis elements if the EDO has 7-12 tones
   3. or 6 basis elements if the EDO has 13-27 tones
2. Add prime 2 to the subgroup
3. Do whichever of the following things results in the most simple consonances being available and the least out of tune consonances being listed (use your own discretion to decide):
   1. Add the next 4-5 prime harmonics to be approximated within 15 cents
   2. Add the next 4-5 prime harmonics with absolute error between -a¢ and +b¢ where (a+b) is no bigger than 50. Make a and b anything you like that fits the rule, try to make them as small as possible while still including most important intervals
   3. Add the 4-5 smallest odd harmonics and/or taxicab-2 intervals to be approximated within 15 cents, avoiding any that include an already-added basis element as a factor

EDOs with 28 to 52 tones/octave

1. The subgroup should have 7 basis elements
2. Primes 2, 3, 5, 7 and 11 must be added to the subgroup
3. If any primes 3, 5, 7 or 11 have more than 40% relative error, then they should be made a dual prime
4. If there are more than 2 dual-primes, then only the 2 lowest dual-primes should be kept dual, and the rest made single again
5. If there are still spots left open, then they should be filled by every prime 13 and up which the EDO approximates with less than 35% relative error, preferencing the lowest primes first, until all spots are filled

EDOs with 53 to 71 tones/octave

1. The subgroup should have 8 basis elements
2. Primes 2, 3, 5, 7 and 11 must be added to the subgroup
3. If any primes 3, 5, 7 or 11 have more than 40% relative error, then they should be made a dual prime
4. If there are more than 3 dual-primes, then only the 3 lowest dual-primes should be kept dual, and the rest made single again
5. If there are still spots left open, then they should be filled by every prime 13 and up which the EDO approximates with less than 35% relative error, preferencing the lowest primes first, until all spots are filled

EDOs with 72 to 98 tones/octave

1. The subgroup should have 9 basis elements
2. Primes 2, 3, 5, 7, 11 and 13 must be added to the subgroup
3. If any primes 3, 5, 7, 11 or 13 have more than 40% relative error, then they should be made a dual prime
4. If there are more than 4 dual-primes, then only the 4 lowest dual-primes should be kept dual, and the rest made single again
5. If there are still spots left open, then they should be filled by every prime 17 and up which the EDO approximates with less than 35% relative error, preferencing the lowest primes first, until all spots are filled

EDOs with 99 or more tones/octave

1. The subgroup should have 11 basis elements
2. Add primes 2, 3, 5, 7, 11, 13, 17, 19 and 23 to the subgroup
3. Add the next two smallest primes with <35% relative error after 23 to the subgroup
4. If any primes 23 or lower have >40% relative error, then they should be made a dual prime
5. If there are now more than 11 basis elements, then the primes should be removed one by one starting with the highest and getting lower until there are 11 basis elements left
6. If a dual-prime is the last one to be removed, and this causes there to be only 10 basis elements left, then add back the smallest non-dual prime that was removed (if no non-dual primes were removed, add the next smallest prime with <35% relative error that's not already in the subgroup)

List of subgroups by EDO

Size categories taken from my human EDO size categorization (HUECAT).

For the purposes of this list, if prime N is mapped to A steps in an EDO, then “<N“ means N but mapped to A-1 steps, and “N>” means N but mapped to A+1 steps.

Picnic EDOs    (1-4)

3 basis elements

• 1edo: 2 • 127 • 129    (comp)
• 2edo: 2 • ⁷/₅ • ¹⁷/₃    (nth-b; 15th)
• 3edo: 2 • 5 • ¹⁹/₃    (nth-b; 3rd)
• 4edo: 2 • ⁵/₃ • ⁴¹/₂₉    (nth-b; 15th)

Birthday EDOs    (5-19)

3 basis elements

• 5edo: 2 • 3 • 7    (no-n)
• 6edo: 2 • 9 • 5    (comp)

5 basis elements

• 7edo: 2 • 3 • 5 • 11 • 29    (-44cen+1)
• 8edo: 2 • ⁵/₃ • ¹¹/₃ • ¹³/₅ • 19    (nth-b; 15th)
• 9edo: 2 • 5 • ⁷/₃ • 11 • ¹³/₇    (nth-b; 21st)
• 10edo: 2 • 3 • 7 • 13 • 17   (no-n)
• 11edo: 2 • 9 • 15 • 7 • 11    (comp)
• 12edo: 2 • 3 • 5 • 17 • 19    (no-n)

6 basis elements

• 13edo: 2 • 9 • 5 • 21 • 11 • 13    (comp)
• 14edo: 2 • 3 • <5 • 7 • 11 • 17    (-44cen+1)
• 15edo: 2 • 3 • 5 • 7 • 11 • 23    (no-n)
• 16edo: 2 • 3 • 5 • 7 • 11 • 13   (-28cen+7)
• 17edo: 2 • 3 • 5> • 7 • 11 • 13    (-1cen+38)
• 18edo: 2 • 3 • 5 • 7 • 13 • 17   (-1cen+32)
• 19edo: 2 • 3 • 5 • 7 • 11 • 13    (lim)

Carousel EDOs    (20-34)

6 basis elements

• 20edo: 2 • 3 • 5> • 7 • 11 • 13    (-12cen+34)
• 21edo: 2 • 3 • 5 • 7 • 17 • 19    (no-n)
• 22edo: 2 • 3 • 5 • 7 • 11 • 17    (no-n)
• 23edo: 2 • 9 • 15 • 21 • 33 • 13    (comp)
• 24edo: 2 • 3 • 5 • 7 • 11 • 13    (lim)
• 25edo: 2 • 3 • 5 • 7 • 17 • 19    (no-n)
• 26edo: 2 • 3 • 5 • 7 • 11 • 13    (lim)
• 27edo: 2 • 3 • 5 • 7 • 11 • 13    (lim)

7 basis elements

• 28edo: 2 • 3 • 5 • 7 • 11 • 13 • 19    (no-n)
• 29edo: 2 • 3 • 5 • 7 • 11 • 13 • 19    (no-n)
• 30edo: 2 • 3+ • 3- • 5 • 7 • 11 • 13    (dual)
• 31edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 32edo: 2 • 3 • 5 • 7 • 11 • 17 • 19    (no-n)
• 33edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 34edo: 2 • 3 • 5 • 7+ • 7- • 11 • 13    (dual)

Schoolbus EDOs    (35-54)

7 basis elements

• 35edo: 2 • 3+ • 3- • 5 • 7 • 11 • 17    (dual)
• 36edo: 2 • 3 • 5+ • 5- • 7 • 11+ • 11-    (dual)
• 37edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 38edo: 2 • 3 • 5 • 7 • 11+ • 11- • 13    (dual)
• 39edo: 2 • 3 • 5+ • 5- • 7+ • 7- • 11    (dual)
• 40edo: 2 • 3+ • 3- • 5 • 7 • 11 • 13    (dual)
• 41edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 42edo: 2 • 3+ • 3- • 5+ • 5- • 7 • 11    (dual)
• 43edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 44edo: 2 • 3 • 5 • 7+ • 7- • 11 • 13    (dual)
• 45edo: 2 • 3 • 5+ • 5- • 7 • 11 • 17    (dual)
• 46edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 47edo: 2 • 3+ • 3- • 5 • 7 • 11+ • 11-    (dual)
• 48edo: 2 • 3 • 5+ • 5- • 7 • 11 • 13    (dual)
• 49edo: 2 • 3 • 5 • 7+ • 7- • 11+ • 11-    (dual)
• 50edo: 2 • 3 • 5 • 7 • 11 • 13 • 17    (lim)
• 51edo: 2 • 3 • 5+ • 5- • 7 • 11+ • 11-    (dual)
• 52edo: 2 • 3+ • 3- • 5 • 7 • 11 • 19    (dual)

8 basis elements

• 53edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 54edo: 2 • 3+ • 3- • 5+ • 5- • 7+ • 7- • 11    (dual)

Double-decker EDOs    (55-74)

8 basis elements

• 55edo: 2 • 3 • 5 • 7+ • 7- • 11 • 17 • 23    (dual)
• 56edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 57edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 58edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 59edo: 2 • 3+ • 3- • 5 • 7 • 11 • 13 • 17    (dual)
• 60edo: 2 • 3 • 5 • 7+ • 7- • 11+ • 11- • 13    (dual)
• 61edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 62edo: 2 • 3 • 5 • 7 • 11+ • 11- • 29 • 31    (dual)
• 63edo: 2 • 3 • 5 • 7 • 11 • 13 • 23 • 29    (no-n)
• 64edo: 2 • 3+ • 3- • 5+ • 5- • 7 • 11+ • 11-    (dual)
• 65edo: 2 • 3 • 5 • 7+ • 7- • 11 • 19 • 23    (dual)
• 66edo: 2 • 3+ • 3- • 5 • 7 • 11 • 13 • 17    (dual)
• 67edo: 2 • 3 • 5+ • 5- • 7 • 11 • 13 • 17    (dual)
• 68edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 69edo: 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
• 70edo: 2 • 3 • 5+ • 5- • 7+ • 7- • 11 • 13    (dual)
• 71edo: 2 • 3+ • 3- • 5 • 7 • 11 • 13 • 17    (dual)

9 basis elements

• 72edo: 2 • 3 • 5 • 7 • 11 • 13+ • 13- • 17 • 19    (dual)
• 73edo: 2 • 3 • 5+ • 5- • 7 • 11+ • 11- • 13 • 19    (dual)
• 74edo: 2 • 3 • 5 • 7 • 11 • 13 • 19 • 23 • 31    (no-n)

Notation of dual-3 EDOs

Most EDO notation systems, including the near-universal ups and downs notation, are built upon chain-of-fifths notation. How then should an EDO be notated if it’s dual-fifth, i.e. it has two mappings of 3: 3+ and 3-?

The most straightforward solution is to just choose whichever 3 is closer to just 3/1, and pretend that’s the "real 3" for notation purposes. Treat the other 3 as just another prime, like 5 or 7. In most cases, I advise to do that.

If you happen to be mainly using an EDO as a tuning for one specific non-dual regular temperament like meantone, mavila, etc., then pretend that temperament’s mapping of 3 is the ‘real’ one for the purpose of notation, and pretend the other 3 is just like any other larger prime.

Of course, this results in multiple notation systems for the same EDO, since different people use different temperaments or none at all, but that’s already the case. All of those notation systems already exist, I’m not adding any new ones, I’m just saying that the ones we already have all have a valid place and it’s okay to use one some day and another some other day on a project-by-project basis.

As long as you name and briefly explain your notation system at the start of your score, use whatever system you want. Use whichever one works in practice for you and the musicians collaborating with you. Invent one, if the existing ones don’t work. It’s fine. Not everything has to be standardized and homogenized.

Because I’m personally a fan of mixing and matching multiple temperaments, and other things that aren’t temperaments like approximated JI scales, MOS scales, MODMOS & inflected MOS scales and even randomly generated scales, I usually like to go with the first option: ups and downs notation, in particular using whichever 3 is closest to just for its chain of fifths, and the other 3 being treated as just another available prime like 5, 7 or 11.

Interpreting 12edo as a 2.3.5.17.19 system

Intervals

0c (0 cents)

Behaves like: 1:1.

100c

Depending on context, behaves like: 20:19, 19:18, 18:17, 17:16 or 16:15.

200c

Depending on context, behaves like: 10:9, 19:17, 9:8 or 17:15.

300c

Depending on context, behaves like: 20:17, 32:27, 19:16 or 6:5.

400c

Depending on context, behaves like: 5:4, 34:27, 24:19 or 19:15.

500c

Depending on context, behaves like: 4:3 or 27:20.

600c

Depending on context, behaves like: 24:17, 17:12 or 27:19.

700c

Behaves like: 3:2.

800c

Depending on context, behaves like: 30:19, 19:12, 27:17 or 8:5.

900c

Depending on context, behaves like: 5:3, 32:19, 27:16 or 17:10.

1000c

Depending on context, behaves like: 30:17, 16:9 or 9:5.

1100c

Depending on context, behaves like: 15:8, 32:17, 17:9, 36:19 or 19:10.

1200c

Behaves like: 2:1.

Chords

Common chords

My interpretation of what the just harmonies are, hiding behind common practice chords.

Note names and common practice titles assume C is the tonic but of course you can transpose to any other key.

Harmonies from inversions of the chord are in italics.

Major chord

Just harmonies approximated:

• 4:5:6
• 2:3:5
• 3:4:5

Note numbers: 0, 4, 7

Note names: C, E, G

Common practice title: C

Minor chord

Just harmonies:

• 10:12:15
• 16:19:24
• 12:16:19

Note numbers: 0, 3, 7

Note names: C, Eb, G

Common practice title: Cm

Diminished chord

Just harmonies:

• 17:20:24
• 10:12:17
• 12:17:20

Note numbers: 0, 3, 6

Note names: C, Eb, Gb

Common practice title: Cdim

Major seventh chord

Just harmonies:

• 8:10:12:15
• 20:25:30:38
• 15:19:20:25

Note numbers: 0, 4, 7, 11

Note names: C, E, G, B

Common practice title: Cmaj7

Minor seventh chord

Just harmonies:

• 10:12:15:18
• 9:10:12:15

Note numbers: 0, 3, 7, 10

Note names: C, Eb, G, Bb

Common practice title: Cmin7

Dominant seventh chord

Just harmonies:

• 20:25:30:36
• 15:18:20:25

Note numbers: 0, 4, 7, 10

Note names: C, E, G, Bb

Common practice title: C7

Sus2 chord

Just harmonies:

• 8:9:12
• 18:20:27
• 6:8:9

Note numbers: 0, 2, 7

Note names: C, D, G

Common practice title: Csus2

Sus4 chord

Just harmonies:

• 6:8:9
• 20:25:27

Note numbers: 0, 5, 7

Note names: C, F, G

Common practice title: Csus4

Augmented chord

Just harmonies:

• 12:15:19
• 15:18:20
• 9:10:15

Note numbers: 0, 4, 8

Note names: C, E, G#

Common practice title: Caug

Dominant seventh chord

Just harmonies:

• 20:25:30:36
• 15:18:20:25

Note numbers: 0, 4, 7, 10

Note names: C, E, G, Bb

Common practice title: C7

Major ninth chord

Just harmonies:

• 8:10:12:15:18
• 8:9:10:12:15

Note numbers: 0, 4, 7, 11, 14

Note names: C, E, G, B, D

Common practice title: Cmaj9

Minor ninth chord

Just harmonies:

• 40:48:60:72:85

Note numbers: 0, 3, 7, 10, 14

Note names: C, Eb, G, Bb, D

Common practice title: Cmin9

Dominant ninth chord

Just harmonies:

• 40:50:60:72:85

Note numbers: 0, 4, 7, 10, 14

Note names: C, E, G, Bb, D

Common practice title: C9

Major eleventh chord

Just harmonies:

• 24:30:36:45:54:64
• 24:28:30:32:36:45

Note numbers: 0, 4, 7, 11, 14, 17

Note names: C, E, G, B, D, F

Common practice title: Cmaj11

Minor eleventh chord

Just harmonies:

• 40:48:60:72:85:108
• 40:48:54:60:72:85

Note numbers: 0, 3, 7, 10, 14, 17

Note names: C, Eb, G, Bb, D, F

Common practice title: Cmin11

Dominant eleventh chord

Just harmonies:

• 40:50:60:72:85:108
• 40:50:54:60:72:85

Note numbers: 0, 4, 7, 10, 14, 17

Note names: C, E, G, Bb, D, F

Common practice title: C11

Parent chords

My list of in my opinion the most harmonious 'parent chords' in 12edo, which you can use as palettes to build novel and pretty smaller chords. Choose one of these chords, take any subset of 2 or more notes from it, and you will make another, also harmonious chord.

These chords work particularly well if you drop the root note down an octave, better mimicking the shape of the harmonic series. (For example you can play "chord 0-12-15-19-20-21-22-23" instead of "chord 0-3-7-8-9-10-11").

Note names and common practice titles assume C is the tonic but of course you can transpose to any other key.

The over-9 parent chord of 12edo

Approximated just harmony: 9:10:12:16:17

Note numbers: 0, 2, 5, 10, 11

(With octave): 0, 12, 15, 17, 22, 23

Note names: C, D, F, A#/Bb, B

Common practice title: Dm7#5/C add(b6)

The over-10 parent chord of 12edo

Just harmony: 10:12:15:16:17:18:19

Note numbers: 0, 3, 7, 8, 9, 10, 11

(With octave): 0, 12, 15, 19, 20, 21, 22, 23

Note names: C, D#/Eb, G, G#/Ab, A, A#/Bb, B

The over-12 parent chord of 12edo

Just harmony: 12:15:16:17:18:19:20

Note numbers: 0, 4, 5, 6, 7, 8, 9

(With octave): 0, 12, 16, 17, 18, 19, 20, 21

Note names: C, E, F, F#/Gb, G, G#/Ab, A

The over-15 parent chord of 12edo

Just harmony: 15:16:17:18:19:20:24:27

Note numbers: 1, 2, 3, 4, 5, 8, 10

(With octave): 0, 12, 13, 14, 15, 16, 17, 20, 22

Note names: C, C#/Db, D, D#/Eb, E, F, G#/Ab, A#/Bb

The over-16 parent chord of 12edo

Just harmony: 16:17:18:19:20:24:30

Note numbers: 0, 1, 2, 3, 4, 7, 11

(With octave): 0, 12, 13, 14, 15, 16, 19, 23

Note names: C, C#/Db, D, D#/Eb, E, G, B

Common practice title: Cmaj9 add(m3,m9)

The over-17 parent chord of 12edo

Just harmony: 17:18:19:20:24:27:30:32

Note numbers: 0, 1, 2, 3, 6, 8, 10, 11

(With octave): 0, 12, 13, 14, 15, 18, 20, 22, 23

Note names: C, C#/Db, D, D#/Eb, F#/Gb, G#/Ab, A#/Bb, B

The over-18 parent chord of 12edo

Just harmony: 18:19:20:24:27:30:32:34

Note numbers: 0, 1, 2, 5, 7, 9, 10, 11

(With octave): 0, 12, 13, 14, 17, 19, 21, 22, 23

Note names: C, C#/Db, D, F, G, A, A#/Bb, B

The over-19 parent chord of 12edo

Just harmony: 19:20:24:27:30:32:36

Note numbers: 0, 1, 4, 6, 8, 9, 11

(With octave): 0, 12, 13, 16, 18, 20, 21, 23

Note names: C, C#/Db, E, F#/Gb, G#/Ab, A, B

The over-20 parent chord of 12edo

Just harmony: 20:24:25:27:30:32:34:36:38

Note numbers: 0, 3, 4, 5, 7, 8, 9, 10, 11

(With octave): 0, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23

Note names: C, D#/Eb, E, F, G, G#/Ab, A, A#/Bb, B

The over-24 parent chord of 12edo

Just harmony: 24:27:30:32:34:36:38:40

Note numbers: 0, 2, 4, 5, 6, 7, 8, 9

(With octave): 0, 12, 14, 16, 17, 18, 19, 20, 21

Note names: C, D, E, F, F#/Gb, G, G#/Ab, A