User:BudjarnLambeth/How I choose a subgroup for an EDO

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 a real wiki page.

This page is only opinion, not fact.

This user page details how I personally assign each EDO to a subgroup of just intonation… This changes all the time because I can’t make up my mind.

Types of subgroups

WIP: I plan to heavily rewrite this section after carefully studying the new User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup.

Procedure for choosing a subgroup

WIP: I plan to heavily rewrite this section after carefully studying the new User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup.

List of subgroups by EDO

WIP: I plan to heavily rewrite this section after carefully studying the new User:Dummy index/Heuristics for picking a nonstandard basis of JI subgroup.

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.

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