User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning

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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

How to choose a type

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.

EDOs with 1 to 6 tones/octave

  1. The subgroup should have 3 basis elements
  2. If the EDO approximates 3 or more primes 11 or lower within 15 cents, then choose the best 3 and use those as its subgroup
  3. If it approximates less than 3 such primes, then include all the ones it does approximate, and fill the remaining spots with odd harmonics smaller than 40 that it approximates within 15 cents (giving preference to the lowest harmonics first)
  4. If there are aren't enough of those to fill all 3 spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes and intervals that are approximated more precisely (use direction to decide which to value more on an edo-by-edo basis)
  5. If there are still spots left open, fill them with the smallest composite harmonics of any size that are approximated within 15 cents

EDOs with 7 to 12 tones/octave

  1. The subgroup should have 5 basis elements
  2. If the EDO approximates any primes 11 or lower within 15 cents, then add all of those to its subgroup
  3. If there are still spots left over, and the EDO does not approximate prime 3, then check if it approximates any harmonics 6, 9, 12 or 15 within 15 cents. If so, add them all to the subgroup starting with the smallest until all spots are filled or all harmonics have been added.
  4. If there are still spots left over, if the EDO does not approximate one or both of 5 or 3, check if it approximates 5/3 within 15 cents and if so, add that as a basis element
  5. If there are still spots left over, if the EDO does not approximate one or both of 7 or 3, check if it approximates 7/3 within 15 cents and if so, add that as a basis element
  6. If there are still spots left over, if the EDO does not approximate one or both of 7 or 5, check if it approximates 7/5 within 15 cents and if so, add that as a basis element
  7. Do the same as above for 11/3, then 11/5, then 11/7
  8. If there are still spots left over, check if the EDO approximates any primes 13, 17, 19 or 23 within 15 cents, if so, then add all of those to its subgroup giving preference to the lowest ones and to the ones approximated most closely (use discretion to choose which of those things to weight more heavily) until all spots have been filled or all primes have been added
  9. If there are still spots left open, choose either (A) or (B) at your discretion on an edo-by-edo basis, or combine both for edos where that makes sense:
    1. (A) fill the remaining spots with odd harmonics smaller than 40 that it the edo approximates within 15 cents (giving preference to harmonics that are multiples of simple harmonics like 21 or 33)
    2. (B) fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes and intervals that are approximated more precisely (use direction to decide which to value more on an edo-by-edo basis)
  10. If there are still spots left open, fill them with the smallest remaining integer harmonics of any size that are approximated within 15 cents

EDOs with 13 to 27 tones/octave

  1. The subgroup should have 6 basis elements
  2. If the EDO approximates any primes 13 or lower within 15 cents, then add all of those to its subgroup
  3. If there are still spots left over, and the EDO does not approximate prime 3, then check if it approximates any harmonics 6, 9, 12 or 15 within 15 cents. If so, add them all to the subgroup starting with the smallest until all spots are filled or all harmonics have been added.
  4. If there are still spots left over, if the EDO does not approximate one or both of 5 or 3, check if it approximates 5/3 within 15 cents and if so, add that as a basis element
  5. If there are still spots left over, if the EDO does not approximate one or both of 7 or 3, check if it approximates 7/3 within 15 cents and if so, add that as a basis element
  6. If there are still spots left over, if the EDO does not approximate one or both of 7 or 5, check if it approximates 7/5 within 15 cents and if so, add that as a basis element
  7. Do the same as above for 11/3, then 11/5, then 11/7, then 13/3, then 13/5, then 13/7, then 13/11
  8. If there are still spots left over, check if the EDO approximates any primes 17, 19 or 23 within 15 cents, if so, then add all of those to its subgroup giving preference to the lowest ones and to the ones approximated most closely (use discretion to choose which of those things to weight more heavily) until all spots have been filled or all primes have been added
  9. If there are still spots left open, choose either (A) or (B) at your discretion on an edo-by-edo basis, or combine both for edos where that makes sense:
    1. (A) fill the remaining spots with odd harmonics smaller than 40 that it the edo approximates within 15 cents (giving preference to harmonics that are multiples of simple harmonics like 21 or 33)
    2. (B) fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes and intervals that are approximated more precisely (use direction to decide which to value more on an edo-by-edo basis)
  10. If there are still spots left open, fill them with the smallest remaining integer harmonics of any size that are approximated within 15 cents

EDOs with 28 to 52 tones/octave

  1. The subgroup should have 7 basis elements
  2. Primes 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 two 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 or more tones/octave

  1. The subgroup should have 8 basis elements
  2. Primes 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 two 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

Subgroups by EDO size

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

Picnic EDOs (1-4)

  • 1edo: 2.125.127 (comp)
  • 2edo: 2.7/5.17/3 (nth-b) (15th)
  • 3edo: 2.5.17/3 (nth-b) (3rd)
  • 4edo: 2.5/3.7/5 (nth-b) (15th)

Birthday EDOs (5-19)

  • 5edo: 2.3.7 (no-n)
  • 6edo: 2.9.5 (comp)
  • 7edo: 2.3.11/3.11/5.13 (nth-b) (15th)
  • 8edo: 2.5/3.11/3.13/5.19 (nth-b) (15th)
  • 9edo: 2.5.7/3.11.13/7 (nth-b) (3rd)
  • 10edo: 2.3.7.13.17 (no-n)
  • 11edo: 2.9.15.7.11 (comp)
  • 12edo: 2.3.5.17.19 (no-n)
  • 13edo: 2.9.5.11.13.17 (comp)
  • 14edo: 2.3.7/5.9/5.11/5.13 (nth-b) (5th)
  • 15edo: 2.3.5.7.11 (lim)
  • 16edo: 2.5.7.13.19 (no-n)
  • 17edo: 2.3.7.11.13 (no-n)
  • 18edo: 2.9.5.7/3.11 (nth-b) (3rd)
  • 19edo: 2.3.5.7.11.13 (lim)

Carousel EDOs (20-34)

  • 20edo: 2.3.7.11.13.17 (no-n)
  • 21edo: 2.3.5.7.13.17 (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)
  • 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)

  • 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)
  • 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)

(May complete later.)

Subgroups by subgroup type

Full prime limit

  • 15edo: 2.3.5.7.11 (lim)
  • 19edo: 2.3.5.7.11.13 (lim)
  • 24edo: 2.3.5.7.11.13 (lim)
  • 26edo: 2.3.5.7.11.13 (lim)
  • 27edo: 2.3.5.7.11.13 (lim)
  • 31edo: 2.3.5.7.11.13.17 (lim)
  • 33edo: 2.3.5.7.11.13.17 (lim)
  • 37edo: 2.3.5.7.11.13.17 (lim)
  • 41edo: 2.3.5.7.11.13.17 (lim)
  • 43edo: 2.3.5.7.11.13.17 (lim)
  • 46edo: 2.3.5.7.11.13.17 (lim)
  • 50edo: 2.3.5.7.11.13.17 (lim)
  • 53edo: 2.3.5.7.11.13.17.19 (lim)

No-n

  • 5edo: 2.3.7 (no-n)
  • 10edo: 2.3.7.13.17 (no-n)
  • 12edo: 2.3.5.17.19 (no-n)
  • 16edo: 2.5.7.13.19 (no-n)
  • 17edo: 2.3.7.11.13 (no-n)
  • 20edo: 2.3.7.11.13.17 (no-n)
  • 21edo: 2.3.5.7.13.17 (no-n)
  • 25edo: 2.3.5.7.17.19 (no-n)
  • 22edo: 2.3.5.7.11.17 (no-n)
  • 28edo: 2.3.5.7.11.13.19 (no-n)
  • 29edo: 2.3.5.7.11.13.19 (no-n)
  • 32edo: 2.3.5.7.11.17.19 (no-n)

Dual-n

  • 30edo: 2.3+.3-.5.7.11.13 (dual)
  • 34edo: 2.3.5.7+.7-.11.13 (dual)
  • 35edo: 2.3+.3-.5.7.11.17 (dual)
  • 36edo: 2.3.5+.5-.7.11+.11- (dual)
  • 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)
  • 42edo: 2.3+.3-.5+.5-.7.11 (dual)
  • 44edo: 2.3.5.7+.7-.11.13 (dual)
  • 45edo: 2.3.5+.5-.7.11.17 (dual)
  • 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)
  • 51edo: 2.3.5+.5-.7.11+.11- (dual)
  • 52edo: 2.3+.3-.5.7.11.19 (dual)
  • 54edo: 2.3+.3-.5+.5-.7+.7-.11 (dual)

Equalizer

No edos really fit this category.

Other composite

  • 6edo: 2.9.5 (comp)
  • 11edo: 2.9.15.7.11 (comp)
  • 13edo: 2.9.5.11.13.17 (comp)
  • 23edo: 2.9.15.21.33.13 (comp)

Nth-basis

Interestingly, all of these can be seen as 15th-basis. It might just be because in EDOs 2 is always pure, and 3 and 5 are the next simplest harmonics, so it just makes sense for them to show up as simple subgroup denominators.

  • 7edo: 2.3.11/3.11/5.13 (nth-b) (15th)
  • 8edo: 2.11/3.13/5.19 (nth-b) (15th)
  • 9edo: 2.5.7/3.11 (nth-b) (3rd)
  • 14edo: 2.3.7/5.9/5.11/5.13 (nth-b) (5th)
  • 18edo: 2.9.5.7/3.11 (nth-b) (3rd)

Other fractional

No edos really fit this category.

(Technically any fractional subgroup can be said to be nth-basis, but if it were something absurdly big like 200th-basis, then it would belong in this category, not nth-basis, for the purpose of this list.

But, there aren't any edos where that kind of subgroup makes sense hence this category being empty.)

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

These chords work particularly well if you drop the root note down an octave, better mimicking the shape of the harmonic series.

(e.g you can play "chord 0-15-19-20-21-22-23" instead of "chord 0-3-7-8-9-10-11")

You can also of course take any subset of 2 or more notes from one of these chords to make another, also harmonious chord.

Notes and names here assume C is the tonic but of course you can transpose to any other key.


Chord 0-3-8-10

(Approximate) just harmony: 5:6:8:9

Notes: C, D#/Eb, G#/Ab, A#/Bb

Name: Cm7#5


Chord 0-5-7-9

Just harmony: 6:8:9:10

Notes: C, F, G, A

Name: Fadd9/C


Chord 0-2-4-7-11

Just harmony 8:9:10:12:15

Notes: C, D, E, G, B

Name: Cmaj9


Chord 0-2-5-10-11

Just harmony: 9:10:12:16:17

Notes: C, D, F, A#/Bb, B

Name: Dm7#5/C add(b6)


Chord 0-3-7-8-9-10-11

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

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


Chord 0-4-5-6-7-8-9

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

Notes: C, E, F, F#/Gb, G, G#/Ab, A


Chord 0-1-2-3-4-5-8-10

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

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


Chord 0-1-2-3-4-7-11

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

Notes: C, C#/Db, D, D#/Eb, E, G, B

Name: Cmaj9 add(m3,m9)


Chord 0-1-2-3-6-8-10-11

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

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


Chord 0-1-2-5-7-9-10-11

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

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


Chord 0-1-4-6-8-9-11

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

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


Chord 0-3-4-5-7-8-9-10-11

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

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

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