Xenharmonic Wiki

User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning

< User:BudjarnLambeth
Revision as of 09:47, 29 December 2024 by BudjarnLambeth (talk | contribs) (→Interpreting 12edo as a 2.3.5.17.19 system)
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

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.

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

List of subgroups by EDO

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

Picnic EDOs    (1-4)

3 basis elements
  • 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)

3 basis elements
  • 5edo: 2 • 3 • 7    (no-n)
  • 6edo: 2 • 9 • 5    (comp)
5 basis elements
  • 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; 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 • 11 • 13 • 17    (comp)
  • 14edo: 2 • 3 • 7/5 • 9/5 • 11/5 • 13    (nth-b; 5th)
  • 15edo: 2 • 3 • 5 • 7 • 11 • 23    (no-n)
  • 16edo: 2 • 27 • 5 • 7 • 13 • 19    (comp)
  • 17edo: 2 • 3 • 7 • 11 • 13 • 19    (no-n)
  • 18edo: 2 • 9 • 5 • 7/3 • 11 • 13/7    (nth-b; 21st)
  • 19edo: 2 • 3 • 5 • 7 • 11 • 13    (lim)

Carousel EDOs    (20-34)

6 basis elements
  • 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)
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)

  • 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)

(May complete later.)

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

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

Just harmony: 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

Just harmony: 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

Just harmony: 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

Just harmony: 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

Just harmony: 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

Just harmony: 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

Just harmony: 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

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

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

Retrieved from "https://en.xen.wiki/index.php?title=User:BudjarnLambeth/12edo_as_a_2.3.5.17.19_tuning&oldid=173651"