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User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning

< User:BudjarnLambeth
Revision as of 03:05, 30 December 2024 by BudjarnLambeth (talk | contribs) (→Chords: add common chords)
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 11-limit composite harmonics smaller than 60 that it approximates within 15 cents (giving preference to harmonics with lower prime factors first and excluding powers of two)
  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

EDOs with 7 to 12 tones/octave

  1. The subgroup should have 5 basis elements
  2. Add prime 2 to the subgroup
  3. If 3 is approximated within 15 cents, add 3 to the subgroup
    1. If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any)
    2. Optionally, add the second-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous one (if any)
  4. If 5 is approximated within 15 cents, add 5 to the subgroup
    1. If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any)
    2. If this is the same as a number already added, just keep that one
  5. If 7 is approximated within 15 cents, add 7 to the subgroup
    1. If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any)
    2. If this is the same as a number already added, just keep that one
  6. If 11 is approximated within 15 cents, add 11 to the subgroup
  7. If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup
  8. If there are still spots free, then add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any)
    1. If this is the same as a number already added, just keep that one
  9. If any composite basis elements now in the subgroup share no common factors with any other element in the subgroup, remove them
  10. If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first)
  11. If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes
  12. Optionally, replace any one basis element with any composite harmonic 60 or smaller, that shares factors in common with at least 2 other basis elements, is approximated within 15 cents, and has not yet been added

EDOs with 13 to 27 tones/octave

  1. The subgroup should have 6 basis elements
  2. Add prime 2 to the subgroup
  3. If 3 is approximated within 15 cents, add 3 to the subgroup
    1. If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any)
    2. Optionally, add the second-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous one (if any)
  4. If 5 is approximated within 15 cents, add 5 to the subgroup
    1. If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any)
    2. If this is the same as a number already added, just keep that one
  5. If 7 is approximated within 15 cents, add 7 to the subgroup
    1. If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any)
    2. If this is the same as a number already added, just keep that one
  6. If 11 is approximated within 15 cents, add 11 to the subgroup
  7. If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup
  8. If there are still spots free, then add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any)
    1. If this is the same as a number already added, just keep that one
  9. If any composite basis elements now in the subgroup share no common factors with any other element in the subgroup, remove them
  10. If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first)
  11. If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes
  12. Optionally, replace any one basis element with any composite harmonic 60 or smaller, that shares factors in common with at least 2 other basis elements, is approximated within 15 cents, and has not yet been added

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 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 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 three 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 117 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 four 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 118 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 <40% 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 <40% relative error that's not already in the subgroup)

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 • 15 • 7 • 13   (comp)
  • 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 • 25 • 35 • 45 • 55    (comp)
  • 15edo: 2 • 3 • 5 • 7 • 11 • 23    (no-n)
  • 16edo: 2 • 5 • 7 • 13 • 27 • 45   (comp)
  • 17edo: 2 • 3 • 35 • 7 • 11 • 13    (comp)
  • 18edo: 2 • 5 • 9 • 11 • 21 • 33   (comp)
  • 19edo: 2 • 3 • 5 • 7 • 11 • 13    (lim)

Carousel EDOs    (20-34)

6 basis elements
  • 20edo: 2 • 3 • 15 • 7 • 11 • 13    (comp)
  • 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 • 5 • 7 • 11 • 13 • 17 • 19    (lim)
  • 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.


Major chord

Just harmonies approximated:

  • 4:5:6

Note numbers: 0, 4, 7

Note names: C, E, G

Common practice title: C


Minor chord

Just harmonies:

  • 10:12:15
  • 16:19:24

Note numbers: 0, 3, 7

Note names: C, Eb, G

Common practice title: Cm


Diminished chord

Just harmonies:

  • 17:20:24

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

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

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

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

Note numbers: 0, 2, 7

Note names: C, D, G

Common practice title: Csus2


Sus4 chord

Just harmonies:

  • 6:8:9
  • 20:27:30

Note numbers: 0, 5, 7

Note names: C, F, G

Common practice title: Csus4


Augmented chord

Just harmonies:

  • 12:15:19
  • 15:18:20

Note numbers: 0, 4, 8

Note names: C, E, G#

Common practice title: Caug


Dominant seventh chord

Just harmonies:

  • 20:25:30:36

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

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

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

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

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 0-15-19-20-21-22-23" 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-5 parent chord of 12edo

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

Note numbers: 0, 3, 8, 10

(With octave): 0, 12, 15, 20, 22

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

Common practice title: Cm7#5


The over-6 parent chord of 12edo

Just harmony: 6:8:9:10

Note numbers: 0, 5, 7, 9

(With octave): 0, 12, 17, 19, 21

Note names: C, F, G, A

Common practice title: Fadd9/C


The over-8 parent chord of 12edo

Just harmony 8:9:10:12:15

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

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

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

Common practice title: Cmaj9


The over-9 parent chord of 12edo

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

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