User:BudjarnLambeth/The intervals I see as important
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The following are all the 16-integer-limit consonant intervals available in an octave, simplified to their simplest form (by taking them to a higher octave to simplify the fraction), then sorted with most consonant (ie mathematically simplest) first.
(^in a higher octave) HYPERCONSONANCES 2/1 (octave) 3/1 (perf 5th^) CONSONANCES 5/1 (maj 3rd^) 4/3 (perf 4th) 5/3 (maj 6th) 7/1 (submin 7th^) 9/1 (large maj 2nd^) 7/3 (submin 3rd^) AMBISONANCES 6/5 (min 3rd) 11/1 (undec 4th^) 7/5 (small tritone) 9/5 (min 7th) 11/3 (neu 7th^) 13/1 (neu 6th^) 8/7 (supmaj 2nd) DISSONANCES 9/7 (supmaj 3rd) 13/3 (large tridec neu 2nd^) 15/1 (maj 7th^) 11/5 (large undec neu 2nd) 10/7 (large tritone) 10/9 (small maj 2nd) 11/7 (undec submin 6th) 13/5 (tridec maj 3rd^) 12/7 (supmaj 6th) 11/9 (undec neu 3rd) 13/7 (tridec maj 7th) 13/9 (tridec tritone) 15/7 (large minor 2nd^) 14/9 (sept submin 6th) 12/11 (small undec neu 2nd) 13/11 (tridec min 3rd) 14/11 (undec maj 3rd) 16/9 (Pythag min 7th) 15/11 (pentdec 4th) 16/11 (undec sub5th) 14/13 (small tridec neu 2nd) 15/13 (tridec semi4th) 16/13 (tridec neu 3rd) 16/15 (small neu 2nd)
Any general-purpose tuning system should approximate all hyperconsonances within 10 cents or less, all consonances within 20 cents, and all ambisonances within 30 cents. It should also try to have as few notes as possible that do not approximate any of those three categories of interval, because every note that isn't approximating one of them is a wolf interval that adds a 'bitter' taste to the tuning, and a tuning cannot survive very much bitterness.
Table of EDOs by general-purpose efficacy (1–53edo)
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This rates EDOs for general purposes. It doesn't take into account specialist (but still very valid) use cases like inharmonic timbres, dual-fifths usage, subgroup usage, or taking carefully chosen subsets (eg blackjack). EDOs that look bad on this table might still be very good for one or more of those specialist uses.
Table created using Claude (I gave it a PDF of the EDO harmonic tables; my prompt is listed below the table). It almost certainly made errors so don't use this as a source for the numbers (instead look at the relevant EDO pages directly, they're linked in the left column).
*denotes EDOs where an especially high % of their intervals are especially consonant
| EDO | Est. no. of dissonances | Hyperconsonances | Consonances | Ambisonances | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2/1 perf oct |
3/1 perf 5th^ |
5/1 maj 3rd^ |
4/3 perf 4th |
5/3 maj 6th |
7/1 submin 7th^ |
9/1 lg maj 2nd^ |
7/3 submin 3rd^ |
6/5 min 3rd |
11/1 undec 4th^ |
7/5 sm tritone |
9/5 min 7th |
11/3 neu 7th^ |
13/1 neu 6th^ |
8/7 supmaj 2nd | ||
| 1edo | 0 | 0.0¢ ✓✓ | 498.0¢ ✗ | 386.0¢ ✗ | 498.0¢ ✗ | 884.0¢ ✗ | 231.0¢ ✗ | 204.0¢ ✗ | 267.0¢ ✗ | 884.0¢ ✗ | 551.0¢ ✗ | 617.0¢ ✗ | 182.0¢ ✗ | 1049.0¢ ✗ | 359.0¢ ✗ | 231.0¢ ✗ |
| 2edo | 0 | 0.0¢ ✓✓ | 102.0¢ ✗ | 214.0¢ ✗ | 102.0¢ ✗ | 316.0¢ ✗ | 231.0¢ ✗ | 204.0¢ ✗ | 333.0¢ ✗ | 316.0¢ ✗ | 49.0¢ ✗ | 17.0¢ ✓✓ | 418.0¢ ✗ | 151.0¢ ✗ | 241.0¢ ✗ | 231.0¢ ✗ |
| 3edo | 1 | 0.0¢ ✓✓ | 98.0¢ ✗ | 14.0¢ ✓ | 98.0¢ ✗ | 84.0¢ ✗ | 169.0¢ ✗ | 196.0¢ ✗ | 267.0¢ ✗ | 84.0¢ ✗ | 151.0¢ ✗ | 183.0¢ ✗ | 182.0¢ ✗ | 249.0¢ ✗ | 41.0¢ ✗ | 169.0¢ ✗ |
| 4edo | 0 | 0.0¢ ✓✓ | 102.0¢ ✗ | 86.0¢ ✗ | 102.0¢ ✗ | 16.0¢ ✓ | 69.0¢ ✗ | 96.0¢ ✗ | 33.0¢ ✗ | 16.0¢ ✓✓ | 49.0¢ ✗ | 17.0¢ ✓✓ | 182.0¢ ✗ | 151.0¢ ✗ | 59.0¢ ✗ | 69.0¢ ✗ |
| 5edo | 1 | 0.0¢ ✓✓ | 18.0¢ ✗ | 94.0¢ ✗ | 18.0¢ ✓ | 76.0¢ ✗ | 9.0¢ ✓✓ | 36.0¢ ✗ | 27.0¢ ✗ | 76.0¢ ✗ | 71.0¢ ✗ | 103.0¢ ✗ | 58.0¢ ✗ | 89.0¢ ✗ | 119.0¢ ✗ | 9.0¢ ✓✓ |
| 6edo | 1 | 0.0¢ ✓✓ | 98.0¢ ✗ | 13.7¢ ✓ | 98.0¢ ✗ | 84.3¢ ✗ | 31.2¢ ✗ | 3.9¢ ✓✓ | 66.8¢ ✗ | 84.3¢ ✗ | 48.7¢ ✗ | 17.5¢ ✓✓ | 17.6¢ ✓✓ | 49.3¢ ✗ | 40.5¢ ✗ | 31.2¢ ✗ |
| 7edo | 1 | 0.0¢ ✓✓ | 16.2¢ ✗ | 43.5¢ ✗ | 16.2¢ ✓ | 27.3¢ ✗ | 59.7¢ ✗ | 32.5¢ ✗ | 75.9¢ ✗ | 27.3¢ ✓ | 37.0¢ ✗ | 103.2¢ ✗ | 11.0¢ ✓✓ | 20.8¢ ✓ | 16.6¢ ✓✓ | 59.7¢ ✗ |
| 8edo | 4 | 0.0¢ ✓✓ | 48.0¢ ✗ | 63.7¢ ✗ | 48.0¢ ✗ | 15.7¢ ✓ | 68.8¢ ✗ | 53.9¢ ✗ | 116.8¢ ✗ | 15.7¢ ✓✓ | 48.7¢ ✗ | 132.5¢ ✗ | 117.6¢ ✗ | 0.7¢ ✓✓ | 59.5¢ ✗ | 68.8¢ ✗ |
| 9edo | 4 | 0.0¢ ✓✓ | 35.3¢ ✗ | 13.7¢ ✓ | 35.3¢ ✗ | 49.0¢ ✗ | 35.5¢ ✗ | 62.8¢ ✗ | 0.2¢ ✓✓ | 49.0¢ ✗ | 18.0¢ ✓✓ | 49.2¢ ✗ | 49.1¢ ✗ | 17.3¢ ✓✓ | 40.5¢ ✗ | 35.5¢ ✗ |
| 10edo | 4* | 0.0¢ ✓✓ | 18.0¢ ✗ | 26.3¢ ✗ | 18.0¢ ✓ | 44.3¢ ✗ | 8.8¢ ✓✓ | 36.1¢ ✗ | 26.8¢ ✗ | 44.3¢ ✗ | 48.7¢ ✗ | 17.5¢ ✓✓ | 62.4¢ ✗ | 30.7¢ ✗ | 0.5¢ ✓✓ | 8.8¢ ✓✓ |
| 11edo | 6 | 0.0¢ ✓✓ | 47.4¢ ✗ | 50.0¢ ✗ | 47.4¢ ✗ | 97.4¢ ✗ | 13.0¢ ✓ | 14.3¢ ✓ | 60.4¢ ✗ | 97.4¢ ✗ | 5.9¢ ✓✓ | 37.0¢ ✗ | 35.7¢ ✗ | 41.5¢ ✗ | 32.2¢ ✗ | 13.0¢ ✓✓ |
| 12edo | 3* | 0.0¢ ✓✓ | 2.0¢ ✓✓ | 13.7¢ ✓ | 2.0¢ ✓✓ | 15.7¢ ✓ | 31.2¢ ✗ | 3.9¢ ✓✓ | 33.2¢ ✗ | 15.7¢ ✓✓ | 48.7¢ ✗ | 17.5¢ ✓✓ | 17.6¢ ✓✓ | 50.7¢ ✗ | 40.5¢ ✗ | 31.2¢ ✗ |
| 13edo | 6 | 0.0¢ ✓✓ | 36.5¢ ✗ | 17.1¢ ✓ | 36.5¢ ✗ | 53.6¢ ✗ | 45.7¢ ✗ | 19.3¢ ✓ | 82.2¢ ✗ | 53.6¢ ✗ | 2.5¢ ✓✓ | 28.6¢ ✓ | 2.2¢ ✓✓ | 34.0¢ ✗ | 9.8¢ ✓✓ | 45.7¢ ✗ |
| 14edo | 8 | 0.0¢ ✓✓ | 16.2¢ ✗ | 42.3¢ ✗ | 16.2¢ ✓ | 58.5¢ ✗ | 26.0¢ ✗ | 32.5¢ ✗ | 9.8¢ ✓✓ | 58.5¢ ✗ | 37.0¢ ✗ | 68.3¢ ✗ | 74.8¢ ✗ | 20.8¢ ✓ | 16.6¢ ✓✓ | 26.0¢ ✓ |
| 15edo | 4* | 0.0¢ ✓✓ | 18.0¢ ✗ | 13.7¢ ✓ | 18.0¢ ✓ | 4.3¢ ✓✓ | 8.8¢ ✓✓ | 36.1¢ ✗ | 26.8¢ ✗ | 4.3¢ ✓✓ | 8.7¢ ✓✓ | 22.5¢ ✓ | 22.4¢ ✓ | 9.3¢ ✓✓ | 39.5¢ ✗ | 8.8¢ ✓✓ |
| 16edo | 6 | 0.0¢ ✓✓ | 27.0¢ ✗ | 11.3¢ ✓ | 27.0¢ ✗ | 15.7¢ ✓ | 6.2¢ ✓✓ | 21.1¢ ✗ | 33.2¢ ✗ | 15.7¢ ✓✓ | 26.3¢ ✓ | 17.5¢ ✓✓ | 32.4¢ ✗ | 0.7¢ ✓✓ | 15.5¢ ✓✓ | 6.2¢ ✓✓ |
| 17edo | 7 | 0.0¢ ✓✓ | 3.9¢ ✓✓ | 33.4¢ ✗ | 3.9¢ ✓✓ | 37.3¢ ✗ | 19.4¢ ✓ | 7.9¢ ✓✓ | 15.5¢ ✓ | 37.3¢ ✗ | 13.4¢ ✓✓ | 52.8¢ ✗ | 41.3¢ ✗ | 9.5¢ ✓✓ | 6.5¢ ✓✓ | 19.4¢ ✓✓ |
| 18edo | 8 | 0.0¢ ✓✓ | 31.4¢ ✗ | 13.7¢ ✓ | 31.4¢ ✗ | 17.7¢ ✓ | 31.2¢ ✗ | 3.9¢ ✓✓ | 0.2¢ ✓✓ | 17.7¢ ✓✓ | 18.0¢ ✓✓ | 17.5¢ ✓✓ | 17.6¢ ✓✓ | 49.4¢ ✗ | 26.1¢ ✓ | 31.2¢ ✗ |
| 19edo | 5* | 0.0¢ ✓✓ | 7.2¢ ✓ | 7.4¢ ✓✓ | 7.2¢ ✓✓ | 0.2¢ ✓✓ | 21.5¢ ✗ | 14.4¢ ✓ | 14.3¢ ✓ | 0.2¢ ✓✓ | 17.1¢ ✓✓ | 14.1¢ ✓✓ | 7.0¢ ✓✓ | 24.3¢ ✓ | 19.5¢ ✓✓ | 21.5¢ ✓ |
| 20edo | 11 | 0.0¢ ✓✓ | 18.0¢ ✗ | 26.3¢ ✗ | 18.0¢ ✓ | 44.3¢ ✗ | 8.8¢ ✓✓ | 23.9¢ ✗ | 26.8¢ ✗ | 44.3¢ ✗ | 11.3¢ ✓✓ | 17.5¢ ✓✓ | 2.4¢ ✓✓ | 29.3¢ ✓ | 0.5¢ ✓✓ | 8.8¢ ✓✓ |
| 21edo | 10 | 0.0¢ ✓✓ | 16.2¢ ✗ | 13.7¢ ✓ | 16.2¢ ✓ | 29.9¢ ✗ | 2.6¢ ✓✓ | 24.7¢ ✗ | 18.8¢ ✓ | 29.9¢ ✓ | 20.1¢ ✓ | 11.1¢ ✓✓ | 11.0¢ ✓✓ | 36.3¢ ✗ | 16.6¢ ✓✓ | 2.6¢ ✓✓ |
| 22edo | 6* | 0.0¢ ✓✓ | 7.1¢ ✓ | 4.5¢ ✓✓ | 7.1¢ ✓✓ | 11.6¢ ✓ | 13.0¢ ✓ | 14.3¢ ✓ | 5.9¢ ✓✓ | 11.6¢ ✓✓ | 5.9¢ ✓✓ | 17.5¢ ✓✓ | 18.8¢ ✓✓ | 13.0¢ ✓✓ | 22.3¢ ✓ | 13.0¢ ✓✓ |
| 23edo | 15 | 0.0¢ ✓✓ | 23.7¢ ✗ | 21.1¢ ✗ | 23.7¢ ✗ | 2.6¢ ✓✓ | 22.5¢ ✗ | 4.8¢ ✓✓ | 46.2¢ ✗ | 2.6¢ ✓✓ | 22.6¢ ✓ | 43.6¢ ✗ | 25.9¢ ✓ | 46.3¢ ✗ | 5.7¢ ✓✓ | 22.5¢ ✓ |
| 24edo | 10* | 0.0¢ ✓✓ | 2.0¢ ✓✓ | 13.7¢ ✓ | 2.0¢ ✓✓ | 15.7¢ ✓ | 18.8¢ ✓ | 3.9¢ ✓✓ | 16.8¢ ✓ | 15.7¢ ✓✓ | 1.3¢ ✓✓ | 17.5¢ ✓✓ | 17.6¢ ✓✓ | 0.7¢ ✓✓ | 9.5¢ ✓✓ | 18.8¢ ✓✓ |
| 25edo | 14 | 0.0¢ ✓✓ | 18.0¢ ✗ | 2.3¢ ✓✓ | 18.0¢ ✓ | 20.3¢ ✗ | 8.8¢ ✓✓ | 11.9¢ ✓ | 26.8¢ ✗ | 20.3¢ ✓ | 23.3¢ ✓ | 6.5¢ ✓✓ | 9.6¢ ✓✓ | 41.3¢ ✗ | 23.5¢ ✓ | 8.8¢ ✓✓ |
| 26edo | 11* | 0.0¢ ✓✓ | 9.6¢ ✓ | 17.1¢ ✓ | 9.6¢ ✓✓ | 7.5¢ ✓✓ | 0.4¢ ✓✓ | 19.3¢ ✓ | 10.0¢ ✓✓ | 7.5¢ ✓✓ | 2.5¢ ✓✓ | 17.5¢ ✓✓ | 2.2¢ ✓✓ | 12.1¢ ✓✓ | 9.8¢ ✓✓ | 0.4¢ ✓✓ |
| 27edo | 12* | 0.0¢ ✓✓ | 9.2¢ ✓ | 13.7¢ ✓ | 9.2¢ ✓✓ | 4.5¢ ✓✓ | 9.0¢ ✓✓ | 18.3¢ ✓ | 0.2¢ ✓✓ | 4.5¢ ✓✓ | 18.0¢ ✓✓ | 4.7¢ ✓✓ | 4.6¢ ✓✓ | 27.2¢ ✓ | 3.9¢ ✓✓ | 9.0¢ ✓✓ |
| 28edo | 15 | 0.0¢ ✓✓ | 16.2¢ ✗ | 0.6¢ ✓✓ | 16.2¢ ✓ | 15.6¢ ✓ | 16.9¢ ✓ | 10.4¢ ✓ | 33.1¢ ✗ | 15.6¢ ✓✓ | 5.8¢ ✓✓ | 17.5¢ ✓✓ | 11.0¢ ✓✓ | 22.0¢ ✓ | 16.6¢ ✓✓ | 16.9¢ ✓✓ |
| 29edo | 14* | 0.0¢ ✓✓ | 1.5¢ ✓✓ | 13.9¢ ✓ | 1.5¢ ✓✓ | 15.4¢ ✓ | 17.1¢ ✓ | 3.0¢ ✓✓ | 18.6¢ ✓ | 15.4¢ ✓✓ | 13.4¢ ✓✓ | 3.2¢ ✓✓ | 16.9¢ ✓✓ | 14.9¢ ✓✓ | 12.9¢ ✓✓ | 17.1¢ ✓✓ |
| 30edo | 18 | 0.0¢ ✓✓ | 18.0¢ ✗ | 13.7¢ ✓ | 18.0¢ ✓ | 4.3¢ ✓✓ | 8.8¢ ✓✓ | 3.9¢ ✓✓ | 26.8¢ ✗ | 4.3¢ ✓✓ | 8.7¢ ✓✓ | 22.5¢ ✓ | 17.6¢ ✓✓ | 9.3¢ ✓✓ | 0.5¢ ✓✓ | 8.8¢ ✓✓ |
| 31edo | 16* | 0.0¢ ✓✓ | 5.2¢ ✓ | 0.8¢ ✓✓ | 5.2¢ ✓✓ | 6.0¢ ✓✓ | 1.1¢ ✓✓ | 10.4¢ ✓ | 4.1¢ ✓✓ | 6.0¢ ✓✓ | 9.4¢ ✓✓ | 1.9¢ ✓✓ | 11.2¢ ✓✓ | 4.2¢ ✓✓ | 11.1¢ ✓✓ | 1.1¢ ✓✓ |
| 32edo | 19 | 0.0¢ ✓✓ | 10.5¢ ✗ | 11.3¢ ✓ | 10.5¢ ✓ | 21.8¢ ✗ | 6.2¢ ✓✓ | 16.4¢ ✓ | 4.3¢ ✓✓ | 21.8¢ ✓ | 11.2¢ ✓✓ | 17.5¢ ✓✓ | 5.1¢ ✓✓ | 0.7¢ ✓✓ | 15.5¢ ✓✓ | 6.2¢ ✓✓ |
| 33edo | 21 | 0.0¢ ✓✓ | 11.0¢ ✗ | 13.7¢ ✓ | 11.0¢ ✓ | 24.7¢ ✗ | 13.0¢ ✓ | 14.3¢ ✓ | 24.0¢ ✗ | 24.7¢ ✓ | 5.9¢ ✓✓ | 0.7¢ ✓✓ | 0.6¢ ✓✓ | 5.1¢ ✓✓ | 4.2¢ ✓✓ | 13.0¢ ✓✓ |
| 34edo | 19* | 0.0¢ ✓✓ | 3.9¢ ✓✓ | 1.9¢ ✓✓ | 3.9¢ ✓✓ | 2.0¢ ✓✓ | 15.9¢ ✓ | 7.9¢ ✓✓ | 19.8¢ ✓ | 2.0¢ ✓✓ | 13.4¢ ✓✓ | 17.8¢ ✓✓ | 6.0¢ ✓✓ | 9.5¢ ✓✓ | 6.5¢ ✓✓ | 15.9¢ ✓✓ |
| 35edo | 21 | 0.0¢ ✓✓ | 16.2¢ ✗ | 9.2¢ ✓✓ | 16.2¢ ✓ | 7.0¢ ✓✓ | 8.8¢ ✓✓ | 1.8¢ ✓✓ | 7.4¢ ✓✓ | 7.0¢ ✓✓ | 2.7¢ ✓✓ | 0.4¢ ✓✓ | 11.0¢ ✓✓ | 13.5¢ ✓✓ | 16.6¢ ✓✓ | 8.8¢ ✓✓ |
| 36edo | 21* | 0.0¢ ✓✓ | 2.0¢ ✓✓ | 13.7¢ ✓ | 2.0¢ ✓✓ | 15.7¢ ✓ | 2.2¢ ✓✓ | 3.9¢ ✓✓ | 0.2¢ ✓✓ | 15.7¢ ✓✓ | 15.3¢ ✓✓ | 15.9¢ ✓✓ | 17.6¢ ✓✓ | 17.3¢ ✓✓ | 7.2¢ ✓✓ | 2.2¢ ✓✓ |
| 37edo | 23 | 0.0¢ ✓✓ | 11.6¢ ✗ | 2.9¢ ✓✓ | 11.6¢ ✓ | 8.7¢ ✓✓ | 4.1¢ ✓✓ | 9.3¢ ✓✓ | 7.5¢ ✓✓ | 8.7¢ ✓✓ | 0.0¢ ✓✓ | 1.2¢ ✓✓ | 12.2¢ ✓✓ | 11.6¢ ✓✓ | 2.7¢ ✓✓ | 4.1¢ ✓✓ |
| 38edo | 23 | 0.0¢ ✓✓ | 7.2¢ ✓ | 7.4¢ ✓✓ | 7.2¢ ✓✓ | 0.2¢ ✓✓ | 10.1¢ ✓ | 14.4¢ ✓ | 17.3¢ ✓ | 0.2¢ ✓✓ | 14.5¢ ✓✓ | 17.5¢ ✓✓ | 7.0¢ ✓✓ | 7.3¢ ✓✓ | 12.1¢ ✓✓ | 10.1¢ ✓✓ |
| 39edo | 25 | 0.0¢ ✓✓ | 5.7¢ ✓ | 13.7¢ ✓ | 5.7¢ ✓✓ | 8.0¢ ✓✓ | 15.0¢ ✓ | 11.5¢ ✓ | 20.7¢ ✗ | 8.0¢ ✓✓ | 2.5¢ ✓✓ | 28.7¢ ✓ | 2.2¢ ✓✓ | 3.2¢ ✓✓ | 9.8¢ ✓✓ | 15.0¢ ✓✓ |
| 40edo | 26 | 0.0¢ ✓✓ | 12.0¢ ✗ | 3.7¢ ✓✓ | 12.0¢ ✓ | 15.7¢ ✓ | 8.8¢ ✓✓ | 6.1¢ ✓✓ | 3.2¢ ✓✓ | 15.7¢ ✓✓ | 11.3¢ ✓✓ | 12.5¢ ✓✓ | 2.4¢ ✓✓ | 0.7¢ ✓✓ | 0.5¢ ✓✓ | 8.8¢ ✓✓ |
| 41edo | 26* | 0.0¢ ✓✓ | 0.5¢ ✓✓ | 5.8¢ ✓✓ | 0.5¢ ✓✓ | 6.3¢ ✓✓ | 3.0¢ ✓✓ | 1.0¢ ✓✓ | 3.5¢ ✓✓ | 6.3¢ ✓✓ | 4.8¢ ✓✓ | 2.8¢ ✓✓ | 6.8¢ ✓✓ | 4.3¢ ✓✓ | 8.3¢ ✓✓ | 3.0¢ ✓✓ |
| 42edo | 28 | 0.0¢ ✓✓ | 12.3¢ ✗ | 13.7¢ ✓ | 12.3¢ ✓ | 1.4¢ ✓✓ | 2.6¢ ✓✓ | 3.9¢ ✓✓ | 9.7¢ ✓✓ | 1.4¢ ✓✓ | 8.5¢ ✓✓ | 11.1¢ ✓✓ | 17.6¢ ✓✓ | 20.8¢ ✓ | 12.0¢ ✓✓ | 2.6¢ ✓✓ |
| 43edo | 28* | 0.0¢ ✓✓ | 4.3¢ ✓✓ | 4.4¢ ✓✓ | 4.3¢ ✓✓ | 8.7¢ ✓✓ | 7.9¢ ✓✓ | 8.6¢ ✓✓ | 12.2¢ ✓ | 8.7¢ ✓✓ | 6.8¢ ✓✓ | 3.5¢ ✓✓ | 13.0¢ ✓✓ | 11.1¢ ✓✓ | 3.3¢ ✓✓ | 7.9¢ ✓✓ |
| 44edo | 29 | 0.0¢ ✓✓ | 7.1¢ ✓ | 4.5¢ ✓✓ | 7.1¢ ✓✓ | 11.6¢ ✓ | 13.0¢ ✓ | 13.0¢ ✓ | 5.9¢ ✓✓ | 11.6¢ ✓✓ | 5.9¢ ✓✓ | 17.5¢ ✓✓ | 8.5¢ ✓✓ | 13.0¢ ✓✓ | 4.9¢ ✓✓ | 13.0¢ ✓✓ |
| 45edo | 30 | 0.0¢ ✓✓ | 8.6¢ ✓ | 13.0¢ ✓ | 8.6¢ ✓✓ | 4.4¢ ✓✓ | 8.8¢ ✓✓ | 9.4¢ ✓✓ | 0.2¢ ✓✓ | 4.4¢ ✓✓ | 8.7¢ ✓✓ | 4.2¢ ✓✓ | 22.4¢ ✓ | 17.3¢ ✓✓ | 12.8¢ ✓✓ | 8.8¢ ✓✓ |
| 46edo | 31* | 0.0¢ ✓✓ | 2.4¢ ✓✓ | 5.0¢ ✓✓ | 2.4¢ ✓✓ | 2.6¢ ✓✓ | 3.6¢ ✓✓ | 4.8¢ ✓✓ | 6.0¢ ✓✓ | 2.6¢ ✓✓ | 3.5¢ ✓✓ | 8.6¢ ✓✓ | 0.2¢ ✓✓ | 5.9¢ ✓✓ | 5.7¢ ✓✓ | 3.6¢ ✓✓ |
| 47edo | 33 | 0.0¢ ✓✓ | 12.6¢ ✗ | 3.3¢ ✓✓ | 12.6¢ ✓ | 9.3¢ ✓✓ | 1.4¢ ✓✓ | 0.3¢ ✓✓ | 14.0¢ ✓ | 9.3¢ ✓✓ | 10.4¢ ✓✓ | 4.7¢ ✓✓ | 3.6¢ ✓✓ | 23.0¢ ✓ | 2.0¢ ✓✓ | 1.4¢ ✓✓ |
| 48edo | 33* | 0.0¢ ✓✓ | 2.0¢ ✓✓ | 11.3¢ ✓ | 2.0¢ ✓✓ | 9.3¢ ✓✓ | 6.2¢ ✓✓ | 3.9¢ ✓✓ | 8.2¢ ✓✓ | 9.3¢ ✓✓ | 1.3¢ ✓✓ | 17.5¢ ✓✓ | 7.4¢ ✓✓ | 0.7¢ ✓✓ | 9.5¢ ✓✓ | 6.2¢ ✓✓ |
| 49edo | 34 | 0.0¢ ✓✓ | 8.2¢ ✓ | 5.5¢ ✓✓ | 8.2¢ ✓✓ | 2.7¢ ✓✓ | 10.8¢ ✓ | 8.0¢ ✓✓ | 2.6¢ ✓✓ | 2.7¢ ✓✓ | 11.9¢ ✓✓ | 5.3¢ ✓✓ | 13.5¢ ✓✓ | 3.7¢ ✓✓ | 7.9¢ ✓✓ | 10.8¢ ✓✓ |
| 50edo | 35 | 0.0¢ ✓✓ | 6.0¢ ✓ | 2.3¢ ✓✓ | 6.0¢ ✓✓ | 3.7¢ ✓✓ | 8.8¢ ✓✓ | 11.9¢ ✓ | 2.8¢ ✓✓ | 3.7¢ ✓✓ | 0.7¢ ✓✓ | 6.5¢ ✓✓ | 9.6¢ ✓✓ | 6.7¢ ✓✓ | 0.5¢ ✓✓ | 8.8¢ ✓✓ |
| 51edo | 36 | 0.0¢ ✓✓ | 3.9¢ ✓✓ | 9.8¢ ✓✓ | 3.9¢ ✓✓ | 13.7¢ ✓ | 4.1¢ ✓✓ | 7.9¢ ✓✓ | 8.0¢ ✓✓ | 13.7¢ ✓✓ | 10.1¢ ✓✓ | 5.7¢ ✓✓ | 17.7¢ ✓✓ | 14.0¢ ✓✓ | 6.5¢ ✓✓ | 4.1¢ ✓✓ |
| 52edo | 37 | 0.0¢ ✓✓ | 9.6¢ ✓ | 6.0¢ ✓✓ | 9.6¢ ✓✓ | 15.6¢ ✓ | 0.4¢ ✓✓ | 3.8¢ ✓✓ | 10.0¢ ✓✓ | 15.6¢ ✓✓ | 2.5¢ ✓✓ | 5.6¢ ✓✓ | 2.2¢ ✓✓ | 12.1¢ ✓✓ | 9.8¢ ✓✓ | 0.4¢ ✓✓ |
| 53edo | 38^ | 0.0¢ ✓✓ | 0.1¢ ✓✓ | 1.4¢ ✓✓ | 0.1¢ ✓✓ | 1.3¢ ✓✓ | 4.8¢ ✓✓ | 0.1¢ ✓✓ | 4.8¢ ✓✓ | 1.3¢ ✓✓ | 7.9¢ ✓✓ | 6.2¢ ✓✓ | 1.3¢ ✓✓ | 7.8¢ ✓✓ | 2.8¢ ✓✓ | 4.8¢ ✓✓ |
Prompt:
Use the attached PDF to fulfil this request: The following are all the 16-integer-limit consonant intervals available in an octave, simplified to their simplest form (by taking them to a higher octave to simplify the fraction), then sorted with most consonant (ie mathematically simplest) first. (^in a higher octave) HYPERCONSONANCES Within 5 cents error = excellent Within 10 cents = good enough More than 10 cents = poor 2/1 (octave) 3/1 (perf 5th^) CONSONANCES Within 10 cents = excellent Within 20 cents = good enough More than 20 cents = poor 5/1 (maj 3rd^) 4/3 (perf 4th) 5/3 (maj 6th) 7/1 (submin 7th^) 9/1 (large maj 2nd^) 7/3 (submin 3rd^) AMBISONANCES Within 20 cents = excellent Within 30 cents = good enough More than 30 cents = poor 6/5 (min 3rd) 11/1 (undec 4th^) 7/5 (small tritone) 9/5 (min 7th) 11/3 (neu 7th^) 13/1 (neu 6th^) 8/7 (supmaj 2nd) DISSONANCES Any note not within range of one of the above. The total number of notes in a tuning minus 10 is a good heuristic to estimate how many of these there are. Less than 5 of these = excellent 5 to 15 of these = good enough More than 15 of these = poor For 2/1 all pure-octave tunings (eg EDOs) will be excellent. For any interval N/1 you can work it out by reading the absolute error from the Nth harmonic on the tables. For any interval L/M you can figure it out by reading the abs error of L and the abs error of M and then simply adding the two together. Please make a wikitext table which shows how well every hyperconsonance, consonance, and ambisonance is approximated in every EDO from 1edo to 53edo, as well as showing each of their estimated number of dissonances.
2nd prompt:
Correct the numbers provided in the wikitext table. Use the attached PDF to calculate the correct numbers. All the columns N/1 (eg 2/1, 3/1) are already correct. For the other columns, some cells are correct and some not. What you need to do is subtract the error (in cents (¢)) of the two harmonics involved, but don't disregard their direction. For example if harmonic 5 has +10¢ error and harmonic 3 has -9¢ error, then the total error of interval 5/3 is 19¢ (opposite errors add). If harmonic 5 has +10¢ error and harmonic 3 has +9¢ error, then the total error of interval 5/3 is 1¢ (same-direction errors subtract).