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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.
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.
== EDO Consonance Ratings (1–53edo) ==
Table created using Claude, prompt listed below table.
{| class="wikitable sortable" style="text-align:center; font-size:80%;"
|-
! rowspan=2 | EDO !! colspan=2 style="background:#c8f0c8;" | Hyperconsonances !! colspan=6 style="background:#d0e8ff;" | Consonances !! colspan=8 style="background:#fff0c8;" | Ambisonances !! rowspan=2 | Dissonances<br/>(EDO−10)
|-
! style="background:#c8f0c8;" | 2/1<br/><small>perf oct</small>
! style="background:#c8f0c8;" | 3/1<br/><small>perf 5th^</small>
! style="background:#d0e8ff;" | 5/1<br/><small>maj 3rd^</small>
! style="background:#d0e8ff;" | 4/3<br/><small>perf 4th</small>
! style="background:#d0e8ff;" | 5/3<br/><small>maj 6th</small>
! style="background:#d0e8ff;" | 7/1<br/><small>submin 7th^</small>
! style="background:#d0e8ff;" | 9/1<br/><small>lg maj 2nd^</small>
! style="background:#d0e8ff;" | 7/3<br/><small>submin 3rd^</small>
! style="background:#fff0c8;" | 6/5<br/><small>min 3rd</small>
! style="background:#fff0c8;" | 11/1<br/><small>undec 4th^</small>
! style="background:#fff0c8;" | 7/5<br/><small>sm tritone</small>
! style="background:#fff0c8;" | 9/5<br/><small>min 7th</small>
! style="background:#fff0c8;" | 11/3<br/><small>neu 7th^</small>
! style="background:#fff0c8;" | 13/1<br/><small>neu 6th^</small>
! style="background:#fff0c8;" | 8/7<br/><small>supmaj 2nd</small>
! style="background:#fff0c8;" | 13/3<br/><small>lg tri neu 2nd^</small>
|-
| '''1'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 498.0¢ ✗
| style="background:#FF9999;" | 386.0¢ ✗
| style="background:#FF9999;" | 498.0¢ ✗
| style="background:#FF9999;" | 884.0¢ ✗
| style="background:#FF9999;" | 231.0¢ ✗
| style="background:#FF9999;" | 204.0¢ ✗
| style="background:#FF9999;" | 729.0¢ ✗
| style="background:#FF9999;" | 884.0¢ ✗
| style="background:#FF9999;" | 551.0¢ ✗
| style="background:#FF9999;" | 617.0¢ ✗
| style="background:#FF9999;" | 590.0¢ ✗
| style="background:#FF9999;" | 1049.0¢ ✗
| style="background:#FF9999;" | 359.0¢ ✗
| style="background:#FF9999;" | 231.0¢ ✗
| style="background:#FF9999;" | 857.0¢ ✗
| style="background:#90EE90;" | -9
|-
| '''2'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 102.0¢ ✗
| style="background:#FF9999;" | 214.0¢ ✗
| style="background:#FF9999;" | 102.0¢ ✗
| style="background:#FF9999;" | 316.0¢ ✗
| style="background:#FF9999;" | 231.0¢ ✗
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| style="background:#FF9999;" | 333.0¢ ✗
| style="background:#FF9999;" | 316.0¢ ✗
| style="background:#FF9999;" | 49.0¢ ✗
| style="background:#FF9999;" | 445.0¢ ✗
| style="background:#FF9999;" | 418.0¢ ✗
| style="background:#FF9999;" | 151.0¢ ✗
| style="background:#FF9999;" | 241.0¢ ✗
| style="background:#FF9999;" | 231.0¢ ✗
| style="background:#FF9999;" | 343.0¢ ✗
| style="background:#90EE90;" | -8
|-
| '''3'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 98.0¢ ✗
| style="background:#FFD700;" | 14.0¢ ✓
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| style="background:#FF9999;" | 112.0¢ ✗
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| style="background:#FF9999;" | 210.0¢ ✗
| style="background:#FF9999;" | 249.0¢ ✗
| style="background:#FF9999;" | 41.0¢ ✗
| style="background:#FF9999;" | 169.0¢ ✗
| style="background:#FF9999;" | 139.0¢ ✗
| style="background:#90EE90;" | -7
|-
| '''4'''
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| style="background:#FF9999;" | 151.0¢ ✗
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| style="background:#FF9999;" | 69.0¢ ✗
| style="background:#FF9999;" | 161.0¢ ✗
| style="background:#90EE90;" | -6
|-
| '''5'''
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| style="background:#FF9999;" | 18.0¢ ✗
| style="background:#FF9999;" | 94.0¢ ✗
| style="background:#FFD700;" | 18.0¢ ✓
| style="background:#FF9999;" | 112.0¢ ✗
| style="background:#90EE90;" | 9.0¢ ✓✓
| style="background:#FF9999;" | 36.0¢ ✗
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| style="background:#FF9999;" | 112.0¢ ✗
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| style="background:#FF9999;" | 137.0¢ ✗
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| '''6'''
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| style="background:#90EE90;" | -4
|-
| '''7'''
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| '''8'''
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| style="background:#FF9999;" | 48.0¢ ✗
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| '''9'''
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| '''10'''
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| '''11'''
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| style="background:#FF9999;" | 97.4¢ ✗
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| style="background:#FF9999;" | 60.4¢ ✗
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| '''12'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 13.7¢ ✓
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| style="background:#FF9999;" | 31.2¢ ✗
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| '''13'''
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| '''14'''
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| '''15'''
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| style="background:#FFD700;" | 15.7¢ ✓
| style="background:#FFD700;" | 18.8¢ ✓
| style="background:#90EE90;" | 3.9¢ ✓✓
| style="background:#FF9999;" | 20.8¢ ✗
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| style="background:#FF9999;" | 32.5¢ ✗
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| '''25'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#90EE90;" | 2.3¢ ✓✓
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| style="background:#FF9999;" | 26.8¢ ✗
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| style="background:#FF9999;" | 41.3¢ ✗
| style="background:#FFD700;" | 23.5¢ ✓
| style="background:#90EE90;" | 8.8¢ ✓✓
| style="background:#FF9999;" | 41.5¢ ✗
| style="background:#FFD700;" | 15
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| '''26'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 17.1¢ ✓
| style="background:#90EE90;" | 9.6¢ ✓✓
| style="background:#FF9999;" | 26.7¢ ✗
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| style="background:#FFD700;" | 19.3¢ ✓
| style="background:#90EE90;" | 10.0¢ ✓✓
| style="background:#FFD700;" | 26.7¢ ✓
| style="background:#90EE90;" | 2.5¢ ✓✓
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| style="background:#FF9999;" | 36.4¢ ✗
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| style="background:#FF9999;" | 16
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| '''27'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#90EE90;" | 9.2¢ ✓✓
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| style="background:#90EE90;" | 9.0¢ ✓✓
| style="background:#FFD700;" | 18.3¢ ✓
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| style="background:#FFD700;" | 22.9¢ ✓
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| style="background:#FFD700;" | 22.7¢ ✓
| style="background:#FF9999;" | 32.0¢ ✗
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| '''28'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 16.2¢ ✓
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| style="background:#FF9999;" | 33.1¢ ✗
| style="background:#90EE90;" | 16.8¢ ✓✓
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| style="background:#FF9999;" | 18
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| '''29'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 13.9¢ ✓
| style="background:#90EE90;" | 1.5¢ ✓✓
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| style="background:#90EE90;" | 13.4¢ ✓✓
| style="background:#FF9999;" | 31.0¢ ✗
| style="background:#90EE90;" | 16.9¢ ✓✓
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| style="background:#FF9999;" | 19
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| '''30'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 18.0¢ ✗
| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#FFD700;" | 18.0¢ ✓
| style="background:#FF9999;" | 31.7¢ ✗
| style="background:#90EE90;" | 8.8¢ ✓✓
| style="background:#90EE90;" | 3.9¢ ✓✓
| style="background:#FF9999;" | 26.8¢ ✗
| style="background:#FF9999;" | 31.7¢ ✗
| style="background:#90EE90;" | 8.7¢ ✓✓
| style="background:#FFD700;" | 22.5¢ ✓
| style="background:#90EE90;" | 17.6¢ ✓✓
| style="background:#FFD700;" | 26.7¢ ✓
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| style="background:#90EE90;" | 8.8¢ ✓✓
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| style="background:#FF9999;" | 20
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| '''31'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FF9999;" | 21
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| '''32'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 10.5¢ ✗
| style="background:#FFD700;" | 11.3¢ ✓
| style="background:#FFD700;" | 10.5¢ ✓
| style="background:#FF9999;" | 21.8¢ ✗
| style="background:#90EE90;" | 6.2¢ ✓✓
| style="background:#FFD700;" | 16.4¢ ✓
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| style="background:#FFD700;" | 21.8¢ ✓
| style="background:#90EE90;" | 11.2¢ ✓✓
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| style="background:#FF9999;" | 22
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| '''33'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 11.0¢ ✗
| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#FFD700;" | 11.0¢ ✓
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| style="background:#FFD700;" | 13.0¢ ✓
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| style="background:#FF9999;" | 24.0¢ ✗
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| style="background:#FFD700;" | 26.7¢ ✓
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| '''34'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#90EE90;" | 3.9¢ ✓✓
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| '''35'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#90EE90;" | 9.2¢ ✓✓
| style="background:#FFD700;" | 16.2¢ ✓
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| style="background:#90EE90;" | 8.8¢ ✓✓
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| style="background:#FF9999;" | 25
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| '''36'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#90EE90;" | 2.0¢ ✓✓
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| '''37'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 11.6¢ ✗
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| style="background:#FFD700;" | 11.6¢ ✓
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| style="background:#FF9999;" | 27
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| '''38'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#90EE90;" | 12.1¢ ✓✓
| style="background:#90EE90;" | 10.1¢ ✓✓
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| style="background:#FF9999;" | 28
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| '''39'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#90EE90;" | 5.7¢ ✓✓
| style="background:#FFD700;" | 19.4¢ ✓
| style="background:#FFD700;" | 15.0¢ ✓
| style="background:#FFD700;" | 11.5¢ ✓
| style="background:#FF9999;" | 20.7¢ ✗
| style="background:#90EE90;" | 19.4¢ ✓✓
| style="background:#90EE90;" | 2.5¢ ✓✓
| style="background:#FFD700;" | 28.7¢ ✓
| style="background:#FFD700;" | 25.2¢ ✓
| style="background:#90EE90;" | 8.2¢ ✓✓
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| style="background:#FF9999;" | 29
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| '''40'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 12.0¢ ✗
| style="background:#90EE90;" | 3.7¢ ✓✓
| style="background:#FFD700;" | 12.0¢ ✓
| style="background:#FFD700;" | 15.7¢ ✓
| style="background:#90EE90;" | 8.8¢ ✓✓
| style="background:#90EE90;" | 6.1¢ ✓✓
| style="background:#FF9999;" | 20.8¢ ✗
| style="background:#90EE90;" | 15.7¢ ✓✓
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| style="background:#90EE90;" | 8.8¢ ✓✓
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| style="background:#FF9999;" | 30
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| '''41'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#90EE90;" | 0.5¢ ✓✓
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| style="background:#FF9999;" | 31
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| '''42'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 12.3¢ ✗
| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#FFD700;" | 12.3¢ ✓
| style="background:#FF9999;" | 26.0¢ ✗
| style="background:#90EE90;" | 2.6¢ ✓✓
| style="background:#90EE90;" | 3.9¢ ✓✓
| style="background:#FFD700;" | 14.9¢ ✓
| style="background:#FFD700;" | 26.0¢ ✓
| style="background:#90EE90;" | 8.5¢ ✓✓
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| style="background:#FFD700;" | 20.8¢ ✓
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| style="background:#FF9999;" | 32
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| '''43'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#FFD700;" | 12.2¢ ✓
| style="background:#90EE90;" | 8.7¢ ✓✓
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| style="background:#90EE90;" | 7.9¢ ✓✓
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| style="background:#FF9999;" | 33
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| '''44'''
| style="background:#90EE90;" | 0.0¢ ✓✓
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| style="background:#90EE90;" | 7.1¢ ✓✓
| style="background:#FFD700;" | 11.6¢ ✓
| style="background:#FFD700;" | 13.0¢ ✓
| style="background:#FFD700;" | 13.0¢ ✓
| style="background:#FF9999;" | 20.1¢ ✗
| style="background:#90EE90;" | 11.6¢ ✓✓
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| style="background:#90EE90;" | 13.0¢ ✓✓
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| style="background:#FF9999;" | 34
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| '''45'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FFD700;" | 8.6¢ ✓
| style="background:#FFD700;" | 13.0¢ ✓
| style="background:#90EE90;" | 8.6¢ ✓✓
| style="background:#FF9999;" | 21.6¢ ✗
| style="background:#90EE90;" | 8.8¢ ✓✓
| style="background:#90EE90;" | 9.4¢ ✓✓
| style="background:#FFD700;" | 17.4¢ ✓
| style="background:#FFD700;" | 21.6¢ ✓
| style="background:#90EE90;" | 8.7¢ ✓✓
| style="background:#FFD700;" | 21.8¢ ✓
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| style="background:#90EE90;" | 17.3¢ ✓✓
| style="background:#90EE90;" | 12.8¢ ✓✓
| style="background:#90EE90;" | 8.8¢ ✓✓
| style="background:#FFD700;" | 21.4¢ ✓
| style="background:#FF9999;" | 35
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| '''46'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#90EE90;" | 2.4¢ ✓✓
| style="background:#90EE90;" | 5.0¢ ✓✓
| style="background:#90EE90;" | 2.4¢ ✓✓
| style="background:#90EE90;" | 7.4¢ ✓✓
| style="background:#90EE90;" | 3.6¢ ✓✓
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| style="background:#90EE90;" | 6.0¢ ✓✓
| style="background:#90EE90;" | 7.4¢ ✓✓
| style="background:#90EE90;" | 3.5¢ ✓✓
| style="background:#90EE90;" | 8.6¢ ✓✓
| style="background:#90EE90;" | 9.8¢ ✓✓
| style="background:#90EE90;" | 5.9¢ ✓✓
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| style="background:#90EE90;" | 3.6¢ ✓✓
| style="background:#90EE90;" | 8.1¢ ✓✓
| style="background:#FF9999;" | 36
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| '''47'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FF9999;" | 12.6¢ ✗
| style="background:#90EE90;" | 3.3¢ ✓✓
| style="background:#FFD700;" | 12.6¢ ✓
| style="background:#FFD700;" | 15.9¢ ✓
| style="background:#90EE90;" | 1.4¢ ✓✓
| style="background:#90EE90;" | 0.3¢ ✓✓
| style="background:#FFD700;" | 14.0¢ ✓
| style="background:#90EE90;" | 15.9¢ ✓✓
| style="background:#90EE90;" | 10.4¢ ✓✓
| style="background:#90EE90;" | 4.7¢ ✓✓
| style="background:#90EE90;" | 3.6¢ ✓✓
| style="background:#FFD700;" | 23.0¢ ✓
| style="background:#90EE90;" | 2.0¢ ✓✓
| style="background:#90EE90;" | 1.4¢ ✓✓
| style="background:#90EE90;" | 14.6¢ ✓✓
| style="background:#FF9999;" | 37
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| '''48'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#90EE90;" | 2.0¢ ✓✓
| style="background:#FFD700;" | 11.3¢ ✓
| style="background:#90EE90;" | 2.0¢ ✓✓
| style="background:#FFD700;" | 13.3¢ ✓
| style="background:#90EE90;" | 6.2¢ ✓✓
| style="background:#90EE90;" | 3.9¢ ✓✓
| style="background:#90EE90;" | 8.2¢ ✓✓
| style="background:#90EE90;" | 13.3¢ ✓✓
| style="background:#90EE90;" | 1.3¢ ✓✓
| style="background:#90EE90;" | 17.5¢ ✓✓
| style="background:#90EE90;" | 15.2¢ ✓✓
| style="background:#90EE90;" | 3.3¢ ✓✓
| style="background:#90EE90;" | 9.5¢ ✓✓
| style="background:#90EE90;" | 6.2¢ ✓✓
| style="background:#90EE90;" | 11.5¢ ✓✓
| style="background:#FF9999;" | 38
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| '''49'''
| style="background:#90EE90;" | 0.0¢ ✓✓
| style="background:#FFD700;" | 8.2¢ ✓
| style="background:#90EE90;" | 5.5¢ ✓✓
| style="background:#90EE90;" | 8.2¢ ✓✓
| style="background:#FFD700;" | 13.7¢ ✓
| style="background:#FFD700;" | 10.8¢ ✓
| style="background:#90EE90;" | 8.0¢ ✓✓
| style="background:#FFD700;" | 19.0¢ ✓
| style="background:#90EE90;" | 13.7¢ ✓✓
| style="background:#90EE90;" | 11.9¢ ✓✓
| style="background:#90EE90;" | 16.3¢ ✓✓
| style="background:#90EE90;" | 13.5¢ ✓✓
| style="background:#FFD700;" | 20.1¢ ✓
| style="background:#90EE90;" | 7.9¢ ✓✓
| style="background:#90EE90;" | 10.8¢ ✓✓
| style="background:#90EE90;" | 16.1¢ ✓✓
| style="background:#FF9999;" | 39
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Prompt:
<pre>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)
13/3 (large tridec neu 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.</pre>
The PDF I gave Claude was the odd harmonics tables for every EDO 1 to 53,
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