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388edo is the first edo that is [[consistency|distinctly consistent]] through to the [[27-odd-limit]]; it is also consistent through the [[37-odd-limit]].
388edo is the first edo that is [[consistency|distinctly consistent]] through to the [[27-odd-limit]]; it is also consistent through the [[37-odd-limit]].


The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[tricot comma]], {{monzo| 39 -29 3 }}, the [[minortone comma]], {{monzo| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider|raider comma]], {{monzo| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.  
The equal temperament [[tempering out|tempers out]] the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[alphatricot comma]], {{monzo| 39 -29 3 }}, the [[minortone comma]], {{monzo| -16 35 -17 }}, and the [[Very high accuracy temperaments #Raider|raider comma]], {{monzo| 71 -99 31 }}, in the 5-limit, giving a strong tuning. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and [[3025/3024]], [[5632/5625]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit.  


It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap|190 & 198}} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].
It provides the [[optimal patent val]] for the rank-5 cuthbert temperament, which tempers out [[847/845]], the cuthbert comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the {{nowrap| 190 & 198 }} temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 388 factors into {{factorization|388}}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.  
Since 388 factors into {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.  


== Regular temperament properties ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 615 -388 }}
| {{Monzo| 615 -388 }}
| {{mapping| 388 615 }}
| {{Mapping| 388 615 }}
| +0.0337
| +0.0337
| 0.0337
| 0.0337
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
| {{Monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
| {{mapping| 388 615 901 }}
| {{Mapping| 388 615 901 }}
| −0.0633
| −0.0633
| 0.0501
| 0.0501
Line 43: Line 43:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 235298/234375, 2100875/2097152
| 4375/4374, 235298/234375, 2100875/2097152
| {{mapping| 388 615 901 1089 }}
| {{Mapping| 388 615 901 1089 }}
| +0.0224
| +0.0224
| 0.1546
| 0.1546
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5632/5625, 235298/234375
| 3025/3024, 4375/4374, 5632/5625, 235298/234375
| {{mapping| 388 615 901 1089 1342 }}
| {{Mapping| 388 615 901 1089 1342 }}
| +0.0643
| +0.0643
| 0.1617
| 0.1617
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374
| 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374
| {{mapping| 388 615 901 1089 1342 1436 }}
| {{Mapping| 388 615 901 1089 1342 1436 }}
| +0.0216
| +0.0216
| 0.1758
| 0.1758
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700
| 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700
| {{mapping| 388 615 901 1089 1342 1436 1586 }}
| {{Mapping| 388 615 901 1089 1342 1436 1586 }}
| +0.0116
| +0.0116
| 0.1646
| 0.1646
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330
| 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330
| {{mapping| 388 615 901 1089 1342 1436 1586 1648 }}
| {{Mapping| 388 615 901 1089 1342 1436 1586 1648 }}
| +0.0280
| +0.0280
| 0.1600
| 0.1600
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 565.97
| 565.97
| 75/52
| 75/52
| [[Trillium]] / [[pseudotrillium]]
| [[Alphatrillium]] / [[pseudotrillium]]
|-
|-
| 2
| 2
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|-
|-
| 4
| 4
| 123\388<br />(26\388)
| 123\388<br>(26\388)
| 380.41<br />(80.41)
| 380.41<br>(80.41)
| 81/65<br />(22/21)
| 81/65<br>(22/21)
| [[Quasithird]]
| [[Quasithird]]
|-
|-
| 97
| 97
| 161\388<br />(1\388)
| 161\388<br>(1\388)
| 497.938<br />(3.09)
| 497.938<br>(3.09)
| 4/3<br />(?)
| 4/3<br>(?)
| [[Berkelium]]
| [[Berkelium]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Cuthbert]]
[[Category:Cuthbert]]
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