388edo: Difference between revisions

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The '''388 equal divisions of the octave''' ('''388EDO'''), or the '''388(-tone) equal temperament''' ('''388TET''', '''388ET''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 388 [[equal]] parts of 3.0928 [[cent]]s each.  
The '''388 equal divisions of the octave''' ('''388edo'''), or the '''388(-tone) equal temperament''' ('''388tet''', '''388et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 388 [[equal]] parts of 3.0928 [[cent]]s each.  


388EDO is the first EDO that is uniquely [[consistent]] through to the [[27-odd-limit]]; it is also consistent through the 37-odd-limit.
== Theory ==
388edo is the first edo that is uniquely [[consistent]] through to the [[27-odd-limit]]; it is also consistent through the 37-odd-limit.


388EDO 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 comma]], {{monzo| 71 -99 31 }}, in the 5-limit, and provides a tuning with less error than any previous equal temperaments. It tempers out [[4375/4374]] and [[Triwellismic temperaments|235298/234375]] in the 7-limit, and 5632/5625, [[3025/3024]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit. It is the [[optimal patent val]] for cuthbert temperament, which tempers out cuthbert, the 847/845 comma, and for a number of other temperaments tempering out cuthbert, e.g. 198&388. By tempering out cuthbert it [[support]]s the [[cuthbert triad]].
388et 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, and provides a tuning with less error than any previous equal temperaments. It tempers out [[4375/4374]] and 235298/234375 in the 7-limit, and 5632/5625, [[3025/3024]] and [[9801/9800]] in the 11-limit and [[847/845]], [[1001/1000]] and [[4096/4095]] in the 13-limit. It is the [[optimal patent val]] for cuthbert temperament, which tempers out cuthbert, the 847/845 comma, and for a number of other temperaments tempering out cuthbert, e.g. [[neusec]], the 190&198 temperament. By tempering out cuthbert it [[support]]s the [[cuthbert triad]], in addition to [[sinbadmic chords]].


{{Primes in edo|edo=388|columns=11|start=2|prec=3}}
=== Prime harmonics ===
{{Harmonics in equal|388|columns=11}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 615 -388 }}
| [{{val| 388 615 }}]
| +0.0337
| 0.0337
| 1.09
|-
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
| [{{val| 388 615 901 }}]
| -0.0633
| 0.0501
| 1.62
|-
| 2.3.5.7
| 4375/4374, 235298/234375, 2100875/2097152
| [{{val| 388 615 901 1089 }}]
| +0.0224
| 0.1546
| 5.00
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5632/5625, 235298/234375
| [{{val| 388 615 901 1089 1342 }}]
| +0.0643
| 0.1617
| 5.23
|-
| 2.3.5.7.11.13
| 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374
| [{{val| 388 615 901 1089 1342 1436 }}]
| +0.0216
| 0.1758
| 5.68
|-
| 2.3.5.7.11.13.17
| 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700
| [{{val| 388 615 901 1089 1342 1436 1586 }}]
| +0.0116
| 0.1646
| 5.32
|-
| 2.3.5.7.11.13.17.19
| 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330
| [{{val| 388 615 901 1089 1342 1436 1586 1648 }}]
| +0.0280
| 0.1600
| 5.17
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 59\388
| 182.47
| 10/9
| [[Mitonic]]
|-
| 1
| 111\388
| 343.30
| 8000/6561
| [[Raider]]
|-
| 1
| 145\388
| 448.45
| 35/27
| [[Semidimfourth]]
|-
| 1
| 183\388
| 565.97
| 75/52
| [[Trillium]] / [[pseudotrillium]]
|-
| 2
| 23\388
| 71.13
| 25/24
| [[Vishnu]] / [[ananta]]
|-
| 2
| 49\388
| 151.54
| 12/11
| [[Neusec]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Cuthbert]]
[[Category:Cuthbert]]

Revision as of 12:51, 1 March 2022

The 388 equal divisions of the octave (388edo), or the 388(-tone) equal temperament (388tet, 388et) when viewed from a regular temperament perspective, divides the octave into 388 equal parts of 3.0928 cents each.

Theory

388edo is the first edo that is uniquely consistent through to the 27-odd-limit; it is also consistent through the 37-odd-limit.

388et tempers out the vishnuzma, [23 6 -14⟩, the tricot comma, [39 -29 3⟩, the minortone comma, [-16 35 -17⟩, and the raider comma, [71 -99 31⟩, in the 5-limit, and provides a tuning with less error than any previous equal temperaments. It tempers out 4375/4374 and 235298/234375 in the 7-limit, and 5632/5625, 3025/3024 and 9801/9800 in the 11-limit and 847/845, 1001/1000 and 4096/4095 in the 13-limit. It is the optimal patent val for cuthbert temperament, which tempers out cuthbert, the 847/845 comma, and for a number of other temperaments tempering out cuthbert, e.g. neusec, the 190&198 temperament. By tempering out cuthbert it supports the cuthbert triad, in addition to sinbadmic chords.

Prime harmonics

Approximation of prime harmonics in 388edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.11 +0.28 -0.78 -0.80 +0.71 +0.20 -0.61 -0.44 +0.32 -0.71
Relative (%) +0.0 +3.5 +9.2 -25.4 -25.9 +22.9 +6.4 -19.6 -14.2 +10.3 -22.8
Steps
(reduced) 		388
(0) 		615
(227) 		901
(125) 		1089
(313) 		1342
(178) 		1436
(272) 		1586
(34) 		1648
(96) 		1755
(203) 		1885
(333) 		1922
(370)

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [615 -388⟩ [⟨388 615]] +0.0337 0.0337 1.09
2.3.5 [23 6 -14⟩, [39 -29 3⟩ [⟨388 615 901]] -0.0633 0.0501 1.62
2.3.5.7 4375/4374, 235298/234375, 2100875/2097152 [⟨388 615 901 1089]] +0.0224 0.1546 5.00
2.3.5.7.11 3025/3024, 4375/4374, 5632/5625, 235298/234375 [⟨388 615 901 1089 1342]] +0.0643 0.1617 5.23
2.3.5.7.11.13 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374 [⟨388 615 901 1089 1342 1436]] +0.0216 0.1758 5.68
2.3.5.7.11.13.17 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700 [⟨388 615 901 1089 1342 1436 1586]] +0.0116 0.1646 5.32
2.3.5.7.11.13.17.19 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330 [⟨388 615 901 1089 1342 1436 1586 1648]] +0.0280 0.1600 5.17

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 59\388 182.47 10/9 Mitonic
1 111\388 343.30 8000/6561 Raider
1 145\388 448.45 35/27 Semidimfourth
1 183\388 565.97 75/52 Trillium / pseudotrillium
2 23\388 71.13 25/24 Vishnu / ananta
2 49\388 151.54 12/11 Neusec
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