388edo: Difference between revisions

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== Theory ==
== Theory ==
388edo is the first edo that is 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]].


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 [[3025/3024]], [[5632/5625]] 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 the rank-5 cuthbert temperament, which tempers out cuthbert, the [[847/845]] comma, and for a number of other temperaments tempering it out, e.g. [[neusec]], the 190 & 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].
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.  
 
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 190 & 198 temperament. By tempering out cuthbert it [[support]]s [[cuthbert chords]], in addition to [[sinbadmic chords]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|388|columns=12}}
{{Harmonics in equal|388|columns=12}}
=== Subsets and supersets ===
Since 388 factors into {{factorization|388}}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.


== Regular temperament properties ==
== Regular temperament properties ==
Line 23: Line 28:
| 2.3
| 2.3
| {{monzo| 615 -388 }}
| {{monzo| 615 -388 }}
| [{{val| 388 615 }}]
| {{mapping| 388 615 }}
| +0.0337
| +0.0337
| 0.0337
| 0.0337
Line 30: Line 35:
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
| {{monzo| 23 6 -14 }}, {{monzo| 39 -29 3 }}
| [{{val| 388 615 901 }}]
| {{mapping| 388 615 901 }}
| -0.0633
| -0.0633
| 0.0501
| 0.0501
Line 37: Line 42:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 235298/234375, 2100875/2097152
| 4375/4374, 235298/234375, 2100875/2097152
| [{{val| 388 615 901 1089 }}]
| {{mapping| 388 615 901 1089 }}
| +0.0224
| +0.0224
| 0.1546
| 0.1546
Line 44: Line 49:
| 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
| [{{val| 388 615 901 1089 1342 }}]
| {{mapping| 388 615 901 1089 1342 }}
| +0.0643
| +0.0643
| 0.1617
| 0.1617
Line 51: Line 56:
| 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
| [{{val| 388 615 901 1089 1342 1436 }}]
| {{mapping| 388 615 901 1089 1342 1436 }}
| +0.0216
| +0.0216
| 0.1758
| 0.1758
Line 58: Line 63:
| 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
| [{{val| 388 615 901 1089 1342 1436 1586 }}]
| {{mapping| 388 615 901 1089 1342 1436 1586 }}
| +0.0116
| +0.0116
| 0.1646
| 0.1646
Line 65: Line 70:
| 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
| [{{val| 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
| 5.17
| 5.17
|}
|}
* 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[323edo|323]] and followed by [[441edo|441]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 75: Line 81:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 128: Line 134:
| [[Berkelium]]
| [[Berkelium]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


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

Revision as of 16:51, 6 November 2023

← 387edo 388edo 389edo →
Prime factorization 22 × 97
Step size 3.09278 ¢ 
Fifth 227\388 (702.062 ¢)
Semitones (A1:m2) 37:29 (114.4 ¢ : 89.69 ¢)
Consistency limit 37
Distinct consistency limit 27

Template:EDO intro

Theory

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

The equal temperament 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, 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 190 & 198 temperament. By tempering out cuthbert it supports cuthbert chords, 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 37
Error Absolute (¢) +0.00 +0.11 +0.28 -0.78 -0.80 +0.71 +0.20 -0.61 -0.44 +0.32 -0.71 -0.83
Relative (%) +0.0 +3.5 +9.2 -25.4 -25.9 +22.9 +6.4 -19.6 -14.2 +10.3 -22.8 -26.8
Steps
(reduced) 		388
(0) 		615
(227) 		901
(125) 		1089
(313) 		1342
(178) 		1436
(272) 		1586
(34) 		1648
(96) 		1755
(203) 		1885
(333) 		1922
(370) 		2021
(81)

Subsets and supersets

Since 388 factors into 22 × 97, 388edo has subset edos 2, 4, 97, and 194.

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
  • 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past 323 and followed by 441.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* 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
4 123\388
(26\388) 		380.41
(80.41) 		81/65
(22/21) 		Quasithird
97 161\388
(1\388) 		497.938
(3.09) 		4/3
(?) 		Berkelium

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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