388edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
ArrowHead294 (talk | contribs)
15,215 edits
mNo edit summary
Cleanup and update
Line 5: Line 5:
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 ===
Line 13: Line 13:


=== 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 ==
Line 21: Line 21:
! 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
|-
|-
Line 28: Line 28:
|-
|-
| 2.3
| 2.3
| {{monzo| 615 -388 }}
| {{Monzo| 615 -388 }}
| {{mapping| 388 615 }}
| {{Mapping| 388 615 }}
| +0.0337
| +0.0337
| 0.0337
| 0.0337
Line 35: Line 35:
|-
|-
| 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
Line 50: Line 50:
| 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
Line 57: Line 57:
| 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
Line 64: Line 64:
| 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
Line 71: Line 71:
| 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
Line 82: Line 82:
|+ 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
|-
|-
Line 110: Line 110:
| 565.97
| 565.97
| 75/52
| 75/52
| [[Trillium]] / [[pseudotrillium]]
| [[Alphatrillium]] / [[pseudotrillium]]
|-
|-
| 2
| 2
Line 125: Line 125:
|-
|-
| 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]]

Revision as of 15:07, 16 March 2025

← 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

388 equal divisions of the octave (abbreviated 388edo or 388ed2), also called 388-tone equal temperament (388tet) or 388 equal temperament (388et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 388 equal parts of about 3.09 ¢ each. Each step represents a frequency ratio of 21/388, or the 388th root of 2.

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 alphatricot 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 Alphatrillium / 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 distinct

Retrieved from "https://en.xen.wiki/index.php?title=388edo&oldid=186462"