388edo: Difference between revisions
Rework; cleanup; clarify the title row of the rank-2 temp table |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 36: | Line 28: | ||
| {{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 | ||
| 0.0501 | | 0.0501 | ||
| 1.62 | | 1.62 | ||
| Line 74: | Line 66: | ||
| 0.1600 | | 0.1600 | ||
| 5.17 | | 5.17 | ||
{{comma basis end}} | |||
* 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[323edo|323]] and followed by [[441edo|441]]. | * 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 === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 123: | Line 109: | ||
|- | |- | ||
| 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]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
[[Category:Cuthbert]] | [[Category:Cuthbert]] | ||
Revision as of 05:00, 16 November 2024
| ← 387edo | 388edo | 389edo → |
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
| 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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past 323 and followed by 441.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf