388edo: Difference between revisions
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→Rank-2 temperaments: berkelium |
Cleanup and style |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|388}} | |||
== Theory == | == Theory == | ||
388edo is the first edo that is | 388edo is the first edo that is 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 | 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]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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|} | |} | ||
[[Category:Cuthbert]] | [[Category:Cuthbert]] | ||
Revision as of 08:01, 21 May 2023
| ← 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.
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 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 supports cuthbert chords, in addition to sinbadmic chords.
Prime harmonics
| 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
| Periods per 8ve |
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 |
| 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 |