375edo: Difference between revisions
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== Theory == | == Theory == | ||
375edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. It can be used as a temperament in the 2.9.5.7.13.17.19 [[subgroup]], in which it [[tempering out|tempers out]] [[250047/250000]], 589824/588245, and {{monzo| 8 7 -13 }} ([[parakleisma]]). | |||
For the full 7-limit, the 375cd [[val]] is the best, where it tempers out [[1029/1024]] and [[15625/15552]], [[support]]ing [[tritikleismic]]. In the 11-limit it tempers out [[385/384]] and [[441/440]], supporting undecimal tritikleismic. | |||
Using the [[patent val]], it tempers out [[6144/6125]] and 177147/175616, supporting [[aufo]]. In the 11-limit it tempers out [[540/539]] and [[5632/5625]], supporting [[aufic]] and [[persephone]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
375 factors into 3 × 5<sup>3</sup> | Since 375 factors into 3 × 5<sup>3</sup>, 375edo has subset edos {{EDOs| 3, 5, 15, 25, 75, and 125 }}. [[1175edo]], which triples it, gives a good correction to the harmonic 3 and is consistent to the [[15-odd-limit]]. | ||
== 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|Comma List]] | ! rowspan="2" | [[Comma list|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 | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.9 | | 2.9 | ||
|{{monzo|1189 -375}} | | {{monzo| 1189 -375 }} | ||
|{{mapping|375 1189}} | | {{mapping| 375 1189 }} | ||
| -0.1404 | | -0.1404 | ||
| 0.1404 | | 0.1404 | ||
| 4.39 | | 4.39 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
|{{monzo|8 7 -13}}, {{monzo|97 -24 -9}} | | {{monzo| 8 7 -13 }}, {{monzo| 97 -24 -9 }} | ||
|{{mapping|375 1189 871}} | | {{mapping| 375 1189 871 }} | ||
| -0.2208 | | -0.2208 | ||
| 0.1615 | | 0.1615 | ||
| 5.05 | | 5.05 | ||
|- | |- | ||
|2.9.5.7 | | 2.9.5.7 | ||
|250047/250000, | | 250047/250000, 589824/588245, {{monzo| 8 7 -13 }} | ||
|{{mapping|375 1189 871 1053}} | | {{mapping| 375 1189 871 1053 }} | ||
| -0.2345 | | -0.2345 | ||
| 0.1418 | | 0.1418 | ||
| 4.43 | | 4.43 | ||
|} | |} | ||
Revision as of 14:08, 14 January 2024
| ← 374edo | 375edo | 376edo → |
Theory
375edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It can be used as a temperament in the 2.9.5.7.13.17.19 subgroup, in which it tempers out 250047/250000, 589824/588245, and [8 7 -13⟩ (parakleisma).
For the full 7-limit, the 375cd val is the best, where it tempers out 1029/1024 and 15625/15552, supporting tritikleismic. In the 11-limit it tempers out 385/384 and 441/440, supporting undecimal tritikleismic.
Using the patent val, it tempers out 6144/6125 and 177147/175616, supporting aufo. In the 11-limit it tempers out 540/539 and 5632/5625, supporting aufic and persephone.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.16 | +0.89 | +0.77 | +0.89 | -0.92 | +1.07 | -0.27 | +0.64 | +0.09 | -0.38 | -1.07 |
| Relative (%) | -36.1 | +27.7 | +24.2 | +27.8 | -28.7 | +33.5 | -8.4 | +20.1 | +2.7 | -11.9 | -33.6 | |
| Steps (reduced) |
594 (219) |
871 (121) |
1053 (303) |
1189 (64) |
1297 (172) |
1388 (263) |
1465 (340) |
1533 (33) |
1593 (93) |
1647 (147) |
1696 (196) | |
Subsets and supersets
Since 375 factors into 3 × 53, 375edo has subset edos 3, 5, 15, 25, 75, and 125. 1175edo, which triples it, gives a good correction to the harmonic 3 and is consistent to the 15-odd-limit.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [1189 -375⟩ | [⟨375 1189]] | -0.1404 | 0.1404 | 4.39 |
| 2.9.5 | [8 7 -13⟩, [97 -24 -9⟩ | [⟨375 1189 871]] | -0.2208 | 0.1615 | 5.05 |
| 2.9.5.7 | 250047/250000, 589824/588245, [8 7 -13⟩ | [⟨375 1189 871 1053]] | -0.2345 | 0.1418 | 4.43 |