397edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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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]] | ||
! 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]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| {{monzo| -629 397 }} | | {{monzo| -629 397 }} | ||
| {{mapping| 397 629 }} | | {{mapping| 397 629 }} | ||
| 0.2194 | | +0.2194 | ||
| 0.2195 | | 0.2195 | ||
| 7.26 | | 7.26 | ||
| Line 41: | Line 42: | ||
| 129140163/128000000, {{monzo| 55 -1 -23 }} | | 129140163/128000000, {{monzo| 55 -1 -23 }} | ||
| {{mapping| 397 629 922 }} | | {{mapping| 397 629 922 }} | ||
| 0.0618 | | +0.0618 | ||
| 0.2859 | | 0.2859 | ||
| 9.46 | | 9.46 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Cotritone]] (397cd) | | [[Cotritone]] (397cd) | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||
Latest revision as of 12:37, 21 February 2025
| ← 396edo | 397edo | 398edo → |
397 equal divisions of the octave (abbreviated 397edo or 397ed2), also called 397-tone equal temperament (397tet) or 397 equal temperament (397et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 397 equal parts of about 3.02 ¢ each. Each step represents a frequency ratio of 21/397, or the 397th root of 2.
Theory
397edo is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit:
- ⟨397 629 922 1115] (patent val)
- ⟨397 629 922 1114] (397d val)
- ⟨397 629 921 1114] (397cd val)
Using the patent val, it tempers out 129140163/128000000 and [55 -1 -23⟩ in the 5-limit; 6144/6125, 16875/16807 and 129140163/128000000 in the 7-limit; supporting grendel.
Using the 397d val, it tempers out 129140163/128000000 and [55 -1 -23⟩ in the 5-limit; 3136/3125, 420175/419904 and 33756345/33554432 in the 7-limit; supporting sengagen.
Using the 397cd val, it tempers out 390625000/387420489 and [-55 23 8⟩ in the 5-limit; 2401/2400, 390625/387072 and 14348907/14336000 in the 7-limit; supporting cotritone.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.70 | +0.59 | +1.45 | -1.19 | -0.23 | +0.84 | -1.29 | +0.44 | +1.15 | +0.56 |
| Relative (%) | +0.0 | -23.0 | +19.5 | +48.0 | -39.4 | -7.5 | +27.7 | -42.7 | +14.6 | +38.2 | +18.4 | |
| Steps (reduced) |
397 (0) |
629 (232) |
922 (128) |
1115 (321) |
1373 (182) |
1469 (278) |
1623 (35) |
1686 (98) |
1796 (208) |
1929 (341) |
1967 (379) | |
Subsets and supersets
397edo is the 78th prime edo. 1588edo, which quadruples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-629 397⟩ | [⟨397 629]] | +0.2194 | 0.2195 | 7.26 |
| 2.3.5 | 129140163/128000000, [55 -1 -23⟩ | [⟨397 629 922]] | +0.0618 | 0.2859 | 9.46 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 8\397 | 24.18 | 686/675 | Sengagen (397d) |
| 1 | 37\397 | 111.84 | 16/15 | Vavoom |
| 1 | 128\397 | 386.90 | 5/4 | Grendel (397) |
| 1 | 171\397 | 516.88 | 27/20 | Gravity |
| 1 | 193\397 | 583.38 | 7/5 | Cotritone (397cd) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct