397edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|397}} == Theory == 397et is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit: (*) {{val|397 629 922 1115}} (pat..." |
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== Theory == | == Theory == | ||
397et is only consistent to the [[5-odd-limit]], with three mappings possible for the 7-limit: | 397et is only consistent to the [[5-odd-limit]], with three mappings possible for the 7-limit: | ||
* {{val|397 629 922 1115}} (patent val) | |||
* {{val|397 629 922 '''1114'''}} (397d val) | |||
* {{val|397 629 '''921''' '''1114'''}} (397cd val) | |||
Using the patent val, it tempers out [[129140163/128000000]] and {{monzo|55 -1 -23}} in the 5-limit; [[6144/6125]], 16875/16807 and 129140163/128000000 in the 7-limit; [[support]]ing [[grendel]], [[porwell]] and [[mirkwai]]. | Using the patent val, it tempers out [[129140163/128000000]] and {{monzo|55 -1 -23}} in the 5-limit; [[6144/6125]], 16875/16807 and 129140163/128000000 in the 7-limit; [[support]]ing [[grendel]], [[porwell]] and [[mirkwai]]. | ||
Revision as of 09:15, 11 January 2024
| ← 396edo | 397edo | 398edo → |
Theory
397et 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, porwell and mirkwai.
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 and parahemwuer.
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 and breed.
= 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 it is distinct