343edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|343}} == Theory == 343et is only consistent to the 3-odd-limit and the harmonic 3 is about inbetween its steps. Using the patent val, it te..." |
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== Theory == | == Theory == | ||
343edo is only [[consistent]] to the [[3-odd-limit]] since its errors of [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large. To start with, consider the 2.9.15.7 [[subgroup]], where it [[tempering out|tempers out]] 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649. | |||
For the full 7-limit, the 343c [[val]] tempers out [[4375/4374]] and [[5120/5103]], [[support]]ing [[amity]]. The 343cdd val tempers out [[16875/16807]] and 65536/64827. The [[patent val]] tempers out [[10976/10935]] and 390625/387072. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
343 factors into 7<sup>3</sup>, | Since 343 factors into 7<sup>3</sup>, 343edo has [[7edo]] and [[49edo]] as its subsets. [[686edo]], which doubles it, gives a good correction to the harmonics 3 and 5. | ||
== 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|-1087 343}} | | {{monzo| -1087 343 }} | ||
|{{mapping|343 1087}} | | {{mapping| 343 1087 }} | ||
| 0.1569 | | 0.1569 | ||
| 0.1569 | | 0.1569 | ||
| 4.48 | | 4.48 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
|{{monzo|-27 -1 13}}, {{monzo|40 -28 21}} | | {{monzo| -27 -1 13 }}, {{monzo| 40 -28 21 }} | ||
|{{mapping|343 1087 796}} | | {{mapping| 343 1087 796 }} | ||
| 0.3162 | | 0.3162 | ||
| 0.2592 | | 0.2592 | ||
| 7.41 | | 7.41 | ||
|- | |- | ||
|2.9.5.7 | | 2.9.5.7 | ||
|118098/117649, 7381125/7340032, 9765625/9680832 | | 118098/117649, 7381125/7340032, 9765625/9680832 | ||
|{{mapping|343 1087 796 963}} | | {{mapping| 343 1087 796 963 }} | ||
| 0.2130 | | 0.2130 | ||
| 0.2869 | | 0.2869 | ||
| 8.20 | | 8.20 | ||
|} | |} | ||
Revision as of 13:53, 12 January 2024
| ← 342edo | 343edo | 344edo → |
Theory
343edo is only consistent to the 3-odd-limit since its errors of harmonics 3 and 5 are quite large. To start with, consider the 2.9.15.7 subgroup, where it tempers out 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.
For the full 7-limit, the 343c val tempers out 4375/4374 and 5120/5103, supporting amity. The 343cdd val tempers out 16875/16807 and 65536/64827. The patent val tempers out 10976/10935 and 390625/387072.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.25 | -1.47 | +0.27 | -0.99 | +1.45 | -0.88 | -0.22 | +0.00 | -0.14 | +1.52 | +1.46 |
| Relative (%) | +35.8 | -42.1 | +7.7 | -28.4 | +41.5 | -25.1 | -6.3 | +0.0 | -3.9 | +43.5 | +41.8 | |
| Steps (reduced) |
544 (201) |
796 (110) |
963 (277) |
1087 (58) |
1187 (158) |
1269 (240) |
1340 (311) |
1402 (30) |
1457 (85) |
1507 (135) |
1552 (180) | |
Subsets and supersets
Since 343 factors into 73, 343edo has 7edo and 49edo as its subsets. 686edo, which doubles it, gives a good correction to the harmonics 3 and 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1087 343⟩ | [⟨343 1087]] | 0.1569 | 0.1569 | 4.48 |
| 2.9.5 | [-27 -1 13⟩, [40 -28 21⟩ | [⟨343 1087 796]] | 0.3162 | 0.2592 | 7.41 |
| 2.9.5.7 | 118098/117649, 7381125/7340032, 9765625/9680832 | [⟨343 1087 796 963]] | 0.2130 | 0.2869 | 8.20 |