345edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 547 -345 }} | | {{monzo| 547 -345 }} | ||
| {{mapping| 345 547 }} | | {{mapping| 345 547 }} | ||
| | | –0.2062 | ||
| 0.2062 | | 0.2062 | ||
| 5.93 | | 5.93 | ||
| Line 32: | Line 24: | ||
| {{monzo| 3 -18 11 }}, {{monzo| 47 -15 -10 }} | | {{monzo| 3 -18 11 }}, {{monzo| 47 -15 -10 }} | ||
| {{mapping| 345 547 801 }} | | {{mapping| 345 547 801 }} | ||
| | | –0.1050 | ||
| 0.2210 | | 0.2210 | ||
| 6.35 | | 6.35 | ||
| Line 39: | Line 31: | ||
| 5120/5103, 16875/16807, 68359375/68024448 | | 5120/5103, 16875/16807, 68359375/68024448 | ||
| {{mapping| 345 547 801 969 }} | | {{mapping| 345 547 801 969 }} | ||
| | | –0.2220 | ||
| 0.2788 | | 0.2788 | ||
| 8.02 | | 8.02 | ||
| Line 46: | Line 38: | ||
| 540/539, 1375/1372, 5120/5103, 1953125/1940598 | | 540/539, 1375/1372, 5120/5103, 1953125/1940598 | ||
| {{mapping| 345 547 801 969 1194 }} | | {{mapping| 345 547 801 969 1194 }} | ||
| | | –0.2773 | ||
| 0.2728 | | 0.2728 | ||
| 7.84 | | 7.84 | ||
| Line 53: | Line 45: | ||
| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103 | | 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103 | ||
| {{mapping| 345 547 801 969 1194 1277 }} | | {{mapping| 345 547 801 969 1194 1277 }} | ||
| | | –0.2857 | ||
| 0.2497 | | 0.2497 | ||
| 7.18 | | 7.18 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 86: | Line 72: | ||
|- | |- | ||
| 5 | | 5 | ||
| 106\345<br>(32\345) | | 106\345<br />(32\345) | ||
| 368.70<br>(111.30) | | 368.70<br />(111.30) | ||
| 1024/891<br>(16/15) | | 1024/891<br />(16/15) | ||
| [[Quintosec]] (5-limit) | | [[Quintosec]] (5-limit) | ||
{{rank-2 end}} | |||
{{orf}} | |||
[[Category:Kwai]] | [[Category:Kwai]] | ||
Revision as of 03:18, 16 November 2024
| ← 344edo | 345edo | 346edo → |
Theory
345et is only consistent to the 5-odd-limit, though it has a reasonable 13-limit interpretation using the patent val. It tempers out [3 -18 11⟩ (quartonic comma) and [47 -15 -10⟩ (quintosec comma) in the 5-limit; 5120/5103, 16875/16807, 2460375/2458624, and 68359375/68024448 in the 7-limit; 540/539, 1375/1372, 3025/3024, 16384/16335, 19712/19683, 46656/46585, 200704/200475, and 532400/531441 in the 11-limit; and 625/624 and 4225/4224 in the 13-limit. It provides the optimal patent val for 7-limit kwai.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.65 | -0.23 | +1.61 | +1.31 | +1.73 | +1.21 | +0.43 | -0.61 | +1.62 | -1.22 | +1.29 |
| Relative (%) | +18.8 | -6.5 | +46.3 | +37.6 | +49.6 | +34.8 | +12.3 | -17.5 | +46.5 | -35.0 | +37.1 | |
| Steps (reduced) |
547 (202) |
801 (111) |
969 (279) |
1094 (59) |
1194 (159) |
1277 (242) |
1348 (313) |
1410 (30) |
1466 (86) |
1515 (135) |
1561 (181) | |
Subsets and supersets
Since 345 factors into 3 × 5 × 23, 345edo has subset edos 3, 5, 15, 23, 69, and 115.
Regular temperament properties
Template:Comma basis begin |- | 2.3 | [547 -345⟩ | [⟨345 547]] | –0.2062 | 0.2062 | 5.93 |- | 2.3.5 | [3 -18 11⟩, [47 -15 -10⟩ | [⟨345 547 801]] | –0.1050 | 0.2210 | 6.35 |- | 2.3.5.7 | 5120/5103, 16875/16807, 68359375/68024448 | [⟨345 547 801 969]] | –0.2220 | 0.2788 | 8.02 |- | 2.3.5.7.11 | 540/539, 1375/1372, 5120/5103, 1953125/1940598 | [⟨345 547 801 969 1194]] | –0.2773 | 0.2728 | 7.84 |- | 2.3.5.7.11.13 | 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103 | [⟨345 547 801 969 1194 1277]] | –0.2857 | 0.2497 | 7.18 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 13\345
| 45.22
| 250/243
| Quartonic (5-limit)
|-
| 1
| 38\345
| 132.17
| [-38 5 13⟩
| Astro
|-
| 1
| 143\345
| 497.39
| 4/3
| Kwai
|-
| 5
| 106\345
(32\345)
| 368.70
(111.30)
| 1024/891
(16/15)
| Quintosec (5-limit)
Template:Rank-2 end
Template:Orf