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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 345 factors into | Since 345 factors into {{factorisation|345}}, 345edo has subset edos {{EDOs| 3, 5, 15, 23, 69, and 115 }}. | ||
== 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]] (%) | ||
| Line 25: | Line 26: | ||
| {{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 33: | ||
| {{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 40: | ||
| 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 47: | ||
| 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 54: | ||
| 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 | ||
| Line 60: | Line 61: | ||
=== 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 | ||
|- | |- | ||
| Line 86: | Line 88: | ||
|- | |- | ||
| 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) | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
[[Category:Kwai]] | [[Category:Kwai]] | ||
Latest revision as of 13:31, 13 March 2026
| ← 344edo | 345edo | 346edo → |
345 equal divisions of the octave (abbreviated 345edo or 345ed2), also called 345-tone equal temperament (345tet) or 345 equal temperament (345et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 345 equal parts of about 3.48 ¢ each. Each step represents a frequency ratio of 21/345, or the 345th root of 2.
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
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 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 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 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) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct