342edo: Difference between revisions
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== Theory == | == Theory == | ||
342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a | 342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert|hemitert]] temperament, and [[support]]s hemiennealimmal. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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| [[Hemienneadecal]] | | [[Hemienneadecal]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Revision as of 19:37, 15 January 2025
| ← 341edo | 342edo | 343edo → |
Theory
342edo is a very strong 11-limit system. It is, as one would expect, distinctly consistent through the 11-odd-limit, but goes no higher; nonetheless, it is a zeta peak edo. A basis for the 11-limit commas consists of 2401/2400, 3025/3024, 4375/4374 and 32805/32768. It is the optimal patent val for 11-limit hemitert temperament, and supports hemiennealimmal.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.20 | -0.35 | -0.40 | -0.44 | +1.58 | +0.31 | +0.73 | -0.20 | -1.51 | -1.18 |
| Relative (%) | +0.0 | -5.7 | -9.9 | -11.5 | -12.6 | +45.0 | +8.8 | +20.9 | -5.8 | -43.0 | -33.5 | |
| Steps (reduced) |
342 (0) |
542 (200) |
794 (110) |
960 (276) |
1183 (157) |
1266 (240) |
1398 (30) |
1453 (85) |
1547 (179) |
1661 (293) |
1694 (326) | |
Subset and supersets
342 factors as 2 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.
684edo, which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4375/4374, 32805/32768 | [⟨342 542 794 960 1183]] | +0.110 | 0.0556 | 1.59 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712 | [⟨342 542 794 960 1183 1265]] (342f) | +0.178 | 0.1618 | 4.61 |
| 2.3.5.7.11.13 | 625/624, 729/728, 847/845, 1575/1573, 4096/4095 | [⟨342 542 794 960 1183 1266]] (342) | +0.020 | 0.2061 | 5.87 |
- 342et is lower in relative error than any previous equal temperaments in the 11-limit, being the first to beat 270. Not until 612 do we find a better equal temperament in terms of absolute error, and not until 1848 do we find one in terms of relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 11\342 | 38.60 | 45/44 | Hemitert |
| 2 | 5\342 | 17.54 | 99/98 | Poseidon |
| 2 | 50\342 | 175.44 | 448/405 | Bisesqui |
| 2 | 124\342 (47\342) |
435.09 (164.91) |
9/7 (11/10) |
Semisupermajor |
| 2 | 142\342 (29\342) |
498.25 (101.75) |
4/3 (35/33) |
Bipont |
| 3 | 71\342 (43\342) |
249.12 (150.88) |
15/13 (12/11) |
Hemiterm |
| 6 | 97\342 (17\342) |
340.35 (59.65) |
162/133 (88/85) |
Semiseptichrome |
| 6 | 142\342 (28\342) |
498.25 (98.25) |
4/3 (18/17) |
Semiterm |
| 9 | 63\342 (13\342) |
221.05 (45.61) |
25/22 (77/75) |
Quadraennealimmal |
| 18 | 71\342 (5\342) |
249.12 (17.54) |
15/13 (99/98) |
Hemiennealimmal |
| 38 | 142\342 (2\342) |
498.25 (7.02) |
4/3 (225/224) |
Hemienneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct