165edo: Difference between revisions
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165edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with two mappings possible for the 5-limit: {{val| 165 262 383 }} ([[patent val]]) and {{val| 165 '''261''' 383 }} (165b). | 165edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with two mappings possible for the 5-limit: {{val| 165 262 383 }} ([[patent val]]) and {{val| 165 '''261''' 383 }} (165b). | ||
Using the patent val (with a sharp fifth), it tempers out 1638400/1594323 ([[immunity comma]]) and {{monzo| -27 -2 13 }} (ditonma) in the 5-limit; [[4000/3969]], [[65625/65536]], and 84035/82944 in the 7-limit; [[385/384]], 2401/2376, [[3388/3375]], and 6655/6561 in the 11-limit; [[196/195]], [[364/363]], [[676/675]], 3185/3168, and 3200/3159 in the 13-limit. | Using the patent val (with a sharp fifth), it tempers out 1638400/1594323 ([[immunity comma]]) and {{monzo| -27 -2 13 }} ([[ditonma]]) in the 5-limit; [[4000/3969]], [[65625/65536]], and 84035/82944 in the 7-limit; [[385/384]], 2401/2376, [[3388/3375]], and 6655/6561 in the 11-limit; [[196/195]], [[364/363]], [[676/675]], 3185/3168, and 3200/3159 in the 13-limit. | ||
Using the 165b val (with a flat fifth), it tempers out 34171875/33554432 ([[ampersand]]) and 129140163/125000000 in the 5-limit; [[225/224]], [[1029/1024]], and 100442349/97656250 in the 7-limit; 1944/1925, 2187/2156, [[4000/3993]], and 12005/11979 in the 11-limit; [[144/143]], [[351/350]], [[625/624]], [[847/845]], and 9261/9152 in the 13-limit. Using the 165bf val, it tempers out 364/363, 975/968, [[1001/1000]], 1701/1690, and [[1716/1715]] in the 13-limit. | Using the 165b val (with a flat fifth), it tempers out 34171875/33554432 ([[ampersand comma]]) and 129140163/125000000 in the 5-limit; [[225/224]], [[1029/1024]], and 100442349/97656250 in the 7-limit; 1944/1925, 2187/2156, [[4000/3993]], and [[12005/11979]] in the 11-limit; [[144/143]], [[351/350]], [[625/624]], [[847/845]], and 9261/9152 in the 13-limit. Using the 165bf val, it tempers out 364/363, 975/968, [[1001/1000]], 1701/1690, and [[1716/1715]] in the 13-limit. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 165 factors into | Since 165 factors into 3 × 5 × 11, 165edo has subset edos {{EDOs| 3, 5, 11, 15, 33, and 55 }}. [[330edo]], which doubles it, provides good correction for the approximation of harmonic 3. | ||
Revision as of 09:16, 18 January 2025
| ← 164edo | 165edo | 166edo → |
165edo is inconsistent to the 5-odd-limit and higher limits, with two mappings possible for the 5-limit: ⟨165 262 383] (patent val) and ⟨165 261 383] (165b).
Using the patent val (with a sharp fifth), it tempers out 1638400/1594323 (immunity comma) and [-27 -2 13⟩ (ditonma) in the 5-limit; 4000/3969, 65625/65536, and 84035/82944 in the 7-limit; 385/384, 2401/2376, 3388/3375, and 6655/6561 in the 11-limit; 196/195, 364/363, 676/675, 3185/3168, and 3200/3159 in the 13-limit.
Using the 165b val (with a flat fifth), it tempers out 34171875/33554432 (ampersand comma) and 129140163/125000000 in the 5-limit; 225/224, 1029/1024, and 100442349/97656250 in the 7-limit; 1944/1925, 2187/2156, 4000/3993, and 12005/11979 in the 11-limit; 144/143, 351/350, 625/624, 847/845, and 9261/9152 in the 13-limit. Using the 165bf val, it tempers out 364/363, 975/968, 1001/1000, 1701/1690, and 1716/1715 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.50 | -0.86 | -1.55 | -0.27 | +1.41 | +3.11 | +2.64 | -3.14 | +0.67 | +1.95 | -2.82 |
| Relative (%) | +48.1 | -11.8 | -21.4 | -3.8 | +19.4 | +42.7 | +36.3 | -43.1 | +9.2 | +26.8 | -38.8 | |
| Steps (reduced) |
262 (97) |
383 (53) |
463 (133) |
523 (28) |
571 (76) |
611 (116) |
645 (150) |
674 (14) |
701 (41) |
725 (65) |
746 (86) | |
Subsets and supersets
Since 165 factors into 3 × 5 × 11, 165edo has subset edos 3, 5, 11, 15, 33, and 55. 330edo, which doubles it, provides good correction for the approximation of harmonic 3.