165edo: Difference between revisions

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[[{{~~PAGENAME~~}}~~]] is the [[EDO|equal division of the octave]] into 165 parts of~~ {{~~#expr: 1200/165~~}} [[cent]]s each. That's the triple of [[55edo]]!!! It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 5-limit: <165 262 383| (patent val) and <165 261 383| (165b) and <165 261 381| ([[K/N|165/3]] generic meantone val). Using the patent val (with a sharp fifth), it tempers out 1638400/1594323 (immunity comma) and 1220703125/1207959552 (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's 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.

[[~~Category~~:~~Edo~~]]

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 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 ===

=== Subsets and supersets ===

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.

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-[[{{PAGENAME}}]] is the [[EDO|equal division of the octave]] into 165 parts of {{#expr: 1200/165}} [[cent]]s each. That's the triple of [[55edo]]!!! It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 5-limit: &lt;165 262 383| (patent val) and &lt;165 261 383| (165b) and &lt;165 261 381| ([[K/N|165/3]] generic meantone val). Using the patent val (with a sharp fifth), it tempers out 1638400/1594323 (immunity comma) and 1220703125/1207959552 (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's 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.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+edo 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 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 ===
+{{Harmonics in equal|165}}
+=== Subsets and supersets ===
+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.