158edo: Difference between revisions

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'''158edo''' is the [[EDO|equal division of the octave]] into 158 parts of 7.5949 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with five mappings possible for the 13-limit: <158 250 367 444 547 585| (optimal patent val), <158 251 367 444 547 585| (158b), <158 250 366 443 546 584| (158cdef), <158 250 367 443 546 585| (158de), and <158 250 367 443 546 584| (158def). Using the optimal patent val, it tempers out [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; 225/224, 8748/8575, and 40960000/40353607 in the 7-limit; 441/440, 1375/1372, 4000/3993, and 19683/19208 in the 11-limit, supporting the 11-limit [[Marvel temperaments|marvolo temperament]]; 144/143, 640/637, 2025/2002, 2200/2197 and 3159/3125 in the 13-limit. Using the 158b val, it tempers out [[Diaschismic family|diaschisma]], 2048/2025 and |-1 -33 23> in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, 625/624, 640/637, and 1625/1617 in the 13-limit. Using the 158cdef val, it tempers out the [[Magic family|small diesis]], 3125/3072 and the double large green comma, 43046721/41943040 in the 5-limit; 2401/2400, 19683/19600, and 78125/76832 in the 7-limit; 243/242, 441/440, 4000/3993, and 33275/32768 in the 11-limit; 325/324, 975/968, 1287/1280, 1573/1568, and 1875/1859 in the 13-limit, supporting the 13-limit [[Gravity family|harry temperament]]. Using the 158de val, it tempers out 126/125, 33075/32768, and 118098/117649 in the 7-limit; 243/242, 385/384, 1617/1600, and 117649/117128 in the 11-limit; 196/195, 351/350, 1287/1280, 1575/1573, and 4455/4394 in the 13-limit. Using the 158def val, it tempers out 676/675, 847/845, 1573/1568, 1701/1690, and 3159/3136 in the 13-limit.

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

158edo is in[[consistent]] to the [[5-odd-limit]] and higher limits. It can be treated as a 2.9.5.21.33.39 [[subgroup]] temperament, which is every other step of [[316edo]], and [[tempering out|tempers out]] the same commas: [[1716/1715]], [[2080/2079]], [[3025/3024]], [[3136/3125]], [[4096/4095]], [[4225/4224]], [[9801/9800]], etc.

Otherwise, it has five mappings possible for the 13-limit: {{val| 158 250 367 444 547 585 }} ([[patent val]]), {{val| 158 '''251''' 367 444 547 585 }} (158b), {{val| 158 250 '''366''' '''443''' '''546''' '''584''' }} (158cdef), {{val| 158 250 367 '''443''' '''546''' 585 }} (158de), and {{val| 158 250 367 '''443''' '''546''' '''584''' }} (158def).

Using the patent val, it tempers out the [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; [[225/224]], 8748/8575, and 40960000/40353607 in the 7-limit; [[441/440]], 1375/1372, [[4000/3993]], and 19683/19208 in the 11-limit, supporting the 11-limit [[marvolo]] temperament and giving a good tuning; [[144/143]], [[640/637]], 2025/2002, [[2200/2197]] and 3159/3125 in the 13-limit.

Using the 158de val, it tempers out [[126/125]], 33075/32768, and 118098/117649 in the 7-limit; [[243/242]], [[385/384]], 1617/1600, and 117649/117128 in the 11-limit; [[196/195]], [[351/350]], [[1287/1280]], [[1575/1573]], and 4455/4394 in the 13-limit. Using the 158def val, it tempers out [[676/675]], [[847/845]], [[1573/1568]], 1701/1690, and 3159/3136 in the 13-limit.

Using the 158cdef val, it tempers out the [[magic comma]], 3125/3072 and the [[python comma]], 43046721/41943040 in the 5-limit; [[2401/2400]], [[19683/19600]], and 78125/76832 in the 7-limit; 243/242, 441/440, 4000/3993, and 33275/32768 in the 11-limit; [[325/324]], 975/968, 1287/1280, 1573/1568, and 1875/1859 in the 13-limit, supporting the 13-limit [[harry]] temperament.

Using the 158b val, it tempers out the [[diaschisma]], 2048/2025 and {{monzo| -1 -33 23 }} in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, [[625/624]], 640/637, and 1625/1617 in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 158 factors into {{factorization|158}}, 158edo contains [[2edo]] and [[79edo]] as its subsets. 316edo, which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.

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-'''158edo''' is the [[EDO|equal division of the octave]] into 158 parts of 7.5949 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with five mappings possible for the 13-limit: &lt;158 250 367 444 547 585| (optimal patent val), &lt;158 251 367 444 547 585| (158b), &lt;158 250 366 443 546 584| (158cdef), &lt;158 250 367 443 546 585| (158de), and &lt;158 250 367 443 546 584| (158def). Using the optimal patent val, it tempers out [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; 225/224, 8748/8575, and 40960000/40353607 in the 7-limit; 441/440, 1375/1372, 4000/3993, and 19683/19208 in the 11-limit, supporting the 11-limit [[Marvel temperaments|marvolo temperament]]; 144/143, 640/637, 2025/2002, 2200/2197 and 3159/3125 in the 13-limit. Using the 158b val, it tempers out [[Diaschismic family|diaschisma]], 2048/2025 and |-1 -33 23&gt; in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, 625/624, 640/637, and 1625/1617 in the 13-limit. Using the 158cdef val, it tempers out the [[Magic family|small diesis]], 3125/3072 and the double large green comma, 43046721/41943040 in the 5-limit; 2401/2400, 19683/19600, and 78125/76832 in the 7-limit; 243/242, 441/440, 4000/3993, and 33275/32768 in the 11-limit; 325/324, 975/968, 1287/1280, 1573/1568, and 1875/1859 in the 13-limit, supporting the 13-limit [[Gravity family|harry temperament]]. Using the 158de val, it tempers out 126/125, 33075/32768, and 118098/117649 in the 7-limit; 243/242, 385/384, 1617/1600, and 117649/117128 in the 11-limit; 196/195, 351/350, 1287/1280, 1575/1573, and 4455/4394 in the 13-limit. Using the 158def val, it tempers out 676/675, 847/845, 1573/1568, 1701/1690, and 3159/3136 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits. It can be treated as a 2.9.5.21.33.39 [[subgroup]] temperament, which is every other step of [[316edo]], and [[tempering out|tempers out]] the same commas: [[1716/1715]], [[2080/2079]], [[3025/3024]], [[3136/3125]], [[4096/4095]], [[4225/4224]], [[9801/9800]], etc.
+Otherwise, it has five mappings possible for the 13-limit: {{val| 158 250 367 444 547 585 }} ([[patent val]]), {{val| 158 '''251''' 367 444 547 585 }} (158b), {{val| 158 250 '''366''' '''443''' '''546''' '''584''' }} (158cdef), {{val| 158 250 367 '''443''' '''546''' 585 }} (158de), and {{val| 158 250 367 '''443''' '''546''' '''584''' }} (158def).
+Using the patent val, it tempers out the [[Würschmidt comma]], 393216/390625 and 3486784401/3355443200 in the 5-limit; [[225/224]], 8748/8575, and 40960000/40353607 in the 7-limit; [[441/440]], 1375/1372, [[4000/3993]], and 19683/19208 in the 11-limit, supporting the 11-limit [[marvolo]] temperament and giving a good tuning; [[144/143]], [[640/637]], 2025/2002, [[2200/2197]] and 3159/3125 in the 13-limit.
+Using the 158de val, it tempers out [[126/125]], 33075/32768, and 118098/117649 in the 7-limit; [[243/242]], [[385/384]], 1617/1600, and 117649/117128 in the 11-limit; [[196/195]], [[351/350]], [[1287/1280]], [[1575/1573]], and 4455/4394 in the 13-limit. Using the 158def val, it tempers out [[676/675]], [[847/845]], [[1573/1568]], 1701/1690, and 3159/3136 in the 13-limit.
+Using the 158cdef val, it tempers out the [[magic comma]], 3125/3072 and the [[python comma]], 43046721/41943040 in the 5-limit; [[2401/2400]], [[19683/19600]], and 78125/76832 in the 7-limit; 243/242, 441/440, 4000/3993, and 33275/32768 in the 11-limit; [[325/324]], 975/968, 1287/1280, 1573/1568, and 1875/1859 in the 13-limit, supporting the 13-limit [[harry]] temperament.
+Using the 158b val, it tempers out the [[diaschisma]], 2048/2025 and {{monzo| -1 -33 23 }} in the 5-limit; 245/243, 6144/6125 and 2500000/2470629 in the 7-limit; 1331/1323, 1375/1372, 2560/2541, and 4375/4356 in the 11-limit; 364/363, 572/567, [[625/624]], 640/637, and 1625/1617 in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|158}}
+=== Subsets and supersets ===
+Since 158 factors into {{factorization|158}}, 158edo contains [[2edo]] and [[79edo]] as its subsets. 316edo, which doubles it, provides good correction to the approximation to harmonics 3, 7, and 11.