138edo: Difference between revisions

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~~'''~~138edo~~''' is the~~ [[~~EDO~~|~~equal division of the octave~~]] ~~into 138 parts of 8.6957~~ [[~~cent~~]]~~s each~~. It is ~~inconsistent~~ to the 5-limit and higher ~~limit~~, with three mappings possible for the 13-limit: ~~<~~138 219 320 387 477 511| (patent val), ~~<138 218 320 387 477 510~~| ~~(138bf), and <~~138 219 321 388 478 511| (~~138cde~~). Using the patent val, it tempers out ~~the shibboleth comma,~~ 1953125/1889568 and ~~the misty comma,~~ 67108864/66430125 in the 5-limit; 875/864, 1029/1024, and 1647086/1594323 in the 7-limit; 896/891, 1331/1323, 1375/1372, and 2401/2376 in the 11-limit; 196/195, 275/273, and 1575/1573 in the 13-limit. ~~Using the 138bf~~ val~~, it tempers out the~~ [[~~syntonic comma~~]]~~, 81/80 and 2288818359375/2199023255552~~ in the 5-limit~~; 2401/2400~~, 2430/2401, and 9765625/9633792 in the 7-limit; 385/384, 1375/1372, 1944/1925, and 9375/9317 in the 11-limit, supporting the [[Meantone family|cuboctahedra temperament]]; 625/624, 975/968, 1001/1000, and 1188/1183 in the 13-limit. Using the ~~138cde val~~, ~~it tempers~~ out the [[~~Diaschismic family|~~diaschisma]], 2048/2025 and the [[~~Sensipent family|~~sensipent comma]], 78732/78125 ~~in the 5-limit;~~ 1728/1715, 10976/10935, and 250047/250000 in the 7-limit; 176/175, 540/539, 896/891, and 85184/84375 in the 11-limit; 351/350, 352/351, 364/363, 640/637, and 2197/2187 in the 13-limit, supporting the [[~~Diaschismic~~ family|~~echidna~~ temperament]].

Since {{nowrap|138 {{=}} 3 × 46}}, 138edo shares its [[3/2|fifth]] with [[46edo]]. Unlike 46edo, it is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for the 13-limit: {{val| 138 219 320 387 477 511 }} ([[patent val]]), {{val| 138 219 '''321''' '''388''' '''478''' 511 }} (138cde), and {{val| 138 '''218''' 320 387 477 '''510''' }} (138bf). The last mapping uses an alternative flat fifth from [[69edo]].

Using the patent val, it [[tempering out|tempers out]] 1953125/1889568 ([[shibboleth comma]]) and 67108864/66430125 ([[misty comma]]) in the 5-limit; [[875/864]], [[1029/1024]], and 1647086/1594323 in the 7-limit; [[896/891]], 1331/1323, 1375/1372, and 2401/2376 in the 11-limit; [[196/195]], [[275/273]], and [[1575/1573]] in the 13-limit.

The 138cde val is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as 46edo, tempering out the [[diaschisma]], 2048/2025 and the [[sensipent comma]], 78732/78125. However, it tempers out [[1728/1715]], [[10976/10935]], and [[250047/250000]] in the 7-limit; [[176/175]], [[540/539]], [[896/891]], and 85184/84375 in the 11-limit; [[351/350]], [[352/351]], [[364/363]], [[640/637]], and [[2197/2187]] in the 13-limit, [[support]]ing the [[echidna]] temperament and giving an excellent tuning.

The 138bf val is also enfactored in the 5-limit, with the same tuning as 69edo, tempering out the [[syntonic comma]], 81/80 and {{monzo| -41 1 17 }}. However, it tempers out [[2401/2400]], [[2430/2401]], and 9765625/9633792 in the 7-limit; [[385/384]], [[1375/1372]], 1944/1925, and 9375/9317 in the 11-limit, supporting the [[Meantone family #Cuboctahedra|cuboctahedra]] temperament; [[625/624]], 975/968, [[1001/1000]], and [[1188/1183]] in the 13-limit.

138edo can be treated as the 2.7/5.11/5.13/3 subgroup temperament, which tempers out 24192/24167, 1449459/1449175, and 75000000/74942413.

[[Category:~~Edo~~]]

=== Odd harmonics ===

=== Subsets and supersets ===

Since 138 factors into {{factorization|138}}, 138edo has subset edos {{EDOs| 2, 3, 6, 23, 46, and 69 }}.

[[Category:Echidna]]

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-'''138edo''' is the [[EDO|equal division of the octave]] into 138 parts of 8.6957 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with three mappings possible for the 13-limit: &lt;138 219 320 387 477 511| (patent val), &lt;138 218 320 387 477 510| (138bf), and &lt;138 219 321 388 478 511| (138cde). Using the patent val, it tempers out the shibboleth comma, 1953125/1889568 and the misty comma, 67108864/66430125 in the 5-limit; 875/864, 1029/1024, and 1647086/1594323 in the 7-limit; 896/891, 1331/1323, 1375/1372, and 2401/2376 in the 11-limit; 196/195, 275/273, and 1575/1573 in the 13-limit. Using the 138bf val, it tempers out the [[syntonic comma]], 81/80 and 2288818359375/2199023255552 in the 5-limit; 2401/2400, 2430/2401, and 9765625/9633792 in the 7-limit; 385/384, 1375/1372, 1944/1925, and 9375/9317 in the 11-limit, supporting the [[Meantone family|cuboctahedra temperament]]; 625/624, 975/968, 1001/1000, and 1188/1183 in the 13-limit. Using the 138cde val, it tempers out the [[Diaschismic family|diaschisma]], 2048/2025 and the [[Sensipent family|sensipent comma]], 78732/78125 in the 5-limit; 1728/1715, 10976/10935, and 250047/250000 in the 7-limit; 176/175, 540/539, 896/891, and 85184/84375 in the 11-limit; 351/350, 352/351, 364/363, 640/637, and 2197/2187 in the 13-limit, supporting the [[Diaschismic family|echidna temperament]].
+{{Infobox ET}}
+{{ED intro}}
+Since {{nowrap|138 {{=}} 3 × 46}}, 138edo shares its [[3/2|fifth]] with [[46edo]]. Unlike 46edo, it is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for the 13-limit: {{val| 138 219 320 387 477 511 }} ([[patent val]]), {{val| 138 219 '''321''' '''388''' '''478''' 511 }} (138cde), and {{val| 138 '''218''' 320 387 477 '''510''' }} (138bf). The last mapping uses an alternative flat fifth from [[69edo]].
+Using the patent val, it [[tempering out|tempers out]] 1953125/1889568 ([[shibboleth comma]]) and 67108864/66430125 ([[misty comma]]) in the 5-limit; [[875/864]], [[1029/1024]], and 1647086/1594323 in the 7-limit; [[896/891]], 1331/1323, 1375/1372, and 2401/2376 in the 11-limit; [[196/195]], [[275/273]], and [[1575/1573]] in the 13-limit.
+The 138cde val is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as 46edo, tempering out the [[diaschisma]], 2048/2025 and the [[sensipent comma]], 78732/78125. However, it tempers out [[1728/1715]], [[10976/10935]], and [[250047/250000]] in the 7-limit; [[176/175]], [[540/539]], [[896/891]], and 85184/84375 in the 11-limit; [[351/350]], [[352/351]], [[364/363]], [[640/637]], and [[2197/2187]] in the 13-limit, [[support]]ing the [[echidna]] temperament and giving an excellent tuning.
+The 138bf val is also enfactored in the 5-limit, with the same tuning as 69edo, tempering out the [[syntonic comma]], 81/80 and {{monzo| -41 1 17 }}. However, it tempers out [[2401/2400]], [[2430/2401]], and 9765625/9633792 in the 7-limit; [[385/384]], [[1375/1372]], 1944/1925, and 9375/9317 in the 11-limit, supporting the [[Meantone family #Cuboctahedra|cuboctahedra]] temperament; [[625/624]], 975/968, [[1001/1000]], and [[1188/1183]] in the 13-limit.
 edo can be treated as the 2.7/5.11/5.13/3 subgroup temperament, which tempers out 24192/24167, 1449459/1449175, and 75000000/74942413.
-[[Category:Edo]]
+=== Odd harmonics ===
+{{Harmonics in equal|138}}
+=== Subsets and supersets ===
+Since 138 factors into {{factorization|138}}, 138edo has subset edos {{EDOs| 2, 3, 6, 23, 46, and 69 }}.
+[[Category:Echidna]]