194edo: Difference between revisions

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'''194edo''' is the [[EDO|equal division of the octave]] into 194 parts of 6.1856 cents each. 194edo is consistent to the 5-limit, but 194b and 194bc vals are also possible as having a large relative error for the 5-limit. It tempers out 43046721/41943040 (double large green comma) and 1224440064/1220703125 (parakleisma) in the 5-limit. Using the patent val, it tempers out 1728/1715, 78125/76832, and 177147/175000 in the 7-limit; 2187/2156, 2420/2401, 2835/2816, and 4000/3993 in the 11-limit; 351/350, 640/637, 1573/1568, 1625/1617, and 1701/1690 in the 13-limit. Using the 194b val, it tempers out 245/243, 65536/64827, and 78125/76832 in the 7-limit; 385/384, 896/891, 2420/2401, and 9375/9317 in the 11-limit; 275/273, 572/567, 640/637, and 1573/1568 in the 13-limit. Using the 194bc val, it tempers out 2048/2025 (diaschisma) and |0 -41 28> ([[28edt]] no-twos comma) in the 5-limit; 2401/2400, 65536/64827, and 1071875/1062882 in the 7-limit; 176/175, 896/891, 3773/3750, and 6655/6561 in the 11-limit; 325/324, 572/567, 847/845, 1573/1568, and 6656/6561 in the 13-limit. Using the 194d val, it tempers out 225/224, 118098/117649, and 15752961/15625000 in the 7-limit; 385/384, 540/539, 4000/3993, and 3720087/3660250 in the 11-limit; 847/845, 1188/1183, 1875/1859, 4225/4224, and 6561/6500 in the 13-limit.

[[~~Category:Equal divisions of~~ the octave]]

194edo is only [[consistent]] to the [[5-odd-limit]], though harmonics 3 are 5 are about halfway between steps. Harmonic [[7/1|7]] is sharp, causing [[7/6]], [[7/5]], and their [[octave complement]]s to become inconsistent in the [[7-odd-limit]]. It is therefore best to treat 194edo as a 2.9.15.21.11.13.17.19-subgroup system, with all [[35-odd-limit]] intervals in this subgroup having less than 25% relative error.

If the patent val is used, the equal temperament [[tempering out|tempers out]] 43046721/41943040 (lalagu comma) and 1224440064/1220703125 ([[parakleisma]]) in the 5-limit. It tempers out [[1728/1715]], 78125/76832, and 177147/175000 in the 7-limit; 2187/2156, 2420/2401, 2835/2816, and [[4000/3993]] in the 11-limit; [[351/350]], [[640/637]], [[1573/1568]], 1625/1617, and 1701/1690 in the 13-limit. Using the 194b val, it tempers out [[245/243]], 65536/64827, and 78125/76832 in the 7-limit; [[385/384]], [[896/891]], 2420/2401, and 9375/9317 in the 11-limit; [[275/273]], 572/567, 640/637, and 1573/1568 in the 13-limit. Using the 194bc val, it tempers out 2048/2025 ([[diaschisma]]) and {{monzo| 0 -41 28 }} ([[28edt]] no-twos comma) in the 5-limit; [[2401/2400]], 65536/64827, and 1071875/1062882 in the 7-limit; [[176/175]], 896/891, 3773/3750, and 6655/6561 in the 11-limit; [[325/324]], 572/567, [[847/845]], 1573/1568, and 6656/6561 in the 13-limit. Using the 194d val, it tempers out [[225/224]], 118098/117649, and 15752961/15625000 in the 7-limit; 385/384, [[540/539]], 4000/3993, and 3720087/3660250 in the 11-limit; 847/845, [[1188/1183]], 1875/1859, [[4225/4224]], and 6561/6500 in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 194 factors into {{factorization|194}}, 194edo contains [[2edo]] and [[97edo]] as subsets. [[388edo]], which doubles it, corrects harmonics 3, 5, and 7, and makes for a strong high-limit system, being consistent to the [[37-odd-limit]].

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-'''194edo''' is the [[EDO|equal division of the octave]] into 194 parts of 6.1856 cents each. 194edo is consistent to the 5-limit, but 194b and 194bc vals are also possible as having a large relative error for the 5-limit. It tempers out 43046721/41943040 (double large green comma) and 1224440064/1220703125 (parakleisma) in the 5-limit. Using the patent val, it tempers out 1728/1715, 78125/76832, and 177147/175000 in the 7-limit; 2187/2156, 2420/2401, 2835/2816, and 4000/3993 in the 11-limit; 351/350, 640/637, 1573/1568, 1625/1617, and 1701/1690 in the 13-limit. Using the 194b val, it tempers out 245/243, 65536/64827, and 78125/76832 in the 7-limit; 385/384, 896/891, 2420/2401, and 9375/9317 in the 11-limit; 275/273, 572/567, 640/637, and 1573/1568 in the 13-limit. Using the 194bc val, it tempers out 2048/2025 (diaschisma) and |0 -41 28&gt; ([[28edt]] no-twos comma) in the 5-limit; 2401/2400, 65536/64827, and 1071875/1062882 in the 7-limit; 176/175, 896/891, 3773/3750, and 6655/6561 in the 11-limit; 325/324, 572/567, 847/845, 1573/1568, and 6656/6561 in the 13-limit. Using the 194d val, it tempers out 225/224, 118098/117649, and 15752961/15625000 in the 7-limit; 385/384, 540/539, 4000/3993, and 3720087/3660250 in the 11-limit; 847/845, 1188/1183, 1875/1859, 4225/4224, and 6561/6500 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave]]
+edo is only [[consistent]] to the [[5-odd-limit]], though harmonics 3 are 5 are about halfway between steps. Harmonic [[7/1|7]] is sharp, causing [[7/6]], [[7/5]], and their [[octave complement]]s to become inconsistent in the [[7-odd-limit]]. It is therefore best to treat 194edo as a 2.9.15.21.11.13.17.19-subgroup system, with all [[35-odd-limit]] intervals in this subgroup having less than 25% relative error.
+If the patent val is used, the equal temperament [[tempering out|tempers out]] 43046721/41943040 (lalagu comma) and 1224440064/1220703125 ([[parakleisma]]) in the 5-limit. It tempers out [[1728/1715]], 78125/76832, and 177147/175000 in the 7-limit; 2187/2156, 2420/2401, 2835/2816, and [[4000/3993]] in the 11-limit; [[351/350]], [[640/637]], [[1573/1568]], 1625/1617, and 1701/1690 in the 13-limit. Using the 194b val, it tempers out [[245/243]], 65536/64827, and 78125/76832 in the 7-limit; [[385/384]], [[896/891]], 2420/2401, and 9375/9317 in the 11-limit; [[275/273]], 572/567, 640/637, and 1573/1568 in the 13-limit. Using the 194bc val, it tempers out 2048/2025 ([[diaschisma]]) and {{monzo| 0 -41 28 }} ([[28edt]] no-twos comma) in the 5-limit; [[2401/2400]], 65536/64827, and 1071875/1062882 in the 7-limit; [[176/175]], 896/891, 3773/3750, and 6655/6561 in the 11-limit; [[325/324]], 572/567, [[847/845]], 1573/1568, and 6656/6561 in the 13-limit. Using the 194d val, it tempers out [[225/224]], 118098/117649, and 15752961/15625000 in the 7-limit; 385/384, [[540/539]], 4000/3993, and 3720087/3660250 in the 11-limit; 847/845, [[1188/1183]], 1875/1859, [[4225/4224]], and 6561/6500 in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|194}}
+=== Subsets and supersets ===
+Since 194 factors into {{factorization|194}}, 194edo contains [[2edo]] and [[97edo]] as subsets. [[388edo]], which doubles it, corrects harmonics 3, 5, and 7, and makes for a strong high-limit system, being consistent to the [[37-odd-limit]].