256edo: Difference between revisions

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The 256 equal division divides the [[octave]] into 256 equal parts of exactly 4.6875 [[cent]]s each. It is [[contorted]] in the 5-limit, and the error of harmonic 7 leads to inconsistency, which is likely one of the reasons this EDO attracts little interest.

~~== Theory ==~~

256edo is [[enfactoring|enfactored]] in the [[5-limit]] with the same tuning as [[128edo]], and the error of [[harmonic]] [[7/1|7]] leads to in[[consistency]], which is likely one of the reasons this edo attracts little interest. To start with, consider the sharp-tending 256c [[val]] {{val| 256 406 '''595''' 719 886 }}, which [[tempering out|tempers out]] [[2401/2400]], [[3388/3375]], [[5120/5103]], so that it [[support]]s 7-limit [[hemififths]] and [[11-limit]] [[semihemi]]. The [[patent val]] {{val| 256 406 '''594''' 719 886 }} tempers out [[540/539]], 2200/2187, [[4000/3969]], 12005/11979, among others. It is best tuned in the 2.3.7.11 [[subgroup]], in which it is consistent to the [[11-odd-limit]] minus intervals involving [[5/1|5]].

~~{{Primes in edo|256|columns=15}}~~

256edo is ~~good at~~ the ~~2.23.43.47 subgroup. If~~ the error ~~below 40% is considered "good", 256edo can be used~~ to ~~play no-fives 17 limit. If the "prime number obsession" approach found~~ in ~~math circles~~ is ~~applied - then 256edo can be played using~~ the ~~coprime harmonics~~, ~~no matter how bad~~ the ~~approximation~~. ~~Under such rule~~, ~~256edo supports~~ the 2.7.~~13.19~~ subgroup.

In the 2.~~7.13.19 subgroup~~ in ~~the patent val~~, 256edo ~~tempers out 32851/32768~~, and ~~[[support]]s the corresponding 20 & 73 & 256 rank 3 temperament~~.

In the higher limits, it approximates harmonics 23, 43, and 47 quite accurately.

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

=== Odd harmonics ===

=== Subsets and supersets ===

Since 256 factors into {{factorization|256}}, 256edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 64, and 128 }}.

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-The 256 equal division divides the [[octave]] into 256 equal parts of exactly 4.6875 [[cent]]s each. It is [[contorted]] in the 5-limit, and the error of harmonic 7 leads to inconsistency, which is likely one of the reasons this EDO attracts little interest.
+{{Infobox ET}}
+{{ED intro}}
-== Theory ==
+edo is [[enfactoring|enfactored]] in the [[5-limit]] with the same tuning as [[128edo]], and the error of [[harmonic]] [[7/1|7]] leads to in[[consistency]], which is likely one of the reasons this edo attracts little interest. To start with, consider the sharp-tending 256c [[val]] {{val| 256 406 '''595''' 719 886 }}, which [[tempering out|tempers out]] [[2401/2400]], [[3388/3375]], [[5120/5103]], so that it [[support]]s 7-limit [[hemififths]] and [[11-limit]] [[semihemi]]. The [[patent val]] {{val| 256 406 '''594''' 719 886 }} tempers out [[540/539]], 2200/2187, [[4000/3969]], 12005/11979, among others. It is best tuned in the 2.3.7.11 [[subgroup]], in which it is consistent to the [[11-odd-limit]] minus intervals involving [[5/1|5]].
-{{Primes in edo|256|columns=15}}
-edo is good at the 2.23.43.47 subgroup. If the error below 40% is considered "good", 256edo can be used to play no-fives 17 limit. If the "prime number obsession" approach found in math circles is applied - then 256edo can be played using the coprime harmonics, no matter how bad the approximation. Under such rule, 256edo supports the 2.7.13.19 subgroup.
-In the 2.7.13.19 subgroup in the patent val, 256edo tempers out 32851/32768, and [[support]]s the corresponding 20 & 73 & 256 rank 3 temperament.
+In the higher limits, it approximates harmonics 23, 43, and 47 quite accurately.
-[[Category:Equal divisions of the octave]]
+=== Odd harmonics ===
+{{Harmonics in equal|256}}
+=== Subsets and supersets ===
+Since 256 factors into {{factorization|256}}, 256edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 64, and 128 }}.