1536edo: Difference between revisions

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{{EDO intro|1536}} Its adjacent step is known as ''[[heptamu]]'' (seventh MIDI-resolution unit, ''[[7mu]]'', 2<sup>7</sup> = 128 equal divisions of the [[12edo]] semitone). The internal data structure of the 7mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 6 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

1536edo is in[[consistent]] to the [[5-odd-limit]] and ~~higher limits~~, with ~~three mappings possible for 5-limit: {{val~~| ~~1536 2435 3566 }} (~~[[~~patent val~~]]), {{val| 1536 '''2434''' 3566 }} (1536b ~~val,~~ [[~~enfactoring|enfactored]] through the 17-limit, see [[768edo~~]]), and {{val| 1536 2435 '''3567''' }} (1536c val)~~. Using the patent val~~, ~~it tempers out {{monzo~~| ~~71 -36 -6 }} and {{monzo| -32 -34 37 }} in~~ the 5-limit; [[~~703125/702464~~]], [[~~4802000/4782969~~]]~~, and 54975581388800/54936068900769 in the 7-limit. Using the 1536c val, it tempers out~~ 6115295232/6103515625 ([[vishnuzma]]) and {{monzo| 169 -111 3 }} in the 5-limit; [[250047/250000]], 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.

== Theory ==

1536edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. Otherwise it is excellent in approximating harmonics [[7/1|7]], [[9/1|9]], [[13/1|13]], and [[15/1|15]], making it suitable for a 2.9.15.7.13 [[subgroup]] interpretation, with an optional addition of either [[11/1|11]] or [[17/1|17]].

If we do use it for the 5-limit, then {{val| 1536 '''2434''' '''3566''' }} (1536b [[val]]) and {{val| 1536 '''2435''' '''3567''' }} (1536c val) are worth considering, but 1536b is [[enfactoring|enfactored]] through the 17-limit (see [[768edo]]), leaving us with 1536c alone. It [[tempering out|tempers out]] 6115295232/6103515625 ([[vishnuzma]]) and {{monzo| 169 -111 3 }} in the 5-limit; [[250047/250000]], 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 1536 factors into {{factorization|1536}}, 1536edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, and 768 }}.

== See also ==

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 {{EDO intro|1536}} Its adjacent step is known as ''[[heptamu]]'' (seventh MIDI-resolution unit, ''[[7mu]]'', 2<sup>7</sup> = 128 equal divisions of the [[12edo]] semitone). The internal data structure of the 7mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or &minus;) showing the direction of the pitch-bend up or down, and 6 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
-edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for 5-limit: {{val| 1536 2435 3566 }} ([[patent val]]), {{val| 1536 '''2434''' 3566 }} (1536b val, [[enfactoring|enfactored]] through the 17-limit, see [[768edo]]), and {{val| 1536 2435 '''3567''' }} (1536c val). Using the patent val, it tempers out {{monzo| 71 -36 -6 }} and {{monzo| -32 -34 37 }} in the 5-limit; [[703125/702464]], [[4802000/4782969]], and 54975581388800/54936068900769 in the 7-limit. Using the 1536c val, it tempers out 6115295232/6103515625 ([[vishnuzma]]) and {{monzo| 169 -111 3 }} in the 5-limit; [[250047/250000]], 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.
+== Theory ==
+edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. Otherwise it is excellent in approximating harmonics [[7/1|7]], [[9/1|9]], [[13/1|13]], and [[15/1|15]], making it suitable for a 2.9.15.7.13 [[subgroup]] interpretation, with an optional addition of either [[11/1|11]] or [[17/1|17]].
+If we do use it for the 5-limit, then {{val| 1536 '''2434''' '''3566''' }} (1536b [[val]]) and {{val| 1536 '''2435''' '''3567''' }} (1536c val) are worth considering, but 1536b is [[enfactoring|enfactored]] through the 17-limit (see [[768edo]]), leaving us with 1536c alone. It [[tempering out|tempers out]] 6115295232/6103515625 ([[vishnuzma]]) and {{monzo| 169 -111 3 }} in the 5-limit; [[250047/250000]], 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|1536}}
+=== Subsets and supersets ===
+Since 1536 factors into {{factorization|1536}}, 1536edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, and 768 }}.
 == See also ==