1536edo: Difference between revisions

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'''1536EDO''' is the [[EDO|equal division of the octave]] into 1536 parts of exact 0.78125 cents each, which is to say 2(1/1536) as a frequency ratio. Its adjacent step is known as '''Heptamu''' (seventh MIDI-resolution unit, 7mu, 27 = 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.

~~==Intervals==~~

1536EDO is inconsistent to the [[5-limit]] and higher limit, with three mappings possible for 5-limit: <1536 2435 3566| (patent val), <1536 2434 3566| (1536b val, contorted through the 17-limit), and <1536 2435 3567| (1536c val). Using the patent val, it tempers out |71 -36 -6> and |-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 |169 -111 3> in the 5-limit; 250047/250000, 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.

~~==See also==~~

*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

**[[24edo]] (1mu tuning)

**[[48edo]] (2mu tuning)

**[[96edo]] (3mu tuning)

**[[192edo]] (4mu tuning)

**[[384edo]] (5mu tuning)

**[[768edo]] (6mu tuning)

**[[3072edo]] (8mu tuning)

**[[6144edo]] (9mu tuning)

**[[12288edo]] (10mu tuning)

**[[24576edo]] (11mu tuning)

**[[49152edo]] (12mu tuning)

**[[98304edo]] (13mu tuning)

**[[196608edo]] (14mu tuning)

[[~~Category:Edo~~]]

== Theory ==

[[~~Category~~:~~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.

=== As a tuning standard ===

A step of 1536edo is known as a ''[[heptamu]]'' (seventh MIDI-resolution unit, 7mu, {{nowrap|27 {{=}} 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 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.

=== 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 ==

* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

** [[24edo]] (1mu tuning)

** [[48edo]] (2mu tuning)

** [[96edo]] (3mu tuning)

** [[192edo]] (4mu tuning)

** [[384edo]] (5mu tuning)

** [[768edo]] (6mu tuning)

** [[3072edo]] (8mu tuning)

** [[6144edo]] (9mu tuning)

** [[12288edo]] (10mu tuning)

** [[24576edo]] (11mu tuning)

** [[49152edo]] (12mu tuning)

** [[98304edo]] (13mu tuning)

** [[196608edo]] (14mu tuning)

== External links ==

* [http://tonalsoft.com/enc/number/7mu.aspx 7mu / heptamu] on [[Tonalsoft Encyclopedia]]

@@ Line 1: / Line 1: @@
-'''1536EDO''' is the [[EDO|equal division of the octave]] into 1536 parts of exact 0.78125 cents each, which is to say 2<sup>(1/1536)</sup> as a frequency ratio. 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.
+{{Infobox ET}}
-==Intervals==
+{{ED intro}}
-EDO is inconsistent to the [[5-limit]] and higher limit, with three mappings possible for 5-limit: &lt;1536 2435 3566| (patent val), &lt;1536 2434 3566| (1536b val, contorted through the 17-limit), and &lt;1536 2435 3567| (1536c val). Using the patent val, it tempers out |71 -36 -6&gt; and |-32 -34 37&gt; 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 |169 -111 3&gt; in the 5-limit; 250047/250000, 134217728/133984375, and 12111126300875/12050326889856 in the 7-limit.
-==See also==
-*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
-**[[24edo]] (1mu tuning)
-**[[48edo]] (2mu tuning)
-**[[96edo]] (3mu tuning)
-**[[192edo]] (4mu tuning)
-**[[384edo]] (5mu tuning)
-**[[768edo]] (6mu tuning)
-**[[3072edo]] (8mu tuning)
-**[[6144edo]] (9mu tuning)
-**[[12288edo]] (10mu tuning)
-**[[24576edo]] (11mu tuning)
-**[[49152edo]] (12mu tuning)
-**[[98304edo]] (13mu tuning)
-**[[196608edo]] (14mu tuning)
-[[Category:Edo]]
+== Theory ==
-[[Category: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.
+=== As a tuning standard ===
+A step of 1536edo is known as a ''[[heptamu]]'' (seventh MIDI-resolution unit, 7mu, {{nowrap|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 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.
+=== 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 ==
+* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
+** [[24edo]] (1mu tuning)
+** [[48edo]] (2mu tuning)
+** [[96edo]] (3mu tuning)
+** [[192edo]] (4mu tuning)
+** [[384edo]] (5mu tuning)
+** [[768edo]] (6mu tuning)
+** [[3072edo]] (8mu tuning)
+** [[6144edo]] (9mu tuning)
+** [[12288edo]] (10mu tuning)
+** [[24576edo]] (11mu tuning)
+** [[49152edo]] (12mu tuning)
+** [[98304edo]] (13mu tuning)
+** [[196608edo]] (14mu tuning)
+== External links ==
+* [http://tonalsoft.com/enc/number/7mu.aspx 7mu / heptamu] on [[Tonalsoft Encyclopedia]]