128edo: Difference between revisions

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{{~~EDO~~ intro~~|128~~}}It is notable ~~because it is~~ the equal division corresponding to a standard MIDI piano roll of 128 notes.

{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes.

== Theory ==

~~128edo is the [[optimal patent val]] for [[7-limit]] [[Rodan]]~~ temperament~~. It~~ [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.

The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 & 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 & 68}} temperament.

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===

=== ~~Miscellaneous properties~~ ===

=== Subsets and supersets ===

~~Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred|long hundred]]~~, 128edo has ~~a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]]~~.

Since 128 factors into 27, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}.

== Regular temperament properties ==

{| class="wikitable center-~~1 center~~-~~2 center-3~~"

{| class="wikitable center-all left-5"

|+~~Rank~~-2 temperaments

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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|-

| 4

| 53\128 (11\128)

| 53\128 (11\128)

| 496.875 (103.125)

| 496.875 (103.125)

| 4/3

| [[Undim]] (7-limit)

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

* [[~~radon5~~]]

* [[Radon5]]

* [[~~radon11~~]]

* [[Radon11]]

* [[~~radon16~~]]

* [[Radon16]]

~~[[Category:128edo| ]] ~~

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

[[Category:Rodan]]

[[Category:Fourfives]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes.
+{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes.
 == Theory ==
-edo is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.
+The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 &amp; 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 &amp; 68}} temperament.
 See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)
 === Prime harmonics ===
-{{Harmonics in equal|128|columns=11}}
+{{Harmonics in equal|128}}
-=== Miscellaneous properties ===
+=== Subsets and supersets ===
-Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred|long hundred]], 128edo has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].
+Since 128 factors into 2<sup>7</sup>, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}.
 == Regular temperament properties ==
-{| class="wikitable center-1 center-2 center-3"
+{| class="wikitable center-all left-5"
-|+Rank-2 temperaments
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 65: / Line 66: @@
 |-
 | 4
-| 53\128<br>(11\128)
+| 53\128<br />(11\128)
-| 496.875<br>(103.125)
+| 496.875<br />(103.125)
 | 4/3
 | [[Undim]] (7-limit)
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
-* [[radon5]]
+* [[Radon5]]
-* [[radon11]]
+* [[Radon11]]
-* [[radon16]]
+* [[Radon16]]
-[[Category:128edo| ]] <!-- main article -->
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Rodan]]
+[[Category:Fourfives]]