128edo: Difference between revisions

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-The 128 equal division divides the [[Octave|octave]] into 128 equal parts of exactly 9.375 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for [[7-limit|7-limit]] [[Gamelismic_clan|rodan temperament]]. It [[tempering_out|tempers out]] 2109375/2097152 in the [[5-limit|5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the binary (fine) relative cent of [[1edo]].
+{{Infobox ET}}
+{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes.
-=Scales=
+== Theory ==
-[[radon5|radon5]]
+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.
-[[radon11|radon11]]
+See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)
-[[radon16|radon16]]
+=== Prime harmonics ===
+{{Harmonics in equal|128}}
-[https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] - Philipp Gerschlauer
+=== Subsets and supersets ===
-[[Category:128edo]]
+Since 128 factors into 2<sup>7</sup>, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}.
-[[Category:edo]]
-[[Category:rodan]]
+== Regular temperament properties ==
-[[Category:theory]]
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperaments
+|-
+| 1
+| 25\128
+| 234.375
+| 8/7
+| [[Rodan]]
+|-
+| 1
+| 29\128
+| 271.875
+| 75/64
+| [[Orson]]
+|-
+| 1
+| 33\128
+| 309.375
+| 448/375
+| [[Triwell]]
+|-
+| 1
+| 53\128
+| 496.875
+| 4/3
+| [[Undecental]]
+|-
+| 2
+| 13\128
+| 121.875
+| 15/14
+| [[Lagaca]]
+|-
+| 2
+| 15\128
+| 140.625
+| 13/12
+| [[Fifive]]
+|-
+| 4
+| 11\128
+| 103.125
+| 625/588
+| [[Undim]] (7-limit)
+|-
+| 4
+| 15\128
+| 140.625
+| 13/12
+| [[Fourfives]]
+|}
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
+== Scales ==
+* [[Radon5]]
+* [[Radon11]]
+* [[Radon16]]
+[[Category:Rodan]]
+[[Category:Fifive]]
+[[Category:Fourfives]]