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 & 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 & 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
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128<br />(11\128)
| 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]]
* [[Radon11]]
* [[Radon16]]
 
[[Category:Rodan]]
[[Category:Fourfives]]

Latest revision as of 11:18, 11 April 2025

← 127edo 128edo 129edo →
Prime factorization 27
Step size 9.375 ¢ 
Fifth 75\128 (703.125 ¢)
Semitones (A1:m2) 13:9 (121.9 ¢ : 84.38 ¢)
Consistency limit 7
Distinct consistency limit 7

128 equal divisions of the octave (abbreviated 128edo or 128ed2), also called 128-tone equal temperament (128tet) or 128 equal temperament (128et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 128 equal parts of about 9.38 ¢ each. Each step represents a frequency ratio of 21/128, or the 128th root of 2. It is notable for being the equal division corresponding to a standard MIDI piano roll of 128 notes.

Theory

The equal temperament 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 41 & 87 temperament, as well as for 7-limit fourfives, the 60 & 68 temperament.

See also 128 notes per octave on Alto Saxophone (Demo by Philipp Gerschlauer)

Prime harmonics

Approximation of prime harmonics in 128edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.17 -1.94 -3.20 +1.81 +3.22 -1.83 +2.49 -0.15 +1.67 -1.29
Relative (%) +0.0 +12.5 -20.7 -34.1 +19.3 +34.4 -19.5 +26.5 -1.6 +17.8 -13.7
Steps
(reduced) 		128
(0) 		203
(75) 		297
(41) 		359
(103) 		443
(59) 		474
(90) 		523
(11) 		544
(32) 		579
(67) 		622
(110) 		634
(122)

Subsets and supersets

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

Regular temperament properties

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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 27/25 Fifive
4 15\128 140.625 27/25 Fourfives
4 53\128
(11\128) 		496.875
(103.125) 		4/3 Undim (7-limit)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

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