128edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|128}} It is notable for being 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 ==
== Theory ==
128edo [[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 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.  
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)
See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|128|columns=11}}
{{Harmonics in equal|128}}


=== Subsets and supersets ===
=== Subsets and supersets ===
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Rank-2 temperaments by generators
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 15\128
| 15\128
| 140.625
| 140.625
| 27/25
| 13/12
| [[Fifive]]
| [[Fifive]]
|-
| 4
| 11\128
| 103.125
| 625/588
| [[Undim]] (7-limit)
|-
|-
| 4
| 4
| 15\128
| 15\128
| 140.625
| 140.625
| 27/25
| 13/12
| [[Fourfives]]
| [[Fourfives]]
|-
| 4
| 53\128<br>(11\128)
| 496.875<br>(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== Scales ==
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[[Category:Rodan]]
[[Category:Rodan]]
[[Category:Fifive]]
[[Category:Fourfives]]
[[Category:Fourfives]]

Latest revision as of 04:37, 25 August 2026

← 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 13/12 Fifive
4 11\128 103.125 625/588 Undim (7-limit)
4 15\128 140.625 13/12 Fourfives

* In minimal-generator form

Scales

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