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


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 ===
{{Harmonics in equal|128|columns=11}}
=== Miscellaneous properties ===
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]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-1 center-2 center-3"
{| class="wikitable center-1 center-2 center-3"
|+Rank-2 temperaments
|+Rank-2 temperaments
!Periods
! Periods<br>per 8ve
per octave
! Generator<br>(Reduced)
!Generator
! Cents<br>(Reduced)
(reduced)
! Associated<br>Ratio
!Cents
! Temperaments
(reduced)
!Associated
Ratio
!Temperaments
|-
|-
|1
| 1
|25\128
| 25\128
|234.375
| 234.375
|8/7
| 8/7
|[[Rodan]]
| [[Rodan]]
|-
|-
|1
| 1
|29\128
| 29\128
|271.875
| 271.875
|75/64
| 75/64
|[[Orson]]
| [[Orson]]
|-
|-
|1
| 1
|33\128
| 33\128
|309.375
| 309.375
|448/375
| 448/375
|[[Triwell]]
| [[Triwell]]
|-
|-
|1
| 1
|53\128
| 53\128
|496.875
| 496.875
|4/3
| 4/3
|[[Undecental]]
| [[Undecental]]
|-
|-
|2
| 2
|13\128
| 13\128
|121.875
| 121.875
|15/14
| 15/14
|[[Gamelismic clan#Lagaca|Lagaca]]
| [[Lagaca]]
|-
|-
|2
| 2
|15\128
| 15\128
|140.625
| 140.625
|27/25
| 27/25
|[[Fifive]]
| [[Fifive]]
|-
|-
|4
| 4
|15\128
| 15\128
|140.625
| 140.625
|27/25
| 27/25
|[[Fifive family#Fourfives|Fourfives]]
| [[Fourfives]]
|-
|-
|4
| 4
|53\128
| 53\128<br>(11\128)
(11\128)
| 496.875<br>(103.125)
|496.875
| 4/3
 
| [[Undim]] (7-limit)
(103.125)
|4/3
|[[Undim]] (7-limit)
|}
|}


== Scales ==
== Scales ==
* [[radon5]]
* [[radon5]]
* [[radon11]]
* [[radon11]]

Revision as of 14:17, 1 November 2022

← 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

Template:EDO introIt is notable because it is 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.

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)

Miscellaneous properties

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

Regular temperament properties

Rank-2 temperaments
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		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)

Scales

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