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{{Infobox ET}}
{{Infobox ET}}
'''64 equal divisions of the octave''' ('''64edo''') is the tuning system that divides the octave into 64 equal parts of exactly 18.75{{cent}} each.
{{ED intro}}


== Theory ==
== Theory ==
64edo is a [[The Riemann zeta function and tuning #Valley_edos|zeta valley edo]] and is very bad at approximating [[JI]] for its size. It has two options of [[3/2|fifth]] almost equally far from [[just]]. The sharp fifth from the 64b [[val]] is inherited from [[32edo]] and produces a hard [[superpythagorean]] scale, while the slightly more accurate flat fifth from the [[patent val]] is within the [[meantone]]/[[flattone]] range. However bizarrely, the flat fifth does not [[support]] meantone or flattone in its [[patent val]], and instead supports the obscure unnamed {{nowrap|7c & 12c}} (or {{nowrap|19 & 64}}) temperament which reaches [[5/4]] as a double-diminished fourth. In order to interpret it as flattone, the 64cd val must be used.
Still, the patent val [[tempering out|tempers out]] [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament.
64be val is a tuning for the [[beatles]] temperament and for the [[rank-3]] temperaments [[heimlaug]] and [[vili]] in the 17-limit. 64bccc tunes [[dichotic]], although that is an [[exotemperament]]. 64cdf is a tuning for [[vibhu]].
The patent val of 64edo is the first patent val to represent the intervals 5/4, 81/64, 14/11, 9/7, 13/10, 21/16, and 4/3 distinctly, although in 64edo they are not in the correct order.
=== Odd harmonics ===
{{Harmonics in equal|64}}
{{Harmonics in equal|64}}
The patent val tempers out [[648/625]] in the 5-limit and [[225/224]] in the 7-limit, plus [[66/65]], [[121/120]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit. It provides the optimal patent val in the 7-, 11- and 13-limits for the 16&64 temperament, which would perhaps be of more interest if it was lower in badness.
 
=== Octave stretch ===
64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
 
[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.
 
If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
 
[[Ed257/128#64ed257/128|64ed257/128]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.
 
There are also some nearby [[zeta peak index]] (ZPI) tunings which can be used to improve 64edo's JI approximations: 326zpi, 327zpi, 328zpi and 329zpi. The main Zeta peak index page details all four tunings.
 
=== Subsets and supersets ===
64edo is the 6th power of two edo, and it has subset edos {{EDOs| 2, 4, 8, 16, 32 }}. [[128edo]], which doubles it, corrects its approximation to many of the lower harmonics.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2"
{{Interval table}}
|-
 
![[Degree|#]]
== Notation ==
![[Cent]]s
=== Ups and downs notation ===
|-
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
| 0
{{sharpness-sharp3a}}
| 0.00
 
|-
Using [[Helmholtz–Ellis]] accidentals, 64edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:
| 1
{{Sharpness-sharp3}}
| 18.75
Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.
|-
 
| 2
=== Sagittal notation ===
| 37.50
==== Best fifth notation ====
|-
This notation uses the same sagittal sequence as EDOs [[50edo#Sagittal notation|50]], [[57edo#Sagittal notation|57]], and [[71edo#Second-best fifth notation|71b]].
| 3
 
| 56.25
===== Evo flavor =====
|-
<imagemap>
| 4
File:64-EDO_Evo_Sagittal.svg
| 75.00
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| 5
rect 300 0 751 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 93.75
rect 20 80 160 106 [[1053/1024]]
|-
default [[File:64-EDO_Evo_Sagittal.svg]]
| 6
</imagemap>
| 112.50
 
|-
===== Revo flavor =====
| 7
<imagemap>
| 131.25
File:64-EDO_Revo_Sagittal.svg
|-
desc none
| 8
rect 80 0 300 50 [[Sagittal_notation]]
| 150.00
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 160 106 [[1053/1024]]
| 9
default [[File:64-EDO_Revo_Sagittal.svg]]
| 168.75
</imagemap>
|-
 
| 10
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
| 187.50
 
|-
==== Second-best fifth notation ====
| 11
This notation is a superset of the notation for [[32edo#Sagittal notation|32-EDO]].
| 206.25
 
|-
===== Evo flavor =====
| 12
<imagemap>
| 225.00
File:64b_Evo_Sagittal.svg
|-
desc none
| 13
rect 80 0 300 50 [[Sagittal_notation]]
| 243.75
rect 300 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
| 14
default [[File:64b_Evo_Sagittal.svg]]
| 262.50
</imagemap>
|-
 
| 15
===== Revo flavor =====
| 281.25
<imagemap>
|-
File:64b_Revo_Sagittal.svg
| 16
desc none
| 300.00
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 17
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
| 318.75
default [[File:64b_Revo_Sagittal.svg]]
|-
</imagemap>
| 18
 
| 337.50
===== Evo-SZ flavor =====
|-
<imagemap>
| 19
File:64b_Evo-SZ_Sagittal.svg
| 356.25
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| 20
rect 300 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 375.00
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
|-
default [[File:64b_Evo-SZ_Sagittal.svg]]
| 21
</imagemap>
| 393.75
 
|-
== Instruments ==
| 22
* [[Lumatone mapping for 64edo]]
| 412.50
|-
| 23
| 431.25
|-
| 24
| 450.00
|-
| 25
| 468.75
|-
| 26
| 487.50
|-
| 27
| 506.25
|-
| 28
| 525.00
|-
| 29
| 543.75
|-
| 30
| 562.50
|-
| 31
| 581.25
|-
| 32
| 600.00
|-
| 33
| 618.75
|-
| 34
| 637.50
|-
| 35
| 656.25
|-
| 36
| 675.00
|-
| 37
| 693.75
|-
| 38
| 712.50
|-
| 39
| 731.25
|-
| 40
| 750.00
|-
| 41
| 768.75
|-
| 42
| 787.50
|-
| 43
| 806.25
|-
| 44
| 825.00
|-
| 45
| 843.75
|-
| 46
| 862.50
|-
| 47
| 881.25
|-
| 48
| 900.00
|-
| 49
| 918.75
|-
| 50
| 937.50
|-
| 51
| 956.25
|-
| 52
| 975.00
|-
| 53
| 993.75
|-
| 54
| 1012.50
|-
| 55
| 1031.25
|-
| 56
| 1050.00
|-
| 57
| 1068.75
|-
| 58
| 1087.50
|-
| 59
| 1106.25
|-
| 60
| 1125.00
|-
| 61
| 1143.75
|-
| 62
| 1162.50
|-
| 63
| 1181.25
|-
| 64
| 1200.00
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/ag1FbgAuR6I ''microtonal improvisation in 64edo''] (2025)
Retrieved from "https://en.xen.wiki/w/64edo"