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default [[File:64b_Evo-SZ_Sagittal.svg]]
default [[File:64b_Evo-SZ_Sagittal.svg]]
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== Octave stretch or compression ==
64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[ed12|229ed12]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
[[ed6|165ed6]] can also be used: it is similar to 229ed12 but both the improvements and shortcomings are amplified.
If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 7/1 and 11/1 are improved by [[zpi|328zpi]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
[[ed12|230ed12]] can also be used: it is similar to 328zpi but both the improvements and shortcomings are amplified.
What follows is a comparison of stretched- and compressed-octave 64edo tunings.
; [[ed7|179ed7]]
* Octave size: 1204.50{{c}}
Stretching the octave of 64edo by around 4.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 8.99{{c}}. The tuning 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3{{c}}.
{{Harmonics in equal|179|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 179ed7}}
{{Harmonics in equal|179|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 179ed7 (continued)}}
; [[ed6|165ed6]]
* Octave size: 1203.18{{c}}
Stretching the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 9.25{{c}}. The tuning 165ed6 does this.
{{Harmonics in equal|165|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 165ed6}}
{{Harmonics in equal|165|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 165ed6 (continued)}}
; [[ed12|229ed12]]
* Octave size: 1202.29{{c}}
Stretching the octave of 64edo by around 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.17{{c}}. The tuning 229ed12 does this. So does the tuning [[equal tuning|221ed11]] whose octave is identical within 0.1{{c}}.
{{Harmonics in equal|229|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 229ed12}}
{{Harmonics in equal|229|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 229ed12 (continued)}}
; [[zpi|327zpi]]
* Step size: 18.767{{c}}, octave size: 1201.09{{c}}
Stretching the octave of 64edo by around 1{{c}} results in improved primes 3 and 11, but worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 9.23{{c}}. The tuning 327zpi does this.
{{Harmonics in cet|18.767|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 327zpi}}
{{Harmonics in cet|18.767|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 327zpi (continued)}}
; [[WE|64et, 11-limit WE tuning]]
* Step size: 18.755{{c}}, octave size: 1200.32{{c}}
Stretching the octave of 64edo by around a third of a cent results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 8.50{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
{{Harmonics in cet|18.755|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning}}
{{Harmonics in cet|18.755|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning (continued)}}
; 64edo
* Step size: 18.750{{c}}, octave size: 1200.00{{c}}
Pure-octaves 64edo approximates all harmonics up to 16 within 8.21{{c}}. The octave of 64edo's 13-limit [[WE]] tuning differs by only 0.13{{c}} from pure.
{{Harmonics in equal|64|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64edo}}
{{Harmonics in equal|64|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64edo (continued)}}
; [[zpi|328zpi]]
* Step size: 18.721{{c}}, octave size: 1198.14{{c}}
Compressing the octave of 64edo by just under 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.02{{c}}. The tuning 328zpi does this.
{{Harmonics in cet|18.721|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 328zpi}}
{{Harmonics in cet|18.721|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 328zpi (continued)}}
; [[ed7|180ed7]]
* Octave size: 1197.80{{c}}
Compressing the octave of 64edo by just over 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.34{{c}}. The tuning 180ed7 does this.
{{Harmonics in equal|180|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 180ed7}}
{{Harmonics in equal|180|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 180ed7 (continued)}}
; [[ed12|230ed12]]
* Octave size: 1197.07{{c}}
Compressing the octave of 64edo by just under 3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.80{{c}}. The tuning 230ed12 does this.
{{Harmonics in equal|230|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 230ed12}}
{{Harmonics in equal|230|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 230ed12 (continued)}}
; [[ed5|149ed5]]
* Step size: Octave size: 1196.81{{c}}
Compressing the octave of 64edo by just over 3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.12{{c}}. The tuning 149ed5 does this.
{{Harmonics in equal|149|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 149ed5}}
{{Harmonics in equal|149|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 149ed5 (continued)}}


== Instruments ==
== Instruments ==

Revision as of 07:28, 2 September 2025

← 63edo 64edo 65edo →
Prime factorization 26
Step size 18.75 ¢ 
Fifth 37\64 (693.75 ¢)
Semitones (A1:m2) 3:7 (56.25 ¢ : 131.3 ¢)
Dual sharp fifth 38\64 (712.5 ¢) (→ 19\32)
Dual flat fifth 37\64 (693.75 ¢)
Dual major 2nd 11\64 (206.25 ¢)
Consistency limit 3
Distinct consistency limit 3

64 equal divisions of the octave (abbreviated 64edo or 64ed2), also called 64-tone equal temperament (64tet) or 64 equal temperament (64et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 64 equal parts of about 18.8 ¢ each. Each step represents a frequency ratio of 21/64, or the 64th root of 2.

Theory

64edo is a zeta valley edo and is very bad at approximating JI for its size. It has two options of 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 7c & 12c (or 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 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

Approximation of odd harmonics in 64edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -8.21 +7.44 +6.17 +2.34 -7.57 +3.22 -0.77 +7.54 +2.49 -2.03 +9.23
Relative (%) -43.8 +39.7 +32.9 +12.5 -40.4 +17.2 -4.1 +40.2 +13.3 -10.8 +49.2
Steps
(reduced) 		101
(37) 		149
(21) 		180
(52) 		203
(11) 		221
(29) 		237
(45) 		250
(58) 		262
(6) 		272
(16) 		281
(25) 		290
(34)

Subsets and supersets

64edo is the 6th power of two edo, and it has subset edos 2, 4, 8, 16, 32. 128edo, which doubles it, corrects its approximation to many of the lower harmonics.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 37\64) 		Ups and downs notation
(Dual sharp fifth 38\64)
0 0 1/1 D D
1 18.8 ^D, E♭♭♭ ^D, vE♭
2 37.5 vD♯, ^E♭♭♭ ^^D, E♭
3 56.3 30/29, 31/30, 32/31 D♯, vE♭♭ ^3D, ^E♭
4 75 ^D♯, E♭♭ ^4D, ^^E♭
5 93.8 vD𝄪, ^E♭♭ ^5D, ^3E♭
6 112.5 16/15, 31/29 D𝄪, vE♭ v4D♯, ^4E♭
7 131.3 14/13 ^D𝄪, E♭ v3D♯, v5E
8 150 12/11 vD♯𝄪, ^E♭ vvD♯, v4E
9 168.8 32/29 D♯𝄪, vE vD♯, v3E
10 187.5 29/26 E D♯, vvE
11 206.3 ^E, F♭♭ ^D♯, vE
12 225 vE♯, ^F♭♭ E
13 243.8 15/13, 23/20 E♯, vF♭ ^E, vF
14 262.5 ^E♯, F♭ F
15 281.3 20/17 vE𝄪, ^F♭ ^F, vG♭
16 300 19/16 E𝄪, vF ^^F, G♭
17 318.8 F ^3F, ^G♭
18 337.5 17/14, 28/23 ^F, G♭♭♭ ^4F, ^^G♭
19 356.3 16/13 vF♯, ^G♭♭♭ ^5F, ^3G♭
20 375 F♯, vG♭♭ v4F♯, ^4G♭
21 393.8 ^F♯, G♭♭ v3F♯, v5G
22 412.5 19/15 vF𝄪, ^G♭♭ vvF♯, v4G
23 431.3 F𝄪, vG♭ vF♯, v3G
24 450 ^F𝄪, G♭ F♯, vvG
25 468.8 21/16 vF♯𝄪, ^G♭ ^F♯, vG
26 487.5 F♯𝄪, vG G
27 506.2 G ^G, vA♭
28 525 19/14, 23/17 ^G, A♭♭♭ ^^G, A♭
29 543.8 26/19 vG♯, ^A♭♭♭ ^3G, ^A♭
30 562.5 29/21 G♯, vA♭♭ ^4G, ^^A♭
31 581.3 7/5 ^G♯, A♭♭ ^5G, ^3A♭
32 600 vG𝄪, ^A♭♭ v4G♯, ^4A♭
33 618.8 10/7 G𝄪, vA♭ v3G♯, v5A
34 637.5 ^G𝄪, A♭ vvG♯, v4A
35 656.3 19/13 vG♯𝄪, ^A♭ vG♯, v3A
36 675 28/19, 31/21, 34/23 G♯𝄪, vA G♯, vvA
37 693.8 A ^G♯, vA
38 712.5 ^A, B♭♭♭ A
39 731.3 29/19, 32/21 vA♯, ^B♭♭♭ ^A, vB♭
40 750 A♯, vB♭♭ ^^A, B♭
41 768.8 ^A♯, B♭♭ ^3A, ^B♭
42 787.5 30/19 vA𝄪, ^B♭♭ ^4A, ^^B♭
43 806.3 A𝄪, vB♭ ^5A, ^3B♭
44 825 ^A𝄪, B♭ v4A♯, ^4B♭
45 843.7 13/8, 31/19 vA♯𝄪, ^B♭ v3A♯, v5B
46 862.5 23/14, 28/17 A♯𝄪, vB vvA♯, v4B
47 881.2 B vA♯, v3B
48 900 32/19 ^B, C♭♭ A♯, vvB
49 918.8 17/10 vB♯, ^C♭♭ ^A♯, vB
50 937.5 B♯, vC♭ B
51 956.3 26/15 ^B♯, C♭ ^B, vC
52 975 vB𝄪, ^C♭ C
53 993.8 B𝄪, vC ^C, vD♭
54 1012.5 C ^^C, D♭
55 1031.3 29/16 ^C, D♭♭♭ ^3C, ^D♭
56 1050 11/6 vC♯, ^D♭♭♭ ^4C, ^^D♭
57 1068.8 13/7 C♯, vD♭♭ ^5C, ^3D♭
58 1087.5 15/8 ^C♯, D♭♭ v4C♯, ^4D♭
59 1106.3 vC𝄪, ^D♭♭ v3C♯, v5D
60 1125 C𝄪, vD♭ vvC♯, v4D
61 1143.8 29/15, 31/16 ^C𝄪, D♭ vC♯, v3D
62 1162.5 vC♯𝄪, ^D♭ C♯, vvD
63 1181.3 C♯𝄪, vD ^C♯, vD
64 1200 2/1 D D

Notation

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.

Step offset 0 1 2 3 4 5 6 7
Sharp symbol  
         
Flat symbol
           

Using Helmholtz–Ellis accidentals, 64edo can also be notated using alternative ups and downs:

Step offset 0 1 2 3 4 5 6 7
Sharp symbol  
 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 
 		 

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.

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as EDOs 50, 57, and 71b.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation1053/1024

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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.

Second-best fifth notation

This notation is a superset of the notation for 32-EDO.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation
Evo-SZ flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Octave stretch or compression

64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by 229ed12, a compressed-octave version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

165ed6 can also be used: it is similar to 229ed12 but both the improvements and shortcomings are amplified.

If one prefers a stretched-octave, 64edo's approximations of 3/1, 5/1, 7/1 and 11/1 are improved by 328zpi, a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.

230ed12 can also be used: it is similar to 328zpi but both the improvements and shortcomings are amplified.

What follows is a comparison of stretched- and compressed-octave 64edo tunings.

179ed7
  • Octave size: 1204.50 ¢

Stretching the octave of 64edo by around 4.5 ¢ results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 8.99 ¢. The tuning 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3 ¢.

Approximation of harmonics in 179ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.50 -1.11 +8.99 -0.92 +3.39 +0.00 -5.33 -2.22 +3.58 +7.96 +7.88
Relative (%) +23.9 -5.9 +47.8 -4.9 +18.0 +0.0 -28.3 -11.8 +19.0 +42.3 +41.9
Steps
(reduced) 		64
(64) 		101
(101) 		128
(128) 		148
(148) 		165
(165) 		179
(0) 		191
(12) 		202
(23) 		212
(33) 		221
(42) 		229
(50)
Approximation of harmonics in 179ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.05 +4.50 -2.02 -0.83 +7.13 +2.28 +2.78 +8.08 -1.11 -6.37 -8.04 -6.44
Relative (%) +5.6 +23.9 -10.8 -4.4 +37.9 +12.1 +14.8 +42.9 -5.9 -33.8 -42.7 -34.2
Steps
(reduced) 		236
(57) 		243
(64) 		249
(70) 		255
(76) 		261
(82) 		266
(87) 		271
(92) 		276
(97) 		280
(101) 		284
(105) 		288
(109) 		292
(113)
165ed6
  • Octave size: 1203.18 ¢

Stretching the octave of 64edo by around 3 ¢ results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 9.25 ¢. The tuning 165ed6 does this.

Approximation of harmonics in 165ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.18 -3.18 +6.37 -3.95 +0.00 -3.67 -9.25 -6.37 -0.77 +3.42 +3.18
Relative (%) +16.9 -16.9 +33.9 -21.0 +0.0 -19.5 -49.2 -33.9 -4.1 +18.2 +16.9
Steps
(reduced) 		64
(64) 		101
(101) 		128
(128) 		148
(148) 		165
(0) 		179
(14) 		191
(26) 		202
(37) 		212
(47) 		221
(56) 		229
(64)
Approximation of harmonics in 165ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.79 -0.49 -7.14 -6.07 +1.77 -3.18 -2.79 +2.41 -6.86 +6.60 +4.85 +6.37
Relative (%) -20.2 -2.6 -38.0 -32.3 +9.4 -16.9 -14.8 +12.8 -36.5 +35.1 +25.8 +33.9
Steps
(reduced) 		236
(71) 		243
(78) 		249
(84) 		255
(90) 		261
(96) 		266
(101) 		271
(106) 		276
(111) 		280
(115) 		285
(120) 		289
(124) 		293
(128)
229ed12
  • Octave size: 1202.29 ¢

Stretching the octave of 64edo by around 2 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.17 ¢. The tuning 229ed12 does this. So does the tuning 221ed11 whose octave is identical within 0.1 ¢.

Approximation of harmonics in 229ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.29 -4.59 +4.59 -6.01 -2.29 -6.16 +6.88 -9.17 -3.72 +0.35 +0.00
Relative (%) +12.2 -24.4 +24.4 -32.0 -12.2 -32.8 +36.6 -48.8 -19.8 +1.9 +0.0
Steps
(reduced) 		64
(64) 		101
(101) 		128
(128) 		148
(148) 		165
(165) 		179
(179) 		192
(192) 		202
(202) 		212
(212) 		221
(221) 		229
(0)
Approximation of harmonics in 229ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.07 -3.87 +8.19 +9.17 -1.85 -6.88 -6.55 -1.42 +8.04 +2.64 +0.83 +2.29
Relative (%) -37.6 -20.6 +43.6 +48.8 -9.9 -36.6 -34.9 -7.6 +42.8 +14.1 +4.4 +12.2
Steps
(reduced) 		236
(7) 		243
(14) 		250
(21) 		256
(27) 		261
(32) 		266
(37) 		271
(42) 		276
(47) 		281
(52) 		285
(56) 		289
(60) 		293
(64)
327zpi
  • Step size: 18.767 ¢, octave size: 1201.09 ¢

Stretching the octave of 64edo by around 1 ¢ results in improved primes 3 and 11, but worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 9.23 ¢. The tuning 327zpi does this.

Approximation of harmonics in 327zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.09 -6.49 +2.18 -8.80 -5.40 +9.23 +3.26 +5.79 -7.71 -3.81 -4.31
Relative (%) +5.8 -34.6 +11.6 -46.9 -28.8 +49.2 +17.4 +30.9 -41.1 -20.3 -23.0
Step 64 101 128 148 165 180 192 203 212 221 229
Approximation of harmonics in 327zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.25 -8.44 +3.48 +4.35 -6.77 +6.88 +7.11 -6.62 +2.75 -2.72 -4.61 -3.22
Relative (%) +38.6 -45.0 +18.6 +23.2 -36.1 +36.7 +37.9 -35.3 +14.6 -14.5 -24.6 -17.2
Step 237 243 250 256 261 267 272 276 281 285 289 293
64et, 11-limit WE tuning
  • Step size: 18.755 ¢, octave size: 1200.32 ¢

Stretching the octave of 64edo by around a third of a cent results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 8.50 ¢. Its 11-limit WE tuning and 11-limit TE tuning both do this.

Approximation of harmonics in 64et, 11-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.32 -7.70 +0.64 +8.18 -7.38 +7.07 +0.96 +3.35 +8.50 -6.46 -7.06
Relative (%) +1.7 -41.1 +3.4 +43.6 -39.3 +37.7 +5.1 +17.9 +45.3 -34.5 -37.6
Step 64 101 128 149 165 180 192 203 213 221 229
Approximation of harmonics in 64et, 11-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.41 +7.39 +0.48 +1.28 +8.85 +3.67 +3.85 +8.82 -0.63 -6.14 -8.08 -6.74
Relative (%) +23.5 +39.4 +2.6 +6.8 +47.2 +19.6 +20.5 +47.0 -3.3 -32.8 -43.1 -35.9
Step 237 244 250 256 262 267 272 277 281 285 289 293
64edo
  • Step size: 18.750 ¢, octave size: 1200.00 ¢

Pure-octaves 64edo approximates all harmonics up to 16 within 8.21 ¢. The octave of 64edo's 13-limit WE tuning differs by only 0.13 ¢ from pure.

Approximation of harmonics in 64edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 -8.21 +0.00 +7.44 -8.21 +6.17 +0.00 +2.34 +7.44 -7.57 -8.21
Relative (%) +0.0 -43.8 +0.0 +39.7 -43.8 +32.9 +0.0 +12.5 +39.7 -40.4 -43.8
Steps
(reduced) 		64
(0) 		101
(37) 		128
(0) 		149
(21) 		165
(37) 		180
(52) 		192
(0) 		203
(11) 		213
(21) 		221
(29) 		229
(37)
Approximation of harmonics in 64edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.22 +6.17 -0.77 +0.00 +7.54 +2.34 +2.49 +7.44 -2.03 -7.57 +9.23 -8.21
Relative (%) +17.2 +32.9 -4.1 +0.0 +40.2 +12.5 +13.3 +39.7 -10.8 -40.4 +49.2 -43.8
Steps
(reduced) 		237
(45) 		244
(52) 		250
(58) 		256
(0) 		262
(6) 		267
(11) 		272
(16) 		277
(21) 		281
(25) 		285
(29) 		290
(34) 		293
(37)
328zpi
  • Step size: 18.721 ¢, octave size: 1198.14 ¢

Compressing the octave of 64edo by just under 2 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.02 ¢. The tuning 328zpi does this.

Approximation of harmonics in 328zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.86 +7.59 -3.71 +3.12 +5.73 +0.95 -5.57 -3.55 +1.26 +4.74 +3.87
Relative (%) -9.9 +40.5 -19.8 +16.6 +30.6 +5.1 -29.7 -18.9 +6.7 +25.3 +20.7
Step 64 102 128 149 166 180 192 203 213 222 230
Approximation of harmonics in 328zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.65 -0.90 -8.02 -7.42 -0.05 -5.40 -5.40 -0.60 +8.54 +2.89 +0.82 +2.02
Relative (%) -19.5 -4.8 -42.8 -39.7 -0.3 -28.9 -28.9 -3.2 +45.6 +15.4 +4.4 +10.8
Step 237 244 250 256 262 267 272 277 282 286 290 294
180ed7
  • Octave size: 1197.80 ¢

Compressing the octave of 64edo by just over 2 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.34 ¢. The tuning 180ed7 does this.

Approximation of harmonics in 180ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.20 +7.05 -4.39 +2.33 +4.85 +0.00 -6.59 -4.62 +0.13 +3.57 +2.66
Relative (%) -11.7 +37.6 -23.5 +12.4 +25.9 +0.0 -35.2 -24.7 +0.7 +19.1 +14.2
Steps
(reduced) 		64
(64) 		102
(102) 		128
(128) 		149
(149) 		166
(166) 		180
(0) 		192
(12) 		203
(23) 		213
(33) 		222
(42) 		230
(50)
Approximation of harmonics in 180ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.91 -2.20 -9.34 -8.78 -1.44 -6.82 -6.84 -2.06 +7.05 +1.37 -0.72 +0.46
Relative (%) -26.2 -11.7 -49.9 -46.9 -7.7 -36.4 -36.6 -11.0 +37.6 +7.3 -3.9 +2.5
Steps
(reduced) 		237
(57) 		244
(64) 		250
(70) 		256
(76) 		262
(82) 		267
(87) 		272
(92) 		277
(97) 		282
(102) 		286
(106) 		290
(110) 		294
(114)
230ed12
  • Octave size: 1197.07 ¢

Compressing the octave of 64edo by just under 3 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.80 ¢. The tuning 230ed12 does this.

Approximation of harmonics in 230ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.93 +5.87 -5.87 +0.60 +2.93 -2.08 -8.80 -6.97 -2.33 +1.00 +0.00
Relative (%) -15.7 +31.4 -31.4 +3.2 +15.7 -11.1 -47.1 -37.2 -12.5 +5.4 +0.0
Steps
(reduced) 		64
(64) 		102
(102) 		128
(128) 		149
(149) 		166
(166) 		180
(180) 		192
(192) 		203
(203) 		213
(213) 		222
(222) 		230
(0)
Approximation of harmonics in 230ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.64 -5.01 +6.47 +6.97 -4.47 +8.80 +8.72 -5.26 +3.79 -1.93 -4.07 -2.93
Relative (%) -40.9 -26.8 +34.6 +37.2 -23.9 +47.1 +46.6 -28.1 +20.3 -10.3 -21.8 -15.7
Steps
(reduced) 		237
(7) 		244
(14) 		251
(21) 		257
(27) 		262
(32) 		268
(38) 		273
(43) 		277
(47) 		282
(52) 		286
(56) 		290
(60) 		294
(64)
149ed5
  • Step size: Octave size: 1196.81 ¢

Compressing the octave of 64edo by just over 3 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.12 ¢. The tuning 149ed5 does this.

Approximation of harmonics in 149ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.19 +5.45 -6.39 +0.00 +2.26 -2.81 +9.12 -7.79 -3.19 +0.10 -0.93
Relative (%) -17.1 +29.2 -34.2 +0.0 +12.1 -15.0 +48.8 -41.7 -17.1 +0.5 -5.0
Steps
(reduced) 		64
(64) 		102
(102) 		128
(128) 		149
(0) 		166
(17) 		180
(31) 		193
(44) 		203
(54) 		213
(64) 		222
(73) 		230
(81)
Approximation of harmonics in 149ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.61 -6.00 +5.45 +5.92 -5.53 +7.71 +7.61 -6.39 +2.65 -3.09 -5.25 -4.13
Relative (%) -46.0 -32.1 +29.2 +31.7 -29.6 +41.3 +40.7 -34.2 +14.1 -16.5 -28.1 -22.1
Steps
(reduced) 		237
(88) 		244
(95) 		251
(102) 		257
(108) 		262
(113) 		268
(119) 		273
(124) 		277
(128) 		282
(133) 		286
(137) 		290
(141) 		294
(145)

Instruments

Music

Bryan Deister
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