# 65edo

 ← 64edo 65edo 66edo →
Prime factorization 5 × 13
Step size 18.4615¢
Fifth 38\65 (701.538¢)
Semitones (A1:m2) 6:5 (110.8¢ : 92.31¢)
Consistency limit 5
Distinct consistency limit 5

65 equal divisions of the octave (abbreviated 65edo or 65ed2), also called 65-tone equal temperament (65tet) or 65 equal temperament (65et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 65 equal parts of about 18.5 ¢ each. Each step represents a frequency ratio of 21/65, or the 65th root of 2.

## Theory

65et can be characterized as the temperament which tempers out 32805/32768 (schisma), 78732/78125 (sensipent comma), 393216/390625 (würschmidt comma), and [-13 17 -6 (graviton). In the 7-limit, there are two different maps; the first is 65 103 151 182] (65), tempering out 126/125, 245/243 and 686/675, so that it supports sensi, and the second is 65 103 151 183] (65d), tempering out 225/224, 3125/3087, 4000/3969 and 5120/5103, so that it supports garibaldi. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit würschmidt temperament (wurschmidt and worschmidt) these two mappings provide.

65edo approximates the intervals 3/2, 5/4, 11/8, 19/16, 23/16, 31/16 and 47/32 well, so that it does a good job representing the 2.3.5.11.19.23.31.47 just intonation subgroup. To this one may want to add 17/16, 29/16 and 43/32, giving the 47-limit no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of schismic/nestoria that focuses on the very primes that 53edo neglects and that instead elegantly connects primes 7, 13, 37 and 41 to nestoria. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the zeta edo 130edo.

### Prime harmonics

Approximation of prime harmonics in 65edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) +0.00 -0.42 +1.38 -8.83 +2.53 +8.70 +5.81 -2.13 -0.58 +4.27 -0.42 +7.12 -4.45 +5.41 -0.89
Relative (%) +0.0 -2.3 +7.5 -47.8 +13.7 +47.1 +31.5 -11.5 -3.2 +23.1 -2.3 +38.6 -24.1 +29.3 -4.8
Steps
(reduced)
65
(0)
103
(38)
151
(21)
182
(52)
225
(30)
241
(46)
266
(6)
276
(16)
294
(34)
316
(56)
322
(62)
339
(14)
348
(23)
353
(28)
361
(36)

### Subsets and supersets

65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Andrew Heathwaite's composition Rubble: a Xenuke Unfolded.

130edo, which doubles its, corrects its approximation to harmonics 7 and 13.

## Intervals

# Cents Approximate Ratios * Ups and Downs Notation
0 0.00 1/1 P1 D
1 18.46 81/80, 88/87, 93/92, 94/93, 95/94, 96/95, 100/99, 121/120, 115/114, 116/115, 125/124 ^1 ^D
2 36.92 45/44, 46/45, 47/46, 48/47, 55/54, 128/125 ^^1 ^^D
3 55.38 30/29, 31/30, 32/31, 33/32, 34/33 vvm2 vvEb
4 73.85 23/22, 24/23, 25/24, 47/45 vm2 vEb
5 92.31 18/17, 19/18, 20/19, 58/55, 135/128, 256/243 m2 Eb
6 110.77 16/15, 17/16, 33/31 A1/^m2 D#/^Eb
7 129.23 14/13, 27/25, 55/51 v~2 ^^Eb
8 147.69 12/11, 25/23 ~2 vvvE
9 166.15 11/10, 32/29 ^~2 vvE
10 184.62 10/9, 19/17 vM2 vE
11 203.08 9/8, 64/57 M2 E
12 221.54 17/15, 25/22, 33/29, 58/51 ^M2 ^E
13 240.00 23/20, 31/27, 38/33, 54/47, 55/48 ^^M2 ^^E
14 258.46 22/19, 29/25, 36/31, 64/55 vvm3 vvF
15 276.92 20/17, 27/23, 34/29, 75/64 vm3 vF
16 295.38 19/16, 32/27 m3 F
17 313.85 6/5, 55/46 ^m3 ^F
18 332.31 23/19, 40/33 v~3 ^^F
19 350.77 11/9, 27/22, 38/31 ~3 ^^^F
20 369.23 26/21, 47/38, 68/55 ^~3 vvF#
21 387.69 5/4, 64/51 vM3 vF#
22 406.15 19/15, 24/19, 29/23, 34/27, 81/64 M3 F#
23 424.62 23/18, 32/25 ^M3 ^F#
24 443.08 22/17, 31/24, 40/31, 128/99 ^^M3 ^^F#
25 461.54 30/23, 47/36, 72/55 vv4 vvG
26 480.00 29/22, 33/25, 62/47 v4 vG
27 498.46 4/3 P4 G
28 516.92 23/17, 27/20, 31/23 ^4 ^G
29 535.38 15/11, 34/25, 64/47 v~4 ^^G
30 553.85 11/8, 40/29, 62/45 ~4 ^^^G
31 572.31 25/18, 32/23 ^~4/vd5 vvG#/vAb
32 590.77 24/17, 31/22, 38/27, 45/32 vA4/d5 vG#/Ab
33 609.23 17/12, 27/19, 44/31, 64/45 A4/^d5 G#/^Ab
34 627.69 36/25, 23/16 ^A4/v~5 ^G#/^^Ab
35 646.15 16/11, 29/20, 45/31 ~5 vvvA
36 664.62 22/15, 25/17, 47/32 ^~5 vvA
37 683.08 34/23, 40/27, 46/31 v5 vA
38 701.54 3/2 P5 A
39 720.00 44/29, 50/33, 47/31 ^5 ^A
40 738.46 23/15, 55/36, 72/47 ^^5 ^^A
41 756.92 17/11, 48/31, 31/20, 99/64 vvm6 vvBb
42 775.38 25/16, 36/23 vm6 vBb
43 793.85 19/12, 27/17, 30/19, 46/29, 128/81 m6 Bb
44 812.31 8/5, 51/32 ^m6 ^Bb
45 830.77 21/13, 55/34, 76/47 v~6 ^^Bb
46 849.23 18/11, 31/19, 44/27 ~6 vvvB
47 867.69 33/20, 38/23 ^~6 vvB
48 886.15 5/3, 92/55 vM6 vB
49 904.62 27/16, 32/19 M6 B
50 923.08 17/10, 29/17, 46/27, 128/75 ^M6 ^B
51 941.54 19/11, 31/18, 50/29, 55/32 ^^M6 ^^B
52 960.00 33/19, 40/23, 47/27, 54/31, 96/55 vvm7 vvC
53 978.46 30/17, 44/25, 51/29, 58/33 vm7 vC
54 996.92 16/9, 57/32 m7 C
55 1015.38 9/5, 34/19 ^m7 ^C
56 1033.85 20/11, 29/16 v~7 ^^C
57 1052.31 11/6, 46/25 ~7 ^^^C
58 1070.77 13/7, 50/27, 102/55 ^~7 vvC#
59 1089.23 15/8, 32/17, 62/33 vM7 vC#
60 1107.69 17/9, 19/10, 36/19, 55/29, 243/128, 256/135 M7 C#
61 1126.15 23/12, 44/23, 48/25, 90/47 ^M7 ^C#
62 1144.62 29/15, 31/16, 33/17, 60/31, 64/33 ^^M7 ^^C#
63 1163.08 45/23, 47/24, 88/45, 92/47, 108/55, 125/64 vv8 vvD
64 1181.54 87/55, 93/47, 95/48, 99/50, 115/58, 160/81, 184/93, 188/95, 228/115, 240/121, 248/125 v8 vD
65 1200.00 2/1 P8 D

* based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-103 65 [65 103]] +0.131 0.131 0.71
2.3.5 32805/32768, 78732/78125 [65 103 151]] -0.110 0.358 1.94
2.3.5.11 243/242, 4000/3993, 5632/5625 [65 103 151 225]] -0.266 0.410 2.22

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 9\65 166.15 11/10 Squirrel etc.
1 12\65 221.54 25/22 Hemisensi
1 21\65 387.69 5/4 Würschmidt
1 24\65 443.08 162/125 Sensipent
1 27\65 498.46 4/3 Helmholtz / nestoria / photia
1 28\65 516.92 27/20 Larry
5 20\65
(6\65)
369.23
(110.77)
99/80
(16/15)
Quintosec
5 27\65
(1\65)
498.46
(18.46)
4/3
(81/80)
Pental
5 30\65
(4\65)
553.85
(73.85)
11/8
(25/24)
Countdown

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