65edo: Difference between revisions

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→Intervals: added intervals < 600c in extended subgroup (which was extended), specifically, superparticulars and (semi)convergents within the appropriately prime-subgroup-limited 57-odd-limit; please add anything you think is missing
Line 28: Line 28:
| 1
| 1
| 18.46
| 18.46
| 81/80, 100/99, 121/120
| 81/80, 100/99, 121/120, 88/87, 93/92, 94/93, 95/94, 96/95, 115/114, 116/115, 125/124
| ^1
| ^1
| ^D
| ^D
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| 2
| 2
| 36.92
| 36.92
| 45/44, 55/54, 128/125
| 45/44, 46/45, 47/46, 48/47, 55/54, 128/125
| ^^1
| ^^1
| ^^D
| ^^D
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| 3
| 3
| 55.38
| 55.38
| 33/32
| 33/32, 34/33, 30/29, 31/30, 32/31
| vvm2
| vvm2
| vvEb
| vvEb
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| 4
| 4
| 73.85
| 73.85
| 25/24
| 25/24, 24/23, 23/22, 47/45
| vm2
| vm2
| vEb
| vEb
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| 5
| 5
| 92.31
| 92.31
| 135/128, 256/243
| 135/128, 256/243, 18/17, 19/18, 20/19, 58/55
| m2
| m2
| Eb
| Eb
Line 58: Line 58:
| 6
| 6
| 110.77
| 110.77
| 16/15
| 16/15, 17/16, 33/31
| A1/^m2
| A1/^m2
| D#/^Eb
| D#/^Eb
Line 64: Line 64:
| 7
| 7
| 129.23
| 129.23
| 14/13
| 27/25, 14/13, 55/51
| v~2
| v~2
| ^^Eb
| ^^Eb
Line 70: Line 70:
| 8
| 8
| 147.69
| 147.69
| 12/11
| 12/11, 25/23
| ~2
| ~2
| vvvE
| vvvE
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| 9
| 9
| 166.15
| 166.15
| 11/10
| 11/10, 32/29
| ^~2
| ^~2
| vvE
| vvE
Line 88: Line 88:
| 11
| 11
| 203.08
| 203.08
| 9/8
| 9/8, 19/17, 64/57
| M2
| M2
| E
| E
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| 12
| 12
| 221.54
| 221.54
| 25/22
| 25/22, 17/15, 33/29, 58/51
| ^M2
| ^M2
| ^E
| ^E
Line 100: Line 100:
| 13
| 13
| 240.00
| 240.00
| 55/48
| 55/48, 23/20, 31/17, 54/47
| ^^M2
| ^^M2
| ^^E
| ^^E
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| 14
| 14
| 258.46
| 258.46
| 64/55
| 64/55, 22/19, 29/25, 36/31
| vvm3
| vvm3
| vvF
| vvF
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| 15
| 15
| 276.92
| 276.92
| 75/64
| 75/64, 20/17, 27/23, 34/29
| vm3
| vm3
| vF
| vF
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| 16
| 16
| 295.38
| 295.38
| 32/27
| 32/27, 19/16
| m3
| m3
| F
| F
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| 17
| 17
| 313.85
| 313.85
| 6/5
| 6/5, 55/46
| ^m3
| ^m3
| ^F
| ^F
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| 18
| 18
| 332.31
| 332.31
| 40/33
| 40/33, 17/14, 23/19
| v~3
| v~3
| ^^F
| ^^F
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| 19
| 19
| 350.77
| 350.77
| 11/9, 27/22
| 11/9, 27/22, 38/31
| ~3
| ~3
| ^^^F
| ^^^F
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| 20
| 20
| 369.23
| 369.23
| 26/21
| 26/21, 68/55, 47/38
| ^~3
| ^~3
| vvF#
| vvF#
Line 148: Line 148:
| 21
| 21
| 387.69
| 387.69
| 5/4
| 5/4, 64/51
| vM3
| vM3
| vF#
| vF#
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| 22
| 22
| 406.15
| 406.15
| 81/64
| 81/64, 19/15, 24/19, 34/27, 29/23
| M3
| M3
| F#
| F#
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| 23
| 23
| 424.62
| 424.62
| 32/25
| 32/25, 23/18
| ^M3
| ^M3
| ^F#
| ^F#
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| 24
| 24
| 443.08
| 443.08
| 128/99
| 128/99, 22/17, 31/24, 40/31
| ^^M3
| ^^M3
| ^^F#
| ^^F#
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| 25
| 25
| 461.54
| 461.54
| 72/55
| 72/55, 30/23, 47/36
| vv4
| vv4
| vvG
| vvG
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| 26
| 26
| 480.00
| 480.00
| 33/25
| 33/25, 29/22, 62/47
| v4
| v4
| vG
| vG
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| 28
| 28
| 516.92
| 516.92
| 27/20
| 27/20, 23/17, 31/23
| ^4
| ^4
| ^G
| ^G
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| 29
| 29
| 535.38
| 535.38
| 15/11
| 15/11, 34/25, 64/47
| v~4
| v~4
| ^^G
| ^^G
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| 30
| 30
| 553.85
| 553.85
| 11/8
| 11/8, 40/29, 62/45
| ~4
| ~4
| ^^^G
| ^^^G
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| 31
| 31
| 572.31
| 572.31
| 25/18
| 25/18, 32/23
| ^~4/vd5
| ^~4/vd5
| vvG#/vAb
| vvG#/vAb
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| 32
| 32
| 590.77
| 590.77
| 45/32
| 45/32, 38/27, 31/22, 24/17
| vA4/d5
| vA4/d5
| vG#/Ab
| vG#/Ab
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| D
| D
|}
|}
<nowiki>*</nowiki> based on treating 65edo as a 2.3.5.11.13/7 subgroup temperament.  
<nowiki>*</nowiki> based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament.


== Regular temperament properties ==
== Regular temperament properties ==

Revision as of 23:56, 5 January 2023

← 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

The 65 equal divisions of the octave (65edo), or 65(-tone) equal temperament (65tet, 65et) when viewed from a regular temperament perspective, divides the octave into 65 equal parts of about 18.5 cents each.

Theory

65et can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the würschmidt comma. In the 7-limit, there are two different maps; the first is ⟨65 103 151 182], tempering out 126/125, 245/243 and 686/675, so that it supports sensi temperament, 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 temperament. 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 (which 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.

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 Rubble: a Xenuke Unfolded.

Prime harmonics

Script error: No such module "primes_in_edo".

Intervals

Degree Cents Approximate Ratios * Ups and Downs Notation
0 0.00 1/1 P1 D
1 18.46 81/80, 100/99, 121/120, 88/87, 93/92, 94/93, 95/94, 96/95, 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 33/32, 34/33, 30/29, 31/30, 32/31 vvm2 vvEb
4 73.85 25/24, 24/23, 23/22, 47/45 vm2 vEb
5 92.31 135/128, 256/243, 18/17, 19/18, 20/19, 58/55 m2 Eb
6 110.77 16/15, 17/16, 33/31 A1/^m2 D#/^Eb
7 129.23 27/25, 14/13, 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 vM2 vE
11 203.08 9/8, 19/17, 64/57 M2 E
12 221.54 25/22, 17/15, 33/29, 58/51 ^M2 ^E
13 240.00 55/48, 23/20, 31/17, 54/47 ^^M2 ^^E
14 258.46 64/55, 22/19, 29/25, 36/31 vvm3 vvF
15 276.92 75/64, 20/17, 27/23, 34/29 vm3 vF
16 295.38 32/27, 19/16 m3 F
17 313.85 6/5, 55/46 ^m3 ^F
18 332.31 40/33, 17/14, 23/19 v~3 ^^F
19 350.77 11/9, 27/22, 38/31 ~3 ^^^F
20 369.23 26/21, 68/55, 47/38 ^~3 vvF#
21 387.69 5/4, 64/51 vM3 vF#
22 406.15 81/64, 19/15, 24/19, 34/27, 29/23 M3 F#
23 424.62 32/25, 23/18 ^M3 ^F#
24 443.08 128/99, 22/17, 31/24, 40/31 ^^M3 ^^F#
25 461.54 72/55, 30/23, 47/36 vv4 vvG
26 480.00 33/25, 29/22, 62/47 v4 vG
27 498.46 4/3 P4 G
28 516.92 27/20, 23/17, 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 45/32, 38/27, 31/22, 24/17 vA4/d5 vG#/Ab
33 609.23 64/45 A4/^d5 G#/^Ab
34 627.69 36/25 ^A4/v~5 ^G#/^^Ab
35 646.15 16/11 ~5 vvvA
36 664.62 22/15 ^~5 vvA
37 683.08 40/27 v5 vA
38 701.54 3/2 P5 A
39 720.00 50/33 ^5 ^A
40 738.46 55/36 ^^5 ^^A
41 756.92 99/64 vvm6 vvBb
42 775.38 25/16 vm6 vBb
43 793.85 128/81 m6 Bb
44 812.31 8/5 ^m6 ^Bb
45 830.77 21/13 v~6 ^^Bb
46 849.23 18/11, 44/27 ~6 vvvB
47 867.69 33/20 ^~6 vvB
48 886.15 5/3 vM6 vB
49 904.62 27/16 M6 B
50 923.08 128/75 ^M6 ^B
51 941.54 55/32 ^^M6 ^^B
52 960.00 96/55 vvm7 vvC
53 978.46 44/25 vm7 vC
54 996.92 16/9 m7 C
55 1015.38 9/5 ^m7 ^C
56 1033.85 20/11 v~7 ^^C
57 1052.31 11/6 ~7 ^^^C
58 1070.77 13/7 ^~7 vvC#
59 1089.23 15/8 vM7 vC#
60 1107.69 243/128, 256/135 M7 C#
61 1126.15 48/25 ^M7 ^C#
62 1144.62 64/33 ^^M7 ^^C#
63 1163.08 88/45, 108/55, 125/64 vv8 vvD
64 1181.54 160/81, 99/50, 240/121 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 octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 3\65 55.38 33/32 Escapade
1 9\65 166.15 11/10 Squirrel etc.
1 12\65 221.54 25/22 Hemisensi
1 19\65 350.77 11/9 Karadeniz
1 21\65 387.69 5/4 Würschmidt
1 24\65 443.08 162/125 Sensipent
1 27\65 498.46 4/3 Helmholtz / photia
1 28\65 516.92 27/20 Gravity
5 20\65
(6\65) 		369.23
(110.77) 		10125/8192
(16/15) 		Qintosec
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) 		Trisedodge / countdown

Scales

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