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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|65}}
{{ED intro}}


== Theory ==
== Theory ==
65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 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.
65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 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]] [[k*N subgroups|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 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]] [[k*N subgroups|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 ===
=== Prime harmonics ===
Line 11: Line 11:


=== Subsets and supersets ===
=== 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 [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].
65edo contains [[5edo]] and [[13edo]] as subsets. 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 [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].


[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.
[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.
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! #
! #
! [[Cent]]s
! [[Cent]]s
! Approximate ratios*
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>
! colspan="2" | [[Ups and downs notation]]
! colspan="2" | [[Ups and downs notation]]
|-
|-
Line 419: Line 419:
| D
| D
|}
|}
<nowiki />* {{sg|2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}
<references group="note" />


== Notation ==
== Notation ==
=== Ups and downs notation ===
65edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}


===Sagittal notation===
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
 
[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have arrows borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp6}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}
 
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
 
{{sharpness-sharp6-iw}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].
This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].
====Evo flavor====


==== Evo flavor ====
<imagemap>
<imagemap>
File:65-EDO_Evo_Sagittal.svg
File:65-EDO_Evo_Sagittal.svg
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</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:65-EDO_Revo_Sagittal.svg
File:65-EDO_Revo_Sagittal.svg
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</imagemap>
</imagemap>


====Evo-SZ flavor====
==== Evo-SZ flavor ====
 
<imagemap>
<imagemap>
File:65-EDO_Evo-SZ_Sagittal.svg
File:65-EDO_Evo-SZ_Sagittal.svg
Line 463: Line 478:
default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 334
| steps = 65.0158450885860
| step size = 18.4570391781413
| tempered height = 7.813349
| pure height = 7.642373
| integral = 1.269821
| gap = 16.514861
| octave = 1199.70754657919
| consistent = 6
| distinct = 6
}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 32805/32768, 78732/78125
| 32805/32768, 78732/78125
| {{mapping| 65 103 151 }}
| {{mapping| 65 103 151 }}
| &minus;0.110
| −0.110
| 0.358
| 0.358
| 1.94
| 1.94
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| 243/242, 4000/3993, 5632/5625
| 243/242, 4000/3993, 5632/5625
| {{mapping| 65 103 151 225 }}
| {{mapping| 65 103 151 225 }}
| &minus;0.266
| −0.266
| 0.410
| 0.410
| 2.22
| 2.22
Line 548: Line 578:
| 498.46
| 498.46
| 4/3
| 4/3
| [[Helmholtz]] / [[nestoria]] / [[photia]]
| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
|-
|-
| 1
| 1
Line 566: Line 596:
| 498.46<br>(18.46)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Pental (temperament)|Pental]]
| [[Quintile]]
|-
|-
| 5
| 5
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== Scales ==
== Scales ==
* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5
* [[Photia7]]
* [[Photia7]]
* [[Photia12]]
* [[Photia12]]
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== Instruments ==
== Instruments ==
[[Lumatone mapping for 65edo]]
[[Lumatone mapping for 65edo]]
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025).


[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 08:08, 22 July 2025

← 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 (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.

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 5edo and 13edo as subsets. 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[note 1] 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
  1. ↑ Based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament; other approaches are also possible.

Notation

Ups and downs notation

65edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Alternative ups and downs have arrows borrowed from extended Helmholtz–Ellis notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

This notation uses the same sagittal sequence as EDOs 72 and 79.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Approximation to JI

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
334zpi 65.015845 18.457039 7.813349 7.642373 1.269821 16.514861 1199.707547 −0.292453 6 6

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* 		Temperament
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 / 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) 		Quintile
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 distinct

Scales

Instruments

Lumatone mapping for 65edo

Music

Bryan Deister
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