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{{Infobox ET}}
{{Infobox ET}}
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 [[cent]]s each.
{{ED intro}}


== Theory ==
== 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 {{val| 65 103 151 '''182''' }}, [[tempering out]] [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]] temperament, 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]] 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.
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]] and [[31/16]] well, so that it does a good job representing the 2.3.5.11.19.23.31 [[just intonation subgroup]]. To this one may want to add [[17/16]] and [[29/16]], giving the [[31-limit]] no-7's no-13's subgroup 2.3.5.11.17.19.23.29.31. 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]].


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 [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded Rubble: a Xenuke Unfolded].
=== Prime harmonics ===
{{Harmonics in equal|65|intervals=prime|columns=15}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|65|columns=15}}
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.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! [[Degree]]
! #
! [[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]]
|-
|-
| 0
| 0
Line 28: Line 31:
| 1
| 1
| 18.46
| 18.46
| 81/80, 100/99, 121/120
| 81/80, 88/87, 93/92, 94/93, 95/94, 96/95, 100/99, 121/120, 115/114, 116/115, 125/124
| ^1
| ^1
| ^D
| ^D
Line 34: Line 37:
| 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
Line 40: Line 43:
| 3
| 3
| 55.38
| 55.38
| 33/32
| 30/29, 31/30, 32/31, 33/32, 34/33
| vvm2
| vvm2
| vvEb
| vvEb
Line 46: Line 49:
| 4
| 4
| 73.85
| 73.85
| 25/24
| 23/22, 24/23, 25/24, 47/45
| vm2
| vm2
| vEb
| vEb
Line 52: Line 55:
| 5
| 5
| 92.31
| 92.31
| 135/128, 256/243
| 18/17, 19/18, 20/19, 58/55, 135/128, 256/243
| m2
| m2
| Eb
| Eb
Line 58: Line 61:
| 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 67:
| 7
| 7
| 129.23
| 129.23
| 14/13
| 14/13, 27/25, 55/51
| v~2
| v~2
| ^^Eb
| ^^Eb
Line 70: Line 73:
| 8
| 8
| 147.69
| 147.69
| 12/11
| 12/11, 25/23
| ~2
| ~2
| vvvE
| vvvE
Line 76: Line 79:
| 9
| 9
| 166.15
| 166.15
| 11/10
| 11/10, 32/29
| ^~2
| ^~2
| vvE
| vvE
Line 82: Line 85:
| 10
| 10
| 184.62
| 184.62
| 10/9
| 10/9, 19/17
| vM2
| vM2
| vE
| vE
Line 88: Line 91:
| 11
| 11
| 203.08
| 203.08
| 9/8
| 9/8, 64/57
| M2
| M2
| E
| E
Line 94: Line 97:
| 12
| 12
| 221.54
| 221.54
| 25/22
| 17/15, 25/22, 33/29, 58/51
| ^M2
| ^M2
| ^E
| ^E
Line 100: Line 103:
| 13
| 13
| 240.00
| 240.00
| 55/48
| 23/20, 31/27, 38/33, 54/47, 55/48
| ^^M2
| ^^M2
| ^^E
| ^^E
Line 106: Line 109:
| 14
| 14
| 258.46
| 258.46
| 64/55
| 22/19, 29/25, 36/31, 64/55
| vvm3
| vvm3
| vvF
| vvF
Line 112: Line 115:
| 15
| 15
| 276.92
| 276.92
| 75/64
| 20/17, 27/23, 34/29, 75/64
| vm3
| vm3
| vF
| vF
Line 118: Line 121:
| 16
| 16
| 295.38
| 295.38
| 32/27
| 19/16, 32/27
| m3
| m3
| F
| F
Line 124: Line 127:
| 17
| 17
| 313.85
| 313.85
| 6/5
| 6/5, 55/46
| ^m3
| ^m3
| ^F
| ^F
Line 130: Line 133:
| 18
| 18
| 332.31
| 332.31
| 40/33
| 23/19, 40/33
| v~3
| v~3
| ^^F
| ^^F
Line 136: Line 139:
| 19
| 19
| 350.77
| 350.77
| 11/9, 27/22
| 11/9, 27/22, 38/31
| ~3
| ~3
| ^^^F
| ^^^F
Line 142: Line 145:
| 20
| 20
| 369.23
| 369.23
| 26/21
| 26/21, 47/38, 68/55
| ^~3
| ^~3
| vvF#
| vvF#
Line 148: Line 151:
| 21
| 21
| 387.69
| 387.69
| 5/4
| 5/4, 64/51
| vM3
| vM3
| vF#
| vF#
Line 154: Line 157:
| 22
| 22
| 406.15
| 406.15
| 81/64
| 19/15, 24/19, 29/23, 34/27, 81/64
| M3
| M3
| F#
| F#
Line 160: Line 163:
| 23
| 23
| 424.62
| 424.62
| 32/25
| 23/18, 32/25
| ^M3
| ^M3
| ^F#
| ^F#
Line 166: Line 169:
| 24
| 24
| 443.08
| 443.08
| 128/99
| 22/17, 31/24, 40/31, 128/99
| ^^M3
| ^^M3
| ^^F#
| ^^F#
Line 172: Line 175:
| 25
| 25
| 461.54
| 461.54
| 72/55
| 30/23, 47/36, 72/55
| vv4
| vv4
| vvG
| vvG
Line 178: Line 181:
| 26
| 26
| 480.00
| 480.00
| 33/25
| 29/22, 33/25, 62/47
| v4
| v4
| vG
| vG
Line 190: Line 193:
| 28
| 28
| 516.92
| 516.92
| 27/20
| 23/17, 27/20, 31/23
| ^4
| ^4
| ^G
| ^G
Line 196: Line 199:
| 29
| 29
| 535.38
| 535.38
| 15/11
| 15/11, 34/25, 64/47
| v~4
| v~4
| ^^G
| ^^G
Line 202: Line 205:
| 30
| 30
| 553.85
| 553.85
| 11/8
| 11/8, 40/29, 62/45
| ~4
| ~4
| ^^^G
| ^^^G
Line 208: Line 211:
| 31
| 31
| 572.31
| 572.31
| 25/18
| 25/18, 32/23
| ^~4/vd5
| ^~4/vd5
| vvG#/vAb
| vvG#/vAb
Line 214: Line 217:
| 32
| 32
| 590.77
| 590.77
| 45/32
| 24/17, 31/22, 38/27, 45/32
| vA4/d5
| vA4/d5
| vG#/Ab
| vG#/Ab
Line 220: Line 223:
| 33
| 33
| 609.23
| 609.23
| 64/45
| 17/12, 27/19, 44/31, 64/45
| A4/^d5
| A4/^d5
| G#/^Ab
| G#/^Ab
Line 226: Line 229:
| 34
| 34
| 627.69
| 627.69
| 36/25
| 36/25, 23/16
| ^A4/v~5
| ^A4/v~5
| ^G#/^^Ab
| ^G#/^^Ab
Line 232: Line 235:
| 35
| 35
| 646.15
| 646.15
| 16/11
| 16/11, 29/20, 45/31
| ~5
| ~5
| vvvA
| vvvA
Line 238: Line 241:
| 36
| 36
| 664.62
| 664.62
| 22/15
| 22/15, 25/17, 47/32
| ^~5
| ^~5
| vvA
| vvA
Line 244: Line 247:
| 37
| 37
| 683.08
| 683.08
| 40/27
| 34/23, 40/27, 46/31
| v5
| v5
| vA
| vA
Line 256: Line 259:
| 39
| 39
| 720.00
| 720.00
| 50/33
| 44/29, 50/33, 47/31
| ^5
| ^5
| ^A
| ^A
Line 262: Line 265:
| 40
| 40
| 738.46
| 738.46
| 55/36
| 23/15, 55/36, 72/47
| ^^5
| ^^5
| ^^A
| ^^A
Line 268: Line 271:
| 41
| 41
| 756.92
| 756.92
| 99/64
| 17/11, 48/31, 31/20, 99/64
| vvm6
| vvm6
| vvBb
| vvBb
Line 274: Line 277:
| 42
| 42
| 775.38
| 775.38
| 25/16
| 25/16, 36/23
| vm6
| vm6
| vBb
| vBb
Line 280: Line 283:
| 43
| 43
| 793.85
| 793.85
| 128/81
| 19/12, 27/17, 30/19, 46/29, 128/81
| m6
| m6
| Bb
| Bb
Line 286: Line 289:
| 44
| 44
| 812.31
| 812.31
| 8/5
| 8/5, 51/32
| ^m6
| ^m6
| ^Bb
| ^Bb
Line 292: Line 295:
| 45
| 45
| 830.77
| 830.77
| 21/13
| 21/13, 55/34, 76/47
| v~6
| v~6
| ^^Bb
| ^^Bb
Line 298: Line 301:
| 46
| 46
| 849.23
| 849.23
| 18/11, 44/27
| 18/11, 31/19, 44/27
| ~6
| ~6
| vvvB
| vvvB
Line 304: Line 307:
| 47
| 47
| 867.69
| 867.69
| 33/20
| 33/20, 38/23
| ^~6
| ^~6
| vvB
| vvB
Line 310: Line 313:
| 48
| 48
| 886.15
| 886.15
| 5/3
| 5/3, 92/55
| vM6
| vM6
| vB
| vB
Line 316: Line 319:
| 49
| 49
| 904.62
| 904.62
| 27/16
| 27/16, 32/19
| M6
| M6
| B
| B
Line 322: Line 325:
| 50
| 50
| 923.08
| 923.08
| 128/75
| 17/10, 29/17, 46/27, 128/75
| ^M6
| ^M6
| ^B
| ^B
Line 328: Line 331:
| 51
| 51
| 941.54
| 941.54
| 55/32
| 19/11, 31/18, 50/29, 55/32
| ^^M6
| ^^M6
| ^^B
| ^^B
Line 334: Line 337:
| 52
| 52
| 960.00
| 960.00
| 96/55
| 33/19, 40/23, 47/27, 54/31, 96/55
| vvm7
| vvm7
| vvC
| vvC
Line 340: Line 343:
| 53
| 53
| 978.46
| 978.46
| 44/25
| 30/17, 44/25, 51/29, 58/33
| vm7
| vm7
| vC
| vC
Line 346: Line 349:
| 54
| 54
| 996.92
| 996.92
| 16/9
| 16/9, 57/32
| m7
| m7
| C
| C
Line 352: Line 355:
| 55
| 55
| 1015.38
| 1015.38
| 9/5
| 9/5, 34/19
| ^m7
| ^m7
| ^C
| ^C
Line 358: Line 361:
| 56
| 56
| 1033.85
| 1033.85
| 20/11
| 20/11, 29/16
| v~7
| v~7
| ^^C
| ^^C
Line 364: Line 367:
| 57
| 57
| 1052.31
| 1052.31
| 11/6
| 11/6, 46/25
| ~7
| ~7
| ^^^C
| ^^^C
Line 370: Line 373:
| 58
| 58
| 1070.77
| 1070.77
| 13/7
| 13/7, 50/27, 102/55
| ^~7
| ^~7
| vvC#
| vvC#
Line 376: Line 379:
| 59
| 59
| 1089.23
| 1089.23
| 15/8
| 15/8, 32/17, 62/33
| vM7
| vM7
| vC#
| vC#
Line 382: Line 385:
| 60
| 60
| 1107.69
| 1107.69
| 243/128, 256/135
| 17/9, 19/10, 36/19, 55/29, 243/128, 256/135
| M7
| M7
| C#
| C#
Line 388: Line 391:
| 61
| 61
| 1126.15
| 1126.15
| 48/25
| 23/12, 44/23, 48/25, 90/47
| ^M7
| ^M7
| ^C#
| ^C#
Line 394: Line 397:
| 62
| 62
| 1144.62
| 1144.62
| 64/33
| 29/15, 31/16, 33/17, 60/31, 64/33
| ^^M7
| ^^M7
| ^^C#
| ^^C#
Line 400: Line 403:
| 63
| 63
| 1163.08
| 1163.08
| 88/45, 108/55, 125/64
| 45/23, 47/24, 88/45, 92/47, 108/55, 125/64
| vv8
| vv8
| vvD
| vvD
Line 406: Line 409:
| 64
| 64
| 1181.54
| 1181.54
| 160/81, 99/50, 240/121
| 87/55, 93/47, 95/48, 99/50, 115/58, 160/81, 184/93, 188/95, 228/115, 240/121, 248/125
| v8
| v8
| vD
| vD
Line 416: Line 419:
| D
| D
|}
|}
<nowiki>*</nowiki> based on treating 65edo as a 2.3.5.11.13/7 subgroup temperament.  
<references group="note" />
 
== 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}}
 
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]].
 
==== Evo flavor ====
<imagemap>
File:65-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:65-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:65-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:65-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:65-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 639 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
</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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 431: Line 508:
| 2.3
| 2.3
| {{monzo| -103 65 }}
| {{monzo| -103 65 }}
| [{{val| 65 103 }}]
| {{mapping| 65 103 }}
| +0.131
| +0.131
| 0.131
| 0.131
Line 438: Line 515:
| 2.3.5
| 2.3.5
| 32805/32768, 78732/78125
| 32805/32768, 78732/78125
| [{{val| 65 103 151 }}]
| {{mapping| 65 103 151 }}
| -0.110
| −0.110
| 0.358
| 0.358
| 1.94
| 1.94
Line 445: Line 522:
| 2.3.5.11
| 2.3.5.11
| 243/242, 4000/3993, 5632/5625
| 243/242, 4000/3993, 5632/5625
| [{{val| 65 103 151 225 }}]
| {{mapping| 65 103 151 225 }}
| -0.266
| −0.266
| 0.410
| 0.410
| 2.22
| 2.22
Line 453: Line 530:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperaments
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
Line 500: Line 578:
| 498.46
| 498.46
| 4/3
| 4/3
| [[Helmholtz]] / [[photia]]
| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
|-
|-
| 1
| 1
Line 506: Line 584:
| 516.92
| 516.92
| 27/20
| 27/20
| [[Gravity]]
| [[Larry]]
|-
|-
| 5
| 5
| 20\65<br>(6\65)
| 20\65<br>(6\65)
| 369.23<br>(110.77)
| 369.23<br>(110.77)
| 10125/8192<br>(16/15)
| 99/80<br>(16/15)
| [[Qintosec]]
| [[Quintosec]]
|-
|-
| 5
| 5
Line 518: Line 596:
| 498.46<br>(18.46)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Pental]]
| [[Quintile]]
|-
|-
| 5
| 5
Line 524: Line 602:
| 553.85<br>(73.85)
| 553.85<br>(73.85)
| 11/8<br>(25/24)
| 11/8<br>(25/24)
| [[Trisedodge]] / [[countdown]]
| [[Countdown]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== 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]]
* [[Skateboard7]]
== Instruments ==
[[Lumatone mapping for 65edo]]
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025).


[[Category:65edo]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Listen]]
[[Category:Subgroup]]
[[Category:Schismic]]
[[Category:Schismic]]
[[Category:Sensipent]]
[[Category:Sensipent]]
[[Category:Subgroup temperaments]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]

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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