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Line 1: Line 1:
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 5 × 13
{{ED intro}}
| Step size = 18.46154¢
| Fifth = 38\65 ≈ 702¢
| Major 2nd = 11\65 ≈ 203¢
| Minor 2nd = 5\65 ≈ 92¢
| Augmented 1sn = 6\65 ≈ 111¢
}}
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.


== 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 supports [[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]], [[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 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]].
=== Prime harmonics ===
{{Harmonics in equal|65|intervals=prime|columns=15}}


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].
=== 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 [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded''].


=== Prime harmonics ===
[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.
{{Primes in edo|65|columns=11}}


== Intervals ==
== Intervals ==
 
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2"
|-
|-
! [[Degree]]
! #
! [[Cent]]s
! [[Cent]]s
! colspan="2" |[[Ups and Downs Notation]]
! 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]]
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
| P1
| P1
| D
| D
Line 34: Line 31:
| 1
| 1
| 18.46
| 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
| ^1
| ^D
| ^D
Line 39: Line 37:
| 2
| 2
| 36.92
| 36.92
| 45/44, 46/45, 47/46, 48/47, 55/54, 128/125
| ^^1
| ^^1
| ^^D
| ^^D
Line 44: Line 43:
| 3
| 3
| 55.38
| 55.38
| 30/29, 31/30, 32/31, 33/32, 34/33
| vvm2
| vvm2
| vvEb
| vvEb
Line 49: Line 49:
| 4
| 4
| 73.85
| 73.85
| 23/22, 24/23, 25/24, 47/45
| vm2
| vm2
| vEb
| vEb
Line 54: Line 55:
| 5
| 5
| 92.31
| 92.31
| 18/17, 19/18, 20/19, 58/55, 135/128, 256/243
| m2
| m2
| Eb
| Eb
Line 59: Line 61:
| 6
| 6
| 110.77
| 110.77
| 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, 27/25, 55/51
| v~2
| v~2
| ^^Eb
| ^^Eb
Line 69: Line 73:
| 8
| 8
| 147.69
| 147.69
| 12/11, 25/23
| ~2
| ~2
| vvvE
| vvvE
Line 74: Line 79:
| 9
| 9
| 166.15
| 166.15
| 11/10, 32/29
| ^~2
| ^~2
| vvE
| vvE
Line 79: Line 85:
| 10
| 10
| 184.62
| 184.62
| 10/9, 19/17
| vM2
| vM2
| vE
| vE
Line 84: Line 91:
| 11
| 11
| 203.08
| 203.08
| 9/8, 64/57
| M2
| M2
| E
| E
Line 89: Line 97:
| 12
| 12
| 221.54
| 221.54
| 17/15, 25/22, 33/29, 58/51
| ^M2
| ^M2
| ^E
| ^E
Line 94: Line 103:
| 13
| 13
| 240.00
| 240.00
| 23/20, 31/27, 38/33, 54/47, 55/48
| ^^M2
| ^^M2
| ^^E
| ^^E
Line 99: Line 109:
| 14
| 14
| 258.46
| 258.46
| 22/19, 29/25, 36/31, 64/55
| vvm3
| vvm3
| vvF
| vvF
Line 104: Line 115:
| 15
| 15
| 276.92
| 276.92
| 20/17, 27/23, 34/29, 75/64
| vm3
| vm3
| vF
| vF
Line 109: Line 121:
| 16
| 16
| 295.38
| 295.38
| 19/16, 32/27
| m3
| m3
| F
| F
Line 114: Line 127:
| 17
| 17
| 313.85
| 313.85
| 6/5, 55/46
| ^m3
| ^m3
| ^F
| ^F
Line 119: Line 133:
| 18
| 18
| 332.31
| 332.31
| 23/19, 40/33
| v~3
| v~3
| ^^F
| ^^F
Line 124: Line 139:
| 19
| 19
| 350.77
| 350.77
| 11/9, 27/22, 38/31
| ~3
| ~3
| ^^^F
| ^^^F
Line 129: Line 145:
| 20
| 20
| 369.23
| 369.23
| 26/21, 47/38, 68/55
| ^~3
| ^~3
| vvF#
| vvF#
Line 134: Line 151:
| 21
| 21
| 387.69
| 387.69
| 5/4, 64/51
| vM3
| vM3
| vF#
| vF#
Line 139: Line 157:
| 22
| 22
| 406.15
| 406.15
| 19/15, 24/19, 29/23, 34/27, 81/64
| M3
| M3
| F#
| F#
Line 144: Line 163:
| 23
| 23
| 424.62
| 424.62
| 23/18, 32/25
| ^M3
| ^M3
| ^F#
| ^F#
Line 149: Line 169:
| 24
| 24
| 443.08
| 443.08
| 22/17, 31/24, 40/31, 128/99
| ^^M3
| ^^M3
| ^^F#
| ^^F#
Line 154: Line 175:
| 25
| 25
| 461.54
| 461.54
| 30/23, 47/36, 72/55
| vv4
| vv4
| vvG
| vvG
Line 159: Line 181:
| 26
| 26
| 480.00
| 480.00
| 29/22, 33/25, 62/47
| v4
| v4
| vG
| vG
Line 164: Line 187:
| 27
| 27
| 498.46
| 498.46
| 4/3
| P4
| P4
| G
| G
Line 169: Line 193:
| 28
| 28
| 516.92
| 516.92
| 23/17, 27/20, 31/23
| ^4
| ^4
| ^G
| ^G
Line 174: Line 199:
| 29
| 29
| 535.38
| 535.38
| 15/11, 34/25, 64/47
| v~4
| v~4
| ^^G
| ^^G
Line 179: Line 205:
| 30
| 30
| 553.85
| 553.85
| 11/8, 40/29, 62/45
| ~4
| ~4
| ^^^G
| ^^^G
Line 184: Line 211:
| 31
| 31
| 572.31
| 572.31
| 25/18, 32/23
| ^~4/vd5
| ^~4/vd5
| vvG#/vAb
| vvG#/vAb
Line 189: Line 217:
| 32
| 32
| 590.77
| 590.77
| 24/17, 31/22, 38/27, 45/32
| vA4/d5
| vA4/d5
| vG#/Ab
| vG#/Ab
Line 194: Line 223:
| 33
| 33
| 609.23
| 609.23
| 17/12, 27/19, 44/31, 64/45
| A4/^d5
| A4/^d5
| G#/^Ab
| G#/^Ab
Line 199: Line 229:
| 34
| 34
| 627.69
| 627.69
| 36/25, 23/16
| ^A4/v~5
| ^A4/v~5
| ^G#/^^Ab
| ^G#/^^Ab
Line 204: Line 235:
| 35
| 35
| 646.15
| 646.15
| 16/11, 29/20, 45/31
| ~5
| ~5
| vvvA
| vvvA
Line 209: Line 241:
| 36
| 36
| 664.62
| 664.62
| 22/15, 25/17, 47/32
| ^~5
| ^~5
| vvA
| vvA
Line 214: Line 247:
| 37
| 37
| 683.08
| 683.08
| 34/23, 40/27, 46/31
| v5
| v5
| vA
| vA
Line 219: Line 253:
| 38
| 38
| 701.54
| 701.54
| 3/2
| P5
| P5
| A
| A
Line 224: Line 259:
| 39
| 39
| 720.00
| 720.00
| 44/29, 50/33, 47/31
| ^5
| ^5
| ^A
| ^A
Line 229: Line 265:
| 40
| 40
| 738.46
| 738.46
| 23/15, 55/36, 72/47
| ^^5
| ^^5
| ^^A
| ^^A
Line 234: Line 271:
| 41
| 41
| 756.92
| 756.92
| 17/11, 48/31, 31/20, 99/64
| vvm6
| vvm6
| vvBb
| vvBb
Line 239: Line 277:
| 42
| 42
| 775.38
| 775.38
| 25/16, 36/23
| vm6
| vm6
| vBb
| vBb
Line 244: Line 283:
| 43
| 43
| 793.85
| 793.85
| 19/12, 27/17, 30/19, 46/29, 128/81
| m6
| m6
| Bb
| Bb
Line 249: Line 289:
| 44
| 44
| 812.31
| 812.31
| 8/5, 51/32
| ^m6
| ^m6
| ^Bb
| ^Bb
Line 254: Line 295:
| 45
| 45
| 830.77
| 830.77
| 21/13, 55/34, 76/47
| v~6
| v~6
| ^^Bb
| ^^Bb
Line 259: Line 301:
| 46
| 46
| 849.23
| 849.23
| 18/11, 31/19, 44/27
| ~6
| ~6
| vvvB
| vvvB
Line 264: Line 307:
| 47
| 47
| 867.69
| 867.69
| 33/20, 38/23
| ^~6
| ^~6
| vvB
| vvB
Line 269: Line 313:
| 48
| 48
| 886.15
| 886.15
| 5/3, 92/55
| vM6
| vM6
| vB
| vB
Line 274: Line 319:
| 49
| 49
| 904.62
| 904.62
| 27/16, 32/19
| M6
| M6
| B
| B
Line 279: Line 325:
| 50
| 50
| 923.08
| 923.08
| 17/10, 29/17, 46/27, 128/75
| ^M6
| ^M6
| ^B
| ^B
Line 284: Line 331:
| 51
| 51
| 941.54
| 941.54
| 19/11, 31/18, 50/29, 55/32
| ^^M6
| ^^M6
| ^^B
| ^^B
Line 289: Line 337:
| 52
| 52
| 960.00
| 960.00
| 33/19, 40/23, 47/27, 54/31, 96/55
| vvm7
| vvm7
| vvC
| vvC
Line 294: Line 343:
| 53
| 53
| 978.46
| 978.46
| 30/17, 44/25, 51/29, 58/33
| vm7
| vm7
| vC
| vC
Line 299: Line 349:
| 54
| 54
| 996.92
| 996.92
| 16/9, 57/32
| m7
| m7
| C
| C
Line 304: Line 355:
| 55
| 55
| 1015.38
| 1015.38
| 9/5, 34/19
| ^m7
| ^m7
| ^C
| ^C
Line 309: Line 361:
| 56
| 56
| 1033.85
| 1033.85
| 20/11, 29/16
| v~7
| v~7
| ^^C
| ^^C
Line 314: Line 367:
| 57
| 57
| 1052.31
| 1052.31
| 11/6, 46/25
| ~7
| ~7
| ^^^C
| ^^^C
Line 319: Line 373:
| 58
| 58
| 1070.77
| 1070.77
| 13/7, 50/27, 102/55
| ^~7
| ^~7
| vvC#
| vvC#
Line 324: Line 379:
| 59
| 59
| 1089.23
| 1089.23
| 15/8, 32/17, 62/33
| vM7
| vM7
| vC#
| vC#
Line 329: Line 385:
| 60
| 60
| 1107.69
| 1107.69
| 17/9, 19/10, 36/19, 55/29, 243/128, 256/135
| M7
| M7
| C#
| C#
Line 334: Line 391:
| 61
| 61
| 1126.15
| 1126.15
| 23/12, 44/23, 48/25, 90/47
| ^M7
| ^M7
| ^C#
| ^C#
Line 339: Line 397:
| 62
| 62
| 1144.62
| 1144.62
| 29/15, 31/16, 33/17, 60/31, 64/33
| ^^M7
| ^^M7
| ^^C#
| ^^C#
Line 344: Line 403:
| 63
| 63
| 1163.08
| 1163.08
| 45/23, 47/24, 88/45, 92/47, 108/55, 125/64
| vv8
| vv8
| vvD
| vvD
Line 349: Line 409:
| 64
| 64
| 1181.54
| 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
| v8
| vD
| vD
Line 354: Line 415:
| 65
| 65
| 1200.00
| 1200.00
| 2/1
| P8
| P8
| D
| D
|}
<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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -103 65 }}
| {{mapping| 65 103 }}
| +0.131
| 0.131
| 0.71
|-
| 2.3.5
| 32805/32768, 78732/78125
| {{mapping| 65 103 151 }}
| −0.110
| 0.358
| 1.94
|-
| 2.3.5.11
| 243/242, 4000/3993, 5632/5625
| {{mapping| 65 103 151 225 }}
| −0.266
| 0.410
| 2.22
|}
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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 (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
|-
| 1
| 28\65
| 516.92
| 27/20
| [[Larry]]
|-
| 5
| 20\65<br>(6\65)
| 369.23<br>(110.77)
| 99/80<br>(16/15)
| [[Quintosec]]
|-
| 5
| 27\65<br>(1\65)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| [[Quintile]]
|-
| 5
| 30\65<br>(4\65)
| 553.85<br>(73.85)
| 11/8<br>(25/24)
| [[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]]
[[Category:Theory]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Subgroup]]
[[Category:Schismic]]
[[Category:Schismic]]
[[Category:Sensipent]]
[[Category:Sensipent]]
[[Category:Subgroup temperaments]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
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