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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]], [[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]]/[[Chromatic_pairs#Nestoria|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]], [[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, 88/87, 93/92, 94/93, 95/94, 96/95, 115/114, 116/115, 125/124
| 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
| 128/125, 45/44, 46/45, 47/46, 48/47, 55/54
| 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, 34/33, 30/29, 32/31, 31/30
| 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, 24/23, 23/22, 47/45
| 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
| 18/17, 19/18, 20/19, 58/55, 135/128, 256/243
| m2
| m2
| Eb
| Eb
Line 64: Line 67:
| 7
| 7
| 129.23
| 129.23
| 27/25, 14/13, 55/51
| 14/13, 27/25, 55/51
| v~2
| v~2
| ^^Eb
| ^^Eb
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, 19/17, 64/57
| 9/8, 64/57
| M2
| M2
| E
| E
Line 94: Line 97:
| 12
| 12
| 221.54
| 221.54
| 25/22, 17/15, 33/29, 58/51
| 17/15, 25/22, 33/29, 58/51
| ^M2
| ^M2
| ^E
| ^E
Line 100: Line 103:
| 13
| 13
| 240.00
| 240.00
| 55/48, 38/33, 23/20, 31/27, 54/47
| 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
| 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
| 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
| 19/16, 32/27
| m3
| m3
| F
| F
Line 130: Line 133:
| 18
| 18
| 332.31
| 332.31
| 40/33, 17/14, 23/19
| 23/19, 40/33
| v~3
| v~3
| ^^F
| ^^F
Line 142: Line 145:
| 20
| 20
| 369.23
| 369.23
| 26/21, 68/55, 47/38
| 26/21, 47/38, 68/55
| ^~3
| ^~3
| vvF#
| vvF#
Line 154: Line 157:
| 22
| 22
| 406.15
| 406.15
| 81/64, 24/19, 19/15, 34/27, 29/23
| 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
| 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
| 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
| 30/23, 47/36, 72/55
| vv4
| vv4
| vvG
| vvG
Line 178: Line 181:
| 26
| 26
| 480.00
| 480.00
| 33/25, 29/22, 62/47
| 29/22, 33/25, 62/47
| v4
| v4
| vG
| vG
Line 190: Line 193:
| 28
| 28
| 516.92
| 516.92
| 27/20, 23/17, 31/23
| 23/17, 27/20, 31/23
| ^4
| ^4
| ^G
| ^G
Line 214: Line 217:
| 32
| 32
| 590.77
| 590.77
| 45/32, 24/17, 38/27, 31/22
| 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
| 17/12, 27/19, 44/31, 64/45
| A4/^d5
| A4/^d5
| G#/^Ab
| G#/^Ab
Line 232: Line 235:
| 35
| 35
| 646.15
| 646.15
| 16/11, 29/20, 44/31
| 16/11, 29/20, 45/31
| ~5
| ~5
| vvvA
| vvvA
Line 244: Line 247:
| 37
| 37
| 683.08
| 683.08
| 40/27, 23/17, 31/23
| 34/23, 40/27, 46/31
| v5
| v5
| vA
| vA
Line 256: Line 259:
| 39
| 39
| 720.00
| 720.00
| 50/33, 44/29, 47/31
| 44/29, 50/33, 47/31
| ^5
| ^5
| ^A
| ^A
Line 262: Line 265:
| 40
| 40
| 738.46
| 738.46
| 55/36, 23/15, 72/47
| 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
| 17/11, 48/31, 31/20, 99/64
| vvm6
| vvm6
| vvBb
| vvBb
Line 280: Line 283:
| 43
| 43
| 793.85
| 793.85
| 128/81, 19/12, 30/19, 27/17, 46/29
| 19/12, 27/17, 30/19, 46/29, 128/81
| m6
| m6
| Bb
| Bb
Line 298: Line 301:
| 46
| 46
| 849.23
| 849.23
| 18/11, 44/27, 31/19
| 18/11, 31/19, 44/27
| ~6
| ~6
| vvvB
| vvvB
Line 304: Line 307:
| 47
| 47
| 867.69
| 867.69
| 33/20, 28/17, 38/23
| 33/20, 38/23
| ^~6
| ^~6
| vvB
| vvB
Line 322: Line 325:
| 50
| 50
| 923.08
| 923.08
| 128/75, 17/10, 46/27, 29/17
| 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, 50/29, 31/18
| 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, 54/31, 47/27
| 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, 58/33, 51/29
| 30/17, 44/25, 51/29, 58/33
| vm7
| vm7
| vC
| vC
Line 346: Line 349:
| 54
| 54
| 996.92
| 996.92
| 16/9, 34/19
| 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 370: Line 373:
| 58
| 58
| 1070.77
| 1070.77
| 50/27, 13/7, 102/55
| 13/7, 50/27, 102/55
| ^~7
| ^~7
| vvC#
| vvC#
Line 382: Line 385:
| 60
| 60
| 1107.69
| 1107.69
| 243/128, 256/135, 17/9, 36/19, 19/10, 55/29
| 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, 90/47  
| 23/12, 44/23, 48/25, 90/47  
| ^M7
| ^M7
| ^C#
| ^C#
Line 394: Line 397:
| 62
| 62
| 1144.62
| 1144.62
| 64/33, 33/17, 29/15, 31/16, 60/31  
| 29/15, 31/16, 33/17, 60/31, 64/33
| ^^M7
| ^^M7
| ^^C#
| ^^C#
Line 400: Line 403:
| 63
| 63
| 1163.08
| 1163.08
| 125/64, 88/45, 45/23, 47/24, 92/47, 108/55  
| 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, 184/93, 93/47, 188/95, 95/48, 228/115, 115/58, 248/125
| 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 2.3.5.11.13/7.17.19.23.29.31.47 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]]
Retrieved from "https://en.xen.wiki/w/65edo"