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=65 tone equal temperament=
{{Infobox ET}}
{{ED intro}}


'''65edo''' divides the [[Octave|octave]] into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the [[schisma|schisma]], 32805/32768, the [[sensipent_comma|sensipent comma]], 78732/78125, and the [[Wuerschmidt_comma|wuerschmidt comma]]. In the [[7-limit|7-limit]], there are two different maps; the first is <65 103 151 182|, [[tempering_out|tempering out]] 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the [[5-limit|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 [[wuerschmidt_temperament|wuerschmidt temperament]] (wurschmidt and worschmidt) these two mappings provide.
== 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.


65edo approximates the intervals [[3/2|3/2]], [[5/4|5/4]], [[11/8|11/8]] and [[19/16|19/16]] well, so that it does a good job representing the 2.3.5.11.19 [[just_intonation_subgroup|just intonation subgroup]]. To this one may want to add 13/8 and 17/16, giving the [[19-limit|19-limit]] no-sevens subgroup 2.3.5.11.13.17.19. 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 [[130edo|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|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}}


=Intervals=
=== 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''].


{| class="wikitable"
[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13.
 
== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
|-
! | [[Degree|Degree]]
! #
! | Size ([[cent|Cents]])
! [[Cent]]s
! 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]]
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0.0000
| 0.00
| 1/1
| P1
| D
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 18.4615
| 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
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" | 36.9231
| 36.92
| 45/44, 46/45, 47/46, 48/47, 55/54, 128/125
| ^^1
| ^^D
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 55.3846
| 55.38
| 30/29, 31/30, 32/31, 33/32, 34/33
| vvm2
| vvEb
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 73.8462
| 73.85
| 23/22, 24/23, 25/24, 47/45
| vm2
| vEb
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 92.3077
| 92.31
| 18/17, 19/18, 20/19, 58/55, 135/128, 256/243
| m2
| Eb
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 110.7692
| 110.77
| 16/15, 17/16, 33/31
| A1/^m2
| D#/^Eb
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 129.2308
| 129.23
| 14/13, 27/25, 55/51
| v~2
| ^^Eb
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 147.6923
| 147.69
| 12/11, 25/23
| ~2
| vvvE
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 166.1538
| 166.15
| 11/10, 32/29
| ^~2
| vvE
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 184.6154
| 184.62
| 10/9, 19/17
| vM2
| vE
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 203.0769
| 203.08
| 9/8, 64/57
| M2
| E
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 221.5385
| 221.54
| 17/15, 25/22, 33/29, 58/51
| ^M2
| ^E
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 240.0000
| 240.00
| 23/20, 31/27, 38/33, 54/47, 55/48
| ^^M2
| ^^E
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:right;" | 258.4615
| 258.46
| 22/19, 29/25, 36/31, 64/55
| vvm3
| vvF
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:right;" | 276.9231
| 276.92
| 20/17, 27/23, 34/29, 75/64
| vm3
| vF
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:right;" | 295.3846
| 295.38
| 19/16, 32/27
| m3
| F
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:right;" | 313.8462
| 313.85
| 6/5, 55/46
| ^m3
| ^F
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:right;" | 332.3077
| 332.31
| 23/19, 40/33
| v~3
| ^^F
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:right;" | 350.7692
| 350.77
| 11/9, 27/22, 38/31
| ~3
| ^^^F
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:right;" | 369.2308
| 369.23
| 26/21, 47/38, 68/55
| ^~3
| vvF#
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:right;" | 387.6923
| 387.69
| 5/4, 64/51
| vM3
| vF#
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:right;" | 406.1538
| 406.15
| 19/15, 24/19, 29/23, 34/27, 81/64
| M3
| F#
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:right;" | 424.6154
| 424.62
| 23/18, 32/25
| ^M3
| ^F#
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:right;" | 443.0769
| 443.08
| 22/17, 31/24, 40/31, 128/99
| ^^M3
| ^^F#
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:right;" | 461.5385
| 461.54
| 30/23, 47/36, 72/55
| vv4
| vvG
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:right;" | 480.0000
| 480.00
| 29/22, 33/25, 62/47
| v4
| vG
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:right;" | 498.4615
| 498.46
| 4/3
| P4
| G
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:right;" | 516.9231
| 516.92
| 23/17, 27/20, 31/23
| ^4
| ^G
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:right;" | 535.3846
| 535.38
| 15/11, 34/25, 64/47
| v~4
| ^^G
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:right;" | 553.8462
| 553.85
| 11/8, 40/29, 62/45
| ~4
| ^^^G
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:right;" | 572.3077
| 572.31
| 25/18, 32/23
| ^~4/vd5
| vvG#/vAb
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:right;" | 590.7692
| 590.77
| 24/17, 31/22, 38/27, 45/32
| vA4/d5
| vG#/Ab
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:right;" | 609.2308
| 609.23
| 17/12, 27/19, 44/31, 64/45
| A4/^d5
| G#/^Ab
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:right;" | 627.6923
| 627.69
| 36/25, 23/16
| ^A4/v~5
| ^G#/^^Ab
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:right;" | 646.1538
| 646.15
| 16/11, 29/20, 45/31
| ~5
| vvvA
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:right;" | 664.6154
| 664.62
| 22/15, 25/17, 47/32
| ^~5
| vvA
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:right;" | 683.0769
| 683.08
| 34/23, 40/27, 46/31
| v5
| vA
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:right;" | 701.5385
| 701.54
| 3/2
| P5
| A
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:right;" | 720.0000
| 720.00
| 44/29, 50/33, 47/31
| ^5
| ^A
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:right;" | 738.4615
| 738.46
| 23/15, 55/36, 72/47
| ^^5
| ^^A
|-
|-
| style="text-align:center;" | 41
| 41
| style="text-align:right;" | 756.9231
| 756.92
| 17/11, 48/31, 31/20, 99/64
| vvm6
| vvBb
|-
|-
| style="text-align:center;" | 42
| 42
| style="text-align:right;" | 775.3846
| 775.38
| 25/16, 36/23
| vm6
| vBb
|-
|-
| style="text-align:center;" | 43
| 43
| style="text-align:right;" | 793.8462
| 793.85
| 19/12, 27/17, 30/19, 46/29, 128/81
| m6
| Bb
|-
|-
| style="text-align:center;" | 44
| 44
| style="text-align:right;" | 812.3077
| 812.31
| 8/5, 51/32
| ^m6
| ^Bb
|-
|-
| style="text-align:center;" | 45
| 45
| style="text-align:right;" | 830.7692
| 830.77
| 21/13, 55/34, 76/47
| v~6
| ^^Bb
|-
|-
| style="text-align:center;" | 46
| 46
| style="text-align:right;" | 849.2308
| 849.23
| 18/11, 31/19, 44/27
| ~6
| vvvB
|-
|-
| style="text-align:center;" | 47
| 47
| style="text-align:right;" | 867.6923
| 867.69
| 33/20, 38/23
| ^~6
| vvB
|-
|-
| style="text-align:center;" | 48
| 48
| style="text-align:right;" | 886.1538
| 886.15
| 5/3, 92/55
| vM6
| vB
|-
|-
| style="text-align:center;" | 49
| 49
| style="text-align:right;" | 904.6154
| 904.62
| 27/16, 32/19
| M6
| B
|-
|-
| style="text-align:center;" | 50
| 50
| style="text-align:right;" | 923.0769
| 923.08
| 17/10, 29/17, 46/27, 128/75
| ^M6
| ^B
|-
|-
| style="text-align:center;" | 51
| 51
| style="text-align:right;" | 941.5385
| 941.54
| 19/11, 31/18, 50/29, 55/32
| ^^M6
| ^^B
|-
|-
| style="text-align:center;" | 52
| 52
| style="text-align:right;" | 960.0000
| 960.00
| 33/19, 40/23, 47/27, 54/31, 96/55
| vvm7
| vvC
|-
|-
| style="text-align:center;" | 53
| 53
| style="text-align:right;" | 978.4615
| 978.46
| 30/17, 44/25, 51/29, 58/33
| vm7
| vC
|-
|-
| style="text-align:center;" | 54
| 54
| style="text-align:right;" | 996.9231
| 996.92
| 16/9, 57/32
| m7
| C
|-
|-
| style="text-align:center;" | 55
| 55
| style="text-align:right;" | 1015.3846
| 1015.38
| 9/5, 34/19
| ^m7
| ^C
|-
|-
| style="text-align:center;" | 56
| 56
| style="text-align:right;" | 1033.8462
| 1033.85
| 20/11, 29/16
| v~7
| ^^C
|-
|-
| style="text-align:center;" | 57
| 57
| style="text-align:right;" | 1052.3077
| 1052.31
| 11/6, 46/25
| ~7
| ^^^C
|-
|-
| style="text-align:center;" | 58
| 58
| style="text-align:right;" | 1070.7692
| 1070.77
| 13/7, 50/27, 102/55
| ^~7
| vvC#
|-
|-
| style="text-align:center;" | 59
| 59
| style="text-align:right;" | 1089.2308
| 1089.23
| 15/8, 32/17, 62/33
| vM7
| vC#
|-
|-
| style="text-align:center;" | 60
| 60
| style="text-align:right;" | 1107.6923
| 1107.69
| 17/9, 19/10, 36/19, 55/29, 243/128, 256/135
| M7
| C#
|-
|-
| style="text-align:center;" | 61
| 61
| style="text-align:right;" | 1126.1538
| 1126.15
| 23/12, 44/23, 48/25, 90/47
| ^M7
| ^C#
|-
|-
| style="text-align:center;" | 62
| 62
| style="text-align:right;" | 1144.6154
| 1144.62
| 29/15, 31/16, 33/17, 60/31, 64/33
| ^^M7
| ^^C#
|-
|-
| style="text-align:center;" | 63
| 63
| style="text-align:right;" | 1163.0769
| 1163.08
| 45/23, 47/24, 88/45, 92/47, 108/55, 125/64
| vv8
| vvD
|-
|-
| style="text-align:center;" | 64
| 64
| style="text-align:right;" | 1181.5385
| 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
|-
|-
| style="text-align:center;" | 65
| 65
| style="text-align:right;" | 1200.0000
| 1200.00
| 2/1
| P8
| 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 ==
* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[würschmidt]]): 5 3 5 5 3 5 12 5 5 3 5 12 5
* [[Photia7]]
* [[Photia12]]
* [[Skateboard7]]
== Instruments ==
[[Lumatone mapping for 65edo]]


=Scales=
== Music ==
[[photia7|photia7]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025).


[[photia12|photia12]]      [[Category:11/8]]
[[Category:Listen]]
[[Category:13/8]]
[[Category:Schismic]]
[[Category:17/16]]
[[Category:Sensipent]]
[[Category:19/16]]
[[Category:Subgroup temperaments]]
[[Category:3/2]]
[[Category:Würschmidt]]
[[Category:5/4]]
[[Category:65edo]]
[[Category:edo]]
[[Category:intervals]]
[[Category:listen]]
[[Category:schismic]]
[[Category:sensipent]]
[[Category:subgroup]]
[[Category:theory]]
[[Category:wuerschmidt]]
[[Category:wurschmidt]]
Retrieved from "https://en.xen.wiki/w/65edo"