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'''65edo''' divides the [[octave]] into 65 equal parts of 18.4615 cents each.
{{Infobox ET}}
{{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 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]] 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=11}}
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 it, corrects its approximation to harmonics 7 and 13.


== Intervals ==
== Intervals ==
 
{{Todo|inline=1|text=Needs consistency check}}
{| class="wikitable"
{| class="wikitable center-all right-2 left-3"
|-
! #
! [[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]]
|-
| 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
|-
|-
! [[Degree|Degree]]
| 4
![[cent|Cents]]
| 73.85
! colspan="2" |[[Ups and Downs Notation]]
| [[23/22]], [[24/23]], [[25/24]], [[47/45]]
| vm2
| vEb
|-
|-
| style="text-align:center;" | 0
| 5
| style="text-align:right;" | 0.0000
| 92.31
|P1
| [[18/17]], [[19/18]], [[20/19]], [[58/55]], [[135/128]], [[256/243]]
|D
| m2
| Eb
|-
|-
| style="text-align:center;" | 1
| 6
| style="text-align:right;" | 18.4615
| 110.77
|^1
| [[16/15]], [[17/16]], [[33/31]]
|^D
| A1/^m2
| D#/^Eb
|-
|-
| style="text-align:center;" | 2
| 7
| style="text-align:right;" | 36.9231
| 129.23
|^^1
| [[14/13]], [[27/25]], [[55/51]]
|^^D
| v~2
| ^^Eb
|-
|-
| style="text-align:center;" | 3
| 8
| style="text-align:right;" | 55.3846
| 147.69
|vvm2
| [[12/11]], [[25/23]]
|vvEb
| ~2
| vvvE
|-
|-
| style="text-align:center;" | 4
| 9
| style="text-align:right;" | 73.84615
| 166.15
|vm2
| [[11/10]], [[32/29]]
|vEb
| ^~2
| vvE
|-
|-
| style="text-align:center;" | 5
| 10
| style="text-align:right;" | 92.3077
| 184.62
|m2
| [[10/9]], [[19/17]]
|Eb
| vM2
| vE
|-
|-
| style="text-align:center;" | 6
| 11
| style="text-align:right;" | 110.7692
| 203.08
|A1/^m2
| [[9/8]], [[64/57]]
|D#/^Eb
| M2
| E
|-
|-
| style="text-align:center;" | 7
| 12
| style="text-align:right;" | 129.2308
| 221.54
|v~2
| [[17/15]], [[25/22]], [[33/29]], [[58/51]]
|^^Eb
| ^M2
| ^E
|-
|-
| style="text-align:center;" | 8
| 13
| style="text-align:right;" | 147.6923
| 240.00
|~2
| [[23/20]], [[31/27]], [[38/33]], [[54/47]], [[55/48]]
|vvvE
| ^^M2
| ^^E
|-
|-
| style="text-align:center;" | 9
| 14
| style="text-align:right;" | 166.15385
| 258.46
|^~2
| [[22/19]], [[29/25]], [[36/31]], [[64/55]]
|vvE
| vvm3
| vvF
|-
|-
| style="text-align:center;" | 10
| 15
| style="text-align:right;" | 184.6154
| 276.92
|vM2
| [[20/17]], [[27/23]], [[34/29]], [[75/64]]
|vE
| vm3
| vF
|-
|-
| style="text-align:center;" | 11
| 16
| style="text-align:right;" | 203.0769
| 295.38
|M2
| [[19/16]], [[32/27]]
|E
| m3
| F
|-
|-
| style="text-align:center;" | 12
| 17
| style="text-align:right;" | 221.5385
| 313.85
|^M2
| [[6/5]], [[55/46]]
|^E
| ^m3
| ^F
|-
|-
| style="text-align:center;" | 13
| 18
| style="text-align:right;" | 240
| 332.31
|^^M2
| [[23/19]], [[40/33]]
|^^E
| v~3
| ^^F
|-
|-
| style="text-align:center;" | 14
| 19
| style="text-align:right;" | 258.4615
| 350.77
|vvm3
| [[11/9]], [[27/22]], [[38/31]]
|vvF
| ~3
| ^^^F
|-
|-
| style="text-align:center;" | 15
| 20
| style="text-align:right;" | 276.9231
| 369.23
|vm3
| [[26/21]], [[47/38]], [[68/55]]
|vF
| ^~3
| vvF#
|-
|-
| style="text-align:center;" | 16
| 21
| style="text-align:right;" | 295.3846
| 387.69
|m3
| [[5/4]], [[64/51]]
|F
| vM3
| vF#
|-
|-
| style="text-align:center;" | 17
| 22
| style="text-align:right;" | 313.84615
| 406.15
|^m3
| [[19/15]], [[24/19]], [[29/23]], [[34/27]], [[81/64]]
|^F
| M3
| F#
|-
|-
| style="text-align:center;" | 18
| 23
| style="text-align:right;" | 332.3077
| 424.62
|v~3
| [[23/18]], [[32/25]]
|^^F
| ^M3
| ^F#
|-
|-
| style="text-align:center;" | 19
| 24
| style="text-align:right;" | 350.7692
| 443.08
|~3
| [[22/17]], [[31/24]], [[40/31]], [[128/99]]
|^^^F
| ^^M3
| ^^F#
|-
|-
| style="text-align:center;" | 20
| 25
| style="text-align:right;" | 369.2308
| 461.54
|^~3
| [[30/23]], [[47/36]], [[72/55]]
|vvF#
| vv4
| vvG
|-
|-
| style="text-align:center;" | 21
| 26
| style="text-align:right;" | 387.6923
| 480.00
|vM3
| [[29/22]], [[33/25]], [[62/47]]
|vF#
| v4
| vG
|-
|-
| style="text-align:center;" | 22
| 27
| style="text-align:right;" | 406.15385
| 498.46
|M3
| [[4/3]]
|F#
| P4
| G
|-
|-
| style="text-align:center;" | 23
| 28
| style="text-align:right;" | 424.6154
| 516.92
|^M3
| [[23/17]], [[27/20]], [[31/23]]
|^F#
| ^4
| ^G
|-
|-
| style="text-align:center;" | 24
| 29
| style="text-align:right;" | 443.0769
| 535.38
|^^M3
| [[15/11]], [[34/25]], [[64/47]]
|^^F#
| v~4
| ^^G
|-
|-
| style="text-align:center;" | 25
| 30
| style="text-align:right;" | 461.5385
| 553.85
|vv4
| [[11/8]], [[40/29]], [[62/45]]
|vvG
| ~4
| ^^^G
|-
|-
| style="text-align:center;" | 26
| 31
| style="text-align:right;" | 480
| 572.31
|v4
| [[25/18]], [[32/23]]
|vG
| ^~4/vd5
| vvG#/vAb
|-
|-
| style="text-align:center;" | 27
| 32
| style="text-align:right;" | 498.4615
| 590.77
|P4
| [[24/17]], [[31/22]], [[38/27]], [[45/32]]
|G
| vA4/d5
| vG#/Ab
|-
|-
| style="text-align:center;" | 28
| 33
| style="text-align:right;" | 516.9231
| 609.23
|^4
| [[17/12]], [[27/19]], [[44/31]], [[64/45]]
|^G
| A4/^d5
| G#/^Ab
|-
|-
| style="text-align:center;" | 29
| 34
| style="text-align:right;" | 535.3846
| 627.69
|v~4
| [[36/25]], [[23/16]]
|^^G
| ^A4/v~5
| ^G#/^^Ab
|-
|-
| style="text-align:center;" | 30
| 35
| style="text-align:right;" | 553.84615
| 646.15
|~4
| [[16/11]], [[29/20]], [[45/31]]
|^^^G
| ~5
| vvvA
|-
|-
| style="text-align:center;" | 31
| 36
| style="text-align:right;" | 572.3077
| 664.62
|^~4/vd5
| [[22/15]], [[25/17]], [[47/32]]
|vvG#/vAb
| ^~5
| vvA
|-
|-
| style="text-align:center;" | 32
| 37
| style="text-align:right;" | 590.7692
| 683.08
|vA4/d5
| [[34/23]], [[40/27]], [[46/31]]
|vG#/Ab
| v5
| vA
|-
|-
| style="text-align:center;" | 33
| 38
| style="text-align:right;" | 609.2308
| 701.54
|A4/^d5
| [[3/2]]
|G#/^Ab
| P5
| A
|-
|-
| style="text-align:center;" | 34
| 39
| style="text-align:right;" | 627.6923
| 720.00
|^A4/v~5
| [[44/29]], [[50/33]], [[47/31]]
|^G#/^^Ab
| ^5
| ^A
|-
|-
| style="text-align:center;" | 35
| 40
| style="text-align:right;" | 646.1538
| 738.46
|~5
| [[23/15]], [[55/36]], [[72/47]]
|vvvA
| ^^5
| ^^A
|-
|-
| style="text-align:center;" | 36
| 41
| style="text-align:right;" | 664.6154
| 756.92
|^~5
| [[17/11]], [[48/31]], [[31/20]], [[99/64]]
|vvA
| vvm6
| vvBb
|-
|-
| style="text-align:center;" | 37
| 42
| style="text-align:right;" | 683.0769
| 775.38
|v5
| [[25/16]], [[36/23]]
|vA
| vm6
| vBb
|-
|-
| style="text-align:center;" | 38
| 43
| style="text-align:right;" | 701.5385
| 793.85
|P5
| [[19/12]], [[27/17]], [[30/19]], [[46/29]], [[128/81]]
|A
| m6
| Bb
|-
|-
| style="text-align:center;" | 39
| 44
| style="text-align:right;" | 720
| 812.31
|^5
| [[8/5]], [[51/32]]
|^A
| ^m6
| ^Bb
|-
|-
| style="text-align:center;" | 40
| 45
| style="text-align:right;" | 738.4615
| 830.77
|^^5
| [[21/13]], [[55/34]], [[76/47]]
|^^A
| v~6
| ^^Bb
|-
|-
| style="text-align:center;" | 41
| 46
| style="text-align:right;" | 756.9231
| 849.23
|vvm6
| [[18/11]], [[31/19]], [[44/27]]
|vvBb
| ~6
| vvvB
|-
|-
| style="text-align:center;" | 42
| 47
| style="text-align:right;" | 775.3846
| 867.69
|vm6
| [[33/20]], [[38/23]]
|vBb
| ^~6
| vvB
|-
|-
| style="text-align:center;" | 43
| 48
| style="text-align:right;" | 793.84615
| 886.15
|m6
| [[5/3]], [[92/55]]
|Bb
| vM6
| vB
|-
|-
| style="text-align:center;" | 44
| 49
| style="text-align:right;" | 812.3077
| 904.62
|^m6
| [[27/16]], [[32/19]]
|^Bb
| M6
| B
|-
|-
| style="text-align:center;" | 45
| 50
| style="text-align:right;" | 830.7692
| 923.08
|v~6
| [[17/10]], [[29/17]], [[46/27]], [[128/75]]
|^^Bb
| ^M6
| ^B
|-
|-
| style="text-align:center;" | 46
| 51
| style="text-align:right;" | 849.2308
| 941.54
|~6
| [[19/11]], [[31/18]], [[50/29]], [[55/32]]
|vvvB
| ^^M6
| ^^B
|-
|-
| style="text-align:center;" | 47
| 52
| style="text-align:right;" | 867.6923
| 960.00
|^~6
| [[33/19]], [[40/23]], [[47/27]], [[54/31]], [[96/55]]
|vvB
| vvm7
| vvC
|-
|-
| style="text-align:center;" | 48
| 53
| style="text-align:right;" | 886.15385
| 978.46
|vM6
| [[30/17]], [[44/25]], [[51/29]], [[58/33]]
|vB
| vm7
| vC
|-
|-
| style="text-align:center;" | 49
| 54
| style="text-align:right;" | 904.6154
| 996.92
|M6
| [[16/9]], [[57/32]]
|B
| m7
| C
|-
|-
| style="text-align:center;" | 50
| 55
| style="text-align:right;" | 923.0769
| 1015.38
|^M6
| [[9/5]], [[34/19]]
|^B
| ^m7
| ^C
|-
|-
| style="text-align:center;" | 51
| 56
| style="text-align:right;" | 941.5385
| 1033.85
|^^M6
| [[20/11]], [[29/16]]
|^^B
| v~7
| ^^C
|-
|-
| style="text-align:center;" | 52
| 57
| style="text-align:right;" | 960
| 1052.31
|vvm7
| [[11/6]], [[46/25]]
|vvC
| ~7
| ^^^C
|-
|-
| style="text-align:center;" | 53
| 58
| style="text-align:right;" | 978.4615
| 1070.77
|vm7
| [[13/7]], [[50/27]], [[102/55]]
|vC
| ^~7
| vvC#
|-
|-
| style="text-align:center;" | 54
| 59
| style="text-align:right;" | 996.9231
| 1089.23
|m7
| [[15/8]], [[32/17]], [[62/33]]
|C
| vM7
| vC#
|-
|-
| style="text-align:center;" | 55
| 60
| style="text-align:right;" | 1015.3846
| 1107.69
|^m7
| [[17/9]], [[19/10]], [[36/19]], [[55/29]], [[243/128]], [[256/135]]
|^C
| M7
| C#
|-
|-
| style="text-align:center;" | 56
| 61
| style="text-align:right;" | 1033.84615
| 1126.15
|v~7
| [[23/12]], [[44/23]], [[48/25]], [[90/47]]
|^^C
| ^M7
| ^C#
|-
|-
| style="text-align:center;" | 57
| 62
| style="text-align:right;" | 1052.3077
| 1144.62
|~7
| [[29/15]], [[31/16]], [[33/17]], [[60/31]], [[64/33]]
|^^^C
| ^^M7
| ^^C#
|-
|-
| style="text-align:center;" | 58
| 63
| style="text-align:right;" | 1070.7692
| 1163.08
|^~7
| [[45/23]], [[47/24]], [[88/45]], [[92/47]], [[108/55]], [[125/64]]
|vvC#
| vv8
| vvD
|-
|-
| style="text-align:center;" | 59
| 64
| style="text-align:right;" | 1089.2308
| 1181.54
|vM7
| [[87/55]], [[93/47]], [[95/48]], [[99/50]], [[115/58]], [[160/81]], [[184/93]], [[188/95]], [[228/115]], [[240/121]], [[248/125]]
|vC#
| v8
| vD
|-
|-
| style="text-align:center;" | 60
| 65
| style="text-align:right;" | 1107.6923
| 1200.00
|M7
| [[2/1]]
|C#
| P8
| D
|}
<references group="note" />
 
== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp6-szg}}
 
If double arrows are not desirable, arrows can be attached to quartertone accidentals:
{{Sharpness-sharp6-qt-szg}}
 
=== Kite's ups and downs notation ===
65edo can also be notated with [[Kite's ups and downs notation|Kite's 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}}
 
=== 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 ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|65}}
{{Q-odd-limit intervals|65.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 65d val mapping}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| style="text-align:center;" | 61
! rowspan="2" | [[Subgroup]]
| style="text-align:right;" | 1126.15385
! rowspan="2" | [[Comma list]]
|^M7
! rowspan="2" | [[Mapping]]
|^C#
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | 62
! [[TE error|Absolute]] (¢)
| style="text-align:right;" | 1144.6154
! [[TE simple badness|Relative]] (%)
|^^M7
|^^C#
|-
|-
| style="text-align:center;" | 63
| 2.3
| style="text-align:right;" | 1163.0769
| {{monzo| -103 65 }}
|vv8
| {{mapping| 65 103 }}
|vvD
| +0.131
| 0.131
| 0.71
|-
|-
| style="text-align:center;" | 64
| 2.3.5
| style="text-align:right;" | 1181.5385
| 32805/32768, 78732/78125
|v8
| {{mapping| 65 103 151 }}
|vD
| −0.110
| 0.358
| 1.94
|-
|-
| style="text-align:center;" | 65
| 2.3.5.11
| style="text-align:right;" | 1200
| 243/242, 4000/3993, 5632/5625
|P8
| {{mapping| 65 103 151 225 }}
|D
| −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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
== Octave stretch or compression ==
65edo tunes [[primes]] 2, 3, 5 and 11 very well, but its 7 and 13 have two about equally-bad mappings. [[Stretched and compressed tuning|Stretching or shrinking the octave]] of 65edo for improvements in its approximations of [[JI]] therefore depends on which mapping is used: the sharp tending 65d val wants octave shrinking, whereas the flat tending 65f val wants octave stretching; both can be achieved at the cost of relatively little damage to other primes.
Compressed tunings of 65edo that well approximate JI include [[zpi|334zpi]], [[ed5|151ed5]] and [[equal tuning|225ed11]].
Stretched tunings of 65edo that well approximate JI include [[WE|13-lim WE-tuned 65f]] (18.473cET) and [[TE|13-lim TE-tuned 65f]] (18.474cET).


== 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 ==
=== Modern renderings ===
; {{W|Soundgarden}}
* [https://www.youtube.com/shorts/c0eWd7UvNQU ''Black Hole Sun''] (1994) — microtonal cover in 65edo by [[Bryan Deister]] (2026)
; {{W|SunKissed Lola}}
* [https://www.youtube.com/shorts/_qcJ7Q9MkLE ''Pasilyo''] (2022) — microtonal cover in 65edo by [[Bryan Deister]] (2026)
; {{W|Nobuo Uematsu}}
* [https://www.youtube.com/shorts/UJZw9NQuGnY ''Zanarkand''] via ''{{W|Final Fantasy X}}'' (2001) — microtonal cover in 65edo by [[Bryan Deister]] (2026)
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025)
* [https://www.youtube.com/shorts/zxgVvwXnIGQ ''Waltz in 65edo''] (2026)
* [https://www.youtube.com/shorts/OtbEDFhjNkc ''65edo prelude''] (2026)
* [https://www.youtube.com/shorts/pSIT-4VaGD0 ''65edo improv''] (2026)


[[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]]
{{todo| unify precision }}
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