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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 it, corrects its approximation to harmonics 7 and 13.
 
== Intervals ==
{{Todo|inline=1|text=Needs consistency check}}
{| 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
|-
|-
! rowspan="2" | [[Degree|Degree]]
| 4
! colspan="2" | Size
| 73.85
| [[23/22]], [[24/23]], [[25/24]], [[47/45]]
| vm2
| vEb
|-
|-
![[cent|Cents]]
| 5
!7mus
| 92.31
| [[18/17]], [[19/18]], [[20/19]], [[58/55]], [[135/128]], [[256/243]]
| m2
| Eb
|-
|-
| colspan="3" style="text-align:center;" | 0
| 6
| 110.77
| [[16/15]], [[17/16]], [[33/31]]
| A1/^m2
| D#/^Eb
|-
|-
| style="text-align:center;" | 1
| 7
| style="text-align:right;" | 18.4615
| 129.23
|23.6308 (17.A17A<sub>16</sub>)
| [[14/13]], [[27/25]], [[55/51]]
| v~2
| ^^Eb
|-
|-
| style="text-align:center;" | 2
| 8
| style="text-align:right;" | 36.9231
| 147.69
|47.2615 (2F.42F4<sub>16</sub>)
| [[12/11]], [[25/23]]
| ~2
| vvvE
|-
|-
| style="text-align:center;" | 3
| 9
| style="text-align:right;" | 55.3846
| 166.15
|70.8923 (46.E46E<sub>16</sub>)
| [[11/10]], [[32/29]]
| ^~2
| vvE
|-
|-
| style="text-align:center;" | 4
| 10
| style="text-align:right;" | 73.84615
| 184.62
|94.5231 (5E.85E8<sub>16</sub>)
| [[10/9]], [[19/17]]
| vM2
| vE
|-
|-
| style="text-align:center;" | 5
| 11
| style="text-align:right;" | 92.3077
| 203.08
|118.15385 (76.2762<sub>16</sub>)
| [[9/8]], [[64/57]]
| M2
| E
|-
|-
| style="text-align:center;" | 6
| 12
| style="text-align:right;" | 110.7692
| 221.54
|141.7846 (8D.C8DD<sub>16</sub>)
| [[17/15]], [[25/22]], [[33/29]], [[58/51]]
| ^M2
| ^E
|-
|-
| style="text-align:center;" | 7
| 13
| style="text-align:right;" | 129.2308
| 240.00
|165.4154 (A5.6A57<sub>16</sub>)
| [[23/20]], [[31/27]], [[38/33]], [[54/47]], [[55/48]]
| ^^M2
| ^^E
|-
|-
| style="text-align:center;" | 8
| 14
| style="text-align:right;" | 147.6923
| 258.46
|189.04615 (BD.0BD1<sub>16</sub>)
| [[22/19]], [[29/25]], [[36/31]], [[64/55]]
| vvm3
| vvF
|-
|-
| style="text-align:center;" | 9
| 15
| style="text-align:right;" | 166.15385
| 276.92
|212.6769 (D4.AD4B<sub>16</sub>
| [[20/17]], [[27/23]], [[34/29]], [[75/64]]
| vm3
| vF
|-
|-
| style="text-align:center;" | 10
| 16
| style="text-align:right;" | 184.6154
| 295.38
|236.3077 (EC.4EC5<sub>16</sub>)
| [[19/16]], [[32/27]]
| m3
| F
|-
|-
| style="text-align:center;" | 11
| 17
| style="text-align:right;" | 203.0769
| 313.85
|259.9385 (103.F03F<sub>16</sub>)
| [[6/5]], [[55/46]]
| ^m3
| ^F
|-
|-
| style="text-align:center;" | 12
| 18
| style="text-align:right;" | 221.5385
| 332.31
|283.5692 (11B.91B9<sub>16</sub>)
| [[23/19]], [[40/33]]
| v~3
| ^^F
|-
|-
| style="text-align:center;" | 13
| 19
| style="text-align:right;" | 240
| 350.77
|307.2 (133.3333<sub>16</sub>)
| [[11/9]], [[27/22]], [[38/31]]
| ~3
| ^^^F
|-
|-
| style="text-align:center;" | 14
| 20
| style="text-align:right;" | 258.4615
| 369.23
|330.8308 (14A.D4AD<sub>16</sub>)
| [[26/21]], [[47/38]], [[68/55]]
| ^~3
| vvF#
|-
|-
| style="text-align:center;" | 15
| 21
| style="text-align:right;" | 276.9231
| 387.69
|354.4615 (162.7627<sub>16</sub>)
| [[5/4]], [[64/51]]
| vM3
| vF#
|-
|-
| style="text-align:center;" | 16
| 22
| style="text-align:right;" | 295.3846
| 406.15
|378.0923 (17A.17A18<sub>16</sub>)
| [[19/15]], [[24/19]], [[29/23]], [[34/27]], [[81/64]]
| M3
| F#
|-
|-
| style="text-align:center;" | 17
| 23
| style="text-align:right;" | 313.84615
| 424.62
|401.7231 (191.B91C<sub>16</sub>)
| [[23/18]], [[32/25]]
| ^M3
| ^F#
|-
|-
| style="text-align:center;" | 18
| 24
| style="text-align:right;" | 332.3077
| 443.08
|425.35385 (1A9.5A96<sub>16</sub>)
| [[22/17]], [[31/24]], [[40/31]], [[128/99]]
| ^^M3
| ^^F#
|-
|-
| style="text-align:center;" | 19
| 25
| style="text-align:right;" | 350.7692
| 461.54
|448.9846 (1C0.FC1<sub>16</sub>)
| [[30/23]], [[47/36]], [[72/55]]
| vv4
| vvG
|-
|-
| style="text-align:center;" | 20
| 26
| style="text-align:right;" | 369.2308
| 480.00
|472.6154 (1D8.9D8A<sub>16</sub>)
| [[29/22]], [[33/25]], [[62/47]]
| v4
| vG
|-
|-
| style="text-align:center;" | 21
| 27
| style="text-align:right;" | 387.6923
| 498.46
|496.24615 (1F0.3F04<sub>16</sub>)
| [[4/3]]
| P4
| G
|-
|-
| style="text-align:center;" | 22
| 28
| style="text-align:right;" | 406.15385
| 516.92
|519.8769 (207.E07E<sub>16</sub>)
| [[23/17]], [[27/20]], [[31/23]]
| ^4
| ^G
|-
|-
| style="text-align:center;" | 23
| 29
| style="text-align:right;" | 424.6154
| 535.38
|543.5077 (21F.81F8<sub>16</sub>)
| [[15/11]], [[34/25]], [[64/47]]
| v~4
| ^^G
|-
|-
| style="text-align:center;" | 24
| 30
| style="text-align:right;" | 443.0769
| 553.85
|567.1385 (237.2372<sub>16</sub>)
| [[11/8]], [[40/29]], [[62/45]]
| ~4
| ^^^G
|-
|-
| style="text-align:center;" | 25
| 31
| style="text-align:right;" | 461.5385
| 572.31
|590.7692 (24E.C4ED<sub>16</sub>)
| [[25/18]], [[32/23]]
| ^~4/vd5
| vvG#/vAb
|-
|-
| style="text-align:center;" | 26
| 32
| style="text-align:right;" | 480
| 590.77
|614.4 (266.6666<sub>16</sub>)
| [[24/17]], [[31/22]], [[38/27]], [[45/32]]
| vA4/d5
| vG#/Ab
|-
|-
| style="text-align:center;" | 27
| 33
| style="text-align:right;" | 498.4615
| 609.23
|638.0308 (27E07E08<sub>16</sub>)
| [[17/12]], [[27/19]], [[44/31]], [[64/45]]
| A4/^d5
| G#/^Ab
|-
|-
| style="text-align:center;" | 28
| 34
| style="text-align:right;" | 516.9231
| 627.69
|661.6615 (295.A95B<sub>16</sub>)
| [[36/25]], [[23/16]]
| ^A4/v~5
| ^G#/^^Ab
|-
|-
| style="text-align:center;" | 29
| 35
| style="text-align:right;" | 535.3846
| 646.15
|685.2923 (2AD.4AD4<sub>16</sub>)
| [[16/11]], [[29/20]], [[45/31]]
| ~5
| vvvA
|-
|-
| style="text-align:center;" | 30
| 36
| style="text-align:right;" | 553.84615
| 664.62
|708.9231 (2C4.EC4F<sub>16</sub>)
| [[22/15]], [[25/17]], [[47/32]]
| ^~5
| vvA
|-
|-
| style="text-align:center;" | 31
| 37
| style="text-align:right;" | 572.3077
| 683.08
|732.55385 (2DC.8DC9<sub>16</sub>)
| [[34/23]], [[40/27]], [[46/31]]
| v5
| vA
|-
|-
| style="text-align:center;" | 32
| 38
| style="text-align:right;" | 590.7692
| 701.54
|756.1846 (2F4.2F42<sub>16</sub>)
| [[3/2]]
| P5
| A
|-
|-
| style="text-align:center;" | 33
| 39
| style="text-align:right;" | 609.2308
| 720.00
|779.8154 (30B.D0BD<sub>16</sub>)
| [[44/29]], [[50/33]], [[47/31]]
| ^5
| ^A
|-
|-
| style="text-align:center;" | 34
| 40
| style="text-align:right;" | 627.6923
| 738.46
|803.44615 (323.7237<sub>16</sub>)
| [[23/15]], [[55/36]], [[72/47]]
| ^^5
| ^^A
|-
|-
| style="text-align:center;" | 35
| 41
| style="text-align:right;" | 646.1538
| 756.92
|827.0769 (33B.13B1<sub>16</sub>)
| [[17/11]], [[48/31]], [[31/20]], [[99/64]]
| vvm6
| vvBb
|-
|-
| style="text-align:center;" | 36
| 42
| style="text-align:right;" | 664.6154
| 775.38
|850.7077 (352.B52B<sub>16</sub>)
| [[25/16]], [[36/23]]
| vm6
| vBb
|-
|-
| style="text-align:center;" | 37
| 43
| style="text-align:right;" | 683.0769
| 793.85
|874.3385 (36A.56A5<sub>16</sub>)
| [[19/12]], [[27/17]], [[30/19]], [[46/29]], [[128/81]]
| m6
| Bb
|-
|-
| style="text-align:center;" | 38
| 44
| style="text-align:right;" | 701.5385
| 812.31
|897.9692 (381.F81F8<sub>16</sub>)
| [[8/5]], [[51/32]]
| ^m6
| ^Bb
|-
|-
| style="text-align:center;" | 39
| 45
| style="text-align:right;" | 720
| 830.77
|921.6 (399.999A<sub>16</sub>)
| [[21/13]], [[55/34]], [[76/47]]
| v~6
| ^^Bb
|-
|-
| style="text-align:center;" | 40
| 46
| style="text-align:right;" | 738.4615
| 849.23
|945.2308 (3B1.3B13<sub>16</sub>)
| [[18/11]], [[31/19]], [[44/27]]
| ~6
| vvvB
|-
|-
| style="text-align:center;" | 41
| 47
| style="text-align:right;" | 756.9231
| 867.69
|968.8615 (3C8.DC8E<sub>16</sub>)
| [[33/20]], [[38/23]]
| ^~6
| vvB
|-
|-
| style="text-align:center;" | 42
| 48
| style="text-align:right;" | 775.3846
| 886.15
|992.4923 (3E0.7E08<sub>16</sub>)
| [[5/3]], [[92/55]]
| vM6
| vB
|-
|-
| style="text-align:center;" | 43
| 49
| style="text-align:right;" | 793.84615
| 904.62
|1016.1231 (3F8.1F82<sub>16</sub>)
| [[27/16]], [[32/19]]
| M6
| B
|-
|-
| style="text-align:center;" | 44
| 50
| style="text-align:right;" | 812.3077
| 923.08
|1039.75385 (40F.C0FC<sub>16</sub>)
| [[17/10]], [[29/17]], [[46/27]], [[128/75]]
| ^M6
| ^B
|-
|-
| style="text-align:center;" | 45
| 51
| style="text-align:right;" | 830.7692
| 941.54
|1063.3846 (427.6276<sub>16</sub>)
| [[19/11]], [[31/18]], [[50/29]], [[55/32]]
| ^^M6
| ^^B
|-
|-
| style="text-align:center;" | 46
| 52
| style="text-align:right;" | 849.2308
| 960.00
|1087.0154 (43F.03F<sub>16</sub>)
| [[33/19]], [[40/23]], [[47/27]], [[54/31]], [[96/55]]
| vvm7
| vvC
|-
|-
| style="text-align:center;" | 47
| 53
| style="text-align:right;" | 867.6923
| 978.46
|1110.64615 (456.A56A<sub>16</sub>)
| [[30/17]], [[44/25]], [[51/29]], [[58/33]]
| vm7
| vC
|-
|-
| style="text-align:center;" | 48
| 54
| style="text-align:right;" | 886.15385
| 996.92
|1134.2769 (46E.46E4<sub>16</sub>)
| [[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
|}
<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;" | 49
! rowspan="2" | [[Subgroup]]
| style="text-align:right;" | 904.6154
! rowspan="2" | [[Comma list]]
|1157.9077 (485.E85E8<sub>16</sub>)
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | 50
! [[TE error|Absolute]] (¢)
| style="text-align:right;" | 923.0769
! [[TE simple badness|Relative]] (%)
|1181.53845 (49D.89D9<sub>16</sub>)
|-
|-
| style="text-align:center;" | 51
| 2.3
| style="text-align:right;" | 941.5385
| {{monzo| -103 65 }}
|1205.1692 (4B5.2B53<sub>16</sub>)
| {{mapping| 65 103 }}
| +0.131
| 0.131
| 0.71
|-
|-
| style="text-align:center;" | 52
| 2.3.5
| style="text-align:right;" | 960
| 32805/32768, 78732/78125
|1228.8 (4CC.CCCD<sub>16</sub>)
| {{mapping| 65 103 151 }}
| −0.110
| 0.358
| 1.94
|-
|-
| style="text-align:center;" | 53
| 2.3.5.11
| style="text-align:right;" | 978.4615
| 243/242, 4000/3993, 5632/5625
|1252.4308 (4E4.6E47<sub>16</sub>)
| {{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
|-
|-
| style="text-align:center;" | 54
! Periods<br>per 8ve
| style="text-align:right;" | 996.9231
! Generator*
|1276.0615 (4FC.0FC1<sub>16</sub>)
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| style="text-align:center;" | 55
| 1
| style="text-align:right;" | 1015.3846
| 3\65
|1299.6923 (513.B13B<sub>16</sub>)
| 55.38
| 33/32
| [[Escapade]]
|-
|-
| style="text-align:center;" | 56
| 1
| style="text-align:right;" | 1033.84615
| 9\65
|1323.3231 (52B.52B5<sub>16</sub>)
| 166.15
| 11/10
| [[Squirrel]] etc.
|-
|-
| style="text-align:center;" | 57
| 1
| style="text-align:right;" | 1052.3077
| 12\65
|1346.95385 (542,F42F<sub>16</sub>)
| 221.54
| 25/22
| [[Hemisensi]]
|-
|-
| style="text-align:center;" | 58
| 1
| style="text-align:right;" | 1070.7692
| 19\65
|1370.5846 (55A.95A9<sub>16</sub>)
| 350.77
| 11/9
| [[Karadeniz]]
|-
|-
| style="text-align:center;" | 59
| 1
| style="text-align:right;" | 1089.2308
| 21\65
|1394.2154 (572.3723<sub>16</sub>)
| 387.69
| 5/4
| [[Würschmidt]]
|-
|-
| style="text-align:center;" | 60
| 1
| style="text-align:right;" | 1107.6923
| 24\65
|1417.84615 (589.D89E<sub>16</sub>)
| 443.08
| 162/125
| [[Sensipent]]
|-
|-
| style="text-align:center;" | 61
| 1
| style="text-align:right;" | 1126.15385
| 27\65
|1441.4769 (5A1.7A18<sub>16</sub>)
| 498.46
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
|-
|-
| style="text-align:center;" | 62
| 1
| style="text-align:right;" | 1144.6154
| 28\65
|1465.1077 (5B9.1B92<sub>16</sub>)
| 516.92
| 27/20
| [[Larry]]
|-
|-
| style="text-align:center;" | 63
| 5
| style="text-align:right;" | 1163.0769
| 20\65<br>(6\65)
|1488.7385 (5D0.BD0C<sub>16</sub>)
| 369.23<br>(110.77)
| 99/80<br>(16/15)
| [[Quintosec]]
|-
|-
| style="text-align:center;" | 64
| 5
| style="text-align:right;" | 1181.5385
| 27\65<br>(1\65)
|1512.3692 (5E8.5E86<sub>16</sub>)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| [[Quintile]]
|-
|-
| style="text-align:center;" | 65
| 5
| style="text-align:right;" | 1200
| 30\65<br>(4\65)
|1536 (600<sub>16</sub>)
| 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 ==
* 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]]
== 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)


=Scales=
=== 21st century ===
[[photia7|photia7]]
; [[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)


[[photia12|photia12]]     
[[Category:Listen]]
[[Category:11/8]]
[[Category:Schismic]]
[[Category:13/8]]
[[Category:Sensipent]]
[[Category:17/16]]
[[Category:Subgroup temperaments]]
[[Category:19/16]]
[[Category:Würschmidt]]
[[Category:3/2]]
[[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"