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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}}
 
=== 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 26: 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 31: 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 36: Line 43:
| 3
| 3
| 55.38
| 55.38
| 30/29, 31/30, 32/31, 33/32, 34/33
| vvm2
| vvm2
| vvEb
| vvEb
Line 41: Line 49:
| 4
| 4
| 73.85
| 73.85
| 23/22, 24/23, 25/24, 47/45
| vm2
| vm2
| vEb
| vEb
Line 46: 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 51: Line 61:
| 6
| 6
| 110.77
| 110.77
| 16/15, 17/16, 33/31
| A1/^m2
| A1/^m2
| D#/^Eb
| D#/^Eb
Line 56: Line 67:
| 7
| 7
| 129.23
| 129.23
| 14/13, 27/25, 55/51
| v~2
| v~2
| ^^Eb
| ^^Eb
Line 61: Line 73:
| 8
| 8
| 147.69
| 147.69
| 12/11, 25/23
| ~2
| ~2
| vvvE
| vvvE
Line 66: Line 79:
| 9
| 9
| 166.15
| 166.15
| 11/10, 32/29
| ^~2
| ^~2
| vvE
| vvE
Line 71: Line 85:
| 10
| 10
| 184.62
| 184.62
| 10/9, 19/17
| vM2
| vM2
| vE
| vE
Line 76: Line 91:
| 11
| 11
| 203.08
| 203.08
| 9/8, 64/57
| M2
| M2
| E
| E
Line 81: Line 97:
| 12
| 12
| 221.54
| 221.54
| 17/15, 25/22, 33/29, 58/51
| ^M2
| ^M2
| ^E
| ^E
Line 86: Line 103:
| 13
| 13
| 240.00
| 240.00
| 23/20, 31/27, 38/33, 54/47, 55/48
| ^^M2
| ^^M2
| ^^E
| ^^E
Line 91: Line 109:
| 14
| 14
| 258.46
| 258.46
| 22/19, 29/25, 36/31, 64/55
| vvm3
| vvm3
| vvF
| vvF
Line 96: Line 115:
| 15
| 15
| 276.92
| 276.92
| 20/17, 27/23, 34/29, 75/64
| vm3
| vm3
| vF
| vF
Line 101: Line 121:
| 16
| 16
| 295.38
| 295.38
| 19/16, 32/27
| m3
| m3
| F
| F
Line 106: Line 127:
| 17
| 17
| 313.85
| 313.85
| 6/5, 55/46
| ^m3
| ^m3
| ^F
| ^F
Line 111: Line 133:
| 18
| 18
| 332.31
| 332.31
| 23/19, 40/33
| v~3
| v~3
| ^^F
| ^^F
Line 116: Line 139:
| 19
| 19
| 350.77
| 350.77
| 11/9, 27/22, 38/31
| ~3
| ~3
| ^^^F
| ^^^F
Line 121: Line 145:
| 20
| 20
| 369.23
| 369.23
| 26/21, 47/38, 68/55
| ^~3
| ^~3
| vvF#
| vvF#
Line 126: Line 151:
| 21
| 21
| 387.69
| 387.69
| 5/4, 64/51
| vM3
| vM3
| vF#
| vF#
Line 131: Line 157:
| 22
| 22
| 406.15
| 406.15
| 19/15, 24/19, 29/23, 34/27, 81/64
| M3
| M3
| F#
| F#
Line 136: Line 163:
| 23
| 23
| 424.62
| 424.62
| 23/18, 32/25
| ^M3
| ^M3
| ^F#
| ^F#
Line 141: Line 169:
| 24
| 24
| 443.08
| 443.08
| 22/17, 31/24, 40/31, 128/99
| ^^M3
| ^^M3
| ^^F#
| ^^F#
Line 146: Line 175:
| 25
| 25
| 461.54
| 461.54
| 30/23, 47/36, 72/55
| vv4
| vv4
| vvG
| vvG
Line 151: Line 181:
| 26
| 26
| 480.00
| 480.00
| 29/22, 33/25, 62/47
| v4
| v4
| vG
| vG
Line 156: Line 187:
| 27
| 27
| 498.46
| 498.46
| 4/3
| P4
| P4
| G
| G
Line 161: Line 193:
| 28
| 28
| 516.92
| 516.92
| 23/17, 27/20, 31/23
| ^4
| ^4
| ^G
| ^G
Line 166: Line 199:
| 29
| 29
| 535.38
| 535.38
| 15/11, 34/25, 64/47
| v~4
| v~4
| ^^G
| ^^G
Line 171: Line 205:
| 30
| 30
| 553.85
| 553.85
| 11/8, 40/29, 62/45
| ~4
| ~4
| ^^^G
| ^^^G
Line 176: Line 211:
| 31
| 31
| 572.31
| 572.31
| 25/18, 32/23
| ^~4/vd5
| ^~4/vd5
| vvG#/vAb
| vvG#/vAb
Line 181: 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 186: 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 191: Line 229:
| 34
| 34
| 627.69
| 627.69
| 36/25, 23/16
| ^A4/v~5
| ^A4/v~5
| ^G#/^^Ab
| ^G#/^^Ab
Line 196: Line 235:
| 35
| 35
| 646.15
| 646.15
| 16/11, 29/20, 45/31
| ~5
| ~5
| vvvA
| vvvA
Line 201: Line 241:
| 36
| 36
| 664.62
| 664.62
| 22/15, 25/17, 47/32
| ^~5
| ^~5
| vvA
| vvA
Line 206: Line 247:
| 37
| 37
| 683.08
| 683.08
| 34/23, 40/27, 46/31
| v5
| v5
| vA
| vA
Line 211: Line 253:
| 38
| 38
| 701.54
| 701.54
| 3/2
| P5
| P5
| A
| A
Line 216: Line 259:
| 39
| 39
| 720.00
| 720.00
| 44/29, 50/33, 47/31
| ^5
| ^5
| ^A
| ^A
Line 221: Line 265:
| 40
| 40
| 738.46
| 738.46
| 23/15, 55/36, 72/47
| ^^5
| ^^5
| ^^A
| ^^A
Line 226: Line 271:
| 41
| 41
| 756.92
| 756.92
| 17/11, 48/31, 31/20, 99/64
| vvm6
| vvm6
| vvBb
| vvBb
Line 231: Line 277:
| 42
| 42
| 775.38
| 775.38
| 25/16, 36/23
| vm6
| vm6
| vBb
| vBb
Line 236: Line 283:
| 43
| 43
| 793.85
| 793.85
| 19/12, 27/17, 30/19, 46/29, 128/81
| m6
| m6
| Bb
| Bb
Line 241: Line 289:
| 44
| 44
| 812.31
| 812.31
| 8/5, 51/32
| ^m6
| ^m6
| ^Bb
| ^Bb
Line 246: Line 295:
| 45
| 45
| 830.77
| 830.77
| 21/13, 55/34, 76/47
| v~6
| v~6
| ^^Bb
| ^^Bb
Line 251: Line 301:
| 46
| 46
| 849.23
| 849.23
| 18/11, 31/19, 44/27
| ~6
| ~6
| vvvB
| vvvB
Line 256: Line 307:
| 47
| 47
| 867.69
| 867.69
| 33/20, 38/23
| ^~6
| ^~6
| vvB
| vvB
Line 261: Line 313:
| 48
| 48
| 886.15
| 886.15
| 5/3, 92/55
| vM6
| vM6
| vB
| vB
Line 266: Line 319:
| 49
| 49
| 904.62
| 904.62
| 27/16, 32/19
| M6
| M6
| B
| B
Line 271: Line 325:
| 50
| 50
| 923.08
| 923.08
| 17/10, 29/17, 46/27, 128/75
| ^M6
| ^M6
| ^B
| ^B
Line 276: Line 331:
| 51
| 51
| 941.54
| 941.54
| 19/11, 31/18, 50/29, 55/32
| ^^M6
| ^^M6
| ^^B
| ^^B
Line 281: Line 337:
| 52
| 52
| 960.00
| 960.00
| 33/19, 40/23, 47/27, 54/31, 96/55
| vvm7
| vvm7
| vvC
| vvC
Line 286: Line 343:
| 53
| 53
| 978.46
| 978.46
| 30/17, 44/25, 51/29, 58/33
| vm7
| vm7
| vC
| vC
Line 291: Line 349:
| 54
| 54
| 996.92
| 996.92
| 16/9, 57/32
| m7
| m7
| C
| C
Line 296: Line 355:
| 55
| 55
| 1015.38
| 1015.38
| 9/5, 34/19
| ^m7
| ^m7
| ^C
| ^C
Line 301: Line 361:
| 56
| 56
| 1033.85
| 1033.85
| 20/11, 29/16
| v~7
| v~7
| ^^C
| ^^C
Line 306: Line 367:
| 57
| 57
| 1052.31
| 1052.31
| 11/6, 46/25
| ~7
| ~7
| ^^^C
| ^^^C
Line 311: Line 373:
| 58
| 58
| 1070.77
| 1070.77
| 13/7, 50/27, 102/55
| ^~7
| ^~7
| vvC#
| vvC#
Line 316: Line 379:
| 59
| 59
| 1089.23
| 1089.23
| 15/8, 32/17, 62/33
| vM7
| vM7
| vC#
| vC#
Line 321: 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 326: Line 391:
| 61
| 61
| 1126.15
| 1126.15
| 23/12, 44/23, 48/25, 90/47
| ^M7
| ^M7
| ^C#
| ^C#
Line 331: Line 397:
| 62
| 62
| 1144.62
| 1144.62
| 29/15, 31/16, 33/17, 60/31, 64/33
| ^^M7
| ^^M7
| ^^C#
| ^^C#
Line 336: 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 341: 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 346: Line 415:
| 65
| 65
| 1200.00
| 1200.00
| 2/1
| P8
| P8
| D
| 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"
|-
! 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 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 ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025)
* [https://www.youtube.com/shorts/UJZw9NQuGnY ''Zanarkand - Nobuo Uematsu (microtonal cover in 65edo)''] (2026)
* [https://www.youtube.com/shorts/zxgVvwXnIGQ ''Waltz in 65edo''] (2026)
* [https://www.youtube.com/shorts/OtbEDFhjNkc ''65edo prelude''] (2026)
* [https://www.youtube.com/shorts/c0eWd7UvNQU ''Black Hole Sun - Soundgarden (microtonal cover in 65edo)''] (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 }}

Latest revision as of 11:37, 16 May 2026

← 64edo 65edo 66edo →
Prime factorization 5 × 13
Step size 18.4615 ¢ 
Fifth 38\65 (701.538 ¢)
Semitones (A1:m2) 6:5 (110.8 ¢ : 92.31 ¢)
Consistency limit 5
Distinct consistency limit 5

65 equal divisions of the octave (abbreviated 65edo or 65ed2), also called 65-tone equal temperament (65tet) or 65 equal temperament (65et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 65 equal parts of about 18.5 ¢ each. Each step represents a frequency ratio of 21/65, or the 65th root of 2.

Theory

65et can be characterized as the temperament which tempers out 32805/32768 (schisma), 78732/78125 (sensipent comma), 393216/390625 (würschmidt comma), and [-13 17 -6⟩ (graviton). In the 7-limit, there are two different maps; the first is ⟨65 103 151 182] (65), tempering out 126/125, 245/243 and 686/675, so that it supports sensi, and the second is ⟨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 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

Approximation of prime harmonics in 65edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) +0.00 -0.42 +1.38 -8.83 +2.53 +8.70 +5.81 -2.13 -0.58 +4.27 -0.42 +7.12 -4.45 +5.41 -0.89
Relative (%) +0.0 -2.3 +7.5 -47.8 +13.7 +47.1 +31.5 -11.5 -3.2 +23.1 -2.3 +38.6 -24.1 +29.3 -4.8
Steps
(reduced) 		65
(0) 		103
(38) 		151
(21) 		182
(52) 		225
(30) 		241
(46) 		266
(6) 		276
(16) 		294
(34) 		316
(56) 		322
(62) 		339
(14) 		348
(23) 		353
(28) 		361
(36)

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 Rubble: a Xenuke Unfolded.

130edo, which doubles its, corrects its approximation to harmonics 7 and 13.

Intervals

# Cents Approximate ratios[note 1] 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
4 73.85 23/22, 24/23, 25/24, 47/45 vm2 vEb
5 92.31 18/17, 19/18, 20/19, 58/55, 135/128, 256/243 m2 Eb
6 110.77 16/15, 17/16, 33/31 A1/^m2 D#/^Eb
7 129.23 14/13, 27/25, 55/51 v~2 ^^Eb
8 147.69 12/11, 25/23 ~2 vvvE
9 166.15 11/10, 32/29 ^~2 vvE
10 184.62 10/9, 19/17 vM2 vE
11 203.08 9/8, 64/57 M2 E
12 221.54 17/15, 25/22, 33/29, 58/51 ^M2 ^E
13 240.00 23/20, 31/27, 38/33, 54/47, 55/48 ^^M2 ^^E
14 258.46 22/19, 29/25, 36/31, 64/55 vvm3 vvF
15 276.92 20/17, 27/23, 34/29, 75/64 vm3 vF
16 295.38 19/16, 32/27 m3 F
17 313.85 6/5, 55/46 ^m3 ^F
18 332.31 23/19, 40/33 v~3 ^^F
19 350.77 11/9, 27/22, 38/31 ~3 ^^^F
20 369.23 26/21, 47/38, 68/55 ^~3 vvF#
21 387.69 5/4, 64/51 vM3 vF#
22 406.15 19/15, 24/19, 29/23, 34/27, 81/64 M3 F#
23 424.62 23/18, 32/25 ^M3 ^F#
24 443.08 22/17, 31/24, 40/31, 128/99 ^^M3 ^^F#
25 461.54 30/23, 47/36, 72/55 vv4 vvG
26 480.00 29/22, 33/25, 62/47 v4 vG
27 498.46 4/3 P4 G
28 516.92 23/17, 27/20, 31/23 ^4 ^G
29 535.38 15/11, 34/25, 64/47 v~4 ^^G
30 553.85 11/8, 40/29, 62/45 ~4 ^^^G
31 572.31 25/18, 32/23 ^~4/vd5 vvG#/vAb
32 590.77 24/17, 31/22, 38/27, 45/32 vA4/d5 vG#/Ab
33 609.23 17/12, 27/19, 44/31, 64/45 A4/^d5 G#/^Ab
34 627.69 36/25, 23/16 ^A4/v~5 ^G#/^^Ab
35 646.15 16/11, 29/20, 45/31 ~5 vvvA
36 664.62 22/15, 25/17, 47/32 ^~5 vvA
37 683.08 34/23, 40/27, 46/31 v5 vA
38 701.54 3/2 P5 A
39 720.00 44/29, 50/33, 47/31 ^5 ^A
40 738.46 23/15, 55/36, 72/47 ^^5 ^^A
41 756.92 17/11, 48/31, 31/20, 99/64 vvm6 vvBb
42 775.38 25/16, 36/23 vm6 vBb
43 793.85 19/12, 27/17, 30/19, 46/29, 128/81 m6 Bb
44 812.31 8/5, 51/32 ^m6 ^Bb
45 830.77 21/13, 55/34, 76/47 v~6 ^^Bb
46 849.23 18/11, 31/19, 44/27 ~6 vvvB
47 867.69 33/20, 38/23 ^~6 vvB
48 886.15 5/3, 92/55 vM6 vB
49 904.62 27/16, 32/19 M6 B
50 923.08 17/10, 29/17, 46/27, 128/75 ^M6 ^B
51 941.54 19/11, 31/18, 50/29, 55/32 ^^M6 ^^B
52 960.00 33/19, 40/23, 47/27, 54/31, 96/55 vvm7 vvC
53 978.46 30/17, 44/25, 51/29, 58/33 vm7 vC
54 996.92 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
  1. ↑ Based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament; other approaches are also possible.

Notation

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol
Flat symbol

If double arrows are not desirable, arrows can be attached to quartertone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
Flat symbol

Kite's ups and downs notation

65edo can also be notated with Kite's ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
         
             
Flat symbol          
             

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
 
    
    
    
Flat symbol  
    
    
    

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Sharp symbol
 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 
  
 		 
Flat symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 

Sagittal notation

This notation uses the same sagittal sequence as edos 72 and 79.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 65edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 65edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.417 2.3
9/8, 16/9 0.833 4.5
13/7, 14/13 0.933 5.1
15/8, 16/15 0.962 5.2
11/10, 20/11 1.150 6.2
5/4, 8/5 1.379 7.5
15/11, 22/15 1.566 8.5
5/3, 6/5 1.795 9.7
9/5, 10/9 2.212 12.0
11/8, 16/11 2.528 13.7
11/6, 12/11 2.945 16.0
11/9, 18/11 3.361 18.2
13/11, 22/13 6.175 33.4
11/7, 14/11 7.107 38.5
13/10, 20/13 7.325 39.7
15/13, 26/15 7.741 41.9
9/7, 14/9 7.993 43.3
7/5, 10/7 8.257 44.7
7/6, 12/7 8.409 45.6
15/14, 28/15 8.674 47.0
13/8, 16/13 8.703 47.1
7/4, 8/7 8.826 47.8
13/9, 18/13 8.925 48.3
13/12, 24/13 9.120 49.4
15-odd-limit intervals in 65edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.417 2.3
9/8, 16/9 0.833 4.5
15/8, 16/15 0.962 5.2
11/10, 20/11 1.150 6.2
5/4, 8/5 1.379 7.5
15/11, 22/15 1.566 8.5
5/3, 6/5 1.795 9.7
9/5, 10/9 2.212 12.0
11/8, 16/11 2.528 13.7
11/6, 12/11 2.945 16.0
11/9, 18/11 3.361 18.2
13/11, 22/13 6.175 33.4
13/10, 20/13 7.325 39.7
15/13, 26/15 7.741 41.9
9/7, 14/9 7.993 43.3
7/6, 12/7 8.409 45.6
13/8, 16/13 8.703 47.1
7/4, 8/7 8.826 47.8
13/12, 24/13 9.120 49.4
13/9, 18/13 9.536 51.7
15/14, 28/15 9.788 53.0
7/5, 10/7 10.205 55.3
11/7, 14/11 11.354 61.5
13/7, 14/13 17.529 94.9
15-odd-limit intervals by 65d val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.417 2.3
9/8, 16/9 0.833 4.5
13/7, 14/13 0.933 5.1
15/8, 16/15 0.962 5.2
11/10, 20/11 1.150 6.2
5/4, 8/5 1.379 7.5
15/11, 22/15 1.566 8.5
5/3, 6/5 1.795 9.7
9/5, 10/9 2.212 12.0
11/8, 16/11 2.528 13.7
11/6, 12/11 2.945 16.0
11/9, 18/11 3.361 18.2
13/11, 22/13 6.175 33.4
11/7, 14/11 7.107 38.5
13/10, 20/13 7.325 39.7
15/13, 26/15 7.741 41.9
7/5, 10/7 8.257 44.7
15/14, 28/15 8.674 47.0
13/8, 16/13 8.703 47.1
13/12, 24/13 9.120 49.4
13/9, 18/13 9.536 51.7
7/4, 8/7 9.636 52.2
7/6, 12/7 10.052 54.4
9/7, 14/9 10.469 56.7

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-103 65⟩ [⟨65 103]] +0.131 0.131 0.71
2.3.5 32805/32768, 78732/78125 [⟨65 103 151]] −0.110 0.358 1.94
2.3.5.11 243/242, 4000/3993, 5632/5625 [⟨65 103 151 225]] −0.266 0.410 2.22

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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 / nestoria / photia
1 28\65 516.92 27/20 Larry
5 20\65
(6\65) 		369.23
(110.77) 		99/80
(16/15) 		Quintosec
5 27\65
(1\65) 		498.46
(18.46) 		4/3
(81/80) 		Quintile
5 30\65
(4\65) 		553.85
(73.85) 		11/8
(25/24) 		Countdown

* Octave-reduced form, reduced to the first half-octave, and 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. 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 334zpi, 151ed5 and 225ed11.

Stretched tunings of 65edo that well approximate JI include 13-lim WE-tuned 65f (18.473cET) and 13-lim TE-tuned 65f (18.474cET).

Scales

Instruments

Lumatone mapping for 65edo

Music

Bryan Deister
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