65edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

Lucius Chiaraviglio (talk | contribs)

3,195 edits
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,837 edits
→Theory: ''It is recommended to read the page regular temperament first to understand this section.''
Tag: Reverted
 
(20 intermediate revisions by 7 users not shown)
Line 3: Line 3:


== Theory ==
== Theory ==
''It is recommended to read the page [[regular temperament]] first to understand this section.''
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.
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.


Line 15: Line 13:
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''].
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.
[[130edo]], which doubles it, corrects its approximation to harmonics 7 and 13.


== Intervals ==
== Intervals ==
{{Todo|inline=1|text=Needs consistency check}}
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! #
! #
! [[Cent]]s
! [[Cent]]s
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13/7.17.19.23.29.31.47 subgroup}}</ref>
! 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
| 0.00
| 0.00
| 1/1
| [[1/1]]
| P1
| P1
| D
| D
Line 33: Line 32:
| 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
| [[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 39: Line 38:
| 2
| 2
| 36.92
| 36.92
| 45/44, 46/45, 47/46, 48/47, 55/54, 128/125
| [[45/44]], [[46/45]], [[47/46]], [[48/47]], [[55/54]], [[128/125]]
| ^^1
| ^^1
| ^^D
| ^^D
Line 45: Line 44:
| 3
| 3
| 55.38
| 55.38
| 30/29, 31/30, 32/31, 33/32, 34/33
| [[30/29]], [[31/30]], [[32/31]], [[33/32]], [[34/33]]
| vvm2
| vvm2
| vvEb
| vvEb
Line 51: Line 50:
| 4
| 4
| 73.85
| 73.85
| 23/22, 24/23, 25/24, 47/45
| [[23/22]], [[24/23]], [[25/24]], [[47/45]]
| vm2
| vm2
| vEb
| vEb
Line 57: Line 56:
| 5
| 5
| 92.31
| 92.31
| 18/17, 19/18, 20/19, 58/55, 135/128, 256/243
| [[18/17]], [[19/18]], [[20/19]], [[58/55]], [[135/128]], [[256/243]]
| m2
| m2
| Eb
| Eb
Line 63: Line 62:
| 6
| 6
| 110.77
| 110.77
| 16/15, 17/16, 33/31
| [[16/15]], [[17/16]], [[33/31]]
| A1/^m2
| A1/^m2
| D#/^Eb
| D#/^Eb
Line 69: Line 68:
| 7
| 7
| 129.23
| 129.23
| 14/13, 27/25, 55/51
| [[14/13]], [[27/25]], [[55/51]]
| v~2
| v~2
| ^^Eb
| ^^Eb
Line 75: Line 74:
| 8
| 8
| 147.69
| 147.69
| 12/11, 25/23
| [[12/11]], [[25/23]]
| ~2
| ~2
| vvvE
| vvvE
Line 81: Line 80:
| 9
| 9
| 166.15
| 166.15
| 11/10, 32/29
| [[11/10]], [[32/29]]
| ^~2
| ^~2
| vvE
| vvE
Line 87: Line 86:
| 10
| 10
| 184.62
| 184.62
| 10/9, 19/17
| [[10/9]], [[19/17]]
| vM2
| vM2
| vE
| vE
Line 93: Line 92:
| 11
| 11
| 203.08
| 203.08
| 9/8, 64/57
| [[9/8]], [[64/57]]
| M2
| M2
| E
| E
Line 99: Line 98:
| 12
| 12
| 221.54
| 221.54
| 17/15, 25/22, 33/29, 58/51
| [[17/15]], [[25/22]], [[33/29]], [[58/51]]
| ^M2
| ^M2
| ^E
| ^E
Line 105: Line 104:
| 13
| 13
| 240.00
| 240.00
| 23/20, 31/27, 38/33, 54/47, 55/48
| [[23/20]], [[31/27]], [[38/33]], [[54/47]], [[55/48]]
| ^^M2
| ^^M2
| ^^E
| ^^E
Line 111: Line 110:
| 14
| 14
| 258.46
| 258.46
| 22/19, 29/25, 36/31, 64/55
| [[22/19]], [[29/25]], [[36/31]], [[64/55]]
| vvm3
| vvm3
| vvF
| vvF
Line 117: Line 116:
| 15
| 15
| 276.92
| 276.92
| 20/17, 27/23, 34/29, 75/64
| [[20/17]], [[27/23]], [[34/29]], [[75/64]]
| vm3
| vm3
| vF
| vF
Line 123: Line 122:
| 16
| 16
| 295.38
| 295.38
| 19/16, 32/27
| [[19/16]], [[32/27]]
| m3
| m3
| F
| F
Line 129: Line 128:
| 17
| 17
| 313.85
| 313.85
| 6/5, 55/46
| [[6/5]], [[55/46]]
| ^m3
| ^m3
| ^F
| ^F
Line 135: Line 134:
| 18
| 18
| 332.31
| 332.31
| 23/19, 40/33
| [[23/19]], [[40/33]]
| v~3
| v~3
| ^^F
| ^^F
Line 141: Line 140:
| 19
| 19
| 350.77
| 350.77
| 11/9, 27/22, 38/31
| [[11/9]], [[27/22]], [[38/31]]
| ~3
| ~3
| ^^^F
| ^^^F
Line 147: Line 146:
| 20
| 20
| 369.23
| 369.23
| 26/21, 47/38, 68/55
| [[26/21]], [[47/38]], [[68/55]]
| ^~3
| ^~3
| vvF#
| vvF#
Line 153: Line 152:
| 21
| 21
| 387.69
| 387.69
| 5/4, 64/51
| [[5/4]], [[64/51]]
| vM3
| vM3
| vF#
| vF#
Line 159: Line 158:
| 22
| 22
| 406.15
| 406.15
| 19/15, 24/19, 29/23, 34/27, 81/64
| [[19/15]], [[24/19]], [[29/23]], [[34/27]], [[81/64]]
| M3
| M3
| F#
| F#
Line 165: Line 164:
| 23
| 23
| 424.62
| 424.62
| 23/18, 32/25
| [[23/18]], [[32/25]]
| ^M3
| ^M3
| ^F#
| ^F#
Line 171: Line 170:
| 24
| 24
| 443.08
| 443.08
| 22/17, 31/24, 40/31, 128/99
| [[22/17]], [[31/24]], [[40/31]], [[128/99]]
| ^^M3
| ^^M3
| ^^F#
| ^^F#
Line 177: Line 176:
| 25
| 25
| 461.54
| 461.54
| 30/23, 47/36, 72/55
| [[30/23]], [[47/36]], [[72/55]]
| vv4
| vv4
| vvG
| vvG
Line 183: Line 182:
| 26
| 26
| 480.00
| 480.00
| 29/22, 33/25, 62/47
| [[29/22]], [[33/25]], [[62/47]]
| v4
| v4
| vG
| vG
Line 189: Line 188:
| 27
| 27
| 498.46
| 498.46
| 4/3
| [[4/3]]
| P4
| P4
| G
| G
Line 195: Line 194:
| 28
| 28
| 516.92
| 516.92
| 23/17, 27/20, 31/23
| [[23/17]], [[27/20]], [[31/23]]
| ^4
| ^4
| ^G
| ^G
Line 201: Line 200:
| 29
| 29
| 535.38
| 535.38
| 15/11, 34/25, 64/47
| [[15/11]], [[34/25]], [[64/47]]
| v~4
| v~4
| ^^G
| ^^G
Line 207: Line 206:
| 30
| 30
| 553.85
| 553.85
| 11/8, 40/29, 62/45
| [[11/8]], [[40/29]], [[62/45]]
| ~4
| ~4
| ^^^G
| ^^^G
Line 213: Line 212:
| 31
| 31
| 572.31
| 572.31
| 25/18, 32/23
| [[25/18]], [[32/23]]
| ^~4/vd5
| ^~4/vd5
| vvG#/vAb
| vvG#/vAb
Line 219: Line 218:
| 32
| 32
| 590.77
| 590.77
| 24/17, 31/22, 38/27, 45/32
| [[24/17]], [[31/22]], [[38/27]], [[45/32]]
| vA4/d5
| vA4/d5
| vG#/Ab
| vG#/Ab
Line 225: Line 224:
| 33
| 33
| 609.23
| 609.23
| 17/12, 27/19, 44/31, 64/45
| [[17/12]], [[27/19]], [[44/31]], [[64/45]]
| A4/^d5
| A4/^d5
| G#/^Ab
| G#/^Ab
Line 231: Line 230:
| 34
| 34
| 627.69
| 627.69
| 36/25, 23/16
| [[36/25]], [[23/16]]
| ^A4/v~5
| ^A4/v~5
| ^G#/^^Ab
| ^G#/^^Ab
Line 237: Line 236:
| 35
| 35
| 646.15
| 646.15
| 16/11, 29/20, 45/31
| [[16/11]], [[29/20]], [[45/31]]
| ~5
| ~5
| vvvA
| vvvA
Line 243: Line 242:
| 36
| 36
| 664.62
| 664.62
| 22/15, 25/17, 47/32
| [[22/15]], [[25/17]], [[47/32]]
| ^~5
| ^~5
| vvA
| vvA
Line 249: Line 248:
| 37
| 37
| 683.08
| 683.08
| 34/23, 40/27, 46/31
| [[34/23]], [[40/27]], [[46/31]]
| v5
| v5
| vA
| vA
Line 255: Line 254:
| 38
| 38
| 701.54
| 701.54
| 3/2
| [[3/2]]
| P5
| P5
| A
| A
Line 261: Line 260:
| 39
| 39
| 720.00
| 720.00
| 44/29, 50/33, 47/31
| [[44/29]], [[50/33]], [[47/31]]
| ^5
| ^5
| ^A
| ^A
Line 267: Line 266:
| 40
| 40
| 738.46
| 738.46
| 23/15, 55/36, 72/47
| [[23/15]], [[55/36]], [[72/47]]
| ^^5
| ^^5
| ^^A
| ^^A
Line 273: Line 272:
| 41
| 41
| 756.92
| 756.92
| 17/11, 48/31, 31/20, 99/64
| [[17/11]], [[48/31]], [[31/20]], [[99/64]]
| vvm6
| vvm6
| vvBb
| vvBb
Line 279: Line 278:
| 42
| 42
| 775.38
| 775.38
| 25/16, 36/23
| [[25/16]], [[36/23]]
| vm6
| vm6
| vBb
| vBb
Line 285: Line 284:
| 43
| 43
| 793.85
| 793.85
| 19/12, 27/17, 30/19, 46/29, 128/81
| [[19/12]], [[27/17]], [[30/19]], [[46/29]], [[128/81]]
| m6
| m6
| Bb
| Bb
Line 291: Line 290:
| 44
| 44
| 812.31
| 812.31
| 8/5, 51/32
| [[8/5]], [[51/32]]
| ^m6
| ^m6
| ^Bb
| ^Bb
Line 297: Line 296:
| 45
| 45
| 830.77
| 830.77
| 21/13, 55/34, 76/47  
| [[21/13]], [[55/34]], [[76/47]]
| v~6
| v~6
| ^^Bb
| ^^Bb
Line 303: Line 302:
| 46
| 46
| 849.23
| 849.23
| 18/11, 31/19, 44/27
| [[18/11]], [[31/19]], [[44/27]]
| ~6
| ~6
| vvvB
| vvvB
Line 309: Line 308:
| 47
| 47
| 867.69
| 867.69
| 33/20, 38/23
| [[33/20]], [[38/23]]
| ^~6
| ^~6
| vvB
| vvB
Line 315: Line 314:
| 48
| 48
| 886.15
| 886.15
| 5/3, 92/55
| [[5/3]], [[92/55]]
| vM6
| vM6
| vB
| vB
Line 321: Line 320:
| 49
| 49
| 904.62
| 904.62
| 27/16, 32/19
| [[27/16]], [[32/19]]
| M6
| M6
| B
| B
Line 327: Line 326:
| 50
| 50
| 923.08
| 923.08
| 17/10, 29/17, 46/27, 128/75
| [[17/10]], [[29/17]], [[46/27]], [[128/75]]
| ^M6
| ^M6
| ^B
| ^B
Line 333: Line 332:
| 51
| 51
| 941.54
| 941.54
| 19/11, 31/18, 50/29, 55/32
| [[19/11]], [[31/18]], [[50/29]], [[55/32]]
| ^^M6
| ^^M6
| ^^B
| ^^B
Line 339: Line 338:
| 52
| 52
| 960.00
| 960.00
| 33/19, 40/23, 47/27, 54/31, 96/55
| [[33/19]], [[40/23]], [[47/27]], [[54/31]], [[96/55]]
| vvm7
| vvm7
| vvC
| vvC
Line 345: Line 344:
| 53
| 53
| 978.46
| 978.46
| 30/17, 44/25, 51/29, 58/33
| [[30/17]], [[44/25]], [[51/29]], [[58/33]]
| vm7
| vm7
| vC
| vC
Line 351: Line 350:
| 54
| 54
| 996.92
| 996.92
| 16/9, 57/32
| [[16/9]], [[57/32]]
| m7
| m7
| C
| C
Line 357: Line 356:
| 55
| 55
| 1015.38
| 1015.38
| 9/5, 34/19
| [[9/5]], [[34/19]]
| ^m7
| ^m7
| ^C
| ^C
Line 363: Line 362:
| 56
| 56
| 1033.85
| 1033.85
| 20/11, 29/16
| [[20/11]], [[29/16]]
| v~7
| v~7
| ^^C
| ^^C
Line 369: Line 368:
| 57
| 57
| 1052.31
| 1052.31
| 11/6, 46/25
| [[11/6]], [[46/25]]
| ~7
| ~7
| ^^^C
| ^^^C
Line 375: Line 374:
| 58
| 58
| 1070.77
| 1070.77
| 13/7, 50/27, 102/55
| [[13/7]], [[50/27]], [[102/55]]
| ^~7
| ^~7
| vvC#
| vvC#
Line 381: Line 380:
| 59
| 59
| 1089.23
| 1089.23
| 15/8, 32/17, 62/33
| [[15/8]], [[32/17]], [[62/33]]
| vM7
| vM7
| vC#
| vC#
Line 387: Line 386:
| 60
| 60
| 1107.69
| 1107.69
| 17/9, 19/10, 36/19, 55/29, 243/128, 256/135
| [[17/9]], [[19/10]], [[36/19]], [[55/29]], [[243/128]], [[256/135]]
| M7
| M7
| C#
| C#
Line 393: Line 392:
| 61
| 61
| 1126.15
| 1126.15
| 23/12, 44/23, 48/25, 90/47  
| [[23/12]], [[44/23]], [[48/25]], [[90/47]]
| ^M7
| ^M7
| ^C#
| ^C#
Line 399: Line 398:
| 62
| 62
| 1144.62
| 1144.62
| 29/15, 31/16, 33/17, 60/31, 64/33
| [[29/15]], [[31/16]], [[33/17]], [[60/31]], [[64/33]]
| ^^M7
| ^^M7
| ^^C#
| ^^C#
Line 405: Line 404:
| 63
| 63
| 1163.08
| 1163.08
| 45/23, 47/24, 88/45, 92/47, 108/55, 125/64
| [[45/23]], [[47/24]], [[88/45]], [[92/47]], [[108/55]], [[125/64]]
| vv8
| vv8
| vvD
| vvD
Line 411: Line 410:
| 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
| [[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 417: Line 416:
| 65
| 65
| 1200.00
| 1200.00
| 2/1
| [[2/1]]
| P8
| P8
| D
| D
|}
|}
<references group="note" />


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould 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.  
[[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}}
{{Sharpness-sharp6a}}


Half-sharps and half-flats can be used to avoid triple arrows:
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
{{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 ===
=== Ivan Wyschnegradsky's notation ===
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
 
{{Sharpness-sharp6-iw}}
{{sharpness-sharp6-iw}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[72edo#Sagittal notation|72]] and [[79edo#Sagittal notation|79]].
This notation uses the same sagittal sequence as edos [[72edo #Sagittal notation|72]] and [[79edo #Sagittal notation|79]].


==== Evo flavor ====
==== Evo flavor ====
Line 479: Line 479:
default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
default [[File:65-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
</imagemap>
== Approximation to JI ==
== Approximation to JI ==
=== Zeta peak index ===
=== 15-odd-limit interval mappings ===
{{ZPI
{{Q-odd-limit intervals|65}}
| zpi = 334
{{Q-odd-limit intervals|65.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 65d val mapping}}
| 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 ==
Line 579: Line 568:
| 498.46
| 498.46
| 4/3
| 4/3
| [[Helmholtz]] / [[nestoria]] / [[photia]]
| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
|-
|-
| 1
| 1
Line 605: Line 594:
| [[Countdown]]
| [[Countdown]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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 ==
Line 616: Line 612:
[[Lumatone mapping for 65edo]]
[[Lumatone mapping for 65edo]]


== Notes ==
== Music ==
<references group="note" />
=== 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:Listen]]
[[Category:Listen]]

Latest revision as of 06:09, 16 September 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 it, corrects its approximation to harmonics 7 and 13.

Intervals

Todo:

Needs consistency check

# 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

Modern renderings

Soundgarden
SunKissed Lola
Nobuo Uematsu

21st century

Bryan Deister
Retrieved from "https://en.xen.wiki/index.php?title=65edo&oldid=237271"