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{{Infobox ET}}
{{Infobox ET}}
The '''65 equal divisions of the octave''' ('''65edo'''), or '''65(-tone) equal temperament''' ('''65tet''', '''65et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 65 [[equal]] parts of about 18.5 [[cent]]s each.
{{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 [[support]]s [[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]], [[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]]/[[Chromatic_pairs#Nestoria|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 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 ===
=== Prime harmonics ===
{{Harmonics in equal|65|intervals=prime|columns=15}}
{{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''].
[[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 *
! 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 28: Line 32:
| 1
| 1
| 18.46
| 18.46
| 81/80, 100/99, 121/120, 88/87, 93/92, 94/93, 95/94, 96/95, 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 34: Line 38:
| 2
| 2
| 36.92
| 36.92
| 128/125, 45/44, 46/45, 47/46, 48/47, 55/54
| [[45/44]], [[46/45]], [[47/46]], [[48/47]], [[55/54]], [[128/125]]
| ^^1
| ^^1
| ^^D
| ^^D
Line 40: Line 44:
| 3
| 3
| 55.38
| 55.38
| 33/32, 34/33, 30/29, 32/31, 31/30
| [[30/29]], [[31/30]], [[32/31]], [[33/32]], [[34/33]]
| vvm2
| vvm2
| vvEb
| vvEb
Line 46: Line 50:
| 4
| 4
| 73.85
| 73.85
| 25/24, 24/23, 23/22, 47/45
| [[23/22]], [[24/23]], [[25/24]], [[47/45]]
| vm2
| vm2
| vEb
| vEb
Line 52: Line 56:
| 5
| 5
| 92.31
| 92.31
| 135/128, 256/243, 18/17, 19/18, 20/19, 58/55
| [[18/17]], [[19/18]], [[20/19]], [[58/55]], [[135/128]], [[256/243]]
| m2
| m2
| Eb
| Eb
Line 58: 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 64: Line 68:
| 7
| 7
| 129.23
| 129.23
| 27/25, 14/13, 55/51
| [[14/13]], [[27/25]], [[55/51]]
| v~2
| v~2
| ^^Eb
| ^^Eb
Line 70: Line 74:
| 8
| 8
| 147.69
| 147.69
| 12/11, 25/23
| [[12/11]], [[25/23]]
| ~2
| ~2
| vvvE
| vvvE
Line 76: Line 80:
| 9
| 9
| 166.15
| 166.15
| 11/10, 32/29
| [[11/10]], [[32/29]]
| ^~2
| ^~2
| vvE
| vvE
Line 82: Line 86:
| 10
| 10
| 184.62
| 184.62
| 10/9, 19/17
| [[10/9]], [[19/17]]
| vM2
| vM2
| vE
| vE
Line 88: Line 92:
| 11
| 11
| 203.08
| 203.08
| 9/8, 64/57
| [[9/8]], [[64/57]]
| M2
| M2
| E
| E
Line 94: Line 98:
| 12
| 12
| 221.54
| 221.54
| 25/22, 17/15, 33/29, 58/51
| [[17/15]], [[25/22]], [[33/29]], [[58/51]]
| ^M2
| ^M2
| ^E
| ^E
Line 100: Line 104:
| 13
| 13
| 240.00
| 240.00
| 55/48, 38/33, 23/20, 31/27, 54/47
| [[23/20]], [[31/27]], [[38/33]], [[54/47]], [[55/48]]
| ^^M2
| ^^M2
| ^^E
| ^^E
Line 106: Line 110:
| 14
| 14
| 258.46
| 258.46
| 64/55, 22/19, 29/25, 36/31
| [[22/19]], [[29/25]], [[36/31]], [[64/55]]
| vvm3
| vvm3
| vvF
| vvF
Line 112: Line 116:
| 15
| 15
| 276.92
| 276.92
| 75/64, 20/17, 27/23, 34/29
| [[20/17]], [[27/23]], [[34/29]], [[75/64]]
| vm3
| vm3
| vF
| vF
Line 118: Line 122:
| 16
| 16
| 295.38
| 295.38
| 32/27, 19/16
| [[19/16]], [[32/27]]
| m3
| m3
| F
| F
Line 124: Line 128:
| 17
| 17
| 313.85
| 313.85
| 6/5, 55/46
| [[6/5]], [[55/46]]
| ^m3
| ^m3
| ^F
| ^F
Line 130: Line 134:
| 18
| 18
| 332.31
| 332.31
| 40/33, 23/19
| [[23/19]], [[40/33]]
| v~3
| v~3
| ^^F
| ^^F
Line 136: 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 142: Line 146:
| 20
| 20
| 369.23
| 369.23
| 26/21, 68/55, 47/38
| [[26/21]], [[47/38]], [[68/55]]
| ^~3
| ^~3
| vvF#
| vvF#
Line 148: Line 152:
| 21
| 21
| 387.69
| 387.69
| 5/4, 64/51
| [[5/4]], [[64/51]]
| vM3
| vM3
| vF#
| vF#
Line 154: Line 158:
| 22
| 22
| 406.15
| 406.15
| 81/64, 24/19, 19/15, 34/27, 29/23
| [[19/15]], [[24/19]], [[29/23]], [[34/27]], [[81/64]]
| M3
| M3
| F#
| F#
Line 160: Line 164:
| 23
| 23
| 424.62
| 424.62
| 32/25, 23/18
| [[23/18]], [[32/25]]
| ^M3
| ^M3
| ^F#
| ^F#
Line 166: Line 170:
| 24
| 24
| 443.08
| 443.08
| 128/99, 22/17, 31/24, 40/31
| [[22/17]], [[31/24]], [[40/31]], [[128/99]]
| ^^M3
| ^^M3
| ^^F#
| ^^F#
Line 172: Line 176:
| 25
| 25
| 461.54
| 461.54
| 72/55, 30/23, 47/36
| [[30/23]], [[47/36]], [[72/55]]
| vv4
| vv4
| vvG
| vvG
Line 178: Line 182:
| 26
| 26
| 480.00
| 480.00
| 33/25, 29/22, 62/47
| [[29/22]], [[33/25]], [[62/47]]
| v4
| v4
| vG
| vG
Line 184: Line 188:
| 27
| 27
| 498.46
| 498.46
| 4/3
| [[4/3]]
| P4
| P4
| G
| G
Line 190: Line 194:
| 28
| 28
| 516.92
| 516.92
| 27/20, 23/17, 31/23
| [[23/17]], [[27/20]], [[31/23]]
| ^4
| ^4
| ^G
| ^G
Line 196: 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 202: 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 208: 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 214: Line 218:
| 32
| 32
| 590.77
| 590.77
| 45/32, 24/17, 38/27, 31/22
| [[24/17]], [[31/22]], [[38/27]], [[45/32]]
| vA4/d5
| vA4/d5
| vG#/Ab
| vG#/Ab
Line 220: Line 224:
| 33
| 33
| 609.23
| 609.23
| 64/45, 17/12, 27/19, 44/31
| [[17/12]], [[27/19]], [[44/31]], [[64/45]]
| A4/^d5
| A4/^d5
| G#/^Ab
| G#/^Ab
Line 226: 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 232: 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 238: 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 244: Line 248:
| 37
| 37
| 683.08
| 683.08
| 40/27, 34/23, 46/31
| [[34/23]], [[40/27]], [[46/31]]
| v5
| v5
| vA
| vA
Line 250: Line 254:
| 38
| 38
| 701.54
| 701.54
| 3/2
| [[3/2]]
| P5
| P5
| A
| A
Line 256: Line 260:
| 39
| 39
| 720.00
| 720.00
| 50/33, 44/29, 47/31
| [[44/29]], [[50/33]], [[47/31]]
| ^5
| ^5
| ^A
| ^A
Line 262: Line 266:
| 40
| 40
| 738.46
| 738.46
| 55/36, 23/15, 72/47
| [[23/15]], [[55/36]], [[72/47]]
| ^^5
| ^^5
| ^^A
| ^^A
Line 268: Line 272:
| 41
| 41
| 756.92
| 756.92
| 99/64, 17/11, 48/31, 31/20
| [[17/11]], [[48/31]], [[31/20]], [[99/64]]
| vvm6
| vvm6
| vvBb
| vvBb
Line 274: Line 278:
| 42
| 42
| 775.38
| 775.38
| 25/16, 36/23
| [[25/16]], [[36/23]]
| vm6
| vm6
| vBb
| vBb
Line 280: Line 284:
| 43
| 43
| 793.85
| 793.85
| 128/81, 19/12, 30/19, 27/17, 46/29
| [[19/12]], [[27/17]], [[30/19]], [[46/29]], [[128/81]]
| m6
| m6
| Bb
| Bb
Line 286: Line 290:
| 44
| 44
| 812.31
| 812.31
| 8/5, 51/32
| [[8/5]], [[51/32]]
| ^m6
| ^m6
| ^Bb
| ^Bb
Line 292: 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 298: Line 302:
| 46
| 46
| 849.23
| 849.23
| 18/11, 44/27, 31/19
| [[18/11]], [[31/19]], [[44/27]]
| ~6
| ~6
| vvvB
| vvvB
Line 304: Line 308:
| 47
| 47
| 867.69
| 867.69
| 33/20, 38/23
| [[33/20]], [[38/23]]
| ^~6
| ^~6
| vvB
| vvB
Line 310: Line 314:
| 48
| 48
| 886.15
| 886.15
| 5/3, 92/55
| [[5/3]], [[92/55]]
| vM6
| vM6
| vB
| vB
Line 316: Line 320:
| 49
| 49
| 904.62
| 904.62
| 27/16, 32/19
| [[27/16]], [[32/19]]
| M6
| M6
| B
| B
Line 322: Line 326:
| 50
| 50
| 923.08
| 923.08
| 128/75, 17/10, 46/27, 29/17
| [[17/10]], [[29/17]], [[46/27]], [[128/75]]
| ^M6
| ^M6
| ^B
| ^B
Line 328: Line 332:
| 51
| 51
| 941.54
| 941.54
| 55/32, 19/11, 50/29, 31/18
| [[19/11]], [[31/18]], [[50/29]], [[55/32]]
| ^^M6
| ^^M6
| ^^B
| ^^B
Line 334: Line 338:
| 52
| 52
| 960.00
| 960.00
| 96/55, 33/19, 40/23, 54/31, 47/27
| [[33/19]], [[40/23]], [[47/27]], [[54/31]], [[96/55]]
| vvm7
| vvm7
| vvC
| vvC
Line 340: Line 344:
| 53
| 53
| 978.46
| 978.46
| 44/25, 30/17, 58/33, 51/29
| [[30/17]], [[44/25]], [[51/29]], [[58/33]]
| vm7
| vm7
| vC
| vC
Line 346: Line 350:
| 54
| 54
| 996.92
| 996.92
| 16/9, 57/32
| [[16/9]], [[57/32]]
| m7
| m7
| C
| C
Line 352: Line 356:
| 55
| 55
| 1015.38
| 1015.38
| 9/5, 34/19
| [[9/5]], [[34/19]]
| ^m7
| ^m7
| ^C
| ^C
Line 358: Line 362:
| 56
| 56
| 1033.85
| 1033.85
| 20/11, 29/16
| [[20/11]], [[29/16]]
| v~7
| v~7
| ^^C
| ^^C
Line 364: Line 368:
| 57
| 57
| 1052.31
| 1052.31
| 11/6, 46/25
| [[11/6]], [[46/25]]
| ~7
| ~7
| ^^^C
| ^^^C
Line 370: Line 374:
| 58
| 58
| 1070.77
| 1070.77
| 50/27, 13/7, 102/55
| [[13/7]], [[50/27]], [[102/55]]
| ^~7
| ^~7
| vvC#
| vvC#
Line 376: 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 382: Line 386:
| 60
| 60
| 1107.69
| 1107.69
| 243/128, 256/135, 17/9, 36/19, 19/10, 55/29
| [[17/9]], [[19/10]], [[36/19]], [[55/29]], [[243/128]], [[256/135]]
| M7
| M7
| C#
| C#
Line 388: Line 392:
| 61
| 61
| 1126.15
| 1126.15
| 48/25, 23/12, 44/23, 90/47  
| [[23/12]], [[44/23]], [[48/25]], [[90/47]]
| ^M7
| ^M7
| ^C#
| ^C#
Line 394: Line 398:
| 62
| 62
| 1144.62
| 1144.62
| 64/33, 33/17, 29/15, 31/16, 60/31  
| [[29/15]], [[31/16]], [[33/17]], [[60/31]], [[64/33]]
| ^^M7
| ^^M7
| ^^C#
| ^^C#
Line 400: Line 404:
| 63
| 63
| 1163.08
| 1163.08
| 125/64, 88/45, 45/23, 47/24, 92/47, 108/55  
| [[45/23]], [[47/24]], [[88/45]], [[92/47]], [[108/55]], [[125/64]]
| vv8
| vv8
| vvD
| vvD
Line 406: Line 410:
| 64
| 64
| 1181.54
| 1181.54
| 160/81, 99/50, 240/121, 87/55, 184/93, 93/47, 188/95, 95/48, 228/115, 115/58, 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 412: Line 416:
| 65
| 65
| 1200.00
| 1200.00
| 2/1
| [[2/1]]
| P8
| P8
| D
| D
|}
|}
<nowiki>*</nowiki> based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament.
<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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 431: Line 498:
| 2.3
| 2.3
| {{monzo| -103 65 }}
| {{monzo| -103 65 }}
| [{{val| 65 103 }}]
| {{mapping| 65 103 }}
| +0.131
| +0.131
| 0.131
| 0.131
Line 438: Line 505:
| 2.3.5
| 2.3.5
| 32805/32768, 78732/78125
| 32805/32768, 78732/78125
| [{{val| 65 103 151 }}]
| {{mapping| 65 103 151 }}
| -0.110
| −0.110
| 0.358
| 0.358
| 1.94
| 1.94
Line 445: Line 512:
| 2.3.5.11
| 2.3.5.11
| 243/242, 4000/3993, 5632/5625
| 243/242, 4000/3993, 5632/5625
| [{{val| 65 103 151 225 }}]
| {{mapping| 65 103 151 225 }}
| -0.266
| −0.266
| 0.410
| 0.410
| 2.22
| 2.22
Line 453: Line 520:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
Line 500: Line 568:
| 498.46
| 498.46
| 4/3
| 4/3
| [[Helmholtz]] / [[photia]]
| [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]]
|-
|-
| 1
| 1
Line 506: Line 574:
| 516.92
| 516.92
| 27/20
| 27/20
| [[Gravity]]
| [[Larry]]
|-
|-
| 5
| 5
| 20\65<br>(6\65)
| 20\65<br>(6\65)
| 369.23<br>(110.77)
| 369.23<br>(110.77)
| 10125/8192<br>(16/15)
| 99/80<br>(16/15)
| [[Qintosec]]
| [[Quintosec]]
|-
|-
| 5
| 5
Line 518: Line 586:
| 498.46<br>(18.46)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Pental]]
| [[Quintile]]
|-
|-
| 5
| 5
Line 524: Line 592:
| 553.85<br>(73.85)
| 553.85<br>(73.85)
| 11/8<br>(25/24)
| 11/8<br>(25/24)
| [[Trisedodge]] / [[countdown]]
| [[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]]
* [[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:Listen]]
[[Category:Listen]]
[[Category:Subgroup]]
[[Category:Schismic]]
[[Category:Schismic]]
[[Category:Sensipent]]
[[Category:Sensipent]]
[[Category:Subgroup temperaments]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
Retrieved from "https://en.xen.wiki/w/65edo"