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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|94}}
{{ED intro}}


== Theory ==
== Theory ==
Line 7: Line 7:
Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.
Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.


94edo can be thought of as two sets of [[47edo]] offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while it dramatically improves on prime 3, as well as primes 11, 19, and 23 to a lesser degree.
=== As a tuning of other temperaments ===
 
=== Significance of cassandra ===
94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it.
94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it.


=== Other temperament properties ===
The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].
The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].


Line 19: Line 16:
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|94|columns=11}}
{{Harmonics in equal|94|columns=11}}
=== Subsets and supersets ===
Since 94 factors into primes as {{nowrap| 2 × 47 }}, 94edo contains [[2edo]] and [[47edo]] as subset edos. It can be thought of as two sets of 47edo offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while dramatically improving on prime 3, as well as primes 11, 19, and 23 to a lesser degree.


== Intervals ==
== Intervals ==
Line 25: Line 25:
Assuming [[23-limit]] [[patent val]] {{val| 94 149 218 264 325 348 384 399 425 }}, here is a table of intervals as approximated by [[94edo]] steps, and their corresponding 13-limit well-ordered extended diatonic interval names. 'S/s' indicates alteration by the septimal comma, [[64/63]]; 'K/k' indicates alteration by the syntonic comma, [[81/80]]; 'U/u' by the undecimal quartertone, [[33/32]]; 'L/l' by pentacircle comma, [[896/891]]; 'O/o' by [[45/44]]; 'R/r' by the rastma, [[243/242]]; 'T/t' by the tridecimal quartertone, [[1053/1024]]; and finally, 'H/h', by [[40/39]]. Capital letters alter downward, lowercase alter upwards. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'K' down from augmented and major, or up from diminished and minor intervals are also added in brackets, along with their associated (5-limit) intervals.
Assuming [[23-limit]] [[patent val]] {{val| 94 149 218 264 325 348 384 399 425 }}, here is a table of intervals as approximated by [[94edo]] steps, and their corresponding 13-limit well-ordered extended diatonic interval names. 'S/s' indicates alteration by the septimal comma, [[64/63]]; 'K/k' indicates alteration by the syntonic comma, [[81/80]]; 'U/u' by the undecimal quartertone, [[33/32]]; 'L/l' by pentacircle comma, [[896/891]]; 'O/o' by [[45/44]]; 'R/r' by the rastma, [[243/242]]; 'T/t' by the tridecimal quartertone, [[1053/1024]]; and finally, 'H/h', by [[40/39]]. Capital letters alter downward, lowercase alter upwards. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'K' down from augmented and major, or up from diminished and minor intervals are also added in brackets, along with their associated (5-limit) intervals.


{| class="wikitable"
{| class="wikitable center-5"
|+ style="font-size: 105%;" | 94edo [[SKULO interval names#WOFED interval names|WOFED interval names]]
|-
|-
! Step
! Step
Line 32: Line 31:
! 13-limit
! 13-limit
! 23-limit
! 23-limit
! Short-form WOFED
![[Ups and downs notation|Ups and downs]]
! Short-form [[SKULO interval names#WOFED interval names|WOFED]]
! Long-form WOFED
! Long-form WOFED
! Diatonic
! Diatonic
|-
|0
|0
|1/1
|
|{{UDnote|step=0}}
|
|
|
|-
|-
| 1
| 1
Line 40: Line 49:
| 896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98)
| 896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98)
| 85/84
| 85/84
|{{UDnote|step=1}}
| L1, R1
| L1, R1
| large unison, rastma
| large unison, rastma
|  
|
|-
|-
| 2
| 2
| 25.532
| 25.532
| 81/80, 64/63, (50/49)
| 81/80, 64/63, (50/49)
|  
|
|{{UDnote|step=2}}
| K1, S1
| K1, S1
| komma, super unison  
| komma, super unison  
|  
|
|-
|-
| 3
| 3
Line 56: Line 67:
| 45/44, 40/39, (250/243, 49/48)
| 45/44, 40/39, (250/243, 49/48)
| 46/45
| 46/45
|{{UDnote|step=3}}
| O1, H1
| O1, H1
| on unison, hyper unison
| on unison, hyper unison
|  
|
|-
|-
| 4
| 4
| 51.064
| 51.064
| 33/32, (128/125, 36/35, 35/34, 34/33)
| 33/32, (128/125, 36/35, 35/34, 34/33)
|  
|
|{{UDnote|step=4}}
| U1, T1, hm2
| U1, T1, hm2
| uber unison, tall unison, hypo minor second
| uber unison, tall unison, hypo minor second
|  
|
|-
|-
| 5
| 5
| 63.830
| 63.830
| 28/27, 729/704, 27/26, (25/24)
| 28/27, 729/704, 27/26, (25/24)
|  
|
|{{UDnote|step=5}}
| sm2, uA1, tA1, (kkA1)
| sm2, uA1, tA1, (kkA1)
| sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison)
| sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison)
Line 80: Line 94:
| 22/21, (648/625, 26/25)
| 22/21, (648/625, 26/25)
| 23/22, 24/23
| 23/22, 24/23
|{{UDnote|step=6}}
| lm2, oA1
| lm2, oA1
| little minor second, off augmented unison
| little minor second, off augmented unison
|  
|
|-
|-
| 7
| 7
Line 88: Line 103:
| 256/243, 135/128, (21/20)
| 256/243, 135/128, (21/20)
| 19/18, 20/19
| 19/18, 20/19
|{{UDnote|step=7}}
| m2, kA1
| m2, kA1
| minor second, komma-down augmented unison
| minor second, komma-down augmented unison
Line 96: Line 112:
| 128/121, (35/33)
| 128/121, (35/33)
| 17/16, 18/17
| 17/16, 18/17
|{{UDnote|step=8}}
| Rm2, rA1
| Rm2, rA1
| rastmic minor second, rastmic augmented unison
| rastmic minor second, rastmic augmented unison
|  
|
|-
|-
| 9
| 9
| 114.894
| 114.894
| 16/15, (15/14)
| 16/15, (15/14)
|  
|
|{{UDnote|step=9}}
| Km2, A1
| Km2, A1
| classic minor second, augmented unison
| classic minor second, augmented unison
Line 111: Line 129:
| 127.660
| 127.660
| 320/297, 189/176, (14/13)
| 320/297, 189/176, (14/13)
|  
|
|{{UDnote|step=10}}
| Om2, LA1
| Om2, LA1
| oceanic minor second, large augmented unison
| oceanic minor second, large augmented unison
|  
|
|-
|-
| 11
| 11
Line 120: Line 139:
| 88/81, 13/12, 243/224, (27/25)
| 88/81, 13/12, 243/224, (27/25)
| 25/23, 38/35
| 25/23, 38/35
|{{UDnote|step=11}}
| n2, Tm2, SA1, (KKm2)
| n2, Tm2, SA1, (KKm2)
| lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
| lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
|  
|
|-
|-
| 12
| 12
Line 128: Line 148:
| 12/11, (35/32)
| 12/11, (35/32)
| 23/21
| 23/21
|{{UDnote|step=12}}
| N2, tM2, HA1
| N2, tM2, HA1
| greater netral second, tiny major second, hyper augmented unison
| greater netral second, tiny major second, hyper augmented unison
Line 135: Line 156:
| 165.957
| 165.957
| 11/10
| 11/10
|  
|
|{{UDnote|step=13}}
| oM2
| oM2
| off major second
| off major second
|  
|
|-
|-
| 14
| 14
Line 144: Line 166:
| 10/9
| 10/9
| 21/19
| 21/19
|{{UDnote|step=14}}
| kM2
| kM2
| komma-down major second
| komma-down major second
Line 152: Line 175:
| 121/108, (49/44, 39/35)
| 121/108, (49/44, 39/35)
| 19/17
| 19/17
|{{UDnote|step=15}}
| rM2
| rM2
| rastmic major second
| rastmic major second
|  
|
|-
|-
| 16
| 16
| 204.255
| 204.255
| 9/8
| 9/8
|  
|
|{{UDnote|step=16}}
| M2
| M2
| major second
| major second
Line 168: Line 193:
| 112/99, (25/22)
| 112/99, (25/22)
| 17/15, 26/23
| 17/15, 26/23
|{{UDnote|step=17}}
| LM2
| LM2
| large major second
| large major second
|  
|
|-
|-
| 18
| 18
| 229.787
| 229.787
| 8/7
| 8/7
|  
|
|{{UDnote|step=18}}
| SM2
| SM2
| super major second
| super major second
Line 184: Line 211:
| 15/13
| 15/13
| 23/20, 38/33
| 23/20, 38/33
|{{UDnote|step=19}}
| HM2
| HM2
| hyper major second
| hyper major second
|  
|
|-
|-
| 20
| 20
Line 192: Line 220:
| 52/45
| 52/45
| 22/19
| 22/19
|{{UDnote|step=20}}
| hm3
| hm3
| hypo minor third
| hypo minor third
|  
|
|-
|-
| 21
| 21
| 268.085
| 268.085
| 7/6, (75/64)
| 7/6, (75/64)
|  
|
|{{UDnote|step=21}}
| sm3, (kkA2)
| sm3, (kkA2)
| sub minor third, (classic augmented second)
| sub minor third, (classic augmented second)
Line 208: Line 238:
| 33/28
| 33/28
| 20/17, 27/23
| 20/17, 27/23
|{{UDnote|step=22}}
| lm3
| lm3
| little minor third
| little minor third
|  
|
|-
|-
| 23
| 23
Line 216: Line 247:
| 32/27, (25/21, 13/11)
| 32/27, (25/21, 13/11)
| 19/16
| 19/16
|{{UDnote|step=23}}
| m3
| m3
| minor third
| minor third
Line 223: Line 255:
| 306.383
| 306.383
| 144/121, (81/70)
| 144/121, (81/70)
|  
|
|{{UDnote|step=24}}
| Rm3
| Rm3
| rastmic minor third
| rastmic minor third
|  
|
|-
|-
| 25
| 25
| 319.149
| 319.149
| 6/5
| 6/5
|  
|
|{{UDnote|step=25}}
| Km3
| Km3
| classic minor third
| classic minor third
Line 240: Line 274:
| 40/33
| 40/33
| 17/14, 23/19
| 17/14, 23/19
|{{UDnote|step=26}}
| Om3
| Om3
| on minor third
| on minor third
|  
|
|-
|-
| 27
| 27
Line 248: Line 283:
| 11/9, 39/32, (243/200, 60/49)
| 11/9, 39/32, (243/200, 60/49)
| 28/23
| 28/23
|{{UDnote|step=27}}
| n3, Tm3
| n3, Tm3
| lesser neutral third, tall minor third
| lesser neutral third, tall minor third
Line 255: Line 291:
| 357.447
| 357.447
| 27/22, 16/13, (100/81,49/40)
| 27/22, 16/13, (100/81,49/40)
|  
|
|{{UDnote|step=28}}
| N3, tM3
| N3, tM3
| greater neutral third, tiny major third
| greater neutral third, tiny major third
Line 264: Line 301:
| 99/80, (26/21)
| 99/80, (26/21)
| 21/17
| 21/17
|{{UDnote|step=29}}
| oM3
| oM3
| off major third
| off major third
|  
|
|-
|-
| 30
| 30
| 382.979
| 382.979
| 5/4
| 5/4
|  
|
|{{UDnote|step=30}}
| kM3
| kM3
| classic major third
| classic major third
Line 279: Line 318:
| 395.745
| 395.745
| 121/96, (34/27)
| 121/96, (34/27)
|  
|
|{{UDnote|step=31}}
| rM3
| rM3
| rastmic major third
| rastmic major third
|  
|
|-
|-
| 32
| 32
Line 288: Line 328:
| 81/64, (33/26)
| 81/64, (33/26)
| 19/15, 24/19
| 19/15, 24/19
|{{UDnote|step=32}}
| M3
| M3
| major third
| major third
Line 296: Line 337:
| 14/11
| 14/11
| 23/18
| 23/18
|{{UDnote|step=33}}
| LM3
| LM3
| large major third
| large major third
|  
|
|-
|-
| 34
| 34
| 434.043
| 434.043
| 9/7, (32/25)
| 9/7, (32/25)
|  
|
|{{UDnote|step=34}}
| SM3, (KKd4)
| SM3, (KKd4)
| super major third, (classic diminished fourth)
| super major third, (classic diminished fourth)
Line 312: Line 355:
| 135/104, (35/27)
| 135/104, (35/27)
| 22/17
| 22/17
|{{UDnote|step=35}}
| HM3
| HM3
| hyper major third
| hyper major third
Line 320: Line 364:
| 13/10
| 13/10
| 17/13, 30/23
| 17/13, 30/23
|{{UDnote|step=36}}
| h4
| h4
| hypo fourth
| hypo fourth
|  
|
|-
|-
| 37
| 37
Line 328: Line 373:
| 21/16
| 21/16
| 25/19, 46/35
| 25/19, 46/35
|{{UDnote|step=37}}
| s4
| s4
| sub fourth
| sub fourth
Line 335: Line 381:
| 485.106
| 485.106
| 297/224
| 297/224
|  
|
|{{UDnote|step=38}}
| l4
| l4
| little fourth
| little fourth
|  
|
|-
|-
| 39
| 39
| 497.872
| 497.872
| 4/3
| 4/3
|  
|
|{{UDnote|step=39}}
| P4
| P4
| perfect fourth
| perfect fourth
Line 351: Line 399:
| 510.638
| 510.638
| 162/121, (35/26)
| 162/121, (35/26)
|  
|
|{{UDnote|step=40}}
| R4
| R4
| rastmic fourth
| rastmic fourth
|  
|
|-
|-
| 41
| 41
Line 360: Line 409:
| 27/20
| 27/20
| 19/14, 23/17
| 19/14, 23/17
|{{UDnote|step=41}}
| K4
| K4
| komma-up fourth
| komma-up fourth
Line 368: Line 418:
| 15/11
| 15/11
| 34/25
| 34/25
|{{UDnote|step=42}}
| O4
| O4
| on fourth
| on fourth
|  
|
|-
|-
| 43
| 43
Line 376: Line 427:
| 11/8
| 11/8
| 26/19
| 26/19
|{{UDnote|step=43}}
| U4, T4
| U4, T4
| uber/undecimal fourth, tall fourth
| uber/undecimal fourth, tall fourth
Line 383: Line 435:
| 561.702
| 561.702
| 18/13, (25/18)
| 18/13, (25/18)
|  
|
|{{UDnote|step=44}}
| tA4, uA4, (kkA4)
| tA4, uA4, (kkA4)
| tiny augmented fourth, unter augmented fourth, (classic augmented fourth)
| tiny augmented fourth, unter augmented fourth, (classic augmented fourth)
Line 392: Line 445:
| 88/63
| 88/63
| 32/23, 46/33
| 32/23, 46/33
|{{UDnote|step=45}}
| ld5, oA4
| ld5, oA4
| little diminished fifth, off augmented fourth
| little diminished fifth, off augmented fourth
|  
|
|-
|-
| 46
| 46
Line 400: Line 454:
| 45/32, (7/5)
| 45/32, (7/5)
| 38/27
| 38/27
|{{UDnote|step=46}}
| kA4
| kA4
| komma-down augmented fourth
| komma-down augmented fourth
Line 408: Line 463:
| 363/256, 512/363, (99/70)
| 363/256, 512/363, (99/70)
| 17/12, 24/17
| 17/12, 24/17
|{{UDnote|step=47}}
| rA4, Rd5
| rA4, Rd5
| rastmic augmented fourth, rastmic diminished fifth
| rastmic augmented fourth, rastmic diminished fifth
|  
|
|-
|-
| 48
| 48
Line 416: Line 472:
| 64/45, (10/7)
| 64/45, (10/7)
| 27/19
| 27/19
|{{UDnote|step=48}}
| Kd5
| Kd5
| komma-up diminished fifth
| komma-up diminished fifth
Line 424: Line 481:
| 63/44
| 63/44
| 23/16, 33/23
| 23/16, 33/23
|{{UDnote|step=49}}
| LA4, Od5
| LA4, Od5
| large augmented fourth, off diminished fifth
| large augmented fourth, off diminished fifth
|  
|
|-
|-
| 50
| 50
| 638.298
| 638.298
| 13/9, (36/25)
| 13/9, (36/25)
|  
|
|{{UDnote|step=50}}
| Td5, Ud5, (KKd5)
| Td5, Ud5, (KKd5)
| tall diminished fifth, uber diminished fifth, (classic diminished fifth)
| tall diminished fifth, uber diminished fifth, (classic diminished fifth)
Line 440: Line 499:
| 16/11
| 16/11
| 19/13
| 19/13
|{{UDnote|step=51}}
| u5, t5
| u5, t5
| unter/undecimal fifth, tiny fifth
| unter/undecimal fifth, tiny fifth
Line 448: Line 508:
| 22/15
| 22/15
| 25/17
| 25/17
|{{UDnote|step=52}}
| o5
| o5
| off fifth
| off fifth
|  
|
|-
|-
| 53
| 53
Line 456: Line 517:
| 40/27
| 40/27
| 28/19, 34/23
| 28/19, 34/23
|{{UDnote|step=53}}
| k5
| k5
| komma-down fifth
| komma-down fifth
Line 463: Line 525:
| 689.362
| 689.362
| 121/81, (52/35)
| 121/81, (52/35)
|  
|
|{{UDnote|step=54}}
| r5
| r5
| rastmic fifth
| rastmic fifth
|  
|
|-
|-
| 55
| 55
| 702.128
| 702.128
| 3/2
| 3/2
|  
|
|{{UDnote|step=55}}
| P5
| P5
| perfect fifth
| perfect fifth
Line 479: Line 543:
| 714.894
| 714.894
| 448/297
| 448/297
|  
|
|{{UDnote|step=56}}
| L5
| L5
| large fifth
| large fifth
|  
|
|-
|-
| 57
| 57
Line 488: Line 553:
| 32/21
| 32/21
| 38/25, 35/23
| 38/25, 35/23
|{{UDnote|step=57}}
| S5
| S5
| super fifth
| super fifth
Line 496: Line 562:
| 20/13
| 20/13
| 26/17, 23/15
| 26/17, 23/15
|{{UDnote|step=58}}
| H5
| H5
| hyper fifth
| hyper fifth
|  
|
|-
|-
| 59
| 59
Line 504: Line 571:
| 208/135
| 208/135
| 17/11
| 17/11
|{{UDnote|step=59}}
| hm6
| hm6
| hypo minor sixth
| hypo minor sixth
Line 510: Line 578:
| 60
| 60
| 765.957
| 765.957
| 14/9, (128/75)
| 14/9, (25/16)
|  
|
|{{UDnote|step=60}}
| sm6, (kkA5)
| sm6, (kkA5)
| sub minor sixth, (classic augmented fifth)
| sub minor sixth, (classic augmented fifth)
Line 520: Line 589:
| 11/7
| 11/7
| 36/23
| 36/23
|{{UDnote|step=61}}
| lm6
| lm6
| little minor sixth
| little minor sixth
|  
|
|-
|-
| 62
| 62
Line 528: Line 598:
| 128/81
| 128/81
| 19/12, 30/19
| 19/12, 30/19
|{{UDnote|step=62}}
| m6
| m6
| minor sixth
| minor sixth
Line 536: Line 607:
| 192/121
| 192/121
| 27/17
| 27/17
|{{UDnote|step=63}}
| Rm6
| Rm6
| rastmic minor sixth
| rastmic minor sixth
|  
|
|-
|-
| 64
| 64
| 817.021
| 817.021
| 8/5
| 8/5
|  
|
|{{UDnote|step=64}}
| Km6
| Km6
| classic minor sixth
| classic minor sixth
Line 552: Line 625:
| 160/99, (21/13)
| 160/99, (21/13)
| 34/21
| 34/21
|{{UDnote|step=65}}
| Om6
| Om6
| on minor sixth
| on minor sixth
|  
|
|-
|-
| 66
| 66
| 842.553
| 842.553
| 44/27, 13/8, (81/50, 80/49)
| 44/27, 13/8, (81/50, 80/49)
|  
|
|{{UDnote|step=66}}
| n6, Tm6
| n6, Tm6
| less neutral sixth, tall minor sixth
| less neutral sixth, tall minor sixth
Line 568: Line 643:
| 18/11, 64/39, (400/243, 49/30)
| 18/11, 64/39, (400/243, 49/30)
| 23/14
| 23/14
|{{UDnote|step=67}}
| N6, tM6
| N6, tM6
| greater neutral sixth, tiny minor sixth
| greater neutral sixth, tiny minor sixth
Line 576: Line 652:
| 33/20
| 33/20
| 28/17, 38/23
| 28/17, 38/23
|{{UDnote|step=68}}
| oM6
| oM6
| off major sixth
| off major sixth
|  
|
|-
|-
| 69
| 69
| 880.851
| 880.851
| 5/3
| 5/3
|  
|
|{{UDnote|step=69}}
| kM6
| kM6
| classic major sixth
| classic major sixth
Line 591: Line 669:
| 893.617
| 893.617
| 121/72
| 121/72
|  
|
|{{UDnote|step=70}}
| rM6
| rM6
| rastmic major sixth
| rastmic major sixth
|  
|
|-
|-
| 71
| 71
Line 600: Line 679:
| 27/16, (42/35, 22/13)
| 27/16, (42/35, 22/13)
| 32/19
| 32/19
|{{UDnote|step=71}}
| M6
| M6
| major sixth
| major sixth
Line 608: Line 688:
| 56/33
| 56/33
| 17/10, 46/27
| 17/10, 46/27
|{{UDnote|step=72}}
| LM6
| LM6
| large major sixth
| large major sixth
|  
|
|-
|-
| 73
| 73
| 931.915
| 931.915
| 12/7, 128/75
| 12/7, (128/75)
|  
|
|{{UDnote|step=73}}
| SM6, (KKd7)
| SM6, (KKd7)
| super major sixth (classic diminished seventh)
| super major sixth (classic diminished seventh)
Line 624: Line 706:
| 45/26
| 45/26
| 19/11
| 19/11
|{{UDnote|step=74}}
| HM6
| HM6
| hyper major sixth
| hyper major sixth
|  
|
|-
|-
| 75
| 75
Line 632: Line 715:
| 26/15
| 26/15
| 40/23, 33/19
| 40/23, 33/19
|{{UDnote|step=75}}
| hm7
| hm7
| hypo minor seventh
| hypo minor seventh
|  
|
|-
|-
| 76
| 76
| 970.213
| 970.213
| 7/4
| 7/4
|  
|
|{{UDnote|step=76}}
| sm7
| sm7
| sub minor seventh
| sub minor seventh
Line 648: Line 733:
| 99/56, (44/25)
| 99/56, (44/25)
| 30/17, 23/13
| 30/17, 23/13
|{{UDnote|step=77}}
| lm7
| lm7
| little minor seventh
| little minor seventh
|  
|
|-
|-
| 78
| 78
| 995.745
| 995.745
| 16/9
| 16/9
|  
|
|{{UDnote|step=78}}
| m7
| m7
| minor seventh
| minor seventh
Line 664: Line 751:
| 216/121
| 216/121
| 34/19
| 34/19
|{{UDnote|step=79}}
| Rm7
| Rm7
| rastmic minor seventh
| rastmic minor seventh
|  
|
|-
|-
| 80
| 80
Line 672: Line 760:
| 9/5
| 9/5
| 38/21
| 38/21
|{{UDnote|step=80}}
| Km7
| Km7
| classic minor seventh
| classic minor seventh
Line 679: Line 768:
| 1034.043
| 1034.043
| 20/11
| 20/11
|  
|
|{{UDnote|step=81}}
| Om7
| Om7
| on minor seventh
| on minor seventh
|  
|
|-
|-
| 82
| 82
Line 688: Line 778:
| 11/6, (64/35)
| 11/6, (64/35)
| 42/23
| 42/23
|{{UDnote|step=82}}
| n7, Tm7, hd8
| n7, Tm7, hd8
| less neutral seventh, tall minor seventh, hypo diminished octave
| less neutral seventh, tall minor seventh, hypo diminished octave
Line 696: Line 787:
| 81/44, 24/13, (50/27)
| 81/44, 24/13, (50/27)
| 46/25, 35/19
| 46/25, 35/19
|{{UDnote|step=83}}
| N7, tM7, sd8, (kkM7)
| N7, tM7, sd8, (kkM7)
| greater neutral seventh, tiny major seventh, sub diminished octave, (2-comma down major seventh)
| greater neutral seventh, tiny major seventh, sub diminished octave, (2-comma down major seventh)
|  
|
|-
|-
| 84
| 84
| 1072.340
| 1072.340
| 297/160, 144/91, (13/7)
| 297/160, 144/91, (13/7)
|  
|
|{{UDnote|step=84}}
| oM7, ld8
| oM7, ld8
| off major seventh, little diminished octave
| off major seventh, little diminished octave
|  
|
|-
|-
| 85
| 85
| 1085.106
| 1085.106
| 15/8, (28/15)
| 15/8, (28/15)
|  
|
|{{UDnote|step=85}}
| kM7, d8
| kM7, d8
| classic major seventh, diminished octave
| classic major seventh, diminished octave
Line 720: Line 814:
| 121/64
| 121/64
| 32/17, 17/9
| 32/17, 17/9
|{{UDnote|step=86}}
| rM7, Rd8
| rM7, Rd8
| rastmic major seventh, rastmic diminished octave
| rastmic major seventh, rastmic diminished octave
|  
|
|-
|-
| 87
| 87
Line 728: Line 823:
| 243/128, 256/135, (40/21)
| 243/128, 256/135, (40/21)
| 36/19, 19/10
| 36/19, 19/10
|{{UDnote|step=87}}
| M7, Kd8
| M7, Kd8
| major seventh, komma-up diminished octave
| major seventh, komma-up diminished octave
Line 736: Line 832:
| 21/11, (25/13)
| 21/11, (25/13)
| 44/23, 23/12
| 44/23, 23/12
|{{UDnote|step=88}}
| LM7, Od8
| LM7, Od8
| large major seventh, on diminished octave
| large major seventh, on diminished octave
|  
|
|-
|-
| 89
| 89
| 1136.170
| 1136.170
| 27/14, 52/27, (48/25)
| 27/14, 52/27, (48/25)
|  
|
|{{UDnote|step=89}}
| SM7, Td8, Ud8, (KKd8)
| SM7, Td8, Ud8, (KKd8)
| super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave)
| super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave)
Line 752: Line 850:
| 64/33, (35/18, 68/35, 33/17)
| 64/33, (35/18, 68/35, 33/17)
| 33/17
| 33/17
|{{UDnote|step=90}}
| u8, t8, HM7
| u8, t8, HM7
| unter octave, tiny octave, hyper major seventh
| unter octave, tiny octave, hyper major seventh
|  
|
|-
|-
| 91
| 91
Line 760: Line 859:
| 88/45, 39/20
| 88/45, 39/20
| 45/23
| 45/23
|{{UDnote|step=91}}
| o8, h8
| o8, h8
| off octave, hypo octave
| off octave, hypo octave
|  
|
|-
|-
| 92
| 92
| 1174.468
| 1174.468
| 160/81, 63/32, (49/25)
| 160/81, 63/32, (49/25)
|  
|
|{{UDnote|step=92}}
| k8, s8
| k8, s8
| komma-down octave, sub octave
| komma-down octave, sub octave
|  
|
|-
|-
| 93
| 93
| 1187.234
| 1187.234
| 891/448, 484/243, (486/245, 99/50, 196/99)
| 891/448, 484/243, (486/245, 99/50, 196/99)
|  
|
|{{UDnote|step=93}}
| l8, r8
| l8, r8
| little octave, octave - rastma
| little octave, octave - rastma
|  
|
|-
|-
| 94
| 94
| 1200.000
| 1200.000
| 2/1
| 2/1
|  
|
|{{UDnote|step=94}}
| P8
| P8
| perfect octave
| perfect octave
Line 791: Line 894:
There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context.
There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context.


The perfect fifth has three, or perhaps even five, functional options, each differing by one step. Although in most timbres only the central perfect fifth at 702.128 cents sounds consonant and stable, the lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.
The perfect fifth has three, or perhaps even five, functional options, each differing by one step. The lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.


Every odd-numbered interval can generate the entire tuning of 94edo except for the 600-cent [[tritone]] (47\94), which divides the octave exactly in half.
Every odd-numbered interval can generate the entire tuning of 94edo except for the 600-cent [[tritone]] (47\94), which divides the octave exactly in half.
Line 799: Line 902:
While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys.
While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys.


== Approximation to JI ==
== Notation ==
=== Zeta peak index ===
94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equating to 9 edosteps and the limma to 7 edosteps.
{| class="wikitable center-all"
{| class="wikitable" style="text-align: center;"
!Degree
!−9
!−8
!−7
!−6
!−5
!−4
!−3
!−2
!−1
!0
!+1
!+2
!+3
!+4
!+5
!+6
!+7
!+8
!+9
|-
|-
! colspan="3" | Tuning
!Evo
! colspan="3" | Strength
|{{sagittal|b}}
! colspan="2" | Closest edo
|{{sagittal|b}}{{sagittal|~|(}}
! colspan="2" | Integer limit
|{{sagittal|b}}{{sagittal|/|}}
|{{sagittal|b}}{{sagittal|(|(}}
|{{sagittal|b}}{{sagittal|/|\}}
| rowspan="2" |{{sagittal|\!/}}
| rowspan="2" |{{sagittal|(!(}}
| rowspan="2" |{{sagittal|\!}}
| rowspan="2" |{{sagittal|~!(}}
| rowspan="2" |{{sagittal||//|}}
| rowspan="2" |{{sagittal|~|(}}
| rowspan="2" |{{sagittal|/|}}
| rowspan="2" |{{sagittal|(|(}}
| rowspan="2" |{{sagittal|/|\}}
|{{sagittal|#}}{{sagittal|\!/}}
|{{sagittal|#}}{{sagittal|(!(}}
|{{sagittal|#}}{{sagittal|\!}}
|{{sagittal|#}}{{sagittal|~!(}}
|{{sagittal|#}}
|-
|-
! ZPI
!Revo
! Steps per octave
|{{sagittal|\!!/}}
! Step size (cents)
|{{sagittal|(!!(}}
! Height
|{{sagittal|!!/}}
! Integral
|{{sagittal|~!!(}}
! Gap
|{{sagittal|(!)}}
! Edo
|{{sagittal|(|)}}
! Octave (cents)
|{{sagittal|~||(}}
! Consistent
|{{sagittal|||\}}
! Distinct
|{{sagittal|(||(}}
|-
|{{sagittal|/||\}}
| [[532zpi]]
| 93.9836761074943
| 12.7681747480009
| 8.806201
| 1.394050
| 17.832744
| 94edo
| 1200.20842631208
| 24
| 15
|}
|}


Line 981: Line 1,110:
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit.
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit.


* {{nowrap|46 & 94}} {{multival| 8 30 -18 -4 -28 8 -24 2 … }}
* {{nowrap|46 & 94}}  
* {{nowrap|68 & 94}} {{multival| 20 28 2 -10 24 20 34 52 … }}
* {{nowrap|68 & 94}}  
* {{nowrap|53 & 94}} {{multival| 1 -8 -14 23 20 -46 -3 -35 … }} (one garibaldi)
* {{nowrap|53 & 94}} (one garibaldi)
* {{nowrap|41 & 94}} {{multival| 1 -8 -14 23 20 48 -3 -35 … }} (another garibaldi, only differing in the mappings of 17 and 23)
* {{nowrap|41 & 94}} (another garibaldi, only differing in the mappings of 17 and 23)
* {{nowrap|135 & 94}} {{multival| 1 -8 -14 23 20 48 -3 59 … }} (another garibaldi)
* {{nowrap|135 & 94}} (another garibaldi)
* {{nowrap|130 & 94}} {{multival| 6 -48 10 -50 26 6 -18 -22 … }} (a pogo extension)
* {{nowrap|130 & 94}} (a pogo extension)
* {{nowrap|58 & 94}} {{multival| 6 46 10 44 26 6 -18 -22 … }} (a supers extension)
* {{nowrap|58 & 94}} (a supers extension)
* {{nowrap|50 & 94}} {{multival| 24 -4 40 -12 10 24 22 6 … }}
* {{nowrap|50 & 94}}  
* {{nowrap|72 & 94}} {{multival| 12 -2 20 -6 52 12 -36 -44 … }} (a gizzard extension)
* {{nowrap|72 & 94}} (a gizzard extension)
* {{nowrap|80 & 94}} {{multival| 18 44 30 38 -16 18 40 28 … }}
* {{nowrap|80 & 94}}  
* 94 solo {{multival| 12 -2 20 -6 -42 12 -36 -44 … }} (a rank one temperament!)
* 94 solo (a rank one temperament!)


Temperaments to which 94et can be detempered:  
Temperaments to which 94et can be detempered:  


* [[Satin]] ({{nowrap|94 & 311}}) {{multival| 3 70 -42 69 -34 50 85 83 … }}
* [[Satin]] ({{nowrap|94 & 311}})  
* {{nowrap|94 & 422}} {{multival| 8 124 -18 90 -28 102 164 96 … }}
* {{nowrap|94 & 422}}  


== Scales ==
== Scales ==
Line 1,003: Line 1,132:
* [[Garibaldi12]]
* [[Garibaldi12]]
* [[Garibaldi17]]
* [[Garibaldi17]]
== Instruments ==
94edo can be played on the Lumatone, although due to the sheer number of notes it does require compromises in either the range or gamut:
* [[Lumatone mapping for 94edo]]
One can also use a [[skip fretting]] system:
* [[Skip fretting system 94 7 16]]


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Zx4xbJhXmgc ''microtonal improvisation in 94edo''] (2025)
; [[Cam Taylor]]
; [[Cam Taylor]]
* [https://archive.org/details/41-94edo09sept2017 4 Improvisations Saturday 9th September 2017]
* [https://archive.org/details/41-94edo09sept2017 4 Improvisations Saturday 9th September 2017]
Retrieved from "https://en.xen.wiki/w/94edo"