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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]].


94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.
94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.


===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 ==
{{See also | Table of 94edo intervals }}
{{See also | Table of 94edo 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.  
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"
 
|+ style="font-size: 105%;" | 94edo [[SKULO interval names#WOFED interval names|WOFED interval names]]
{| class="wikitable center-5"
|-
|-
! Step
! Step
Line 31: 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 39: 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 55: 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 79: 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 87: 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 95: 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 110: 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 119: 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 127: 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 134: Line 156:
| 165.957
| 165.957
| 11/10
| 11/10
|  
|
|{{UDnote|step=13}}
| oM2
| oM2
| off major second
| off major second
|  
|
|-
|-
| 14
| 14
Line 143: Line 166:
| 10/9
| 10/9
| 21/19
| 21/19
|{{UDnote|step=14}}
| kM2
| kM2
| komma-down major second
| komma-down major second
Line 151: 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 167: 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 183: 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 191: 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 207: 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 215: 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 222: 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 239: 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 247: 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 254: 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 263: 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 278: 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 287: 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 295: 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 311: 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 319: 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 327: Line 373:
| 21/16
| 21/16
| 25/19, 46/35
| 25/19, 46/35
|{{UDnote|step=37}}
| s4
| s4
| sub fourth
| sub fourth
Line 334: 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 350: 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 359: 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 367: Line 418:
| 15/11
| 15/11
| 34/25
| 34/25
|{{UDnote|step=42}}
| O4
| O4
| on fourth
| on fourth
|  
|
|-
|-
| 43
| 43
Line 375: 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 382: 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 391: 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 399: 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 407: 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 415: 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 423: 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 439: 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 447: Line 508:
| 22/15
| 22/15
| 25/17
| 25/17
|{{UDnote|step=52}}
| o5
| o5
| off fifth
| off fifth
|  
|
|-
|-
| 53
| 53
Line 455: 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 462: 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 478: Line 543:
| 714.894
| 714.894
| 448/297
| 448/297
|  
|
|{{UDnote|step=56}}
| L5
| L5
| large fifth
| large fifth
|  
|
|-
|-
| 57
| 57
Line 487: Line 553:
| 32/21
| 32/21
| 38/25, 35/23
| 38/25, 35/23
|{{UDnote|step=57}}
| S5
| S5
| super fifth
| super fifth
Line 495: 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 503: Line 571:
| 208/135
| 208/135
| 17/11
| 17/11
|{{UDnote|step=59}}
| hm6
| hm6
| hypo minor sixth
| hypo minor sixth
Line 509: 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 519: Line 589:
| 11/7
| 11/7
| 36/23
| 36/23
|{{UDnote|step=61}}
| lm6
| lm6
| little minor sixth
| little minor sixth
|  
|
|-
|-
| 62
| 62
Line 527: Line 598:
| 128/81
| 128/81
| 19/12, 30/19
| 19/12, 30/19
|{{UDnote|step=62}}
| m6
| m6
| minor sixth
| minor sixth
Line 535: 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 551: 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 567: 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 575: 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 590: Line 669:
| 893.617
| 893.617
| 121/72
| 121/72
|  
|
|{{UDnote|step=70}}
| rM6
| rM6
| rastmic major sixth
| rastmic major sixth
|  
|
|-
|-
| 71
| 71
Line 599: 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 607: 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 623: Line 706:
| 45/26
| 45/26
| 19/11
| 19/11
|{{UDnote|step=74}}
| HM6
| HM6
| hyper major sixth
| hyper major sixth
|  
|
|-
|-
| 75
| 75
Line 631: 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 647: 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 663: Line 751:
| 216/121
| 216/121
| 34/19
| 34/19
|{{UDnote|step=79}}
| Rm7
| Rm7
| rastmic minor seventh
| rastmic minor seventh
|  
|
|-
|-
| 80
| 80
Line 671: Line 760:
| 9/5
| 9/5
| 38/21
| 38/21
|{{UDnote|step=80}}
| Km7
| Km7
| classic minor seventh
| classic minor seventh
Line 678: Line 768:
| 1034.043
| 1034.043
| 20/11
| 20/11
|  
|
|{{UDnote|step=81}}
| Om7
| Om7
| on minor seventh
| on minor seventh
|  
|
|-
|-
| 82
| 82
Line 687: 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 695: 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 719: 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 727: 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 735: 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 751: 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 759: 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 790: 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 797: Line 901:


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.
== Notation ==
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" style="text-align: center;"
!Degree
!−9
!−8
!−7
!−6
!−5
!−4
!−3
!−2
!−1
!0
!+1
!+2
!+3
!+4
!+5
!+6
!+7
!+8
!+9
|-
!Evo
|{{sagittal|b}}
|{{sagittal|b}}{{sagittal|~|(}}
|{{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|#}}
|-
!Revo
|{{sagittal|\!!/}}
|{{sagittal|(!!(}}
|{{sagittal|!!/}}
|{{sagittal|~!!(}}
|{{sagittal|(!)}}
|{{sagittal|(|)}}
|{{sagittal|~||(}}
|{{sagittal|||\}}
|{{sagittal|(||(}}
|{{sagittal|/||\}}
|}


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| 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
| 2.3
| {{monzo| 149 -94 }}
| {{monzo| 149 -94 }}
| {{mapping| 94 149 }}
| {{mapping| 94 149 }}
| &minus;0.054
| −0.054
| 0.054
| 0.054
| 0.43
| 0.43
Line 856: Line 1,027:
| 0.637
| 0.637
| 4.99
| 4.99
{{comma basis end}}
|}
* 94et is lower in relative error than any previous equal temperaments in the 23-limit, and the next equal temperament that does better in this subgroup is [[190edo|190g]].  
* 94et is lower in relative error than any previous equal temperaments in the 23-limit, and the next equal temperament that does better in this subgroup is [[190edo|190g]].  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| 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
| 1
Line 927: Line 1,105:
| 11/8
| 11/8
| [[Kleischismic]]
| [[Kleischismic]]
{{rank-2 end}}
|}
{{orf}}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


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.


* 46&amp;94 {{multival| 8 30 -18 -4 -28 8 -24 2 … }}
* {{nowrap|46 &amp; 94}}  
* 68&amp;94 {{multival| 20 28 2 -10 24 20 34 52 … }}
* {{nowrap|68 &amp; 94}}  
* 53&amp;94 {{multival| 1 -8 -14 23 20 -46 -3 -35 … }} (one garibaldi)
* {{nowrap|53 &amp; 94}} (one garibaldi)
* 41&amp;94 {{multival| 1 -8 -14 23 20 48 -3 -35 … }} (another garibaldi, only differing in the mappings of 17 and 23)
* {{nowrap|41 &amp; 94}} (another garibaldi, only differing in the mappings of 17 and 23)
* 135&amp;94 {{multival| 1 -8 -14 23 20 48 -3 59 … }} (another garibaldi)
* {{nowrap|135 &amp; 94}} (another garibaldi)
* 130&amp;94 {{multival| 6 -48 10 -50 26 6 -18 -22 … }} (a pogo extension)
* {{nowrap|130 &amp; 94}} (a pogo extension)
* 58&amp;94 {{multival| 6 46 10 44 26 6 -18 -22 … }} (a supers extension)
* {{nowrap|58 &amp; 94}} (a supers extension)
* 50&amp;94 {{multival| 24 -4 40 -12 10 24 22 6 … }}
* {{nowrap|50 &amp; 94}}  
* 72&amp;94 {{multival| 12 -2 20 -6 52 12 -36 -44 … }} (a gizzard extension)
* {{nowrap|72 &amp; 94}} (a gizzard extension)
* 80&amp;94 {{multival| 18 44 30 38 -16 18 40 28 … }}
* {{nowrap|80 &amp; 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]] (94&amp;311) {{multival| 3 70 -42 69 -34 50 85 83 … }}
* [[Satin]] ({{nowrap|94 & 311}})  
* 94&amp;422 {{multival| 8 124 -18 90 -28 102 164 96 … }}
* {{nowrap|94 & 422}}  


== Scales ==
== Scales ==
Line 954: 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]

Latest revision as of 00:26, 16 August 2025

← 93edo 94edo 95edo →
Prime factorization 2 × 47
Step size 12.766 ¢ 
Fifth 55\94 (702.128 ¢)
(semiconvergent)
Semitones (A1:m2) 9:7 (114.9 ¢ : 89.36 ¢)
Consistency limit 23
Distinct consistency limit 13

94 equal divisions of the octave (abbreviated 94edo or 94ed2), also called 94-tone equal temperament (94tet) or 94 equal temperament (94et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 94 equal parts of about 12.8 ¢ each. Each step represents a frequency ratio of 21/94, or the 94th root of 2.

Theory

94edo is a remarkable all-around utility tuning system, good from low prime limit to very high prime limit situations. It is the first edo to be consistent through the 23-odd-limit, and no other edo is so consistent until 282 and 311 make their appearance.

Its step size is close to that of 144/143, which is consistently represented in this tuning system.

As a tuning of other temperaments

94edo can also be thought of as the "sum" of 41edo and 53edo (41 + 53 = 94), both of which are not only known for their approximation of Pythagorean tuning, but also support a variety of schismatic temperament known as cassandra (which is itself a variety of 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 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.

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.

94edo is an excellent edo for Carlos Beta scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.

Prime harmonics

Approximation of prime harmonics in 94edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.17 -3.33 +1.39 -2.38 +2.03 -2.83 -3.90 -2.74 +4.47 +3.90
Relative (%) +0.0 +1.4 -26.1 +10.9 -18.7 +15.9 -22.2 -30.5 -21.5 +35.0 +30.6
Steps
(reduced) 		94
(0) 		149
(55) 		218
(30) 		264
(76) 		325
(43) 		348
(66) 		384
(8) 		399
(23) 		425
(49) 		457
(81) 		466
(90)

Subsets and supersets

Since 94 factors into primes as 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

See also: Table of 94edo intervals

Assuming 23-limit patent 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.

Step Cents 13-limit 23-limit Ups and downs Short-form WOFED Long-form WOFED Diatonic
0 0 1/1 D
1 12.766 896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98) 85/84 ^D, ^3E♭♭ L1, R1 large unison, rastma
2 25.532 81/80, 64/63, (50/49) ^^D, ^4E♭♭ K1, S1 komma, super unison
3 38.298 45/44, 40/39, (250/243, 49/48) 46/45 ^3D, v4E♭ O1, H1 on unison, hyper unison
4 51.064 33/32, (128/125, 36/35, 35/34, 34/33) ^4D, v3E♭ U1, T1, hm2 uber unison, tall unison, hypo minor second
5 63.830 28/27, 729/704, 27/26, (25/24) v4D♯, vvE♭ sm2, uA1, tA1, (kkA1) sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison) dd3
6 76.596 22/21, (648/625, 26/25) 23/22, 24/23 v3D♯, vE♭ lm2, oA1 little minor second, off augmented unison
7 89.362 256/243, 135/128, (21/20) 19/18, 20/19 vvD♯, E♭ m2, kA1 minor second, komma-down augmented unison m2
8 102.128 128/121, (35/33) 17/16, 18/17 vD♯, ^E♭ Rm2, rA1 rastmic minor second, rastmic augmented unison
9 114.894 16/15, (15/14) D♯, ^^E♭ Km2, A1 classic minor second, augmented unison A1
10 127.660 320/297, 189/176, (14/13) ^D♯, ^3E♭ Om2, LA1 oceanic minor second, large augmented unison
11 140.426 88/81, 13/12, 243/224, (27/25) 25/23, 38/35 ^^D♯, ^4E♭ n2, Tm2, SA1, (KKm2) lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
12 153.191 12/11, (35/32) 23/21 ^3D♯, v4E N2, tM2, HA1 greater netral second, tiny major second, hyper augmented unison ddd4
13 165.957 11/10 ^4D♯, v3E oM2 off major second
14 178.723 10/9 21/19 v4D𝄪, vvE kM2 komma-down major second d3
15 191.489 121/108, (49/44, 39/35) 19/17 v3D𝄪, vE rM2 rastmic major second
16 204.255 9/8 E M2 major second M2
17 217.021 112/99, (25/22) 17/15, 26/23 ^E, ^3F♭ LM2 large major second
18 229.787 8/7 ^^E, ^4F♭ SM2 super major second AA1
19 242.553 15/13 23/20, 38/33 ^3E, v4F HM2 hyper major second
20 255.319 52/45 22/19 ^4E, v3F hm3 hypo minor third
21 268.085 7/6, (75/64) v4E♯, vvF sm3, (kkA2) sub minor third, (classic augmented second) dd4
22 280.851 33/28 20/17, 27/23 v3E♯, vF lm3 little minor third
23 293.617 32/27, (25/21, 13/11) 19/16 F m3 minor third m3
24 306.383 144/121, (81/70) ^F, ^3G♭♭ Rm3 rastmic minor third
25 319.149 6/5 ^^F, ^4G♭♭ Km3 classic minor third A2
26 331.915 40/33 17/14, 23/19 ^3F, v4G♭ Om3 on minor third
27 344.681 11/9, 39/32, (243/200, 60/49) 28/23 ^4F, v3G♭ n3, Tm3 lesser neutral third, tall minor third AAA1
28 357.447 27/22, 16/13, (100/81,49/40) v4F♯, vvG♭ N3, tM3 greater neutral third, tiny major third ddd5
29 370.213 99/80, (26/21) 21/17 v3F♯, vG♭ oM3 off major third
30 382.979 5/4 vvF♯, G♭ kM3 classic major third d4
31 395.745 121/96, (34/27) vF♯, ^G♭ rM3 rastmic major third
32 408.511 81/64, (33/26) 19/15, 24/19 F♯, ^^G♭ M3 major third M3
33 421.277 14/11 23/18 ^F♯, ^3G♭ LM3 large major third
34 434.043 9/7, (32/25) ^^F♯, ^4G♭ SM3, (KKd4) super major third, (classic diminished fourth) AA2
35 446.809 135/104, (35/27) 22/17 ^3F♯, v4G HM3 hyper major third ddd6
36 459.574 13/10 17/13, 30/23 ^4F♯, v3G h4 hypo fourth
37 472.340 21/16 25/19, 46/35 v4F𝄪, vvG s4 sub fourth dd5
38 485.106 297/224 v3F𝄪, vG l4 little fourth
39 497.872 4/3 G P4 perfect fourth P4
40 510.638 162/121, (35/26) ^G, ^3A♭♭ R4 rastmic fourth
41 523.404 27/20 19/14, 23/17 ^^G, ^4A♭♭ K4 komma-up fourth A3
42 536.170 15/11 34/25 ^3G, v4A♭ O4 on fourth
43 548.936 11/8 26/19 ^4G, v3A♭ U4, T4 uber/undecimal fourth, tall fourth AAA2
44 561.702 18/13, (25/18) v4G♯, vvA♭ tA4, uA4, (kkA4) tiny augmented fourth, unter augmented fourth, (classic augmented fourth) dd6
45 574.468 88/63 32/23, 46/33 v3G♯, vA♭ ld5, oA4 little diminished fifth, off augmented fourth
46 587.234 45/32, (7/5) 38/27 vvG♯, A♭ kA4 komma-down augmented fourth d5
47 600.000 363/256, 512/363, (99/70) 17/12, 24/17 vG♯, ^A♭ rA4, Rd5 rastmic augmented fourth, rastmic diminished fifth
48 612.766 64/45, (10/7) 27/19 G♯, ^^A♭ Kd5 komma-up diminished fifth A4
49 625.532 63/44 23/16, 33/23 ^G♯, ^3A♭ LA4, Od5 large augmented fourth, off diminished fifth
50 638.298 13/9, (36/25) ^^G♯, ^4A♭ Td5, Ud5, (KKd5) tall diminished fifth, uber diminished fifth, (classic diminished fifth) AA3
51 651.064 16/11 19/13 ^3G♯, v4A u5, t5 unter/undecimal fifth, tiny fifth ddd7
52 663.830 22/15 25/17 ^4G♯, v3A o5 off fifth
53 676.596 40/27 28/19, 34/23 v4G𝄪, vvA k5 komma-down fifth d6
54 689.362 121/81, (52/35) v3G𝄪, vA r5 rastmic fifth
55 702.128 3/2 A P5 perfect fifth P5
56 714.894 448/297 ^A, ^3B♭♭ L5 large fifth
57 727.660 32/21 38/25, 35/23 ^^A, ^4B♭♭ S5 super fifth AA4
58 740.426 20/13 26/17, 23/15 ^3A, v4B♭ H5 hyper fifth
59 753.191 208/135 17/11 ^4A, v3B♭ hm6 hypo minor sixth AAA3
60 765.957 14/9, (25/16) v4A♯, vvB♭ sm6, (kkA5) sub minor sixth, (classic augmented fifth) dd7
61 778.723 11/7 36/23 v3A♯, vB♭ lm6 little minor sixth
62 791.489 128/81 19/12, 30/19 vvA♯, B♭ m6 minor sixth m6
63 804.255 192/121 27/17 vA♯, ^B♭ Rm6 rastmic minor sixth
64 817.021 8/5 A♯, ^^B♭ Km6 classic minor sixth A5
65 829.787 160/99, (21/13) 34/21 ^A♯, ^3B♭ Om6 on minor sixth
66 842.553 44/27, 13/8, (81/50, 80/49) ^^A♯, ^4B♭ n6, Tm6 less neutral sixth, tall minor sixth AAA4
67 855.319 18/11, 64/39, (400/243, 49/30) 23/14 ^3A♯, v4B N6, tM6 greater neutral sixth, tiny minor sixth ddd8
68 868.085 33/20 28/17, 38/23 ^4A♯, v3B oM6 off major sixth
69 880.851 5/3 v4A𝄪, vvB kM6 classic major sixth d7
70 893.617 121/72 v3A𝄪, vB rM6 rastmic major sixth
71 906.383 27/16, (42/35, 22/13) 32/19 B M6 major sixth M6
72 919.149 56/33 17/10, 46/27 ^B, ^3C♭ LM6 large major sixth
73 931.915 12/7, (128/75) ^^B, ^4C♭ SM6, (KKd7) super major sixth (classic diminished seventh) AA5
74 944.681 45/26 19/11 ^3B, v4C HM6 hyper major sixth
75 957.447 26/15 40/23, 33/19 ^4B, v3C hm7 hypo minor seventh
76 970.213 7/4 v4B♯, vvC sm7 sub minor seventh dd8
77 982.979 99/56, (44/25) 30/17, 23/13 v3B♯, vC lm7 little minor seventh
78 995.745 16/9 C m7 minor seventh m7
79 1008.511 216/121 34/19 ^C, ^3D♭♭ Rm7 rastmic minor seventh
80 1021.277 9/5 38/21 ^^C, ^4D♭♭ Km7 classic minor seventh A6
81 1034.043 20/11 ^3C, v4D♭ Om7 on minor seventh
82 1046.809 11/6, (64/35) 42/23 ^4C, v3D♭ n7, Tm7, hd8 less neutral seventh, tall minor seventh, hypo diminished octave AAA5
83 1059.574 81/44, 24/13, (50/27) 46/25, 35/19 v4C♯, vvD♭ N7, tM7, sd8, (kkM7) greater neutral seventh, tiny major seventh, sub diminished octave, (2-comma down major seventh)
84 1072.340 297/160, 144/91, (13/7) v3C♯, vD♭ oM7, ld8 off major seventh, little diminished octave
85 1085.106 15/8, (28/15) vvC♯, D♭ kM7, d8 classic major seventh, diminished octave d8
86 1097.872 121/64 32/17, 17/9 vC♯, ^D♭ rM7, Rd8 rastmic major seventh, rastmic diminished octave
87 1110.638 243/128, 256/135, (40/21) 36/19, 19/10 C♯, ^^D♭ M7, Kd8 major seventh, komma-up diminished octave M7
88 1123.404 21/11, (25/13) 44/23, 23/12 ^C♯, ^3D♭ LM7, Od8 large major seventh, on diminished octave
89 1136.170 27/14, 52/27, (48/25) ^^C♯, ^4D♭ SM7, Td8, Ud8, (KKd8) super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave) AA6
90 1148.936 64/33, (35/18, 68/35, 33/17) 33/17 ^3C♯, v4D u8, t8, HM7 unter octave, tiny octave, hyper major seventh
91 1161.702 88/45, 39/20 45/23 ^4C♯, v3D o8, h8 off octave, hypo octave
92 1174.468 160/81, 63/32, (49/25) v4C𝄪, vvD k8, s8 komma-down octave, sub octave
93 1187.234 891/448, 484/243, (486/245, 99/50, 196/99) v3C𝄪, vD l8, r8 little octave, octave - rastma
94 1200.000 2/1 D P8 perfect octave P8

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. 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.

The regular major second divisible into 16 equal parts can be helpful for realising some of the subtle tunings of Ancient Greek tetrachordal theory, Indian raga and Turkish maqam, though it has not been used historically as a division in those musical cultures.

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.

Notation

94edo can be notated in Sagittal using the Athenian extension, with the apotome equating to 9 edosteps and the limma to 7 edosteps.

Degree −9 −8 −7 −6 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9
Evo ⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠⁠ ⁠ ⁠ ⁠
Revo ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠ ⁠

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [149 -94⟩ [⟨94 149]] −0.054 0.054 0.43
2.3.5 32805/32768, 9765625/9565938 [⟨94 149 218]] +0.442 0.704 5.52
2.3.5.7 225/224, 3125/3087, 118098/117649 [⟨94 149 218 264]] +0.208 0.732 5.74
2.3.5.7.11 225/224, 385/384, 1331/1323, 2200/2187 [⟨94 149 218 264 325]] +0.304 0.683 5.35
2.3.5.7.11.13 225/224, 275/273, 325/324, 385/384, 1331/1323 [⟨94 149 218 264 325 348]] +0.162 0.699 5.48
2.3.5.7.11.13.17 170/169, 225/224, 275/273, 289/288, 325/324, 385/384 [⟨94 149 218 264 325 348 384]] +0.238 0.674 5.28
2.3.5.7.11.13.17.19 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384 [⟨94 149 218 264 325 348 384 399]] +0.323 0.669 5.24
2.3.5.7.11.13.17.19.23 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322 [⟨94 149 218 264 325 348 384 399 425]] +0.354 0.637 4.99
  • 94et is lower in relative error than any previous equal temperaments in the 23-limit, and the next equal temperament that does better in this subgroup is 190g.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 3\94 38.30 49/48 Slender
1 5\94 63.83 25/24 Betic
1 11\94 140.43 243/224 Tsaharuk / quanic
1 13\94 165.96 11/10 Tertiaschis
1 19\94 242.55 147/128 Septiquarter
1 39\94 497.87 4/3 Garibaldi / cassandra
2 2\94 25.53 64/63 Ketchup
2 11\94 140.43 27/25 Fifive
2 30\94 382.98 5/4 Wizard / gizzard
2 34\94 434.04 9/7 Pogo / supers
2 43\94 548.94 11/8 Kleischismic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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.

  • 46 & 94
  • 68 & 94
  • 53 & 94 (one garibaldi)
  • 41 & 94 (another garibaldi, only differing in the mappings of 17 and 23)
  • 135 & 94 (another garibaldi)
  • 130 & 94 (a pogo extension)
  • 58 & 94 (a supers extension)
  • 50 & 94
  • 72 & 94 (a gizzard extension)
  • 80 & 94
  • 94 solo (a rank one temperament!)

Temperaments to which 94et can be detempered:

  • Satin (94 & 311)
  • 94 & 422

Scales

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:

One can also use a skip fretting system:

Music

Bryan Deister
Cam Taylor
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