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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 47
{{ED intro}}
| Step size = 12.766¢
| Fifth = 55\94 = 702.128¢
| Major 2nd = 16\94 = 204¢
| Minor 2nd = 7\94 = 89¢
| Augmented 1sn = 9\94 = 115¢
}}
The '''94 equal divisions of the octave''' ('''94edo'''), or the '''94(-tone) equal temperament''' ('''94tet''', '''94et''') when viewed from a [[regular temperament]] perspective, results from dividing the [[octave]] into 94 [[equal]]ly-sized steps, where each step is 12.766 [[cent]]s.


== Theory ==
== 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 [[282edo|282]] and [[311edo|311]] make their appearance.
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 [[282edo|282]] and [[311edo|311]] make their appearance.


The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].
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]] {{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.
 
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 ===
=== Prime harmonics ===
{{Primes in edo|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 ==
{{main| 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.
 
{| class="wikitable  center-5"
|-
! Step
! Cents
! 13-limit
! 23-limit
![[Ups and downs notation|Ups and downs]]
! Short-form [[SKULO interval names#WOFED interval names|WOFED]]
! Long-form WOFED
! Diatonic
|-
|0
|0
|1/1
|
|{{UDnote|step=0}}
|
|
|
|-
| 1
| 12.766
| 896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98)
| 85/84
|{{UDnote|step=1}}
| L1, R1
| large unison, rastma
|
|-
| 2
| 25.532
| 81/80, 64/63, (50/49)
|
|{{UDnote|step=2}}
| K1, S1
| komma, super unison
|
|-
| 3
| 38.298
| 45/44, 40/39, (250/243, 49/48)
| 46/45
|{{UDnote|step=3}}
| O1, H1
| on unison, hyper unison
|
|-
| 4
| 51.064
| 33/32, (128/125, 36/35, 35/34, 34/33)
|
|{{UDnote|step=4}}
| U1, T1, hm2
| uber unison, tall unison, hypo minor second
|
|-
| 5
| 63.830
| 28/27, 729/704, 27/26, (25/24)
|
|{{UDnote|step=5}}
| 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
|{{UDnote|step=6}}
| lm2, oA1
| little minor second, off augmented unison
|
|-
| 7
| 89.362
| 256/243, 135/128, (21/20)
| 19/18, 20/19
|{{UDnote|step=7}}
| m2, kA1
| minor second, komma-down augmented unison
| m2
|-
| 8
| 102.128
| 128/121, (35/33)
| 17/16, 18/17
|{{UDnote|step=8}}
| Rm2, rA1
| rastmic minor second, rastmic augmented unison
|
|-
| 9
| 114.894
| 16/15, (15/14)
|
|{{UDnote|step=9}}
| Km2, A1
| classic minor second, augmented unison
| A1
|-
| 10
| 127.660
| 320/297, 189/176, (14/13)
|
|{{UDnote|step=10}}
| Om2, LA1
| oceanic minor second, large augmented unison
|
|-
| 11
| 140.426
| 88/81, 13/12, 243/224, (27/25)
| 25/23, 38/35
|{{UDnote|step=11}}
| 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
|{{UDnote|step=12}}
| N2, tM2, HA1
| greater netral second, tiny major second, hyper augmented unison
| ddd4
|-
| 13
| 165.957
| 11/10
|
|{{UDnote|step=13}}
| oM2
| off major second
|
|-
| 14
| 178.723
| 10/9
| 21/19
|{{UDnote|step=14}}
| kM2
| komma-down major second
| d3
|-
| 15
| 191.489
| 121/108, (49/44, 39/35)
| 19/17
|{{UDnote|step=15}}
| rM2
| rastmic major second
|
|-
| 16
| 204.255
| 9/8
|
|{{UDnote|step=16}}
| M2
| major second
| M2
|-
| 17
| 217.021
| 112/99, (25/22)
| 17/15, 26/23
|{{UDnote|step=17}}
| LM2
| large major second
|
|-
| 18
| 229.787
| 8/7
|
|{{UDnote|step=18}}
| SM2
| super major second
| AA1
|-
| 19
| 242.553
| 15/13
| 23/20, 38/33
|{{UDnote|step=19}}
| HM2
| hyper major second
|
|-
| 20
| 255.319
| 52/45
| 22/19
|{{UDnote|step=20}}
| hm3
| hypo minor third
|
|-
| 21
| 268.085
| 7/6, (75/64)
|
|{{UDnote|step=21}}
| sm3, (kkA2)
| sub minor third, (classic augmented second)
| dd4
|-
| 22
| 280.851
| 33/28
| 20/17, 27/23
|{{UDnote|step=22}}
| lm3
| little minor third
|
|-
| 23
| 293.617
| 32/27, (25/21, 13/11)
| 19/16
|{{UDnote|step=23}}
| m3
| minor third
| m3
|-
| 24
| 306.383
| 144/121, (81/70)
|
|{{UDnote|step=24}}
| Rm3
| rastmic minor third
|
|-
| 25
| 319.149
| 6/5
|
|{{UDnote|step=25}}
| Km3
| classic minor third
| A2
|-
| 26
| 331.915
| 40/33
| 17/14, 23/19
|{{UDnote|step=26}}
| Om3
| on minor third
|
|-
| 27
| 344.681
| 11/9, 39/32, (243/200, 60/49)
| 28/23
|{{UDnote|step=27}}
| n3, Tm3
| lesser neutral third, tall minor third
| AAA1
|-
| 28
| 357.447
| 27/22, 16/13, (100/81,49/40)
|
|{{UDnote|step=28}}
| N3, tM3
| greater neutral third, tiny major third
| ddd5
|-
| 29
| 370.213
| 99/80, (26/21)
| 21/17
|{{UDnote|step=29}}
| oM3
| off major third
|
|-
| 30
| 382.979
| 5/4
|
|{{UDnote|step=30}}
| kM3
| classic major third
| d4
|-
| 31
| 395.745
| 121/96, (34/27)
|
|{{UDnote|step=31}}
| rM3
| rastmic major third
|
|-
| 32
| 408.511
| 81/64, (33/26)
| 19/15, 24/19
|{{UDnote|step=32}}
| M3
| major third
| M3
|-
| 33
| 421.277
| 14/11
| 23/18
|{{UDnote|step=33}}
| LM3
| large major third
|
|-
| 34
| 434.043
| 9/7, (32/25)
|
|{{UDnote|step=34}}
| SM3, (KKd4)
| super major third, (classic diminished fourth)
| AA2
|-
| 35
| 446.809
| 135/104, (35/27)
| 22/17
|{{UDnote|step=35}}
| HM3
| hyper major third
| ddd6
|-
| 36
| 459.574
| 13/10
| 17/13, 30/23
|{{UDnote|step=36}}
| h4
| hypo fourth
|
|-
| 37
| 472.340
| 21/16
| 25/19, 46/35
|{{UDnote|step=37}}
| s4
| sub fourth
| dd5
|-
| 38
| 485.106
| 297/224
|
|{{UDnote|step=38}}
| l4
| little fourth
|
|-
| 39
| 497.872
| 4/3
|
|{{UDnote|step=39}}
| P4
| perfect fourth
| P4
|-
| 40
| 510.638
| 162/121, (35/26)
|
|{{UDnote|step=40}}
| R4
| rastmic fourth
|
|-
| 41
| 523.404
| 27/20
| 19/14, 23/17
|{{UDnote|step=41}}
| K4
| komma-up fourth
| A3
|-
| 42
| 536.170
| 15/11
| 34/25
|{{UDnote|step=42}}
| O4
| on fourth
|
|-
| 43
| 548.936
| 11/8
| 26/19
|{{UDnote|step=43}}
| U4, T4
| uber/undecimal fourth, tall fourth
| AAA2
|-
| 44
| 561.702
| 18/13, (25/18)
|
|{{UDnote|step=44}}
| tA4, uA4, (kkA4)
| tiny augmented fourth, unter augmented fourth, (classic augmented fourth)
| dd6
|-
| 45
| 574.468
| 88/63
| 32/23, 46/33
|{{UDnote|step=45}}
| ld5, oA4
| little diminished fifth, off augmented fourth
|
|-
| 46
| 587.234
| 45/32, (7/5)
| 38/27
|{{UDnote|step=46}}
| kA4
| komma-down augmented fourth
| d5
|-
| 47
| 600.000
| 363/256, 512/363, (99/70)
| 17/12, 24/17
|{{UDnote|step=47}}
| rA4, Rd5
| rastmic augmented fourth, rastmic diminished fifth
|
|-
| 48
| 612.766
| 64/45, (10/7)
| 27/19
|{{UDnote|step=48}}
| Kd5
| komma-up diminished fifth
| A4
|-
| 49
| 625.532
| 63/44
| 23/16, 33/23
|{{UDnote|step=49}}
| LA4, Od5
| large augmented fourth, off diminished fifth
|
|-
| 50
| 638.298
| 13/9, (36/25)
|
|{{UDnote|step=50}}
| Td5, Ud5, (KKd5)
| tall diminished fifth, uber diminished fifth, (classic diminished fifth)
| AA3
|-
| 51
| 651.064
| 16/11
| 19/13
|{{UDnote|step=51}}
| u5, t5
| unter/undecimal fifth, tiny fifth
| ddd7
|-
| 52
| 663.830
| 22/15
| 25/17
|{{UDnote|step=52}}
| o5
| off fifth
|
|-
| 53
| 676.596
| 40/27
| 28/19, 34/23
|{{UDnote|step=53}}
| k5
| komma-down fifth
| d6
|-
| 54
| 689.362
| 121/81, (52/35)
|
|{{UDnote|step=54}}
| r5
| rastmic fifth
|
|-
| 55
| 702.128
| 3/2
|
|{{UDnote|step=55}}
| P5
| perfect fifth
| P5
|-
| 56
| 714.894
| 448/297
|
|{{UDnote|step=56}}
| L5
| large fifth
|
|-
| 57
| 727.660
| 32/21
| 38/25, 35/23
|{{UDnote|step=57}}
| S5
| super fifth
| AA4
|-
| 58
| 740.426
| 20/13
| 26/17, 23/15
|{{UDnote|step=58}}
| H5
| hyper fifth
|
|-
| 59
| 753.191
| 208/135
| 17/11
|{{UDnote|step=59}}
| hm6
| hypo minor sixth
| AAA3
|-
| 60
| 765.957
| 14/9, (25/16)
|
|{{UDnote|step=60}}
| sm6, (kkA5)
| sub minor sixth, (classic augmented fifth)
| dd7
|-
| 61
| 778.723
| 11/7
| 36/23
|{{UDnote|step=61}}
| lm6
| little minor sixth
|
|-
| 62
| 791.489
| 128/81
| 19/12, 30/19
|{{UDnote|step=62}}
| m6
| minor sixth
| m6
|-
| 63
| 804.255
| 192/121
| 27/17
|{{UDnote|step=63}}
| Rm6
| rastmic minor sixth
|
|-
| 64
| 817.021
| 8/5
|
|{{UDnote|step=64}}
| Km6
| classic minor sixth
| A5
|-
| 65
| 829.787
| 160/99, (21/13)
| 34/21
|{{UDnote|step=65}}
| Om6
| on minor sixth
|
|-
| 66
| 842.553
| 44/27, 13/8, (81/50, 80/49)
|
|{{UDnote|step=66}}
| n6, Tm6
| less neutral sixth, tall minor sixth
| AAA4
|-
| 67
| 855.319
| 18/11, 64/39, (400/243, 49/30)
| 23/14
|{{UDnote|step=67}}
| N6, tM6
| greater neutral sixth, tiny minor sixth
| ddd8
|-
| 68
| 868.085
| 33/20
| 28/17, 38/23
|{{UDnote|step=68}}
| oM6
| off major sixth
|
|-
| 69
| 880.851
| 5/3
|
|{{UDnote|step=69}}
| kM6
| classic major sixth
| d7
|-
| 70
| 893.617
| 121/72
|
|{{UDnote|step=70}}
| rM6
| rastmic major sixth
|
|-
| 71
| 906.383
| 27/16, (42/35, 22/13)
| 32/19
|{{UDnote|step=71}}
| M6
| major sixth
| M6
|-
| 72
| 919.149
| 56/33
| 17/10, 46/27
|{{UDnote|step=72}}
| LM6
| large major sixth
|
|-
| 73
| 931.915
| 12/7, (128/75)
|
|{{UDnote|step=73}}
| SM6, (KKd7)
| super major sixth (classic diminished seventh)
| AA5
|-
| 74
| 944.681
| 45/26
| 19/11
|{{UDnote|step=74}}
| HM6
| hyper major sixth
|
|-
| 75
| 957.447
| 26/15
| 40/23, 33/19
|{{UDnote|step=75}}
| hm7
| hypo minor seventh
|
|-
| 76
| 970.213
| 7/4
|
|{{UDnote|step=76}}
| sm7
| sub minor seventh
| dd8
|-
| 77
| 982.979
| 99/56, (44/25)
| 30/17, 23/13
|{{UDnote|step=77}}
| lm7
| little minor seventh
|
|-
| 78
| 995.745
| 16/9
|
|{{UDnote|step=78}}
| m7
| minor seventh
| m7
|-
| 79
| 1008.511
| 216/121
| 34/19
|{{UDnote|step=79}}
| Rm7
| rastmic minor seventh
|
|-
| 80
| 1021.277
| 9/5
| 38/21
|{{UDnote|step=80}}
| Km7
| classic minor seventh
| A6
|-
| 81
| 1034.043
| 20/11
|
|{{UDnote|step=81}}
| Om7
| on minor seventh
|
|-
| 82
| 1046.809
| 11/6, (64/35)
| 42/23
|{{UDnote|step=82}}
| 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
|{{UDnote|step=83}}
| 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)
|
|{{UDnote|step=84}}
| oM7, ld8
| off major seventh, little diminished octave
|
|-
| 85
| 1085.106
| 15/8, (28/15)
|
|{{UDnote|step=85}}
| kM7, d8
| classic major seventh, diminished octave
| d8
|-
| 86
| 1097.872
| 121/64
| 32/17, 17/9
|{{UDnote|step=86}}
| rM7, Rd8
| rastmic major seventh, rastmic diminished octave
|
|-
| 87
| 1110.638
| 243/128, 256/135, (40/21)
| 36/19, 19/10
|{{UDnote|step=87}}
| M7, Kd8
| major seventh, komma-up diminished octave
| M7
|-
| 88
| 1123.404
| 21/11, (25/13)
| 44/23, 23/12
|{{UDnote|step=88}}
| LM7, Od8
| large major seventh, on diminished octave
|
|-
| 89
| 1136.170
| 27/14, 52/27, (48/25)
|
|{{UDnote|step=89}}
| 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
|{{UDnote|step=90}}
| u8, t8, HM7
| unter octave, tiny octave, hyper major seventh
|
|-
| 91
| 1161.702
| 88/45, 39/20
| 45/23
|{{UDnote|step=91}}
| o8, h8
| off octave, hypo octave
|
|-
| 92
| 1174.468
| 160/81, 63/32, (49/25)
|
|{{UDnote|step=92}}
| k8, s8
| komma-down octave, sub octave
|
|-
| 93
| 1187.234
| 891/448, 484/243, (486/245, 99/50, 196/99)
|
|{{UDnote|step=93}}
| l8, r8
| little octave, octave - rastma
|
|-
| 94
| 1200.000
| 2/1
|
|{{UDnote|step=94}}
| 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 [[tetrachord]]al 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 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 ==
{| 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]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 33: Line 974:
| 2.3
| 2.3
| {{monzo| 149 -94 }}
| {{monzo| 149 -94 }}
| [{{val| 94 149 }}]
| {{mapping| 94 149 }}
| -0.054
| −0.054
| 0.054
| 0.054
| 0.43
| 0.43
Line 40: Line 981:
| 2.3.5
| 2.3.5
| 32805/32768, 9765625/9565938
| 32805/32768, 9765625/9565938
| [{{val| 94 149 218 }}]
| {{mapping| 94 149 218 }}
| +0.442
| +0.442
| 0.704
| 0.704
Line 47: Line 988:
| 2.3.5.7
| 2.3.5.7
| 225/224, 3125/3087, 118098/117649
| 225/224, 3125/3087, 118098/117649
| [{{val| 94 149 218 264 }}]
| {{mapping| 94 149 218 264 }}
| +0.208
| +0.208
| 0.732
| 0.732
Line 54: Line 995:
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 2200/2187
| 225/224, 385/384, 1331/1323, 2200/2187
| [{{val| 94 149 218 264 325 }}]
| {{mapping| 94 149 218 264 325 }}
| +0.304
| +0.304
| 0.683
| 0.683
Line 61: Line 1,002:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 275/273, 325/324, 385/384, 1331/1323
| 225/224, 275/273, 325/324, 385/384, 1331/1323
| [{{val| 94 149 218 264 325 348 }}]
| {{mapping| 94 149 218 264 325 348 }}
| +0.162
| +0.162
| 0.699
| 0.699
Line 68: Line 1,009:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 170/169, 225/224, 275/273, 289/288, 325/324, 385/384
| 170/169, 225/224, 275/273, 289/288, 325/324, 385/384
| [{{val| 94 149 218 264 325 348 384 }}]
| {{mapping| 94 149 218 264 325 348 384 }}
| +0.238
| +0.238
| 0.674
| 0.674
Line 75: Line 1,016:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384
| 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384
| [{{val| 94 149 218 264 325 348 384 399 }}]
| {{mapping| 94 149 218 264 325 348 384 399 }}
| +0.323
| +0.323
| 0.669
| 0.669
Line 82: Line 1,023:
| 2.3.5.7.11.13.17.19.23
| 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
| 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322
| [{{val| 94 149 218 264 325 348 384 399 425 }}]
| {{mapping| 94 149 218 264 325 348 384 399 425 }}
| +0.354
| +0.354
| 0.637
| 0.637
| 4.99
| 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 [[190edo|190g]].  
94et is lower in relative error than any previous equal temperaments in the 23-limit, and the next ET that does better in this subgroup is 193.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
 
{| class="wikitable center-all left-5"
{| class="wikitable center-all right-3 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator
! Periods<br>per 8ve
! Cents
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
| 1
| 1
| 3\94
| 3\94
| 38.298
| 38.30
| 49/48
| 49/48
| [[Slender]]
| [[Slender]]
Line 107: Line 1,048:
| 1
| 1
| 5\94
| 5\94
| 63.830
| 63.83
| 25/24
| 25/24
| [[Sycamore]] / [[betic]]
| [[Betic]]
|-
|-
| 1
| 1
| 11\94
| 11\94
| 140.426
| 140.43
| 243/224
| 243/224
| [[Tsaharuk]] / [[quanic]]
| [[Tsaharuk]] / [[quanic]]
|-
| 1
| 13\94
| 165.96
| 11/10
| [[Tertiaschis]]
|-
|-
| 1
| 1
| 19\94
| 19\94
| 242.553
| 242.55
| 147/128
| 147/128
| [[Septiquarter]]
| [[Septiquarter]]
Line 125: Line 1,072:
| 1
| 1
| 39\94
| 39\94
| 497.872
| 497.87
| 4/3
| 4/3
| [[Helmholtz]] / [[garibaldi]] / [[cassandra]]
| [[Garibaldi]] / [[cassandra]]
|-
|-
| 2
| 2
| 2\94
| 2\94
| 25.532
| 25.53
| 64/63
| 64/63
| [[Ketchup]]
| [[Ketchup]]
Line 137: Line 1,084:
| 2
| 2
| 11\94
| 11\94
| 140.426
| 140.43
| 27/25
| 27/25
| [[Fifive]]
| [[Fifive]]
Line 143: Line 1,090:
| 2
| 2
| 30\94
| 30\94
| 382.979
| 382.98
| 5/4
| 5/4
| [[Wizard]] / [[gizzard]]
| [[Wizard]] / [[gizzard]]
Line 149: Line 1,096:
| 2
| 2
| 34\94
| 34\94
| 434.043
| 434.04
| 9/7
| 9/7
| [[Pogo]] / [[supers]]
| [[Pogo]] / [[supers]]
Line 155: Line 1,102:
| 2
| 2
| 43\94
| 43\94
| 548.936
| 548.94
| 11/8
| 11/8
| [[Kleischismic]]
| [[Kleischismic]]
|}
|}
<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 ==
* [[Garibaldi5]]
* [[Garibaldi7]]
* [[Garibaldi12]]
* [[Garibaldi12]]
* [[Garibaldi17]]
* [[Garibaldi17]]


Since 94edo has a step of 12.766 cents, it also allows one to use its MOS scales as circulating temperaments and is the first edo to allows one to use a Mohajira, Pajara or Miracle MOS scale a as circulating temperament{{clarify}}.
== Instruments ==
{| class="wikitable"
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:  
|+Circulating temperaments in 94edo
* [[Lumatone mapping for 94edo]]
!Tones
 
!Pattern
One can also use a [[skip fretting]] system:
!L:s
* [[Skip fretting system 94 7 16]]
|-
 
|5
== Music ==
|[[4L 1s]]
; [[Bryan Deister]]
|19:18
* [https://www.youtube.com/shorts/Zx4xbJhXmgc ''microtonal improvisation in 94edo''] (2025)
|-
 
|6
; [[Cam Taylor]]
|[[4L 2s]]
* [https://archive.org/details/41-94edo09sept2017 4 Improvisations Saturday 9th September 2017]
|16:15
* [https://archive.org/details/4194EDOBosanquetAxis18thAug20181FeelingSadButWarmingUp Feeling Sad But Warming Up (in 2 parts)]
|-
* [https://archive.org/details/4191edoPlayingWithThe13Limit Playing with the 13-limit]
|7
 
|[[3L 4s]]
|14:13
|-
|8
|[[6L 2s]]
|12:11
|-
|9
|[[4L 5s]]
|11:10
|-
|10
|[[4L 6s]]
|10:9
|-
|11
|[[6L 5s]]
|9:8
|-
|12
|[[10L 2s]]
| rowspan="2" |8:7
|-
|13
|[[3L 10s]]
|-
|14
|[[10L 4s]]
| rowspan="2" |7:6
|-
|15
|[[4L 11s]]
|-
|16
|14L 2s
| rowspan="3" |6:5
|-
|17
|[[9L 8s]]
|-
|18
|4L 14s
|-
|19
|18L 1s
| rowspan="5" |5:4
|-
|20
|14L 6s
|-
|21
|[[10L 11s]]
|-
|22
|6L 16s
|-
|23
|2L 21s
|-
|24
|22L 2s
| rowspan="8" |4:3
|-
|25
|19L 6s
|-
|26
|16L 10s
|-
|27
|13L 14s
|-
|28
|10L 18s
|-
|29
|7L 22s
|-
|30
|4L 22s
|-
|31
|1L 30s
|-
|32
|30L 2s
| rowspan="15" |3:2
|-
|33
|28L 5s
|-
|34
|26L 8s
|-
|35
|24L 11s
|-
|36
|22L 14s
|-
|37
|20L 17s
|-
|38
|18L 20s
|-
|39
|16L 23s
|-
|40
|14L 26s
|-
|41
|13L 28s
|-
|42
|10L 32s
|-
|43
|8L 35s
|-
|44
|6L 38s
|-
|45
|4L 41s
|-
|46
|2L 44s
|-
|47
|[[47edo]]
|equal
|-
|48
|46L 2s
| rowspan="28" |2:1
|-
|49
|45L 4s
|-
|50
|44L 6s
|-
|51
|43L 8s
|-
|52
|42L 10s
|-
|53
|41L 12s
|-
|54
|40L 14s
|-
|55
|39L 16s
|-
|56
|38L 18s
|-
|57
|37L 20s
|-
|58
|36L 22s
|-
|59
|35L 24s
|-
|60
|34L 26s
|-
|61
|33L 28s
|-
|62
|32L 30s
|-
|63
|31L 32s
|-
|64
|30L 34s
|-
|65
|29L 36s
|-
|66
|28L 38s
|-
|67
|27L 40s
|-
|68
|26L 42s
|-
|69
|25L 44s
|-
|70
|24L 46s
|-
|71
|23L 48s
|-
|72
|22L 50s
|-
|73
|21L 52s
|-
|74
|20L 54s
|-
|75
|19L 56s
|}
[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:94edo| ]] <!-- main article -->
[[Category:94edo| ]] <!-- main article -->
[[Category:Garibaldi]]
[[Category:Garibaldi]]
[[Category:Marvel]]
[[Category:Marvel]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/94edo"