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


== 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.


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. It can also be thought of as the "sum" of [[41edo]] and [[53edo]] (41 + 53 = 94), both of which are known for their approximation of [[Pythagorean tuning]]. Therefore 94edo's fifth is the [[mediant]] of these two tunings' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's.
Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.


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]].
=== 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.
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.
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=== 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]] <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"
{| class="wikitable"
|-
|+94edo [[SKULO interval names#WOFED interval names|WOFED interval names]]
! 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
|-
|-
!Step
|0
!Cents
|0
!13-limit
|1/1
!23-limit
|
!Short-form WOFED
|{{UDnote|step=0}}
!Long-form WOFED
|
!Diatonic
|
|
|-
|-
| 1
| 1
|12.766
| 12.766
|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
|L1, R1
|{{UDnote|step=1}}
|large unison, rastma
| L1, R1
| large unison, rastma
|
|
|-
|-
| 2
| 2
|25.532
| 25.532
|81/80, 64/63, (50/49)
| 81/80, 64/63, (50/49)
|
|
|K1, S1
|{{UDnote|step=2}}
|komma, super unison  
| K1, S1
| komma, super unison  
|
|
|-
|-
|3
| 3
|38.298
| 38.298
|45/44, 40/39, (250/243, 49/48)
| 45/44, 40/39, (250/243, 49/48)
|46/45
| 46/45
|O1, H1
|{{UDnote|step=3}}
|on unison, hyper unison
| O1, H1
| 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)
|
|
|U1, T1, hm2
|{{UDnote|step=4}}
|uber unison, tall unison, hypo minor second
| U1, T1, hm2
| 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)
|
|
|sm2, uA1, tA1, (kkA1)
|{{UDnote|step=5}}
|sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison)
| sm2, uA1, tA1, (kkA1)
|dd3
| sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison)
| dd3
|-
|-
| 6
| 6
|76.596
| 76.596
|22/21, (648/625, 26/25)
| 22/21, (648/625, 26/25)
|23/22, 24/23
| 23/22, 24/23
|lm2, oA1
|{{UDnote|step=6}}
|little minor second, off augmented unison
| lm2, oA1
| little minor second, off augmented unison
|
|
|-
|-
|7
| 7
|89.362
| 89.362
|256/243, 135/128, (21/20)
| 256/243, 135/128, (21/20)
|19/18, 20/19
| 19/18, 20/19
|m2, kA1
|{{UDnote|step=7}}
|minor second, komma-down augmented unison
| m2, kA1
|m2
| minor second, komma-down augmented unison
| m2
|-
|-
|8
| 8
|102.128
| 102.128
|128/121, (35/33)
| 128/121, (35/33)
|17/16, 18/17
| 17/16, 18/17
|Rm2, rA1
|{{UDnote|step=8}}
|rastmic minor second, rastmic augmented unison
| Rm2, rA1
| rastmic minor second, rastmic augmented unison
|
|
|-
|-
|9
| 9
|114.894
| 114.894
|16/15, (15/14)
| 16/15, (15/14)
|
|
|Km2, A1
|{{UDnote|step=9}}
|classic minor second, augmented unison
| Km2, A1
|A1
| classic minor second, augmented unison
| A1
|-
|-
|10
| 10
|127.660
| 127.660
|320/297, 189/176, (14/13)
| 320/297, 189/176, (14/13)
|
|
|Om2, LA1
|{{UDnote|step=10}}
|oceanic minor second, large augmented unison
| Om2, LA1
| oceanic minor second, large augmented unison
|
|
|-
|-
|11
| 11
|140.426
| 140.426
|88/81, 13/12, 243/224, (27/25)
| 88/81, 13/12, 243/224, (27/25)
|25/23, 38/35
| 25/23, 38/35
|n2, Tm2, SA1, (KKm2)
|{{UDnote|step=11}}
|lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
| n2, Tm2, SA1, (KKm2)
| lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
|
|
|-
|-
|12
| 12
|153.191
| 153.191
|12/11, (35/32)
| 12/11, (35/32)
|23/21
| 23/21
|N2, tM2, HA1
|{{UDnote|step=12}}
|greater netral second, tiny major second, hyper augmented unison
| N2, tM2, HA1
|ddd4
| greater netral second, tiny major second, hyper augmented unison
| ddd4
|-
|-
|13
| 13
| 165.957
| 165.957
|11/10
| 11/10
|
|
|oM2
|{{UDnote|step=13}}
|off major second
| oM2
| off major second
|
|
|-
|-
|14
| 14
|178.723
| 178.723
|10/9
| 10/9
|21/19
| 21/19
|kM2
|{{UDnote|step=14}}
|komma-down major second
| kM2
|d3
| komma-down major second
| d3
|-
|-
|15
| 15
| 191.489
| 191.489
|121/108, (49/44, 39/35)
| 121/108, (49/44, 39/35)
|19/17
| 19/17
|rM2
|{{UDnote|step=15}}
|rastmic major second
| rM2
| rastmic major second
|
|
|-
|-
|16
| 16
|204.255
| 204.255
|9/8
| 9/8
|
|
|M2
|{{UDnote|step=16}}
|major second
| M2
|M2
| major second
| M2
|-
|-
|17
| 17
|217.021
| 217.021
|112/99, (25/22)
| 112/99, (25/22)
|17/15, 26/23
| 17/15, 26/23
|LM2
|{{UDnote|step=17}}
|large major second
| LM2
| large major second
|
|
|-
|-
|18
| 18
|229.787
| 229.787
|8/7
| 8/7
|
|
|SM2
|{{UDnote|step=18}}
|super major second
| SM2
|AA1
| super major second
| AA1
|-
|-
|19
| 19
|242.553
| 242.553
|15/13
| 15/13
|23/20, 38/33
| 23/20, 38/33
|HM2
|{{UDnote|step=19}}
|hyper major second
| HM2
| hyper major second
|
|
|-
|-
| 20
| 20
|255.319
| 255.319
|52/45
| 52/45
|22/19
| 22/19
|hm3
|{{UDnote|step=20}}
|hypo minor third
| hm3
| hypo minor third
|
|
|-
|-
|21
| 21
|268.085
| 268.085
|7/6, (75/64)
| 7/6, (75/64)
|
|
|sm3, (kkA2)
|{{UDnote|step=21}}
|sub minor third, (classic augmented second)
| sm3, (kkA2)
|dd4
| sub minor third, (classic augmented second)
| dd4
|-
|-
| 22
| 22
|280.851
| 280.851
|33/28
| 33/28
|20/17, 27/23
| 20/17, 27/23
|lm3
|{{UDnote|step=22}}
|little minor third
| lm3
| little minor third
|
|
|-
|-
| 23
| 23
|293.617
| 293.617
|32/27, (25/21, 13/11)
| 32/27, (25/21, 13/11)
|19/16
| 19/16
|m3
|{{UDnote|step=23}}
|minor third
| m3
|m3
| minor third
| m3
|-
|-
|24
| 24
| 306.383
| 306.383
|144/121, (81/70)
| 144/121, (81/70)
|
|
|Rm3
|{{UDnote|step=24}}
|rastmic minor third
| Rm3
| rastmic minor third
|
|
|-
|-
|25
| 25
|319.149
| 319.149
|6/5
| 6/5
|
|
|Km3
|{{UDnote|step=25}}
|classic minor third
| Km3
|A2
| classic minor third
| A2
|-
|-
|26
| 26
|331.915
| 331.915
|40/33
| 40/33
|17/14, 23/19
| 17/14, 23/19
|Om3
|{{UDnote|step=26}}
|on minor third
| Om3
| on minor third
|
|
|-
|-
|27
| 27
|344.681
| 344.681
|11/9, 39/32, (243/200, 60/49)
| 11/9, 39/32, (243/200, 60/49)
|28/23
| 28/23
|n3, Tm3
|{{UDnote|step=27}}
|lesser neutral third, tall minor third
| n3, Tm3
|AAA1
| lesser neutral third, tall minor third
| AAA1
|-
|-
|28
| 28
|357.447
| 357.447
|27/22, 16/13, (100/81,49/40)
| 27/22, 16/13, (100/81,49/40)
|
|
|N3, tM3
|{{UDnote|step=28}}
|greater neutral third, tiny major third
| N3, tM3
|ddd5
| greater neutral third, tiny major third
| ddd5
|-
|-
|29
| 29
|370.213
| 370.213
|99/80, (26/21)
| 99/80, (26/21)
|21/17
| 21/17
|oM3
|{{UDnote|step=29}}
|off major third
| oM3
| off major third
|
|
|-
|-
| 30
| 30
|382.979
| 382.979
|5/4
| 5/4
|
|
|kM3
|{{UDnote|step=30}}
|classic major third
| kM3
|d4
| classic major third
| d4
|-
|-
|31
| 31
|395.745
| 395.745
|121/96, (34/27)
| 121/96, (34/27)
|
|
|rM3
|{{UDnote|step=31}}
|rastmic major third
| rM3
| rastmic major third
|
|
|-
|-
|32
| 32
| 408.511
| 408.511
|81/64, (33/26)
| 81/64, (33/26)
|19/15, 24/19
| 19/15, 24/19
|M3
|{{UDnote|step=32}}
|major third
| M3
|M3
| major third
| M3
|-
|-
|33
| 33
|421.277
| 421.277
|14/11
| 14/11
|23/18
| 23/18
|LM3
|{{UDnote|step=33}}
|large major third
| LM3
| large major third
|
|
|-
|-
|34
| 34
|434.043
| 434.043
|9/7, (32/25)
| 9/7, (32/25)
|
|
|SM3, (KKd4)
|{{UDnote|step=34}}
|super major third, (classic diminished fourth)
| SM3, (KKd4)
|AA2
| super major third, (classic diminished fourth)
| AA2
|-
|-
| 35
| 35
| 446.809
| 446.809
|135/104, (35/27)
| 135/104, (35/27)
|22/17
| 22/17
|HM3
|{{UDnote|step=35}}
|hyper major third
| HM3
|ddd6
| hyper major third
| ddd6
|-
|-
|36
| 36
|459.574
| 459.574
|13/10
| 13/10
|17/13, 30/23
| 17/13, 30/23
|h4
|{{UDnote|step=36}}
|hypo fourth
| h4
| hypo fourth
|
|
|-
|-
|37
| 37
| 472.340
| 472.340
|21/16
| 21/16
|25/19, 46/35
| 25/19, 46/35
|s4
|{{UDnote|step=37}}
|sub fourth
| s4
|dd5
| sub fourth
| dd5
|-
|-
|38
| 38
|485.106
| 485.106
|297/224
| 297/224
|
|
|l4
|{{UDnote|step=38}}
|little fourth
| l4
| little fourth
|
|
|-
|-
| 39
| 39
|497.872
| 497.872
|4/3
| 4/3
|
|
|P4
|{{UDnote|step=39}}
|perfect fourth
| P4
|P4
| perfect fourth
| P4
|-
|-
| 40
| 40
|510.638
| 510.638
|162/121, (35/26)
| 162/121, (35/26)
|
|
|R4
|{{UDnote|step=40}}
|rastmic fourth
| R4
| rastmic fourth
|
|
|-
|-
|41
| 41
|523.404
| 523.404
|27/20
| 27/20
|19/14, 23/17
| 19/14, 23/17
|K4
|{{UDnote|step=41}}
|komma-up fourth
| K4
|A3
| komma-up fourth
| A3
|-
|-
|42
| 42
|536.170
| 536.170
|15/11
| 15/11
|34/25
| 34/25
|O4
|{{UDnote|step=42}}
|on fourth
| O4
| on fourth
|
|
|-
|-
|43
| 43
|548.936
| 548.936
|11/8
| 11/8
|26/19
| 26/19
|U4, T4
|{{UDnote|step=43}}
|uber/undecimal fourth, tall fourth
| U4, T4
|AAA2
| uber/undecimal fourth, tall fourth
| AAA2
|-
|-
|44
| 44
|561.702
| 561.702
|18/13, (25/18)
| 18/13, (25/18)
|
|
|tA4, uA4, (kkA4)
|{{UDnote|step=44}}
|tiny augmented fourth, unter augmented fourth, (classic augmented fourth)
| tA4, uA4, (kkA4)
|dd6
| tiny augmented fourth, unter augmented fourth, (classic augmented fourth)
| dd6
|-
|-
|45
| 45
|574.468
| 574.468
|88/63
| 88/63
|32/23, 46/33
| 32/23, 46/33
|ld5, oA4
|{{UDnote|step=45}}
|little diminished fifth, off augmented fourth
| ld5, oA4
| little diminished fifth, off augmented fourth
|
|
|-
|-
| 46
| 46
|587.234
| 587.234
|45/32, (7/5)
| 45/32, (7/5)
|38/27
| 38/27
|kA4
|{{UDnote|step=46}}
|komma-down augmented fourth
| kA4
|d5
| komma-down augmented fourth
| d5
|-
|-
| 47
| 47
|600.000
| 600.000
|363/256, 512/363, (99/70)
| 363/256, 512/363, (99/70)
|17/12, 24/17
| 17/12, 24/17
|rA4, Rd5
|{{UDnote|step=47}}
|rastmic augmented fourth, rastmic diminished fifth
| rA4, Rd5
| rastmic augmented fourth, rastmic diminished fifth
|
|
|-
|-
|48
| 48
|612.766
| 612.766
|64/45, (10/7)
| 64/45, (10/7)
|27/19
| 27/19
|Kd5
|{{UDnote|step=48}}
|komma-up diminished fifth
| Kd5
|A4
| komma-up diminished fifth
| A4
|-
|-
|49
| 49
|625.532
| 625.532
|63/44
| 63/44
|23/16, 33/23
| 23/16, 33/23
|LA4, Od5
|{{UDnote|step=49}}
|large augmented fourth, off diminished fifth
| LA4, Od5
| large augmented fourth, off diminished fifth
|
|
|-
|-
| 50
| 50
|638.298
| 638.298
|13/9, (36/25)
| 13/9, (36/25)
|
|
|Td5, Ud5, (KKd5)
|{{UDnote|step=50}}
|tall diminished fifth, uber diminished fifth, (classic diminished fifth)
| Td5, Ud5, (KKd5)
|AA3
| tall diminished fifth, uber diminished fifth, (classic diminished fifth)
| AA3
|-
|-
|51
| 51
|651.064
| 651.064
|16/11
| 16/11
|19/13
| 19/13
|u5, t5
|{{UDnote|step=51}}
|unter/undecimal fifth, tiny fifth
| u5, t5
|ddd7
| unter/undecimal fifth, tiny fifth
| ddd7
|-
|-
| 52
| 52
|663.830
| 663.830
|22/15
| 22/15
|25/17
| 25/17
|o5
|{{UDnote|step=52}}
|off fifth
| o5
| off fifth
|
|
|-
|-
|53
| 53
|676.596
| 676.596
|40/27
| 40/27
|28/19, 34/23
| 28/19, 34/23
|k5
|{{UDnote|step=53}}
|komma-down fifth
| k5
|d6
| komma-down fifth
| d6
|-
|-
|54
| 54
| 689.362
| 689.362
|121/81, (52/35)
| 121/81, (52/35)
|
|
|r5
|{{UDnote|step=54}}
|rastmic fifth
| r5
| rastmic fifth
|
|
|-
|-
|55
| 55
|702.128
| 702.128
|3/2
| 3/2
|
|
|P5
|{{UDnote|step=55}}
|perfect fifth
| P5
|P5
| perfect fifth
| P5
|-
|-
|56
| 56
|714.894
| 714.894
|448/297
| 448/297
|
|
|L5
|{{UDnote|step=56}}
|large fifth
| L5
| large fifth
|
|
|-
|-
|57
| 57
| 727.660
| 727.660
|32/21
| 32/21
|38/25, 35/23
| 38/25, 35/23
|S5
|{{UDnote|step=57}}
|super fifth
| S5
|AA4
| super fifth
| AA4
|-
|-
|58
| 58
|740.426
| 740.426
|20/13
| 20/13
|26/17, 23/15
| 26/17, 23/15
|H5
|{{UDnote|step=58}}
|hyper fifth
| H5
| hyper fifth
|
|
|-
|-
|59
| 59
|753.191
| 753.191
|208/135
| 208/135
|17/11
| 17/11
|hm6
|{{UDnote|step=59}}
|hypo minor sixth
| hm6
|AAA3
| hypo minor sixth
| AAA3
|-
|-
|60
| 60
|765.957
| 765.957
|14/9, (128/75)
| 14/9, (25/16)
|
|
|sm6, (kkA5)
|{{UDnote|step=60}}
|sub minor sixth, (classic augmented fifth)
| sm6, (kkA5)
|dd7
| sub minor sixth, (classic augmented fifth)
| dd7
|-
|-
| 61
| 61
|778.723
| 778.723
|11/7
| 11/7
|36/23
| 36/23
|lm6
|{{UDnote|step=61}}
|little minor sixth
| lm6
| little minor sixth
|
|
|-
|-
|62
| 62
|791.489
| 791.489
|128/81
| 128/81
|19/12, 30/19
| 19/12, 30/19
|m6
|{{UDnote|step=62}}
|minor sixth
| m6
|m6
| minor sixth
| m6
|-
|-
|63
| 63
|804.255
| 804.255
|192/121
| 192/121
|27/17
| 27/17
|Rm6
|{{UDnote|step=63}}
|rastmic minor sixth
| Rm6
| rastmic minor sixth
|
|
|-
|-
|64
| 64
|817.021
| 817.021
|8/5
| 8/5
|
|
|Km6
|{{UDnote|step=64}}
|classic minor sixth
| Km6
|A5
| classic minor sixth
| A5
|-
|-
|65
| 65
|829.787
| 829.787
|160/99, (21/13)
| 160/99, (21/13)
|34/21
| 34/21
|Om6
|{{UDnote|step=65}}
|on minor sixth
| Om6
| 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)
|
|
|n6, Tm6
|{{UDnote|step=66}}
|less neutral sixth, tall minor sixth
| n6, Tm6
|AAA4
| less neutral sixth, tall minor sixth
| AAA4
|-
|-
|67
| 67
| 855.319
| 855.319
|18/11, 64/39, (400/243, 49/30)
| 18/11, 64/39, (400/243, 49/30)
|23/14
| 23/14
|N6, tM6
|{{UDnote|step=67}}
|greater neutral sixth, tiny minor sixth
| N6, tM6
|ddd8
| greater neutral sixth, tiny minor sixth
| ddd8
|-
|-
|68
| 68
| 868.085
| 868.085
|33/20
| 33/20
|28/17, 38/23
| 28/17, 38/23
|oM6
|{{UDnote|step=68}}
|off major sixth
| oM6
| off major sixth
|
|
|-
|-
|69
| 69
|880.851
| 880.851
|5/3
| 5/3
|
|
|kM6
|{{UDnote|step=69}}
|classic major sixth
| kM6
|d7
| classic major sixth
| d7
|-
|-
|70
| 70
|893.617
| 893.617
|121/72
| 121/72
|
|
|rM6
|{{UDnote|step=70}}
|rastmic major sixth
| rM6
| rastmic major sixth
|
|
|-
|-
|71
| 71
|906.383
| 906.383
|27/16, (42/35, 22/13)
| 27/16, (42/35, 22/13)
|32/19
| 32/19
|M6
|{{UDnote|step=71}}
|major sixth
| M6
|M6
| major sixth
| M6
|-
|-
|72
| 72
|919.149
| 919.149
|56/33
| 56/33
|17/10, 46/27
| 17/10, 46/27
|LM6
|{{UDnote|step=72}}
|large major sixth
| LM6
| large major sixth
|
|
|-
|-
|73
| 73
|931.915
| 931.915
|12/7, 128/75
| 12/7, (128/75)
|
|
|SM6, (KKd7)
|{{UDnote|step=73}}
|super major sixth (classic diminished seventh)
| SM6, (KKd7)
|AA5
| super major sixth (classic diminished seventh)
| AA5
|-
|-
|74
| 74
|944.681
| 944.681
|45/26
| 45/26
|19/11
| 19/11
|HM6
|{{UDnote|step=74}}
|hyper major sixth
| HM6
| hyper major sixth
|
|
|-
|-
|75
| 75
|957.447
| 957.447
|26/15
| 26/15
|40/23, 33/19
| 40/23, 33/19
|hm7
|{{UDnote|step=75}}
|hypo minor seventh
| hm7
| hypo minor seventh
|
|
|-
|-
|76
| 76
|970.213
| 970.213
|7/4
| 7/4
|
|
|sm7
|{{UDnote|step=76}}
|sub minor seventh
| sm7
|dd8
| sub minor seventh
| dd8
|-
|-
|77
| 77
|982.979
| 982.979
|99/56, (44/25)
| 99/56, (44/25)
|30/17, 23/13
| 30/17, 23/13
|lm7
|{{UDnote|step=77}}
|little minor seventh
| lm7
| little minor seventh
|
|
|-
|-
|78
| 78
|995.745
| 995.745
|16/9
| 16/9
|
|
|m7
|{{UDnote|step=78}}
|minor seventh
| m7
|m7
| minor seventh
| m7
|-
|-
| 79
| 79
|1008.511
| 1008.511
|216/121
| 216/121
|34/19
| 34/19
|Rm7
|{{UDnote|step=79}}
|rastmic minor seventh
| Rm7
| rastmic minor seventh
|
|
|-
|-
|80
| 80
|1021.277
| 1021.277
|9/5
| 9/5
|38/21
| 38/21
|Km7
|{{UDnote|step=80}}
|classic minor seventh
| Km7
|A6
| classic minor seventh
| A6
|-
|-
|81
| 81
|1034.043
| 1034.043
|20/11
| 20/11
|
|
|Om7
|{{UDnote|step=81}}
|on minor seventh
| Om7
| on minor seventh
|
|
|-
|-
|82
| 82
| 1046.809
| 1046.809
|11/6, (64/35)
| 11/6, (64/35)
|42/23
| 42/23
|n7, Tm7, hd8
|{{UDnote|step=82}}
|less neutral seventh, tall minor seventh, hypo diminished octave
| n7, Tm7, hd8
|AAA5
| less neutral seventh, tall minor seventh, hypo diminished octave
| AAA5
|-
|-
|83
| 83
|1059.574
| 1059.574
|81/44, 24/13, (50/27)
| 81/44, 24/13, (50/27)
|46/25, 35/19
| 46/25, 35/19
|N7, tM7, sd8, (kkM7)
|{{UDnote|step=83}}
|greater neutral seventh, tiny major seventh, sub diminished octave, (2-comma down major seventh)
| N7, tM7, sd8, (kkM7)
| 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)
|
|
|oM7, ld8
|{{UDnote|step=84}}
|off major seventh, little diminished octave
| oM7, ld8
| off major seventh, little diminished octave
|
|
|-
|-
|85
| 85
|1085.106
| 1085.106
|15/8, (28/15)
| 15/8, (28/15)
|
|
|kM7, d8
|{{UDnote|step=85}}
|classic major seventh, diminished octave
| kM7, d8
|d8
| classic major seventh, diminished octave
| d8
|-
|-
|86
| 86
|1097.872
| 1097.872
|121/64
| 121/64
|32/17, 17/9
| 32/17, 17/9
|rM7, Rd8
|{{UDnote|step=86}}
|rastmic major seventh, rastmic diminished octave
| rM7, Rd8
| rastmic major seventh, rastmic diminished octave
|
|
|-
|-
|87
| 87
|1110.638
| 1110.638
|243/128, 256/135, (40/21)
| 243/128, 256/135, (40/21)
|36/19, 19/10
| 36/19, 19/10
|M7, Kd8
|{{UDnote|step=87}}
|major seventh, komma-up diminished octave
| M7, Kd8
|M7
| major seventh, komma-up diminished octave
| M7
|-
|-
|88
| 88
|1123.404
| 1123.404
|21/11, (25/13)
| 21/11, (25/13)
|44/23, 23/12
| 44/23, 23/12
|LM7, Od8
|{{UDnote|step=88}}
|large major seventh, on diminished octave
| LM7, Od8
| large major seventh, on diminished octave
|
|
|-
|-
|89
| 89
|1136.170
| 1136.170
|27/14, 52/27, (48/25)
| 27/14, 52/27, (48/25)
|
|
|SM7, Td8, Ud8, (KKd8)
|{{UDnote|step=89}}
|super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave)
| SM7, Td8, Ud8, (KKd8)
|AA6
| super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave)
| AA6
|-
|-
|90
| 90
|1148.936
| 1148.936
|64/33, (35/18, 68/35, 33/17)
| 64/33, (35/18, 68/35, 33/17)
|33/17
| 33/17
|u8, t8, HM7
|{{UDnote|step=90}}
|unter octave, tiny octave, hyper major seventh
| u8, t8, HM7
| unter octave, tiny octave, hyper major seventh
|
|
|-
|-
|91
| 91
| 1161.702
| 1161.702
|88/45, 39/20
| 88/45, 39/20
|45/23
| 45/23
|o8, h8
|{{UDnote|step=91}}
|off octave, hypo octave
| o8, h8
| off octave, hypo octave
|
|
|-
|-
|92
| 92
|1174.468
| 1174.468
|160/81, 63/32, (49/25)
| 160/81, 63/32, (49/25)
|
|
|k8, s8
|{{UDnote|step=92}}
|komma-down octave, sub octave
| k8, s8
| 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)
|
|
|l8, r8
|{{UDnote|step=93}}
|little octave, octave - rastma
| l8, r8
| little octave, octave - rastma
|
|
|-
|-
|94
| 94
|1200.000
| 1200.000
|2/1
| 2/1
|
|
|P8
|{{UDnote|step=94}}
|perfect octave
| P8
|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.
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 791: 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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 805: 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 812: 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 819: 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 826: 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 833: 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 840: 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 847: 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 854: 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 equal temperament that does better in this subgroup is 193.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator*
! Cents
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
Line 880: Line 1,050:
| 63.83
| 63.83
| 25/24
| 25/24
| [[Sycamore]] / [[betic]]
| [[Betic]]
|-
|-
| 1
| 1
Line 904: Line 1,074:
| 497.87
| 497.87
| 4/3
| 4/3
| [[Helmholtz]] / [[garibaldi]] / [[cassandra]]
| [[Garibaldi]] / [[cassandra]]
|-
|-
| 2
| 2
Line 936: Line 1,106:
| [[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 ==
Line 961: 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]
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