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94edo: Difference between revisions

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== 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 well-rounded 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.


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.


=== As a tuning of other temperaments ===
=== 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.
94edo is the sum of [[41edo]] and [[53edo]], both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperaments]], like [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]], and tempering together the [[81/80|syntonic]], [[Septimal comma|septimal]], and [[pythagorean comma]] into the same interval.


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's fifth is the [[mediant]] of these two edos' fifths; it is ever so slightly sharp of just and only a hair 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 non-garibaldi schismatic notable edos in the patent val are [[118edo|118]], [[159edo|159]], [[171edo|171]], [[224edo|224]], and [[460edo|460]].
 
The list of 23-limit commas it tempers out is huge (see below), 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.
Line 31: Line 33:
! 13-limit
! 13-limit
! 23-limit
! 23-limit
![[Ups and downs notation|Ups and downs]]
! [[Ups and downs notation|Ups and downs]]
! Short-form [[SKULO interval names#WOFED interval names|WOFED]]
! Short-form [[SKULO interval names#WOFED interval names|WOFED]]
! Long-form WOFED
! Long-form WOFED
Line 38: Line 40:
|0
|0
|0
|0
|1/1
|[[1/1]]
|
|
|{{UDnote|step=0}}
|{{UDnote|step=0}}
Line 47: Line 49:
| 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]]
|{{UDnote|step=1}}
|{{UDnote|step=1}}
| L1, R1
| L1, R1
Line 56: Line 58:
| 2
| 2
| 25.532
| 25.532
| 81/80, 64/63, (50/49)
| [[531441/524288]], [[81/80]], [[64/63]], ([[50/49]])
|
|
|{{UDnote|step=2}}
|{{UDnote|step=2}}
Line 65: Line 67:
| 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]]
|{{UDnote|step=3}}
|{{UDnote|step=3}}
| O1, H1
| O1, H1
Line 74: Line 76:
| 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}}
|{{UDnote|step=4}}
Line 83: Line 85:
| 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}}
|{{UDnote|step=5}}
Line 92: Line 94:
| 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]]
|{{UDnote|step=6}}
|{{UDnote|step=6}}
| lm2, oA1
| lm2, oA1
Line 101: Line 103:
| 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]]
|{{UDnote|step=7}}
|{{UDnote|step=7}}
| m2, kA1
| m2, kA1
Line 110: Line 112:
| 8
| 8
| 102.128
| 102.128
| 128/121, (35/33)
| [[128/121]], ([[35/33]])
| 17/16, 18/17
| [[17/16]], [[18/17]]
|{{UDnote|step=8}}
|{{UDnote|step=8}}
| Rm2, rA1
| Rm2, rA1
Line 119: Line 121:
| 9
| 9
| 114.894
| 114.894
| 16/15, (15/14)
| [[2187/2048]], [[16/15]], ([[15/14]])
|
|
|{{UDnote|step=9}}
|{{UDnote|step=9}}
Line 128: Line 130:
| 10
| 10
| 127.660
| 127.660
| 320/297, 189/176, (14/13)
| [[320/297]], [[189/176]], ([[14/13]])
|
|
|{{UDnote|step=10}}
|{{UDnote|step=10}}
Line 137: Line 139:
| 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]]
|{{UDnote|step=11}}
|{{UDnote|step=11}}
| n2, Tm2, SA1, (KKm2)
| n2, Tm2, SA1, (KKm2)
Line 146: Line 148:
| 12
| 12
| 153.191
| 153.191
| 12/11, (35/32)
| [[12/11]], ([[35/32]])
| 23/21
| [[23/21]]
|{{UDnote|step=12}}
|{{UDnote|step=12}}
| N2, tM2, HA1
| N2, tM2, HA1
Line 155: Line 157:
| 13
| 13
| 165.957
| 165.957
| 11/10
| [[11/10]]
|
|
|{{UDnote|step=13}}
|{{UDnote|step=13}}
Line 164: Line 166:
| 14
| 14
| 178.723
| 178.723
| 10/9
| [[10/9]]
| 21/19
| [[21/19]]
|{{UDnote|step=14}}
|{{UDnote|step=14}}
| kM2
| kM2
Line 173: Line 175:
| 15
| 15
| 191.489
| 191.489
| 121/108, (49/44, 39/35)
| [[121/108]], ([[49/44]], [[39/35]])
| 19/17
| [[19/17]]
|{{UDnote|step=15}}
|{{UDnote|step=15}}
| rM2
| rM2
Line 182: Line 184:
| 16
| 16
| 204.255
| 204.255
| 9/8
| [[9/8]]
|
|
|{{UDnote|step=16}}
|{{UDnote|step=16}}
Line 191: Line 193:
| 17
| 17
| 217.021
| 217.021
| 112/99, (25/22)
| [[112/99]], ([[25/22]])
| 17/15, 26/23
| [[17/15]], [[26/23]]
|{{UDnote|step=17}}
|{{UDnote|step=17}}
| LM2
| LM2
Line 200: Line 202:
| 18
| 18
| 229.787
| 229.787
| 8/7
| [[8/7]]
|
|
|{{UDnote|step=18}}
|{{UDnote|step=18}}
Line 209: Line 211:
| 19
| 19
| 242.553
| 242.553
| 15/13
| [[15/13]]
| 23/20, 38/33
| [[23/20]], [[38/33]]
|{{UDnote|step=19}}
|{{UDnote|step=19}}
| HM2
| HM2
Line 218: Line 220:
| 20
| 20
| 255.319
| 255.319
| 52/45
| [[52/45]]
| 22/19
| [[22/19]]
|{{UDnote|step=20}}
|{{UDnote|step=20}}
| hm3
| hm3
Line 227: Line 229:
| 21
| 21
| 268.085
| 268.085
| 7/6, (75/64)
| [[7/6]], ([[75/64]])
|
|
|{{UDnote|step=21}}
|{{UDnote|step=21}}
Line 236: Line 238:
| 22
| 22
| 280.851
| 280.851
| 33/28
| [[33/28]]
| 20/17, 27/23
| [[20/17]], [[27/23]]
|{{UDnote|step=22}}
|{{UDnote|step=22}}
| lm3
| lm3
Line 245: Line 247:
| 23
| 23
| 293.617
| 293.617
| 32/27, (25/21, 13/11)
| [[32/27]], ([[25/21]], [[13/11]])
| 19/16
| [[19/16]]
|{{UDnote|step=23}}
|{{UDnote|step=23}}
| m3
| m3
Line 254: Line 256:
| 24
| 24
| 306.383
| 306.383
| 144/121, (81/70)
| [[144/121]], ([[81/70]])
|
|
|{{UDnote|step=24}}
|{{UDnote|step=24}}
Line 263: Line 265:
| 25
| 25
| 319.149
| 319.149
| 6/5
| [[6/5]]
|
|
|{{UDnote|step=25}}
|{{UDnote|step=25}}
Line 272: Line 274:
| 26
| 26
| 331.915
| 331.915
| 40/33
| [[40/33]]
| 17/14, 23/19
| [[17/14]], [[23/19]]
|{{UDnote|step=26}}
|{{UDnote|step=26}}
| Om3
| Om3
Line 281: Line 283:
| 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]]
|{{UDnote|step=27}}
|{{UDnote|step=27}}
| n3, Tm3
| n3, Tm3
Line 290: Line 292:
| 28
| 28
| 357.447
| 357.447
| 27/22, 16/13, (100/81,49/40)
| [[27/22]], [[16/13]], ([[100/81]],[[49/40]])
|
|
|{{UDnote|step=28}}
|{{UDnote|step=28}}
Line 299: Line 301:
| 29
| 29
| 370.213
| 370.213
| 99/80, (26/21)
| [[99/80]], ([[26/21]])
| 21/17
| [[21/17]]
|{{UDnote|step=29}}
|{{UDnote|step=29}}
| oM3
| oM3
Line 308: Line 310:
| 30
| 30
| 382.979
| 382.979
| 5/4
| [[8192/6561]], [[5/4]]
|
|
|{{UDnote|step=30}}
|{{UDnote|step=30}}
Line 317: Line 319:
| 31
| 31
| 395.745
| 395.745
| 121/96, (34/27)
| [[121/96]], ([[34/27]])
|
|
|{{UDnote|step=31}}
|{{UDnote|step=31}}
Line 326: Line 328:
| 32
| 32
| 408.511
| 408.511
| 81/64, (33/26)
| [[81/64]], ([[33/26]])
| 19/15, 24/19
| [[19/15]], [[24/19]]
|{{UDnote|step=32}}
|{{UDnote|step=32}}
| M3
| M3
Line 335: Line 337:
| 33
| 33
| 421.277
| 421.277
| 14/11
| [[14/11]]
| 23/18
| [[23/18]]
|{{UDnote|step=33}}
|{{UDnote|step=33}}
| LM3
| LM3
Line 344: Line 346:
| 34
| 34
| 434.043
| 434.043
| 9/7, (32/25)
| [[9/7]], ([[32/25]])
|
|
|{{UDnote|step=34}}
|{{UDnote|step=34}}
Line 353: Line 355:
| 35
| 35
| 446.809
| 446.809
| 135/104, (35/27)
| [[135/104]], ([[35/27]])
| 22/17
| [[22/17]]
|{{UDnote|step=35}}
|{{UDnote|step=35}}
| HM3
| HM3
Line 362: Line 364:
| 36
| 36
| 459.574
| 459.574
| 13/10
| [[13/10]]
| 17/13, 30/23
| [[17/13]], [[30/23]]
|{{UDnote|step=36}}
|{{UDnote|step=36}}
| h4
| h4
Line 371: Line 373:
| 37
| 37
| 472.340
| 472.340
| 21/16
| [[21/16]]
| 25/19, 46/35
| [[25/19]], [[46/35]]
|{{UDnote|step=37}}
|{{UDnote|step=37}}
| s4
| s4
Line 380: Line 382:
| 38
| 38
| 485.106
| 485.106
| 297/224
| [[297/224]]
|
|
|{{UDnote|step=38}}
|{{UDnote|step=38}}
Line 389: Line 391:
| 39
| 39
| 497.872
| 497.872
| 4/3
| [[4/3]]
|
|
|{{UDnote|step=39}}
|{{UDnote|step=39}}
Line 398: Line 400:
| 40
| 40
| 510.638
| 510.638
| 162/121, (35/26)
| [[162/121]], ([[35/26]])
|
|
|{{UDnote|step=40}}
|{{UDnote|step=40}}
Line 407: Line 409:
| 41
| 41
| 523.404
| 523.404
| 27/20
| [[27/20]]
| 19/14, 23/17
| [[19/14]], [[23/17]]
|{{UDnote|step=41}}
|{{UDnote|step=41}}
| K4
| K4
Line 416: Line 418:
| 42
| 42
| 536.170
| 536.170
| 15/11
| [[15/11]]
| 34/25
| [[34/25]]
|{{UDnote|step=42}}
|{{UDnote|step=42}}
| O4
| O4
Line 425: Line 427:
| 43
| 43
| 548.936
| 548.936
| 11/8
| [[11/8]]
| 26/19
| [[26/19]]
|{{UDnote|step=43}}
|{{UDnote|step=43}}
| U4, T4
| U4, T4
Line 434: Line 436:
| 44
| 44
| 561.702
| 561.702
| 18/13, (25/18)
| [[18/13]], ([[25/18]])
|
|
|{{UDnote|step=44}}
|{{UDnote|step=44}}
Line 443: Line 445:
| 45
| 45
| 574.468
| 574.468
| 88/63
| [[88/63]]
| 32/23, 46/33
| [[32/23]], [[46/33]]
|{{UDnote|step=45}}
|{{UDnote|step=45}}
| ld5, oA4
| ld5, oA4
Line 452: Line 454:
| 46
| 46
| 587.234
| 587.234
| 45/32, (7/5)
| [[45/32]], ([[7/5]])
| 38/27
| [[38/27]]
|{{UDnote|step=46}}
|{{UDnote|step=46}}
| kA4
| kA4
Line 461: Line 463:
| 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]]
|{{UDnote|step=47}}
|{{UDnote|step=47}}
| rA4, Rd5
| rA4, Rd5
Line 470: Line 472:
| 48
| 48
| 612.766
| 612.766
| 64/45, (10/7)
| [[64/45]], ([[10/7]])
| 27/19
| [[27/19]]
|{{UDnote|step=48}}
|{{UDnote|step=48}}
| Kd5
| Kd5
Line 479: Line 481:
| 49
| 49
| 625.532
| 625.532
| 63/44
| [[63/44]]
| 23/16, 33/23
| [[23/16]], [[33/23]]
|{{UDnote|step=49}}
|{{UDnote|step=49}}
| LA4, Od5
| LA4, Od5
Line 488: Line 490:
| 50
| 50
| 638.298
| 638.298
| 13/9, (36/25)
| [[13/9]], ([[36/25]])
|
|
|{{UDnote|step=50}}
|{{UDnote|step=50}}
Line 497: Line 499:
| 51
| 51
| 651.064
| 651.064
| 16/11
| [[16/11]]
| 19/13
| [[19/13]]
|{{UDnote|step=51}}
|{{UDnote|step=51}}
| u5, t5
| u5, t5
Line 506: Line 508:
| 52
| 52
| 663.830
| 663.830
| 22/15
| [[22/15]]
| 25/17
| [[25/17]]
|{{UDnote|step=52}}
|{{UDnote|step=52}}
| o5
| o5
Line 515: Line 517:
| 53
| 53
| 676.596
| 676.596
| 40/27
| [[40/27]]
| 28/19, 34/23
| [[28/19]], [[34/23]]
|{{UDnote|step=53}}
|{{UDnote|step=53}}
| k5
| k5
Line 524: Line 526:
| 54
| 54
| 689.362
| 689.362
| 121/81, (52/35)
| [[121/81]], ([[52/35]])
|
|
|{{UDnote|step=54}}
|{{UDnote|step=54}}
Line 533: Line 535:
| 55
| 55
| 702.128
| 702.128
| 3/2
| [[3/2]]
|
|
|{{UDnote|step=55}}
|{{UDnote|step=55}}
Line 542: Line 544:
| 56
| 56
| 714.894
| 714.894
| 448/297
| [[448/297]]
|
|
|{{UDnote|step=56}}
|{{UDnote|step=56}}
Line 551: Line 553:
| 57
| 57
| 727.660
| 727.660
| 32/21
| [[32/21]]
| 38/25, 35/23
| [[38/25]], [[35/23]]
|{{UDnote|step=57}}
|{{UDnote|step=57}}
| S5
| S5
Line 560: Line 562:
| 58
| 58
| 740.426
| 740.426
| 20/13
| [[20/13]]
| 26/17, 23/15
| [[26/17]], [[23/15]]
|{{UDnote|step=58}}
|{{UDnote|step=58}}
| H5
| H5
Line 569: Line 571:
| 59
| 59
| 753.191
| 753.191
| 208/135
| [[208/135]]
| 17/11
| [[17/11]]
|{{UDnote|step=59}}
|{{UDnote|step=59}}
| hm6
| hm6
Line 578: Line 580:
| 60
| 60
| 765.957
| 765.957
| 14/9, (25/16)
| [[14/9]], ([[25/16]])
|
|
|{{UDnote|step=60}}
|{{UDnote|step=60}}
Line 587: Line 589:
| 61
| 61
| 778.723
| 778.723
| 11/7
| [[11/7]]
| 36/23
| [[36/23]]
|{{UDnote|step=61}}
|{{UDnote|step=61}}
| lm6
| lm6
Line 596: Line 598:
| 62
| 62
| 791.489
| 791.489
| 128/81
| [[128/81]]
| 19/12, 30/19
| [[19/12]], [[30/19]]
|{{UDnote|step=62}}
|{{UDnote|step=62}}
| m6
| m6
Line 605: Line 607:
| 63
| 63
| 804.255
| 804.255
| 192/121
| [[192/121]]
| 27/17
| [[27/17]]
|{{UDnote|step=63}}
|{{UDnote|step=63}}
| Rm6
| Rm6
Line 614: Line 616:
| 64
| 64
| 817.021
| 817.021
| 8/5
| [[8/5]]
|
|
|{{UDnote|step=64}}
|{{UDnote|step=64}}
Line 623: Line 625:
| 65
| 65
| 829.787
| 829.787
| 160/99, (21/13)
| [[160/99]], ([[21/13]])
| 34/21
| [[34/21]]
|{{UDnote|step=65}}
|{{UDnote|step=65}}
| Om6
| Om6
Line 632: Line 634:
| 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}}
|{{UDnote|step=66}}
Line 641: Line 643:
| 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]]
|{{UDnote|step=67}}
|{{UDnote|step=67}}
| N6, tM6
| N6, tM6
Line 650: Line 652:
| 68
| 68
| 868.085
| 868.085
| 33/20
| [[33/20]]
| 28/17, 38/23
| [[28/17]], [[38/23]]
|{{UDnote|step=68}}
|{{UDnote|step=68}}
| oM6
| oM6
Line 659: Line 661:
| 69
| 69
| 880.851
| 880.851
| 5/3
| [[5/3]]
|
|
|{{UDnote|step=69}}
|{{UDnote|step=69}}
Line 668: Line 670:
| 70
| 70
| 893.617
| 893.617
| 121/72
| [[121/72]]
|
|
|{{UDnote|step=70}}
|{{UDnote|step=70}}
Line 677: Line 679:
| 71
| 71
| 906.383
| 906.383
| 27/16, (42/35, 22/13)
| [[27/16]], ([[42/35]], [[22/13]])
| 32/19
| [[32/19]]
|{{UDnote|step=71}}
|{{UDnote|step=71}}
| M6
| M6
Line 686: Line 688:
| 72
| 72
| 919.149
| 919.149
| 56/33
| [[56/33]]
| 17/10, 46/27
| [[17/10]], [[46/27]]
|{{UDnote|step=72}}
|{{UDnote|step=72}}
| LM6
| LM6
Line 695: Line 697:
| 73
| 73
| 931.915
| 931.915
| 12/7, (128/75)
| [[12/7]], ([[128/75]])
|
|
|{{UDnote|step=73}}
|{{UDnote|step=73}}
Line 704: Line 706:
| 74
| 74
| 944.681
| 944.681
| 45/26
| [[45/26]]
| 19/11
| [[19/11]]
|{{UDnote|step=74}}
|{{UDnote|step=74}}
| HM6
| HM6
Line 713: Line 715:
| 75
| 75
| 957.447
| 957.447
| 26/15
| [[26/15]]
| 40/23, 33/19
| [[40/23]], [[33/19]]
|{{UDnote|step=75}}
|{{UDnote|step=75}}
| hm7
| hm7
Line 722: Line 724:
| 76
| 76
| 970.213
| 970.213
| 7/4
| [[7/4]]
|
|
|{{UDnote|step=76}}
|{{UDnote|step=76}}
Line 731: Line 733:
| 77
| 77
| 982.979
| 982.979
| 99/56, (44/25)
| [[99/56]], ([[44/25]])
| 30/17, 23/13
| [[30/17]], [[23/13]]
|{{UDnote|step=77}}
|{{UDnote|step=77}}
| lm7
| lm7
Line 740: Line 742:
| 78
| 78
| 995.745
| 995.745
| 16/9
| [[16/9]]
|
|
|{{UDnote|step=78}}
|{{UDnote|step=78}}
Line 749: Line 751:
| 79
| 79
| 1008.511
| 1008.511
| 216/121
| [[216/121]]
| 34/19
| [[34/19]]
|{{UDnote|step=79}}
|{{UDnote|step=79}}
| Rm7
| Rm7
Line 758: Line 760:
| 80
| 80
| 1021.277
| 1021.277
| 9/5
| [[9/5]]
| 38/21
| [[38/21]]
|{{UDnote|step=80}}
|{{UDnote|step=80}}
| Km7
| Km7
Line 767: Line 769:
| 81
| 81
| 1034.043
| 1034.043
| 20/11
| [[20/11]]
|
|
|{{UDnote|step=81}}
|{{UDnote|step=81}}
Line 776: Line 778:
| 82
| 82
| 1046.809
| 1046.809
| 11/6, (64/35)
| [[11/6]], ([[64/35]])
| 42/23
| [[42/23]]
|{{UDnote|step=82}}
|{{UDnote|step=82}}
| n7, Tm7, hd8
| n7, Tm7, hd8
Line 785: Line 787:
| 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]]
|{{UDnote|step=83}}
|{{UDnote|step=83}}
| N7, tM7, sd8, (kkM7)
| N7, tM7, sd8, (kkM7)
Line 794: Line 796:
| 84
| 84
| 1072.340
| 1072.340
| 297/160, 144/91, (13/7)
| [[297/160]], [[144/91]], ([[13/7]])
|
|
|{{UDnote|step=84}}
|{{UDnote|step=84}}
Line 803: Line 805:
| 85
| 85
| 1085.106
| 1085.106
| 15/8, (28/15)
| [[15/8]], ([[28/15]])
|
|
|{{UDnote|step=85}}
|{{UDnote|step=85}}
Line 812: Line 814:
| 86
| 86
| 1097.872
| 1097.872
| 121/64
| [[121/64]]
| 32/17, 17/9
| [[32/17]], [[17/9]]
|{{UDnote|step=86}}
|{{UDnote|step=86}}
| rM7, Rd8
| rM7, Rd8
Line 821: Line 823:
| 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]]
|{{UDnote|step=87}}
|{{UDnote|step=87}}
| M7, Kd8
| M7, Kd8
Line 830: Line 832:
| 88
| 88
| 1123.404
| 1123.404
| 21/11, (25/13)
| [[21/11]], ([[25/13]])
| 44/23, 23/12
| [[44/23]], [[23/12]]
|{{UDnote|step=88}}
|{{UDnote|step=88}}
| LM7, Od8
| LM7, Od8
Line 839: Line 841:
| 89
| 89
| 1136.170
| 1136.170
| 27/14, 52/27, (48/25)
| [[27/14]], [[52/27]], ([[48/25]])
|
|
|{{UDnote|step=89}}
|{{UDnote|step=89}}
Line 848: Line 850:
| 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]]
|{{UDnote|step=90}}
|{{UDnote|step=90}}
| u8, t8, HM7
| u8, t8, HM7
Line 857: Line 859:
| 91
| 91
| 1161.702
| 1161.702
| 88/45, 39/20
| [[88/45]], [[39/20]]
| 45/23
| [[45/23]]
|{{UDnote|step=91}}
|{{UDnote|step=91}}
| o8, h8
| o8, h8
Line 866: Line 868:
| 92
| 92
| 1174.468
| 1174.468
| 160/81, 63/32, (49/25)
| [[160/81]], [[63/32]], ([[49/25]])
|
|
|{{UDnote|step=92}}
|{{UDnote|step=92}}
Line 875: Line 877:
| 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}}
|{{UDnote|step=93}}
Line 884: Line 886:
| 94
| 94
| 1200.000
| 1200.000
| 2/1
| [[2/1]]
|
|
|{{UDnote|step=94}}
|{{UDnote|step=94}}
Line 903: Line 905:


== Notation ==
== Notation ==
94edo can be notated in Sagittal using the Athenian extension, with the chroma equating to 9 edosteps and the limma to 7 edosteps,.
=== Ups and downs notation ===
94edo can be written using [[Kite's ups and downs notation]]. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and that quupflat (quadruple-up flat) is equivalent to quid (quintuple-down):
{{Ups and downs sharpness}}
 
=== Sagittal ===
94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian|Athenian set]], with the apotome equal to 9 edosteps and the limma to 7 edosteps.  
{| class="wikitable" style="text-align: center;"
{| class="wikitable" style="text-align: center;"
!Degree
! colspan="2" |Steps
!−9
!'''0'''
!−8
! 1
!−7
! 2
!−6
! 3
!−5
! 4
!−4
! 5
!−3
! 6
!−2
! 7
!−1
! 8
!0
! '''9'''
!+1
!+2
!+3
!+4
!+5
!+6
!+7
!+8
!+9
|-
|-
! rowspan="2" |Symbol
!Evo
!Evo
|{{sagittal|b}}
| rowspan="2" |<big>{{sagittal||//|}}</big>
|{{sagittal|b}}{{sagittal|~|(}}
| rowspan="2" |<big>{{sagittal|~|(}}</big>
|{{sagittal|b}}{{sagittal|/|}}
| rowspan="2" |<big>{{sagittal|/|}}</big>
|{{sagittal|b}}{{sagittal|(|(}}
| rowspan="2" |<big>{{sagittal|(|(}}</big>
|{{sagittal|b}}{{sagittal|/|\}}
| rowspan="2" |<big>{{sagittal|/|\}}</big>
|{{sagittal|\!/}}
| rowspan="2" |<big>{{sagittal|(|)}}</big>
|{{sagittal|(!(}}
|{{sagittal|(!(}}{{sagittal|#}}
|{{sagittal|\!}}
|{{sagittal|\!}}{{sagittal|#}}
|{{sagittal|~!(}}
|{{sagittal|~!(}}{{sagittal|#}}
|{{sagittal||//|}}
|{{sagittal|~|(}}
|{{sagittal|/|}}
|{{sagittal|(|(}}
|{{sagittal|/|\}}
|{{sagittal|#}}{{sagittal|\!/}}
|{{sagittal|#}}{{sagittal|(!(}}
|{{sagittal|#}}{{sagittal|\!}}
|{{sagittal|#}}{{sagittal|~!(}}
|{{sagittal|#}}
|{{sagittal|#}}
|-
|-
!Revo
!Revo
|{{sagittal|\!!/}}
|<big>{{sagittal|~||(}}</big>
|{{sagittal|(!!(}}
|<big>{{sagittal|||\}}</big>
|{{sagittal|!!/}}
|<big>{{sagittal|(||(}}</big>
|{{sagittal|~!!(}}
|<big>{{sagittal|/||\}}</big>
|{{sagittal|(!)}}
|{{sagittal|\!/}}
|{{sagittal|(!(}}
|{{sagittal|\!}}
|{{sagittal|~!(}}
|{{sagittal||//|}}
|{{sagittal|~|(}}
|{{sagittal|/|}}
|{{sagittal|(|(}}
|{{sagittal|/|\}}
|{{sagittal|(|)}}
|{{sagittal|~||(}}
|{{sagittal|||\}}
|{{sagittal|(||(}}
|{{sagittal|/||\}}
|}
|}
The following enharmonics from the Athenian set are present (comma tempered out):


== Approximation to JI ==
* {{sagittal|//|}} = {{Sagittal|/|)}} = {{Sagittal|/|\}} ([[325/324]], [[352/351]])
=== Zeta peak index ===
* {{sagittal|/|}} = {{sagittal||)}} = {{sagittal||\}} ([[225/224]], [[2200/2187]])
{{ZPI
* {{sagittal|)|(}} = {{sagittal|~|(}} ([[3680721/3670016]])
| zpi = 532
* {{sagittal|(|}} = {{sagittal|(|(}} ([[5120/5103]])
| steps = 93.9836761074943
* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])
| step size = 12.7681747480009
 
| tempered height = 8.806201
See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.
| pure height = 8.585151
 
| integral = 1.394050
The JI chord 16:17:18:19:20:21:22:23:24:25:26:27:28:30 from D would be written D{{sagittal||//|}}:E{{sagittal|(!!(}}:E{{sagittal||//|}}:F{{sagittal||//|}}:F{{sagittal|||\}}:G{{sagittal|\!}}:G{{sagittal|/|\}}:G{{sagittal|~|||(}}:A{{sagittal||//|}}:A{{sagittal|(|)}}:B{{sagittal|(!)}}:B{{sagittal||//|}}:C{{sagittal|\!}}:C{{sagittal|||\}}. Music that doesn't modulate much in the 2.3.5.7.11.13.19 subgroup can be notated by only using {{sagittal|/|}} {{sagittal|/|\}} / {{sagittal|\!}} {{sagittal|\!/}} and their apotome complements; where naturals are used for 3 and 19, {{sagittal|/|}} / {{sagittal|\!}} for 5 and 7, and {{sagittal|/|\}} / {{sagittal|\!/}} for 11 and 13.
| gap = 17.832744
| octave = 1200.20842631208
| consistent = 24
| distinct = 15
}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 997: Line 968:
|-
|-
| 2.3
| 2.3
| {{monzo| 149 -94 }}
| {{Monzo| 149 -94 }}
| {{mapping| 94 149 }}
| {{Mapping| 94 149 }}
| −0.054
| −0.054
| 0.054
| 0.054
Line 1,005: Line 976:
| 2.3.5
| 2.3.5
| 32805/32768, 9765625/9565938
| 32805/32768, 9765625/9565938
| {{mapping| 94 149 218 }}
| {{Mapping| 94 149 218 }}
| +0.442
| +0.442
| 0.704
| 0.704
Line 1,012: Line 983:
| 2.3.5.7
| 2.3.5.7
| 225/224, 3125/3087, 118098/117649
| 225/224, 3125/3087, 118098/117649
| {{mapping| 94 149 218 264 }}
| {{Mapping| 94 149 218 264 }}
| +0.208
| +0.208
| 0.732
| 0.732
Line 1,019: Line 990:
| 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
| {{mapping| 94 149 218 264 325 }}
| {{Mapping| 94 149 218 264 325 }}
| +0.304
| +0.304
| 0.683
| 0.683
Line 1,026: Line 997:
| 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
| {{mapping| 94 149 218 264 325 348 }}
| {{Mapping| 94 149 218 264 325 348 }}
| +0.162
| +0.162
| 0.699
| 0.699
Line 1,033: Line 1,004:
| 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
| {{mapping| 94 149 218 264 325 348 384 }}
| {{Mapping| 94 149 218 264 325 348 384 }}
| +0.238
| +0.238
| 0.674
| 0.674
Line 1,040: Line 1,011:
| 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
| {{mapping| 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 1,047: Line 1,018:
| 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
| {{mapping| 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
Line 1,075: Line 1,046:
| 25/24
| 25/24
| [[Betic]]
| [[Betic]]
|-
| 1
| 7\94
| 89.36
| 21/20
| [[Slithy]]
|-
|-
| 1
| 1
| 11\94
| 11\94
| 140.43
| 140.43
| 243/224
| 13/12
| [[Tsaharuk]] / [[quanic]]
| [[Tsaharuk]] / [[quanic]]
|-
|-
Line 1,093: Line 1,070:
| 147/128
| 147/128
| [[Septiquarter]]
| [[Septiquarter]]
|-
| 1
| 25\94
| 319.15
| 6/5
| [[Dhaivatic]]
|-
|-
| 1
| 1
Line 1,130: Line 1,113:
| [[Kleischismic]]
| [[Kleischismic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


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.
Temperaments to which 94et can be detempered:
* [[Satin]] ({{nowrap| 94 & 217 }})
* [[Gariwizmic]] ({{nowrap| 94 & 176 }})
* {{nowrap| 94 & 422 }}


* {{nowrap|46 &amp; 94}}
=== Commas ===
* {{nowrap|68 &amp; 94}}
94et [[tempering out|tempers out]] the following [[comma]]s using its 23-limit patent [[val]], {{val| 94 149 218 264 325 348 384 399 425 }}.
* {{nowrap|53 &amp; 94}}  (one garibaldi)
* {{nowrap|41 &amp; 94}}  (another garibaldi, only differing in the mappings of 17 and 23)
* {{nowrap|135 &amp; 94}} (another garibaldi)
* {{nowrap|130 &amp; 94}}  (a pogo extension)
* {{nowrap|58 &amp; 94}}  (a supers extension)
* {{nowrap|50 &amp; 94}}
* {{nowrap|72 &amp; 94}}  (a gizzard extension)
* {{nowrap|80 &amp; 94}}
* 94 solo  (a rank one temperament!)


Temperaments to which 94et can be detempered:
{| class="commatable wikitable center-1 center-2 right-3 center-6"
 
! [[Harmonic limit|Prime<br>limit]]
* [[Satin]] ({{nowrap|94 & 311}})  
! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>
* {{nowrap|94 & 422}}  
! [[Cents]]
! [[Monzo]]
! colspan="2" | [[Kite's color notation|Color name]]
! Name(s)
|-
| 3
| <abbr title="36893488147419103232/36472996377170786403">(90 digits)</abbr>
| 16.22
| {{Monzo| 149 -94 }}
| Wa-94
| 94-edo
| [[94-comma]]
|-
| 5
| [[32805/32768|(10 digits)]]
| 1.95
| {{Monzo| -15 8 1 }}
| Layo
| Ly
| [[Schisma]]
|-
| 7
| [[3125/3087]]
| 21.18
| {{Monzo| 0 -2 5 -3 }}
| Triru-aquinyo
| r<sup>3</sup>y<sup>5</sup>
| Gariboh comma
|-
| 7
| [[4000/3969]]
| 13.47
| {{Monzo| 5 -4 3 -2 }}
| Rurutriyo
| rry<sup>3</sup>
| Octagar comma
|-
| 7
| [[225/224]]
| 7.71
| {{monzo| -5 2 2 -1 }}
| Ruyoyo
| ryy
| Marvel comma
|-
| 7
| <abbr title="36893488147419103232/36472996377170786403">(12 digits)</abbr>
| 6.59
| {{monzo| 1 10 0 -6 }}
| Latribiru
| L6r
| Stearnsma
|-
| 7
| [[5120/5103]]
| 5.76
| {{monzo| 10 -6 1 -1 }}
| Saruyo
| sry
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| 3.80
| {{Monzo| 25 -14 0 -1 }}
| Sasaru
| ssr
| [[Garischisma]]
|-
| 11
| [[385/384]]
| 4.50
| {{Monzo| -7 -1 1 1 1 }}
| Lozoyo
| 1ozg
| Keenanisma
|-
| 11
| [[540/539]]
| 3.21
| {{Monzo| 2 3 1 -2 -1 }}
| Lururuyo
| 1urry
| Swetisma
|-
| 11
| [[9801/9800]]
| 0.17
| {{Monzo| -3 4 -2 -2 2 }}
| Bilorugu
| 1oorrgg-2
| Kalisma
|-
| 13
| [[275/273]]
| 12.64
| {{Monzo| 0 -1 2 -1 1 -1 }}
| Thuloruyoyo
| 3u1oryy
| Gassorma
|-
| 13
| [[640/637]]
| 8.13
| {{Monzo| 7 0 1 -2 0 -1 }}
| Thururuyo
| 3urry
| Huntma
|-
| 13
| [[1188/1183]]
| 7.30
| {{Monzo| 2 3 0 -1 1 -2 }}
| Thuthuloru
| 3uu1or
| Kestrel comma
|-
| 13
| [[325/324]]
| 5.34
| {{Monzo| -2 -4 2 0 0 1 }}
| Thoyoyo
| 3oyy
| Marveltwin comma
|-
| 13
| [[352/351]]
| 4.93
| {{Monzo| 5 -3 0 0 1 -1 }}
| Thulo
| 3u1o
| Major minthma
|-
| 13
| [[847/845]]
| 4.09
| {{Monzo| 0 0 -1 1 2 -2 }}
| Thuthulolozogu
| 3uu1oozg
| Cuthbert comma
|-
| 13
| [[729/728]]
| 2.38
| {{Monzo| -3 6 0 -1 0 -1 }}
| Lathuru
| L3ur
| Squbema
|-
| 13
| [[2080/2079]]
| 0.83
| {{Monzo| 5 -3 1 -1 -1 1 }}
| Tholuruyo
| 3o1ury
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| 0.42
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| Sathurugu
| s3urg
| Minisma
|-
| 13
| [[4225/4224]]
| 0.41
| {{Monzo| -7 -1 2 0 -1 2 }}
| Thotholuyoyo
| 3oo1uyy
| Leprechaun comma
|-
| 17
| [[289/288]]
| 6.00
| {{Monzo| 5 -2 0 0 0 0 2 }}
| Soso
| 17oo
| Semitonisma
|-
| 17
| [[715/714]]
| 2.42
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| Sutholoruyo
| 17u3o1ory
| Septendecimal bridge comma
|-
| 19
| [[361/360]]
| 4.80
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| Nonogu
| 19oog2
| Go comma
|-
| 19
| [[513/512]]
| 3.38
| {{Monzo| -9 3 0 0 0 0 0 1 }}
| Lano
| L19o
| Boethius' comma
|-
| 19
| [[1216/1215]]
| 1.42
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| Sanogu
| s19og
| Eratosthenes' comma
|-
| 19
| [[11859211/11859210|(16 digits)]]
| 0.00
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
| <small>Quadno-athoquadlu-azogu</small>
| <small>9o<sup>4</sup>3o1u<sup>4</sup>zg</small>
| Tredekisma
|-
| 23
| [[300/299]]
| 5.78
| {{Monzo| 2 1 2 0 0 -1 0 0 -1 }}
| Twethuthuyoyo
| 23u3uyy
| Major naiadvicema
|-
| 23
| [[323/322]]
| 5.37
| {{Monzo| -1 0 0 -1 0 0 1 1 -1 }}
| Twethunosoru
| 23u19o17or
| Major semivicema
|-
| 23
| [[391/390]]
| 4.43
| {{Monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| Twethosothugu
| 23o17o3ug
| Minor naiadvicema
|-
| 23
| [[460/459]]
| 3.77
| {{Monzo| 2 -3 1 0 0 0 -1 0 1 }}
| Twethosuyo
| 23o17uy
| Scanisma, vicewolf comma
|-
| 23
| [[484/483]]
| 3.58
| {{Monzo| 2 -1 0 -1 2 0 0 0 -1 }}
| Twethuloloru
| 23u1oor
| Pittsburghisma
|}


== Scales ==
== Scales ==
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* [[Garibaldi12]]
* [[Garibaldi12]]
* [[Garibaldi17]]
* [[Garibaldi17]]
* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]]
* [[Overtone scale#Over-3 scales|Mode 12]]: 11 10 9 9 8 8 7 7 7 6 6 6


== Instruments ==
== Instruments ==
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; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Zx4xbJhXmgc ''microtonal improvisation in 94edo''] (2025)
* [https://www.youtube.com/shorts/Zx4xbJhXmgc ''microtonal improvisation in 94edo''] (2025)
* [https://www.youtube.com/shorts/HMD8QFpB2-U ''94edo improv''] (2025)
* [https://www.youtube.com/watch?v=KrPQ_tPsvzQ ''Waltz in 94edo''] (2025)
* [https://www.youtube.com/shorts/KJ5dbF4aH2A ''Twinleaf Town - Pokémon Diamond and Pearl (microtonal cover in 94edo)''] (2026)
; [[Eufalesio]]
* [https://soundcloud.com/eufalesio/expanding-my-horizons?in=eufalesio/sets/microtonal-stuff "Expanding my Horizons"] from [https://soundcloud.com/eufalesio/sets/microtonal-stuff ''Microtonal stuff''] (2022)


; [[Cam Taylor]]
; [[Cam Taylor]]
Line 1,174: Line 1,419:


[[Category:94edo| ]] <!-- main article -->
[[Category:94edo| ]] <!-- main article -->
[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:Garibaldi]]
[[Category:Garibaldi]]
[[Category:Marvel]]
[[Category:Marvel]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/94edo"