94edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Lhearne (talk | contribs)
autopatrolled
1,510 edits
→Intervals: added a table with interval names, work in progress
Lhearne (talk | contribs)
autopatrolled
1,510 edits
→Intervals: 'finished' it. I hope this can be more helpful than confusing
Line 14: Line 14:
== Intervals ==
== Intervals ==
{{Main| Table of 94edo intervals }}
{{Main| Table of 94edo intervals }}
Assuming [[23-limit]] [[patent val]] <94 149 218 264 325 348 384 399 425|, here is a table of intervals as approximated by [[94edo]] steps, and their corresponding 13-limit well-ordered extended diatonic interval names. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'k' (representing 81/80), 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' indicates alteration by the septimal comma, [[64/63]]; 'K' indicates alteration by the syntonic comma, [[81/80]]; 'U' by the undecimal quatertone, [[33/32]]; 'L' by pentacircle comma, [[896/891]]; 'O' by [[45/44]]; 'R' by the rastma, [[243/242]]; 'T' by the tridecimal quartertone, [[1053/1024]]; and finally, 'H', by [[40/39]]. Capital letters alter downward, lowercase alter upwards. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'K' down from augmented and major, or up from diminished and minor intervals are also added in brackets, along with their associated (5-limit) intervals.  
{| class="wikitable"
{| class="wikitable"
|+94edo well-ordered extended diatonic (WED) interval names
|+94edo well-ordered extended diatonic (WED) interval names
Line 28: Line 28:
| 1
| 1
|12.766
|12.766
|896/891, 243/242, (3125/3072, 245/243, 99/98)
|896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98)
|L1, R1
|L1, R1
|
|
|
|large unison, rastma
|85/84
|85/84
|-
|-
Line 39: Line 39:
|K1, S1
|K1, S1
|
|
|
|komma, super unison
|
|
|-
|-
Line 47: Line 47:
|O1, H1
|O1, H1
|
|
|
|on unison, hyper unison
|46/45
|46/45
|-
|-
Line 53: Line 53:
| 51.064
| 51.064
|33/32, (128/125, 36/35)
|33/32, (128/125, 36/35)
|U1, T1
|U1, T1, hm2
|
|
|
|uber unison, tall unison, hypo minor second
|
|
|-
|-
Line 63: Line 63:
|sm2, uA1, tA1, (kkA1)
|sm2, uA1, tA1, (kkA1)
|dd3
|dd3
|
|sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison)
|
|
|-
|-
Line 71: Line 71:
|lm2, oA1
|lm2, oA1
|
|
|
|little minor second, off augmented unison
|
|
|-
|-
Line 79: Line 79:
|m2, kA1
|m2, kA1
|m2
|m2
|
|minor second, komma-down augmented unison
|19/18
|19/18, 20/19
|-
|-
|8
|8
Line 87: Line 87:
|Rm2, rA1
|Rm2, rA1
|
|
|
|rastmic minor second, rastmic augmented unison
|17/16, 18/17
|17/16, 18/17
|-
|-
Line 95: Line 95:
|Km2, A1
|Km2, A1
|A1
|A1
|
|classic minor second, augmented unison
|
|
|-
|-
Line 103: Line 103:
|Om2, LA1
|Om2, LA1
|
|
|
|oceanic minor second, large augmented unison
|
|
|-
|-
Line 109: Line 109:
|140.426
|140.426
|88/81, 13/12, 243/224, (27/25)
|88/81, 13/12, 243/224, (27/25)
|n2, Tm2, SA1, (kkm2)
|n2, Tm2, SA1, (KKm2)
|
|
|
|less neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
|
|
|-
|-
Line 117: Line 117:
|153.191
|153.191
|12/11, (35/32)
|12/11, (35/32)
|N2, HA1
|N2, tM2, HA1
|ddd4
|ddd4
|
|greater netral second, tiny major second, hyper augmented unison
|
|
|-
|-
Line 127: Line 127:
|oM2
|oM2
|
|
|
|off major second
|
|
|-
|-
Line 135: Line 135:
|kM2
|kM2
|d3
|d3
|
|komma-down major second
|
|
|-
|-
Line 143: Line 143:
|rM2
|rM2
|
|
|
|rastmic major second
|19/17
|19/17
|-
|-
Line 151: Line 151:
|M2
|M2
|M2
|M2
|
|major second
|
|
|-
|-
Line 159: Line 159:
|LM2
|LM2
|
|
|
|large major second
|17/15
|17/15
|-
|-
Line 167: Line 167:
|SM2
|SM2
|AA1
|AA1
|
|super major second
|
|
|-
|-
Line 175: Line 175:
|HM2
|HM2
|
|
|
|hyper major second
|23/20
|23/20
|-
|-
Line 183: Line 183:
|hm3
|hm3
|
|
|
|hypo minor third
|22/19
|22/19
|-
|-
Line 191: Line 191:
|sm3, (kkA2)
|sm3, (kkA2)
|dd4
|dd4
|
|sub minor third, (classic augmented second)
|
|
|-
|-
Line 199: Line 199:
|lm3
|lm3
|
|
|
|little minor third
|20/17
|20/17
|-
|-
| 23
| 23
|293.617
|293.617
|32/27, (25/21)
|32/27, (25/21, 13/11)
|m3
|m3
|m3
|m3
|minor third
|
|
|13/11
|-
|-
|24
|24
Line 215: Line 215:
|Rm3
|Rm3
|
|
|
|rastmic minor third
|
|
|-
|-
Line 223: Line 223:
|Km3
|Km3
|A2
|A2
|
|classic minor third
|
|
|-
|-
Line 231: Line 231:
|Om3
|Om3
|
|
|
|on minor third
|17/14, 23/19
|17/14, 23/19
|-
|-
Line 239: Line 239:
|n3, Tm3
|n3, Tm3
|AAA1
|AAA1
|
|lesser neutral third, tall minor third
|
|
|-
|-
Line 247: Line 247:
|N3, tM3
|N3, tM3
|ddd5
|ddd5
|
|greater neutral third, tiny major third
|
|
|-
|-
Line 255: Line 255:
|oM3
|oM3
|
|
|
|off major third
|21/17
|21/17
|-
|-
Line 263: Line 263:
|kM3
|kM3
|d4
|d4
|
|classic major third
|
|
|-
|-
Line 271: Line 271:
|rM3
|rM3
|
|
|
|rastmic major
|
|
|-
|-
Line 279: Line 279:
|M3
|M3
|M3
|M3
|
|major third
|19/15
|19/15, 24/19
|-
|-
|33
|33
Line 287: Line 287:
|LM3
|LM3
|
|
|
|large major third
|23/18
|23/18
|-
|-
Line 295: Line 295:
|SM3, (KKd4)
|SM3, (KKd4)
|AA2
|AA2
|
|super major third, (classic diminished fourth)
|
|
|-
|-
Line 303: Line 303:
|HM3
|HM3
|ddd6
|ddd6
|
|hyper major third
|22/17
|22/17
|-
|-
Line 311: Line 311:
|h4
|h4
|
|
|
|hypo fourth
|
|
|-
|-
Line 319: Line 319:
|s4
|s4
|dd5
|dd5
|
|sub fourth
|
|
|-
|-
Line 327: Line 327:
|l4
|l4
|
|
|
|little fourth
|
|
|-
|-
Line 335: Line 335:
|P4
|P4
|P4
|P4
|
|perfect fourth
|
|
|-
|-
| 40
| 40
|510.638
|510.638
|162/121, (35/36)
|162/121, (35/26)
|R4
|R4
|
|
|
|rastmic fourth
|
|
|-
|-
Line 351: Line 351:
|K4
|K4
|A3
|A3
|
|komma-up fourth
|19/14, 23/17
|19/14, 23/17
|-
|-
Line 359: Line 359:
|O4
|O4
|
|
|
|on fourth
|
|
|-
|-
Line 367: Line 367:
|U4, T4
|U4, T4
|AAA2
|AAA2
|
|uber/undecimal fourth, tall fourth
|
|
|-
|-
|44
|44
|561.702
|561.702
|243/176, 18/13, (25/18)
|18/13, (25/18)
|uA4, tA4, (kkA4)
|tA4, uA4, (kkA4)
|dd6
|dd6
|
|tiny augmented fourth, unter augmented fourth, (lesser classic augmented fourth)
|
|
|-
|-
Line 383: Line 383:
|ld5, oA4
|ld5, oA4
|
|
|
|little diminished fifth, off augmented fourth
|
|
|-
|-
Line 391: Line 391:
|kA4
|kA4
|d5
|d5
|
|greater classic augmented fourth
|
|
|-
|-
Line 399: Line 399:
|rA4, Rd5
|rA4, Rd5
|
|
|
|rastmic augmented fourth, rastmic diminished fifth
|17/12, 24/17
|17/12, 24/17
|-
|-
|48
|48
|612.766
|612.766
|
|64/45, (10/7)
|
|Kd5
|A4
|A4
|
|lesser classic diminished fifth
|
|
|-
|-
|49
|49
|625.532
|625.532
|63/44
|LA4, Od5
|
|
|
|large augmented fourth, off diminished fifth
|
|
|
|
|-
|-
| 50
| 50
|638.298
|638.298
|
|13/9, (36/25)
|
|Td5, Ud5, (KKd5)
|
|AA3
|
|tall diminished fifth, uber diminished fifth, (greater classic diminished fifth)
|
|
|-
|-
|51
|51
|651.064
|651.064
|
|16/11
|
|u5, t5
|ddd7
|ddd7
|
|unter/undecimal fifth, tiny fifth
|
|
|-
|-
| 52
| 52
|663.830
|663.830
|22/15
|o5
|
|
|
|off fifth
|
|
|
|
|-
|-
|53
|53
|676.596
|676.596
|
|40/27
|
|k5
|d6
|d6
|
|komma-down fifth
|
|28/19, 34/23
|-
|-
|54
|54
| 689.362
| 689.362
|121/81, (52/35)
|r5
|
|
|
|rastmic fifth
|
|
|
|
|-
|-
|55
|55
|702.128
|702.128
|
|3/2
|
|P5
|P5
|P5
|
|perfect fifth
|
|
|-
|-
|56
|56
|714.894
|714.894
|448/297
|L5
|
|
|
|large fifth
|
|
|
|
|-
|-
|57
|57
| 727.660
| 727.660
|
|32/21
|
|S5
|AA4
|AA4
|
|super fifth
|
|
|-
|-
|58
|58
|740.426
|740.426
|20/13
|H5
|
|
|
|hyper fifth
|
|
|
|
|-
|-
|59
|59
|753.191
|753.191
|
|208/135
|
|hm6
|
|AAA3
|
|hypo minor sixth
|
|17/11
|-
|-
|60
|60
|765.957
|765.957
|
|14/9, (128/75)
|
|sm6, (kkA5)
|dd7
|dd7
|
|sub minor sixth, (classic augmented fifth)
|
|
|-
|-
| 61
| 61
|778.723
|778.723
|11/7
|lm6
|
|
|
|little minor sixth
|
|36/23
|
|
|-
|-
|62
|62
|791.489
|791.489
|
|128/81
|
|m6
|m6
|m6
|
|minor sixth
|
|19/12, 30/19
|-
|-
|63
|63
|804.255
|804.255
|192/121
|Rm6
|
|
|
|rastmic minor sixth
|
|27/17
|
|
|-
|-
|64
|64
|817.021
|817.021
|
|8/5
|
|Km6
|A5
|A5
|
|classic minor sixth
|
|
|-
|-
|65
|65
|829.787
|829.787
|160/99, (21/13)
|Om6
|
|
|
|on minor sixth
|
|34/21
|
|
|-
|-
|66
|66
|842.553
|842.553
|
|44/27, 13/8, (81/50, 80/49)
|
|n6, Tm6
|AAA4
|AAA4
|
|less neutral sixth, tall minor sixth
|
|
|-
|-
|67
|67
| 855.319
| 855.319
|
|18/11, 64/39, (400/243, 49/30)
|
|N6, tM6
|ddd8
|ddd8
|
|greater neutral sixth, tiny minor sixth
|
|
|-
|-
|68
|68
| 868.085
| 868.085
|33/20
|oM6
|
|
|
|off major sixth
|
|28/17, 38/23
|
|
|-
|-
|69
|69
|880.851
|880.851
|
|5/3
|
|kM6
|d7
|d7
|
|classic major sixth
|
|
|-
|-
|70
|70
|893.617
|893.617
|121/72
|rM6
|
|
|
|rastmic major sixth
|
|
|
|
|-
|-
|71
|71
|906.383
|906.383
|
|27/16, (42/35, 22/13)
|
|M6
|M6
|M6
|
|major sixth
|
|
|-
|-
|72
|72
|919.149
|919.149
|56/33
|LM6
|
|
|
|large major sixth
|
|17/10
|
|
|-
|-
|73
|73
|931.915
|931.915
|
|12/7, 128/75
|
|SM6, (KKd7)
|AA5
|AA5
|
|super major sixth (classic diminished seventh)
|
|
|-
|-
|74
|74
|944.681
|944.681
|45/26
|HM6
|
|
|
|hyper major sixth
|
|19/11
|
|
|-
|-
|75
|75
|957.447
|957.447
|26/15
|hm7
|
|
|
|hypo minor seventh
|
|40/23
|
|
|-
|-
|76
|76
|970.213
|970.213
|
|7/4
|
|sm7
|dd8
|dd8
|
|sub minor seventh
|
|
|-
|-
|77
|77
|982.979
|982.979
|99/56, (44/25)
|lm7
|
|
|
|little minor seventh
|
|30/17
|
|
|-
|-
|78
|78
|995.745
|995.745
|
|16/9
|
|m7
|m7
|m7
|
|minor seventh
|
|
|-
|-
| 79
| 79
|1008.511
|1008.511
|216/121
|Rm7
|
|
|
|rastmic minor seventh
|
|34/19
|
|
|-
|-
|80
|80
|1021.277
|1021.277
|
|9/5
|
|Km7
|A6
|A6
|
|classic minor seventh
|
|
|-
|-
|81
|81
|1034.043
|1034.043
|20/11
|Om7
|
|
|
|on minor seventh
|
|
|
|
|-
|-
|82
|82
| 1046.809
| 1046.809
|
|11/6, (64/35)
|
|n7, Tm7, hd8
|AAA5
|AAA5
|
|less neutral seventh, tall minor seventh, hypo diminished octave
|
|
|-
|-
|83
|83
|1059.574
|1059.574
|81/44, 24/13, (50/27)
|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)
|oM7, ld8
|
|
|
|off major seventh, little diminished octave
|
|
|
|
|-
|-
|85
|85
|1085.106
|1085.106
|
|15/8, (28/15)
|
|kM7, d8
|d8
|d8
|
|classic major seventh, diminished octave
|
|
|-
|-
|86
|86
|1097.872
|1097.872
|121/64
|rM7, Rd8
|
|
|
|rastmic major seventh, rastmic diminished octave
|
|32/17, 17/9
|
|
|-
|-
|87
|87
|1110.638
|1110.638
|
|243/128, 256/135, (40/21)
|
|M7, Kd8
|M7
|M7
|
|major seventh, komma-up diminished octave
|
|36/19, 19/10
|-
|-
|88
|88
|1123.404
|1123.404
|21/11
|LM7, Od8
|
|
|
|large major seventh, on diminished octave
|
|
|
|
|-
|-
|89
|89
|1136.170
|1136.170
|
|27/14, 52/27, (48/25)
|
|SM7, Td8, Ud8, (KKd8)
|AA6
|AA6
|
|super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave)
|
|
|-
|-
|90
|90
|1148.936
|1148.936
|64/33, (35/18)
|u8, t8, HM7
|
|
|
|unter octave, tiny octave, hyper major seventh
|
|33/17
|
|
|-
|-
|91
|91
| 1161.702
| 1161.702
|88/45, 39/20
|o8, h8
|
|
|
|off octave, hypo octave
|
|45/23
|
|
|-
|-
|92
|92
|1174.468
|1174.468
|160/81, 63/32, (49/25)
|k8, s8
|
|
|
|komma-down octave, sub octave
|
|
|
|
|-
|-
|93
|93
|1187.234
|1187.234
|891/448, 484/243, (486/245, 99/50, 196/99)
|l8, r8
|
|
|
|little octave, octave - rastma
|
|
|
|
|-
|-
|94
|94
|1200.000
|1200.000
|
|2/1
|
|P8
|P1
|P8
|
|perfect octave
|
|
|}
|}

Revision as of 03:53, 20 April 2023

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

Template:EDO intro

Theory

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

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.

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

Prime harmonics

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

Intervals

Main article: Table of 94edo intervals

Assuming 23-limit patent val <94 149 218 264 325 348 384 399 425|, here is a table of intervals as approximated by 94edo steps, and their corresponding 13-limit well-ordered extended diatonic interval names. 'S' indicates alteration by the septimal comma, 64/63; 'K' indicates alteration by the syntonic comma, 81/80; 'U' by the undecimal quatertone, 33/32; 'L' by pentacircle comma, 896/891; 'O' by 45/44; 'R' by the rastma, 243/242; 'T' by the tridecimal quartertone, 1053/1024; and finally, '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.

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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [149 -94⟩ [⟨94 149]] -0.054 0.054 0.43
2.3.5 32805/32768, 9765625/9565938 [⟨94 149 218]] +0.442 0.704 5.52
2.3.5.7 225/224, 3125/3087, 118098/117649 [⟨94 149 218 264]] +0.208 0.732 5.74
2.3.5.7.11 225/224, 385/384, 1331/1323, 2200/2187 [⟨94 149 218 264 325]] +0.304 0.683 5.35
2.3.5.7.11.13 225/224, 275/273, 325/324, 385/384, 1331/1323 [⟨94 149 218 264 325 348]] +0.162 0.699 5.48
2.3.5.7.11.13.17 170/169, 225/224, 275/273, 289/288, 325/324, 385/384 [⟨94 149 218 264 325 348 384]] +0.238 0.674 5.28
2.3.5.7.11.13.17.19 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384 [⟨94 149 218 264 325 348 384 399]] +0.323 0.669 5.24
2.3.5.7.11.13.17.19.23 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322 [⟨94 149 218 264 325 348 384 399 425]] +0.354 0.637 4.99

94et is lower in relative error than any previous equal temperaments in the 23-limit, and the equal temperament that does better in this subgroup is 193.

Rank-2 temperaments

Periods
per 8ve 		Generator Cents Associated
Ratio 		Temperament
1 3\94 38.30 49/48 Slender
1 5\94 63.83 25/24 Sycamore / betic
1 11\94 140.43 243/224 Tsaharuk / quanic
1 13\94 165.96 11/10 Tertiaschis
1 19\94 242.55 147/128 Septiquarter
1 39\94 497.87 4/3 Helmholtz / garibaldi / cassandra
2 2\94 25.53 64/63 Ketchup
2 11\94 140.43 27/25 Fifive
2 30\94 382.98 5/4 Wizard / gizzard
2 34\94 434.04 9/7 Pogo / supers
2 43\94 548.94 11/8 Kleischismic

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

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

Temperaments to which 94et can be detempered:

  • Satin (94&311) ⟨⟨ 3 70 -42 69 -34 50 85 83 … ]]
  • 94&422 ⟨⟨ 8 124 -18 90 -28 102 164 96 … ]]

Scales

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 nohajira, pajara or miracle mos scale a as circulating temperament[clarification needed].

Circulating temperaments in 94edo
Tones Pattern L:s
5 4L 1s 19:18
6 4L 2s 16:15
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 8:7
13 3L 10s
14 10L 4s 7:6
15 4L 11s
16 14L 2s 6:5
17 9L 8s
18 4L 14s
19 18L 1s 5:4
20 14L 6s
21 10L 11s
22 6L 16s
23 2L 21s
24 22L 2s 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 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 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
Retrieved from "https://en.xen.wiki/index.php?title=94edo&oldid=108479"