99edo: Difference between revisions

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Line 24: Line 24:
| 1
| 1
| 7\99
| 7\99
| 84.848
| 84.8484
| 21/20
|21/20
| [[Amicable]]
|[[Amicable]]
|-
|-
| 1
|1
| 16\99
|16\99
| 193.939
|193.9393
| 28/25
|28/25
| [[Hemiwürschmidt]] (99e) / Hemithir (99ef) / Hemiwur (99f)
|[[Hemiwürschmidt]] (99e) / Hemithir (99ef) / Hemiwur (99f)
|-
|-
| 1
|1
| 19\99
|19\99
| 230.303
|230.303
| 8/7
|8/7
| [[Gamera]]
|[[Gamera]]
|-
|-
| 1
|1
| 20\99
|20\99
| 242.424
| 242.4242
| 147/128
|147/128
| [[Septiquarter]]
|[[Septiquarter]]
|-
|-
| 1
|1
| 25\99
|25\99
| 303.030
|303.0303
| 25/21
|25/21
| [[Quinmite]]
|[[Quinmite]]
|-
|-
| 1
| 1
| 27\99
|27\99
| 315.152
|315.1515
| 6/5
|6/5
| [[Parakleismic]] / Parkleismic / Paradigmic (99e)
|[[Parakleismic]] / Parkleismic / Paradigmic (99e)
|-
|-
| 1
|1
| 28\99
|28\99
| 339.394
| 339.3939
| 128/105
|128/105
| [[Amity]] (99ef) / Hitchcock
|[[Amity]] (99ef) / Hitchcock
|-
|-
| 1
|1
| 29\99
|29\99
| 351.515
| 351.5151
| 49/40
|49/40
| [[Hemififths]] (99ef)
|[[Hemififths]] (99ef)
|-
|-
| 1
|1
| 32\99
|32\99
| 387.879
|387.8787
| 5/4
|5/4
| [[Würschmidt]] / Whirrschmidt
|[[Würschmidt]] / Whirrschmidt
|-
|-
| 1
|1
| 37\99
|37\99
| 448.485
|448.4848
| 35/27
|35/27
| [[Semidimfourth]]
|[[Semidimfourth]]
|-
|-
| 3
|3
| 5\99
|5\99
| 60.606
|60.606
| 28/27
|28/27
| [[Chromat]]
|[[Chromat]]
|-
|-
| 3
|3
| 13\99
|13\99
| 157.576
|157.5757
| 35/32
|35/32
| [[Nessafof]]
|[[Nessafof]]
|-
|-
| 3
| 3
| 41\99 (8\99)
|41\99 (8\99)
| (4)96.970
| (4)96.9696
| 4/3 (18/17~19/18)
|4/3 (18/17~19/18)
| [[Misty]]
|[[Misty]]
|-
|-
| 9
| 9
| 4\99
|4\99
| 48.485
|48.4848
| 36/35
|36/35
| [[Ennealimmal]]
|[[Ennealimmal]]
|-
|-
| 11
|11
| 41\99 (4\99)
|41\99 (4\99)
| 496.970 (48.485)
|496.9696 (48.4848)
| 4/3 (36/35)
|4/3 (36/35)
| [[Hendecatonic]]
|[[Hendecatonic]]
|}
|}


== Scales ==
* [[tutone6]]
* [[tutone7]]
* [[tutone13]]
* [[zeus7tri]]
* [[zeus8tri]]


== Music ==
==Scales==
* [http://www.archive.org/details/NonagintaEtNovem Nonaginta et Novem] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/Nonaginta%20et%20Novem.mp3 play]'' by [[Gene Ward Smith]]
*[[tutone6]]
* [http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3 Benny] Smith-Palestrina in [[zeus7tri]]
*[[tutone7]]
*[[tutone13]]
*[[zeus7tri]]
*[[zeus8tri]]
 
 
Since 98edo has a step of 12.1212 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 99edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|20:19
|-
|6
|[[3L 3s]]
|17:16
|-
| 7
|[[1L 6s]]
|15:14
|-
|8
|[[3L 5s]]
|13:12
|-
|9
|[[9edo]]
|equal
|-
|10
|[[9L 1s]]
|10:9
|-
|11
|[[11edo]]
|equal
|-
|12
|[[3L 9s]]
|9:8
|-
|13
|[[8L 5s]]
| rowspan="2" |8:7
|-
|14
|[[1L 13s]]
|-
|15
| [[9L 6s]]
| rowspan="2" | 7:6
|-
|16
|[[3L 13s]]
|-
|17
|[[14L 3s]]
| rowspan="3" |6:5
|-
|18
|9L 9s
|-
|19
|4L 15s
|-
|20
| 19L 1s
| rowspan="5" |5:4
|-
|21
|15L 6s
|-
|22
|11L 11s
|-
|23
| 7L 16s
|-
|24
|3L 21s
|-
|25
|24L 1s
| rowspan="8" |4:3
|-
|26
|21L 5s
|-
|27
|18L 9s
|-
|28
|15L 13s
|-
| 29
|[[12L 17s]]
|-
|30
|9L 21s
|-
|31
|[[6L 25s]]
|-
| 32
|3L 29s
|-
|33
|[[33edo]]
|equal
|-
|34
|31L 3s
| rowspan="16" |3:2
|-
|35
|29L 6s
|-
|36
|27L 9s
|-
|37
|25L 12s
|-
|38
|23L 15s
|-
|39
|21L 18s
|-
|40
|19L 21s
|-
|41
|17L 24s
|-
|42
|15L 27s
|-
|43
|13L 30s
|-
|44
|11L 33s
|-
|45
|9L 36s
|-
|46
|7L 39s
|-
|47
|5L 42s
|-
|48
|3L 45s
|-
|49
|1L 48s
|-
|50
|49L 1s
| rowspan="30" |2:1
|-
|51
|48L 3s
|-
|52
|47L 5s
|-
|53
|46L 7s
|-
|54
|45L 9s
|-
|55
|44L 11s
|-
|56
|43L 13s
|-
|57
|42L 15s
|-
|58
|41L 17s
|-
|59
|40L 19s
|-
|60
|39L 21s
|-
|61
|38L 23s
|-
|62
|37L 25s
|-
|63
|36L 27s
|-
|64
|35L 29s
|-
|65
|34L 31s
|-
|66
|33L 33s
|-
|67
|32L 35s
|-
|68
|31L 37s
|-
|69
|30L 39s
|-
|70
|29L 41s
|-
|71
|28L 43s
|-
|72
|27L 45s
|-
|73
|26L 47s
|-
|74
|25L 49s
|-
|75
|24L 51s
|-
|76
|23L 53s
|-
|77
|22L 55s
|-
|78
|21L 57s
|-
|79
|20L 59s
|}
==Music==
*[http://www.archive.org/details/NonagintaEtNovem Nonaginta et Novem] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/Nonaginta%20et%20Novem.mp3 play]'' by [[Gene Ward Smith]]
*[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3 Benny] Smith-Palestrina in [[zeus7tri]]


== See also ==
==See also==
* [[157edt]]
*[[157edt]]
* [[58edf]]
*[[58edf]]
* [[87edo]], [[94edo]], [[111edo]], similarly sized edos all with consistency in higher harmonics.  
*[[87edo]], [[94edo]], [[111edo]], similarly sized edos all with consistency in higher harmonics.
* [[198edo]], the half-sized edo to reconcile the mappings of 11 and 13.  
*[[198edo]], the half-sized edo to reconcile the mappings of 11 and 13.
* [[105edo]], a similarly sized edo that supports meantone, septimal meantone, undecimal meantone and grosstone
*[[105edo]], a similarly sized edo that supports meantone, septimal meantone, undecimal meantone and grosstone


[[Category:Theory]]
[[Category:Theory]]

Revision as of 03:48, 19 April 2021

99edo is the equal division of the octave into 99 parts of 12.1212 cents each.

Theory

99edo is a very strong 7-limit (and 9 odd limit) temperament, but extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It tempers out 393216/390625 (würschmidt comma) and 1600000/1594323 (amity comma) in the 5-limit; 2401/2400 (breedsma), 3136/3125 (hemimean comma), and 4375/4374 (ragisma) in the 7-limit, supporting hemififths, amity, parakleismic, hemiwürschmidt and ennealimmal temperaments, and is pretty well a perfect tuning for hendecatonic temperament. It has a sound defined by the slight sharpness (1.075, 1.565, 0.871 cents) of its 3, 5, and 7.

Using the patent val, 99edo is the optimal patent val for the rank four temperament tempering out 121/120; zeus, the rank three temperament tempering out 121/120 and 176/175; hemiwür, one of the rank two 11-limit extensions of hemiwürschmidt; and hitchcock (11-limit amity), the rank two temperament which also tempers out 2200/2187. Using the ⟨99 157 230 278 343] (99e) val, it tempers out 896/891, 243/242, 441/440 and 540/539, and is an excellent tuning for the 11-limit version of hemififths temperament. Hence 99 equal divisions, in spite of the fact that it tunes 11 relatively badly, is an important 11-limit tuning in more than one way.

Intervals

See Table of 99edo intervals.

Just approximation

Script error: No such module "primes_in_edo".

Temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator (reduced) Cents (reduced) Associated
ratio 		Temperament
1 7\99 84.8484 21/20 Amicable
1 16\99 193.9393 28/25 Hemiwürschmidt (99e) / Hemithir (99ef) / Hemiwur (99f)
1 19\99 230.303 8/7 Gamera
1 20\99 242.4242 147/128 Septiquarter
1 25\99 303.0303 25/21 Quinmite
1 27\99 315.1515 6/5 Parakleismic / Parkleismic / Paradigmic (99e)
1 28\99 339.3939 128/105 Amity (99ef) / Hitchcock
1 29\99 351.5151 49/40 Hemififths (99ef)
1 32\99 387.8787 5/4 Würschmidt / Whirrschmidt
1 37\99 448.4848 35/27 Semidimfourth
3 5\99 60.606 28/27 Chromat
3 13\99 157.5757 35/32 Nessafof
3 41\99 (8\99) (4)96.9696 4/3 (18/17~19/18) Misty
9 4\99 48.4848 36/35 Ennealimmal
11 41\99 (4\99) 496.9696 (48.4848) 4/3 (36/35) Hendecatonic


Scales


Since 98edo has a step of 12.1212 cents, it also allows one to use its MOS scales as circulating temperaments.

Circulating temperaments in 99edo
Tones Pattern L:s
5 4L 1s 20:19
6 3L 3s 17:16
7 1L 6s 15:14
8 3L 5s 13:12
9 9edo equal
10 9L 1s 10:9
11 11edo equal
12 3L 9s 9:8
13 8L 5s 8:7
14 1L 13s
15 9L 6s 7:6
16 3L 13s
17 14L 3s 6:5
18 9L 9s
19 4L 15s
20 19L 1s 5:4
21 15L 6s
22 11L 11s
23 7L 16s
24 3L 21s
25 24L 1s 4:3
26 21L 5s
27 18L 9s
28 15L 13s
29 12L 17s
30 9L 21s
31 6L 25s
32 3L 29s
33 33edo equal
34 31L 3s 3:2
35 29L 6s
36 27L 9s
37 25L 12s
38 23L 15s
39 21L 18s
40 19L 21s
41 17L 24s
42 15L 27s
43 13L 30s
44 11L 33s
45 9L 36s
46 7L 39s
47 5L 42s
48 3L 45s
49 1L 48s
50 49L 1s 2:1
51 48L 3s
52 47L 5s
53 46L 7s
54 45L 9s
55 44L 11s
56 43L 13s
57 42L 15s
58 41L 17s
59 40L 19s
60 39L 21s
61 38L 23s
62 37L 25s
63 36L 27s
64 35L 29s
65 34L 31s
66 33L 33s
67 32L 35s
68 31L 37s
69 30L 39s
70 29L 41s
71 28L 43s
72 27L 45s
73 26L 47s
74 25L 49s
75 24L 51s
76 23L 53s
77 22L 55s
78 21L 57s
79 20L 59s

Music

See also

  • 157edt
  • 58edf
  • 87edo, 94edo, 111edo, similarly sized edos all with consistency in higher harmonics.
  • 198edo, the half-sized edo to reconcile the mappings of 11 and 13.
  • 105edo, a similarly sized edo that supports meantone, septimal meantone, undecimal meantone and grosstone
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