91edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 90edo 91edo 92edo →
Prime factorization 7 × 13
Step size 13.1868¢ 
Fifth 53\91 (698.901¢)
Semitones (A1:m2) 7:8 (92.31¢ : 105.5¢)
Consistency limit 9
Distinct consistency limit 9

91 equal divisions of the octave (abbreviated 91edo or 91ed2), also called 91-tone equal temperament (91tet) or 91 equal temperament (91et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 91 equal parts of about 13.2 ¢ each. Each step represents a frequency ratio of 21/91, or the 91st root of 2.

Theory

The harmonics 3, 5 and 7 for 91edo are on the flat side, making this a mostly flat system. The equal temperament tempers out 15625/15552 in the 5-limit, 225/224 and 4375/4374 in the 7-limit, 245/242, 385/384 in the 11-limit, and 105/104, 144/143, 196/195 in the 13-limit. It provides the optimal patent val for 11- and 13-limit septimin temperament, and the 13-limit rank-3 tripod temperament, as well as the 11-limit rank-4 temperament tempering out 245/242 and the 13-limit rank-5 temperament tempering out 105/104, or rank-4 tempering out 105/104 and 144/143, or else 105/104 and 196/195 and hence 225/224 also.

Using the 91c val, it is audibly indistinguishable from a closed system of 1/7-comma meantone, with a 5th only 0.018 cents sharper. The chromatic semitone in this scale corresponds to 135/128, the eigenmonzo (unchanged-interval) of 1/7-comma meantone. Being 7 steps, what is also remarkable is that in this instance the chromatic semitone is equal to one step of 13edo. Since 135/128 is also equal to 1/13 of the octave, the 91c val tempers out the aluminium comma in the 5-limit.

The equal temperament also tempers out the [-11 26 -13, the tridecatonic comma, which assigns 10/9 to 2/13 of the octave, and it supports trideci in the 7-limit, tempering out 4375/4374 and 83349/81920. It supports a variant of semaphore temperament which tempers out the [-42 23 2 comma in the 2.3.7 subgroup, and is generated by a 19\91 generator. It is the second highest in a series of four consecutive edos that temper out quartisma ([24 -6 0 1 -5), and as a corollary it is a tuning for the quartkeenlig temperament, which can also act as a stretched 23edo. In the 13-limit, it supports vidar and gives a reasonable tuning for its size.

Odd harmonics

Approximation of odd harmonics in 91edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.05 -3.90 -6.19 -6.11 +2.53 +3.43 +6.24 +0.54 +5.78 +3.94 +4.69
Relative (%) -23.2 -29.5 -46.9 -46.3 +19.2 +26.0 +47.3 +4.1 +43.9 +29.9 +35.6
Steps
(reduced)
144
(53)
211
(29)
255
(73)
288
(15)
315
(42)
337
(64)
356
(83)
372
(8)
387
(23)
400
(36)
412
(48)

Subsets and supersets

91 is the smallest composite number whose composite character is not immediately evident in the decimal system; it is, in fact, the product of 7 and 13. As such, 91edo contains 7edo and 13edo as subsets.

Miscellaneous properties

The concoctic scale for 91edo is 27 steps, where two concoctic neutral thirds make a sharp fifth of 54\91, representing 3/2 in the 91b val.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3 [-144 91 [91 144]] +0.963 0.964 7.31
2.3.5 15625/15552, 43046721/41943040 [91 144 211]] +1.202 0.857 6.49
2.3.5.7 225/224, 4375/4374, 50421/50000 [91 144 211 255]] +1.453 0.860 6.51

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
ratio*
Temperament
1 2\91 26.37 49/48 Sfourth
1 4\91 52.75 33/32 Quartkeenlig (91f)
1 11\91 145.05 49/45 Swetneus (91ef)
1 19\91 250.55 1240029/1048576 Semaphore variant (24 & 91) †
1 20\91 263.74 7/6 Septimin (91)
1 24\91 316.48 6/5 Catakleismic (91f)
1 33\91 435.16 9/7 Supermajor
1 34\91 448.35 35/27 Semidimfourth
1 38\91 501.10 4/3 Python
1 44\91 580.22 7/5 Tritonic
7 38\91
(1\91)
501.10
(13.19)
4/3
(81/80)
Absurdity
13 38\91
(1\91)
501.10
(13.19)
4/3
(265/252)
Trideci (91)
Aluminium (91c)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

derived from scales in the Scales section, official name not decided upon yet.

Interval table

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 13.187 ^D, E♭♭
2 26.374 ^^D, v6E♭
3 39.56 ^3D, v5E♭
4 52.747 32/31, 33/32, 34/33 ^4D, v4E♭
5 65.934 ^5D, v3E♭
6 79.121 23/22 ^6D, vvE♭
7 92.308 39/37 D♯, vE♭
8 105.495 17/16, 33/31 ^D♯, E♭
9 118.681 15/14 ^^D♯, v6E
10 131.868 27/25, 41/38 ^3D♯, v5E
11 145.055 37/34 ^4D♯, v4E
12 158.242 34/31 ^5D♯, v3E
13 171.429 32/29 ^6D♯, vvE
14 184.615 10/9 D𝄪, vE
15 197.802 28/25, 37/33 E
16 210.989 26/23 ^E, F♭
17 224.176 33/29 ^^E, v6F
18 237.363 39/34 ^3E, v5F
19 250.549 37/32 ^4E, v4F
20 263.736 ^5E, v3F
21 276.923 34/29 ^6E, vvF
22 290.11 13/11 E♯, vF
23 303.297 25/21, 31/26 F
24 316.484 6/5 ^F, G♭♭
25 329.67 23/19, 29/24 ^^F, v6G♭
26 342.857 39/32 ^3F, v5G♭
27 356.044 ^4F, v4G♭
28 369.231 ^5F, v3G♭
29 382.418 ^6F, vvG♭
30 395.604 39/31 F♯, vG♭
31 408.791 ^F♯, G♭
32 421.978 37/29 ^^F♯, v6G
33 435.165 9/7 ^3F♯, v5G
34 448.352 22/17, 35/27 ^4F♯, v4G
35 461.538 ^5F♯, v3G
36 474.725 ^6F♯, vvG
37 487.912 F𝄪, vG
38 501.099 G
39 514.286 31/23, 39/29 ^G, A♭♭
40 527.473 ^^G, v6A♭
41 540.659 26/19 ^3G, v5A♭
42 553.846 11/8 ^4G, v4A♭
43 567.033 25/18 ^5G, v3A♭
44 580.22 7/5 ^6G, vvA♭
45 593.407 31/22 G♯, vA♭
46 606.593 ^G♯, A♭
47 619.78 10/7 ^^G♯, v6A
48 632.967 36/25 ^3G♯, v5A
49 646.154 16/11 ^4G♯, v4A
50 659.341 19/13 ^5G♯, v3A
51 672.527 ^6G♯, vvA
52 685.714 G𝄪, vA
53 698.901 A
54 712.088 ^A, B♭♭
55 725.275 ^^A, v6B♭
56 738.462 ^3A, v5B♭
57 751.648 17/11, 37/24 ^4A, v4B♭
58 764.835 14/9 ^5A, v3B♭
59 778.022 ^6A, vvB♭
60 791.209 41/26 A♯, vB♭
61 804.396 ^A♯, B♭
62 817.582 ^^A♯, v6B
63 830.769 ^3A♯, v5B
64 843.956 ^4A♯, v4B
65 857.143 ^5A♯, v3B
66 870.33 38/23 ^6A♯, vvB
67 883.516 5/3 A𝄪, vB
68 896.703 42/25 B
69 909.89 22/13 ^B, C♭
70 923.077 29/17 ^^B, v6C
71 936.264 ^3B, v5C
72 949.451 ^4B, v4C
73 962.637 ^5B, v3C
74 975.824 ^6B, vvC
75 989.011 23/13, 39/22 B♯, vC
76 1002.198 25/14, 41/23 C
77 1015.385 9/5 ^C, D♭♭
78 1028.571 29/16 ^^C, v6D♭
79 1041.758 31/17 ^3C, v5D♭
80 1054.945 ^4C, v4D♭
81 1068.132 ^5C, v3D♭
82 1081.319 28/15 ^6C, vvD♭
83 1094.505 32/17 C♯, vD♭
84 1107.692 ^C♯, D♭
85 1120.879 ^^C♯, v6D
86 1134.066 ^3C♯, v5D
87 1147.253 31/16, 33/17 ^4C♯, v4D
88 1160.44 ^5C♯, v3D
89 1173.626 ^6C♯, vvD
90 1186.813 C𝄪, vD
91 1200 2/1 D

Other notations

Eliora's notation

Eliora, who believes the diatonic way of naming intervals in 91edo is not useful due to the fact that other temperaments and techniques for 91edo are more prominent, proposes a way of naming that merges the factors 7 and 13—7 equidistant notes are named do, re, mi, and 13 are named by some other virtue. The proposition is to use Old Slavic letter names, since no one uses them for naming or in mathematics. The 7 + 13 naming convention can be called a duality notation. Intervals can be named through Latin ones for the 7-note scale, and Greek ones for the 13-note.

Table of intervals in 91edo
# Eliora's Naming System Eliora's Notation Associated Ratio
0 unison
perfect prime
perfect prota
A
Az (А)
1/1
1 major prime
major prota
A#
Az#
1728/1715
2 augmented prota Az##
3 biaugmented prota Az###
4 bidiminished deiteria Buki♭♭♭ 33/32
5 diminished deiteria Buki♭♭
6 minor deiteria Buki♭
7 neutral deiteria Buki (Б) 135/128
8 major deiteria Buki#
9 augmented deiteria Buki##
10 biaugmented deiteria Buki###
11 bidiminished tritia Vedi♭♭♭ 13/12, 12/11
12 diminished tritia Vedi♭♭
13 neutral secunde
minor tritia
B
Vedi♭
11/10
14 neural tritia Vedi (В) 10/9
15 major tritia Vedi# 9/8
16 augmented tritia Vedi##
17 biaugmented tritia Vedi###
18 bidiminished tesseria Glagol♭♭♭ 8/7
19 diminished tesseria Glagol♭♭
20 minor tesseria Glagol♭ 7/6
21 neutral tesseria Glagol (Г)
22 major tesseria Glagol# 13/11
23 augmented tesseria Glagol##
24 biaugmented tesseria Glagol### 6/5
25 bidiminished pemptia Dobro♭♭♭
26 neutral tertie
diminished pemptia
C
Dobro♭♭
11/9
27 major tertie
minor pemptia
C#
Dobro♭
16/13, 27/22
28 neutral pemptia Dobro (Д)
29 major pemptia Dobro# 5/4
30 augmented pemptia Dobro##
31 biaugmented pemptia Dobro###
32 bidiminished hektia Yest♭♭♭ 14/11
33 diminished hektia Yest♭♭ 9/7
34 minor hektia Yest♭
35 neutral hektia Yest (Е)
36 major hektia Yest#
37 augmented hektia Yest##
38 biaugmented hektia Yest### 4/3
39 neutral quarte
bidiminished hebdomia
D
Zhivete♭♭♭
40 diminished hebdomia Zhivete♭♭
41 minor hebdomia Zhivete♭
42 neutral hebdomia Zhivete (Ж) 11/8
43 major hebdomia Zhivete#
44 augmented hebdomia Zhivete## 7/5
45 biaugmented hebdomia Zhivete###
46 bidiminished ogdonia Dzelo♭♭♭
47 diminished ogdonia Dzelo♭♭ 10/7
48 minor ogdonia Dzelo♭
49 neutral ogdonia Dzelo (Ѕ)
52 neutral quinte E 121/81
53 major quinte E# 3/2
54 augmented quinte
diminished ennatia
E##
Zemle♭♭
256/169
55 minor ennatia Zemle♭
56 neutral ennatia Zemle (З)
63 neutral decatia Izhe (И)
64 major decatia
minor sexte
Izhe#
F♭
65 neutral sexte F
70 neutral hendecatia Jerve (Ђ)
77 neutral dodecatia Kako (К)
78 neutral septime G
84 neutral decatotritia Ludi (Л)
91 perfect octave
perfect decatotetartia
A
Az (А)
2/1 exact

Scales

  • Semaphore5: 19 15 19 19 19
  • Semaphore9: 15 4 15 4 15 4 15 15 4
  • Semaphore14: 4 11 4 4 11 4 4 11 4 11 4 4 11 4
  • NaiveMajor[7]: 13 16 10 13 16 13 10
  • NaiveMinor[7]: 13 10 16 13 10 13 16
  • Septimin[9]: 11 9 11 9 11 9 11 9 11
  • SeptiminHijaz[9]: 5 15 11 9 11 9 5 15 11
  • Meantone[12]: 878787887878
  • Meantone43 in 91edo
  • Meantone55 in 91edo
  • NaiveOrwell[13]: 5795797597579
  • ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9
  • HungarianNaiveSurorwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
  • Quartkeenlig[23]: 44444444444444444444443
  • ConcocticSubset[7]: 17 10 17 10 17 10 17
  • ConcocticMaqamSikah: 10 17 17 10 10 17 10

Music

Mercury Amalgam
Chris Vaisvil