91edo

← 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

The 91 equal divisions of the octave (91edo), or 91-tone equal temperament (91tet, 91et) when viewed from a regular temperament perspective, divides the octave into 91 parts of 13.187 cents each.

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. From an aesthetic standpoint, the factoring of 91 represents a kind of "yin-yang" since historically, the number 7 symbolizes luck and 13 misfortune.

Theory

The 3, 5 and 7 for 91 are on the flat side, making this a mostly flat system. It provides the optimal patent val for 11- and 13-limit septimin temperament, and the 13-limit rank three tripod temperament, as well as the 11-limit rank four temperament tempering out 245/242 and the 13-limit rank five temperament tempering out 105/104, or rank four tempering out 105/104 and 144/143, or else 105/104 and 196/195 and hence 225/224 also. It 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.

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 also corresponds to 135/128, the eigenmonzo of 1/7 comma meantone, both being 7 steps.

91edo tempers out the devil's tridecalimma, which equates as stack of 13 10/9s with two octaves. In addition, it tempers out the 5-limit tridecalimma which equates a stack of 13 135/128s with the octave. It also supports the trideci temperament, which in the 7-limit tempers out 4375/4374 and 83349/81920.

It is the second highest it a series of four consecutive edos that temper out quartisma ([24 -6 0 1 -5⟩). 33/32 is quite close to 4 steps of 91edo, and when used as a generator, produces 68 & 91 temperament with the comma basis 385/384, 6250/6237, 67228/66825. Eliora proposes the name quartkeenlig for this temperament, as it tempers out the quartisma by virtue of 5 33/32 being with 7/6, keenanisma (385/384), and liganellus comma (6250/6237). It can also act as a stretched 23edo.

The concoctic scale for 91edo is 27 steps, where two concoctic neutral thirds make a sharp 91b val fifth of 54\91. From a regular temperament theory perspective, there is more than one way to interpret this, as they're all harmonically not very precise. First, in the 13-limit, is to assume that 27\91 is directly equivalent to 16/13 and set a temperament in the 2.3.5.7.16/13 subgroup, which produces a 27 & 91b temperament with the comma basis 91/90, 6272/6075, [84 13 11 -10⟩. Second is to directly take the 27 & 91b val in the 13-limit, which can also be taken using the 27e & 91b.

91edo supports a variant of semaphore temperament which tempers out the 4613015762523/4398046511104 comma in the 2.3.7 subgroup, and is produced by a 19\91 generator.

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
Error	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)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	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

Periods per Octave	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	2\91	26.37	49/48	Sfourth
1	4\91	52.75	33/32	Quartkeenlig
1	19\91	250.55	1240029/1048576	Semaphore (24 & 91 variant) †
1	20\91	263.74	7/6	Septimin
1	24\91	316.48	6/5	Catakleismic
1	33\91	435.16	9/7	Supermajor
1	34\91	448.35	35/27	Semidimfourth
1	44\91	580.22	7/5	Tritonic
7	38\91 (1\91)	501.10 (13.19)	4/3 (81/80)	Absurdity
13	53\91 (4\91)	700.382 (52.74)	3/2 (36/35)	Trideci

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

Intervals

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
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