91edo

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91edo

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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

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)

Miscellaneous properties

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.

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

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
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	Lalagu (12 & 79) (91) Meantone (91c)
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.

Notation

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

See also: 5- to 10-tone scales in 91edo

• 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

• Sadness - Nope (2022) – Bandcamp | YouTube

Chris Vaisvil

• DPRK ISON CHASE (2014) – blog | YouTube