91edo

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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. It is the second highest it a series of four consecutive edos that temper out quartisma ([24 -6 0 1 -5⟩). 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.

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)

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

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	C Az (А)	1/1
1	major prime major prota	C# Az#
2	augmented prota	Az##
3	biaugmented prota	Az###
4	bidiminished deiteria	Buki♭♭♭
5	diminished deiteria	Buki♭♭
6	minor deiteria	Buki♭
7	neutral deiteria	Buki (Б)
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	D 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	E Dobro♭♭	11/9
27	major tertie minor pemptia	E# 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	F 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	G	121/81
53	major quinte	G#	3/2
54	augmented quinte diminished ennatia	G## Zemle♭♭	256/169
55	minor ennatia	Zemle♭
56	neutral ennatia	Zemle (З)
63	neutral decatia	Izhe (И)
64	major decatia minor sexte	Izhe# A♭
65	neutral sexte	A
70	neutral hendecatia	Jerve (Ђ)
77	neutral dodecatia	Kako (К)
78	neutral septime	B
84	neutral decatotritia	Ludi (Л)
91	perfect octave perfect decatotetartia	C Az (А)	2/1 exact

Scales

• NaiveMajor[7]: 13 16 10 13 16 13 10
• NaiveMinor[7]: 13 10 16 13 10 13 16
• HungarianNaiveSurorwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
• Septimin[9]: 11 9 11 9 11 9 11 9 11
• ArabicSeptimin[9]: 5 15 11 9 11 9 5 15 11
• Semaphore5: 19 19 19 19 15
• Semaphore9
• Semaphore14

Music

• DPRK ISON CHASE by Chris Vaisvil
• DPRK ISON CHASE - YouTube
• Sadness - Nope by Mercury Amalgam