91edo

91edo, the 91 equal division 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

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)

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 (117440512/117406179). 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.

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

Table of intervals

Due to 91edo's unusual layout, there isn't one unique way to name notes in it, and due to its size, the diatonic way of naming is starting to lag.

Eliora 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

Step	Interval (duality notation using 7x13 framework)	Name (if unison = 0) (duality notation)	Associated ratio
0	unison, perfect prime, perfect prota	C, Az	1/1
1	major prime, major prota	C#
7	neutral deiteria	Buki
13	neutral secunde	D
14	neural tritia	Vedi
20	minor tesseria	Glagol♭	7/6
21	neutral tesseria	Glagol
26	neutral tertie	E	11/9
27	major tertie, minor pemptia	E#	16/13, 27/22
28	neutral pemptia	Dobro
35	neutral hektia	Yest
39	neutral quarte	F
42	neutral hebdomia	Zhivete
44	augmented hebdomia	Zhivete##	7/5
45	biaugmented hebdomia	Zhivete###	10/7
49	neutral ogdonia	Dzelo
52	neutral quinte	G	121/81
53	major quinte	G#	3/2
54	augmented quinte, diminished ennatia	G##, Zemle♭♭	3/2 II, 256/169
55	minor ennatia	Zemle♭
56	neutral ennatia	Zemle
63	neutral decatia	Izhe
64	major decatia, minor sexte	Izhe#, A♭
65	neutral sexte	A
70	neutral hendecatia	Izhe-yi
77	neutral dodecatia	Jerve
78	neutral septime	B
84	neutral decatotritia	Kako
91	perfect octave, perfect decatotetartia	C, Az	2/1 exact

Scales

• NaiveMajor[7]: 13 16 10 13 16 13 10
• NaiveMinor[7]: 13 10 16 13 10 13 16
• HungarianNaiveOrwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
• Semaphore5
• Semaphore9
• Semaphore14

Music

• DPRK ISON CHASE by Chris Vaisvil
• DPRK ISON CHASE - YouTube

See also

• Wikipedia: 91 (number)