91edo
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.
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. 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.
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 (unchanged-interval) of 1/7-comma meantone, both being 7 steps. What's also remarkable is that in this instance the chromatic semitone is equal to one step of 13edo. Since 135/128 is also equal to 1/13th of the octave, 91c val tempers out the aluminium comma in the 5-limit.
91edo also tempers out the devil's tridecalimma, which assigns 10/9 to 2/13ths of 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⟩), and as a corollary it is a tuning for the quartkeenlig temperament, which 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.
In the 13-limit, 91edo is an optimal ET for vidar temperament coming after 87d val and before 133d val, although it is no longer consistent in the 11-odd-limit.
Odd harmonics
| 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.
Since the number 7 has historically represented luck, and the number 13 has always stood for bad luck, from an aesthetic standpoint, the factoring of 91 represents a kind of "yin-yang", a combination of opposites.
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 (Reduced) |
Cents (Reduced) |
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 (24 & 91 variant) † |
| 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) |
† 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.
| # | 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
- 5- to 10-tone scales in 91edo