94edo-a
I have copied and pasted Igs' template for 15edo-a, and am using that as a framework. I aim to finish the main work on this page in the next week, then I am happy for many more further edits.
-Cam Taylor, 29/09/2018
Contents
Introduction
94 Equal Divisions of the Octave (94edo, also 94-tET or 94-ET) is a tuning which divides the octave into 94 equally-sized steps, where each step represents a frequency ratio of 12.766 cents. As 94 = 2 × 47, 94edo can be thought of as two sets of 47edo offset by one step of 94edo, or 12.766 cents, and it shares with 12edo the familiar 600-cent tritone which splits the octave into two equal parts.
It is notable for providing a very close approximation to Pythagorean tuning as well as providing consistent and accurate approximations to intervals involving primes up to 23. Although a circle of 94 pitches is a lot to deal with, many of the useful intervals come directly from the chain of fifths, and for the most part, 94edo can be treated in a similar manner to 41edo and 53edo, as a schismatic system that can model just intonation fairly well.
It is a remarkable all-around utility temperament, good from low prime limit to very high prime limit situations. It is the first equal temperament to be consistent though the 23-limit, and no other equal temperament is so consistent until 282 and 311 make their appearance.
Intervals
At first sight, 94edo has an overwhelming variety of interval sizes. However, it is helpful to think of the one-step and even two-step intervals as a change in colour, or ratio approximation, but not necessarily of function. In the table below, traditional names are given to intervals in terms of their location in the chain of fifths, e.g. major second at +2 fifths, while other intervals are described in terms of larger and smaller variants. Being an equal division of the octave, 94edo has enharmonically equivalent names for each of its 94 unique intervals, but for now, this should suffice as an introduction to its interval shadings.
Degree | Cents | Extra-Diatonic Category | pythagorean name |
---|---|---|---|
0 | 0 | Perfect unison | unison |
1 | 12.766 | Large unison | diminished diminished diminished diminished diminished diminished fifth |
2 | 25.532 | augmented seventh | |
3 | 38.298 | diminished diminished diminished diminished fourth | |
4 | 51.064 | augmented augmented augmented sixth | |
5 | 63.830 | Small minor second | diminished diminished third |
6 | 76.596 | augmented augmented augmented augmented augmented fifth | |
7 | 89.362 | Minor second | minor second |
8 | 102.128 | Large minor second | diminished diminished diminished diminished diminished diminished sixth |
9 | 114.894 | augmented unison | |
10 | 127.660 | Small middle second | |
11 | 140.426 | ||
12 | 153.191 | Large middle second | |
13 | 165.957 | ||
14 | 178.723 | Small major second | |
15 | 191.489 | ||
16 | 204.255 | Major second | |
17 | 217.021 | Large major second | |
18 | 229.787 | ||
19 | 242.553 | Intermediate
second-third region | |
20 | 255.319 | ||
21 | 268.085 | Small minor third | |
22 | 280.851 | ||
23 | 293.617 | Minor third | |
24 | 306.383 | Large minor third | |
25 | 319.149 | ||
26 | 331.915 | Small middle third | |
27 | 344.681 | ||
28 | 357.447 | Large middle third | |
29 | 370.213 | ||
30 | 382.979 | Small major third | |
31 | 395.745 | ||
32 | 408.511 | Major third | |
33 | 421.277 | Large major third | |
34 | 434.043 | ||
35 | 446.809 | Intermediate
third-fourth region | |
36 | 459.574 | ||
37 | 472.340 | Small fourth | |
38 | 485.106 | ||
39 | 497.872 | Perfect fourth | |
40 | 510.6383 | Large fourth | |
41 | 523.404 | ||
42 | 536.170 | Super fourth | |
43 | 548.936 | ||
44 | 561.702 | Small tritone | |
45 | 574.468 | ||
46 | 587.234 | Tritone | |
47 | 600.000 |
The remaining 47 intervals are the octave-inversions of those given above, and are left as an exercise to the reader.
There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context.
The perfect fifth has three, or perhaps even five, functional options, each differing by one step. Although in most timbres only the central perfect fifth at 702.128 cents sounds consonant and stable, the lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.
Every odd-numbered interval can generate the entire tuning of 94edo, and the only interval that gives a perfect octave when stacked less than 47 times is the 600 cent tritone.
The regular major second divisible into 16 equal parts can be helpful for realising some of the subtle tunings of Ancient Greek tetrachordal theory, Indian Raga and Turkish Makam, though it has not been used historically as a division in those musical cultures.
Notation
Degree | Cents | Relative notation
(extended diatonic) |
(extended
pythagorean) |
Absolute notation |
---|---|---|---|---|
0 | 0 | P1 | P1 | D |
1 | 12.766 | ^1 | ^D | |
2 | 25.532 | /1 | A7 (-1 octave) | /D |
3 | 38.298 | /^1 | /^D | |
4 | 51.064 | //1 | AAA6 (-1 octave) | \vEb |
5 | 63.830 | \m2 | \Eb | |
6 | 76.596 | vm2 | vEb | |
7 | 89.362 | \A1, m2 | m2 | \D#, Eb |
8 | 102.128 | vA1, ^m2 | vD#, ^Eb | |
9 | 114.894 | A1, /m2 | A1 | D#, /Eb |
10 | 127.660 | /^m2 | /^Eb | |
11 | 140.426 | //m2 | //Eb | |
12 | 153.191 | \\M2 | \\E | |
13 | 165.957 | \vM2 | \vE | |
14 | 178.723 | \M2 | d3 | \E |
15 | 191.489 | vE | ||
16 | 204.255 | M2 | M2 | E |
17 | 217.021 | ^E | ||
18 | 229.787 | /M2 | AA1 | /E |
19 | 242.553 | /^M2, \\m3 | /^E, \\F | |
20 | 255.319 | //M2, \vm3 | //E, \vF | |
21 | 268.085 | \\A2, \m3 | \\E#, \F | |
22 | 280.851 | \vA2, vm3 | \vE#, vF | |
23 | 293.617 | \A2, m3 | m3 | \E#, F |
24 | 306.383 | vA2, ^m3 | vE#, ^F | |
25 | 319.149 | /m3 | A2 | E#, /F |
26 | 331.915 | /^m3 | /^F | |
27 | 344.681 | //m3 | //F | |
28 | 357.447 | \\M3 | \\F# | |
29 | 370.213 | \vM3 | \vF# | |
30 | 382.979 | \M3 | d4 | \F#, Gb |
31 | 395.745 | vM3 | vF#, ^Gb | |
32 | 408.511 | M3 | M3 | F# |
33 | 421.277 | ^M3 | ^F# | |
34 | 434.043 | /M3 | AA2 | /F# |
35 | 446.809 | /^M3 | /^F# | |
36 | 459.574 | \v4 | \vG | |
37 | 472.340 | \4 | dd5 | \G |
38 | 485.106 | v4 | vG | |
39 | 497.872 | P4 | P4 | G |
40 | 510.6383 | ^4 | ^G | |
41 | 523.404 | /4 | A3 | /G |
42 | 536.170 | /^4 | /^G | |
43 | 548.936 | //4 | //G | |
44 | 561.702 | \\A4, \d5 | \\G#, \Ab | |
45 | 574.468 | \vA4, vd5 | \vG#, vAb | |
46 | 587.234 | \A4, d5 | d5 | \G#, Ab |
47 | 600.000 | vA4, ^d5 | vG#, ^Ab |
Close–up of Pythagorean notation 94edo:
Cb . B . . . . Dbb . C . B# . . . . Db . C# . B## . . . . D
Ups and Downs Notation
In ups and downs notation, which is fifth-generated, every 15edo note has at least three names.
All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too.0-3-9 = D E A = D2 = "D sus 2", or D F A = Dm = "D minor" (approximate 6:7:9)
0-4-9 = D F^ A = D.^m = "D upminor" (approximate 10:12:15)
0-5-9 = D F#v A = D.v = "D dot down" or "D downmajor" (approximate 4:5:6)
0-6-9 = D G A = D4, or D F# A = D = "D" or "D major" (approximate 14:18:21)
0-3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"
0-5-9-12 = D F#v A C = D7(v3) = "D seven down-three", or D F#v A B = D6(v3) = "D six down-three"
0-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"
0-5-9-14 = D F#v A C#v = D.vM7 = "D downmajor seven"
0-4-9-13 = D F^ A C^ = D.^m7 = "D dot up minor-seven", or D F^ A B^ = D.^m6 = "D dot up minor-six"
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.
While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys.
94edo as a Regular Temperament
The 94 equal temperament, often abbreviated 94-tET, 94-EDO, or 94-ET, is the scale derived by dividing the octave into 94 equally-sized steps, where each step represents a frequency ratio of 12.766 cents. It is a remarkable all-around utility temperament, good from low prime limit to very high prime limit situations. It is the first equal temperament to be consistent though the 23-limit, and no other equal temperament is so consistent until 282 and 311 make their appearance.
The list of 23-limit commas it tempers out is huge, but it's worth noting that it tempers out 32805/32768 and is thus a schismatic system, that it tempers out 225/224 and 385/384 and so is a marvel system, and that it also tempers out 3125/3087, 4000/3969, 5120/5103 and 540/539. It provides the optimal patent val for the rank five temperament tempering out 275/273, and for a number of other temperaments, such as isis.
Also, see the Table of 94edo intervals.
Rank two temperaments[edit | edit source]
Periods
per octave |
Generator | Cents | Associated
ratio |
Temperament |
---|---|---|---|---|
1 | 3\94 | 38.298 | 49/48 | Slender |
1 | 5\94 | 63.830 | 25/24 | Sycamore/betic |
1 | 11\94 | 140.426 | 243/224
13/12 |
Tsaharuk |
1 | 19\94 | 242.553 | 147/128 | Septiquarter |
1 | 39\94 | 497.872 | 4/3 | Schismatic/garibaldi |
2 | 2\94 | 25.532 | 64/63 | Ketchup |
2 | 43\94 | 548.936 | 11/8 | Kleischismic |
2 | 11\94 | 140.426 | 27/25 | Fifive |
2 | 34\94 | 434.043 | 9/7 | Pogo, supers |
2 | 30\94 | 382.979 | 5/4 | Wizard/gizzard |
Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for garibaldi temperament, shows us how to extend it to the 23-limit.
46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||
68&94 <<20 28 2 -10 24 20 34 52 ... ||
53&94 <<1 -8 -14 23 20 -46 -3 -35 ... || (one garibaldi)
41&94 <<1 -8 -14 23 20 48 -3 -35 ... || (another garibaldi, only differing in the mappings of 17 and 23)
135&94 <<1 -8 -14 23 20 48 -3 59 ... || (another garibaldi)
130&94 <<6 -48 10 -50 26 6 -18 -22 ... || (a pogo extension)
58&94 <<6 46 10 44 26 6 -18 -22 ... || (a supers extension)
50&94 <<24 -4 40 -12 10 24 22 6 ... ||
72&94 <<12 -2 20 -6 52 12 -36 -44 ... || (a gizzard extension)
80&94 <<18 44 30 38 -16 18 40 28 ... ||
94 solo <<12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!)
Temperaments for which 94 is a MOS:
311&94 <<3 70 -42 69 -34 50 85 83...||
422&94 <<8 124 -18 90 -28 102 164 96 ... || 15edo may be treated as a regular temperament of 5-, 7-, and 11-limit JI, or as a 2.5.7.11 subgroup temperament. While it does significant damage to the ratios of 3, it offers significant improvement over 12edo in approximating ratios of 5, 7, and 11. As a 5-limit temperament, it is notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive syntonic comma.
todo:replace the svg with a svg suitable for 94edo
Using the 11-limit mapping from the table above leads to the intervals of 15edo representing 11-limit ratios as follows:Selected just intervals by error
The following table shows how some prominent just intervals are represented in 94edo (ordered by absolute error).
Interval, complement | Error (abs., in cents) |
4/3, 3/2 | 0.173 |
9/8, 16/9 | 0.345 |
14/13, 13/7 | 0.639 |
15/11, 22/15 | 0.781 |
11/10, 20/11 | 0.953 |
9/7, 14/9 | 1.042 |
7/6, 12/7 | 1.214 |
8/7, 7/4 | 1.387 |
18/13, 13/9 | 1.680 |
13/12, 24/13 | 1.853 |
16/13, 13/8 | 2.026 |
11/8, 16/11 | 2.382 |
12/11, 11/6 | 2.554 |
11/9, 18/11 | 2.727 |
16/15, 15/8 | 3.162 |
5/4, 8/5 | 3.335 |
6/5, 5/3 | 3.508 |
10/9, 9/5 | 3.680 |
14/11, 11/7 | 3.769 |
13/11, 22/13 | 4.407 |
15/14, 28/15 | 4.549 |
7/5, 10/7 | 4.722 |
15/13, 26/15 | 5.188 |
13/10, 20/13 | 5.361 |
Rank two temperaments
List of 15et rank two temperaments by badness
List of edo-distinct 15et rank two temperaments
Commas
15 EDO tempers out the following commas. (Note: This assumes the 13-limit val < 15 24 35 42 52 56 |.)
Musical Examples
4 Improvisations Saturday 9th September 2017 by Cam Taylor
Feeling Sad But Warming Up (in 2 parts) by Cam Taylor
Playing with the 13-limit by Cam Taylor