# 94edo-a

94edo, the wonderful edo
STILL A WORK IN PROGRESS

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

## 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.

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

15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to some ratios of 7 and 11, but its approximation to most ratios of 3 and 9 are rather off. Its mappings and error for various 11-limit subgroups is shown in the table below.

15edo Subgroup Errors
2.3 <15 24] 8.979801
2.5 <15 35] 6.826357
2.7 <15 42] 4.418738
2.11 <15 52] 4.336492
2.3.5 <15 24 35] 10.742841
2.3.7 <15 24 42] 17.481581
2.3.11 <15 24 52] 16.831238
2.5.7 <15 35 42] 10.509269
2.5.11 <15 35 52] 8.335693
2.7.11 <15 42 52] 8.002641
2.3.5.7 <15 24 35 42] 15.603114
2.3.5.11 <15 24 35 52] 14.693746
2.3.7.11 <15 24 42 52] 18.660367
2.5.7.11 <15 35 42 52] 11.462127
2.3.5.7.11 <15 24 35 42 52] 17.258371
Using the 11-limit mapping from the table above leads to the intervals of 15edo representing 11-limit ratios as follows:
Degree Cents Represented Ratios
0 0 1/1
1 80 25/24, 21/20, 16/15
2 160 11/10, 12/11, 10/9
3 240 8/7, 7/6, 9/8
4 320 6/5, 11/9
5 400 5/4, 14/11
6 480 4/3, 9/7, 21/16
7 560 11/8, 7/5
8 640 16/11, 10/7
9 720 3/2, 14/9, 32/21
10 800 8/5, 11/7
11 880 5/3, 18/11
12 960 7/4, 12/7, 16/9
13 1040 20/11, 11/6, 9/5
14 1120 48/25, 40/21, 15/8
15 1200 2/1

## 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

Periods Period Generator Temperaments
1 15\15 1\15 Nautilus/valentine
1 15\15 2\15 Porcupine/opossum
1 15\15 4\15 Hanson/keemun/orgone
1 15\15 7\15 Progress
3 5\15 1\15 Augmented/augene
3 5\15 2\15 Triforce
5 3\15 1\15 Blackwood/blacksmith

# Commas

15 EDO tempers out the following commas. (Note: This assumes the 13-limit val < 15 24 35 42 52 56 |.)

Rational Monzo Size (Cents) Name 1 Name 2 Name 3
256/243 | 8 -5 > 90.22 Limma Pythagorean Minor 2nd
28/27 | 2 -3 0 1 > 62.96 Septimal Third Tone Small Septimal Chroma
250/243 | 1 -5 3 > 49.17 Maximal Diesis Porcupine Comma
128/125 | 7 0 -3 > 41.06 Diesis Augmented Comma
15625/15552 | -6 -5 6 > 8.11 Kleisma Semicomma Majeur
1029/1000 | -3 1 -3 3 > 49.49 Keega
49/48 | -4 -1 0 2 > 35.70 Slendro Diesis
64/63 | 6 -2 0 -1 > 27.26 Septimal Comma Archytas' Comma Leipziger Komma
64827/64000 | -9 3 -3 4 > 22.23 Squalentine
875/864 | -5 -3 3 1 > 21.90 Keema
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
4000/3969 | 5 -4 3 -2 > 13.47 Octagar
1029/1024 | -10 1 0 3 > 8.43 Gamelisma
6144/6125 | 11 1 -3 -2 > 5.36 Porwell
250047/250000 | -4 6 -6 3 > 0.33 Landscape Comma
100/99 | 2 -2 2 0 -1 > 17.40 Ptolemisma
121/120 | -3 -1 -1 0 2 > 14.37 Biyatisma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
65536/65219 | 16 0 0 -2 -3 > 8.39 Orgonisma
385/384 | -7 -1 1 1 1 > 4.50 Keenanisma
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Wizardharry
3025/3024 | -4 -3 2 -1 2 > 0.57 Lehmerisma
91/90 | -1 -2 -1 1 0 1 > 19.13 Superleap
676/675 | 2 -3 -2 0 0 2 > 2.56 Parizeksma

# 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