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.

step	cents	ups and downs relative notation (partial list, e.g. M2 is also A1 and d4)		ups and downs absolute notation
0	0¢	P1, m2	unison, min 2nd	C# / D / Eb
1	80	^1, ^m2	up-unison, upminor 2nd	C#^ / D^ / Eb^
2	160	vM2	downmajor 2nd	D#v / Ev / Fv / Gbv
3	240	M2, m3	major 2nd, minor 3rd	D# / E / F / Gb
4	320	^m3	upminor 3rd	D#^ / E^ / F^ / Gb^
5	400	vM3	downmajor 3rd	F#v / Gv / Abv
6	480	M3, P4, d5	major 3rd, perfect 4th, dim 5th	F# / G / Ab
7	560	^4, ^d5	up 4th, updim 5th	F#^ / G^ / Ab^
8	640	vA4, v5	downaug 4th, down 5th	G#v / Av / Bbv
9	720	A4, P5, m6	aug 4th, perfect 5th, minor 6th	G# / A / Bb
10	800	^5, ^m6	up 5th, upminor 6th	G#^ / A^ / Bb^
11	880	vA5, vM6	downaug 5th, downmajor 6th	A#v / Bv / Cv / Dbv
12	960	M6, m7	major 6th, minor 7th	A# / B / C / Db
13	1040	^m7	upminor 7th	A#^ / B^ / C^ / Db^
14	1120	vM7, v8	downmajor 7th, down octave	C#v / Dv / Ebv
15	1200	M7, P8	major 7th, octave	C# / D / Eb

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"

0-4-9-12 = D F^ A C = Dm7(^3) = "D minor seven up-three", or D F^ A B = Dm6(^3) = "D minor six up-three"

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 Quanic
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

File:15ed2-001.svg

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

Subgroup	Mapping	Adjusted Error (cents)
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

List of 15et rank two temperaments by badness

List of edo-distinct 15et rank two temperaments

Periods per octave	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

Images