36edo

36edo, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33.333... cents.

36 is a highly composite number, factoring into 2x2x3x3. Since 36 is divisible by 12, it contains the overly-familiar 12edo as a subset. It divides 12edo's 100-cent half step into three microtonal step of approximately 33 cents, which could be called "sixth tones." 36edo also contains 18edo ("third tones") and 9edo ("two-thirds tones") as subsets, not to mention the 6edo whole tone scale, 4edo full-diminished seventh chord, and the 3edo augmented triad, all of which are present in 12edo.

That 36edo contains 12edo as a subset makes in compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, De-quinin'). Three 12edo instruments could play the entire gamut.

As a harmonic temperament

For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 just intonation subgroup, 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called Slendro diesis of around 36 cents, and as 64:63, the so-called septimal comma of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called Septimal third-tone (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the 2*36 subgroup 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as 72edo does in the full 17-limit.

The 36edo patent val, like 12, tempers out 81/80, 128/125 and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, slendric, is well supported by 36edo, it's generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242 and 540/539, and is the optimal patent val for the rank four temperament tempering out 56/55, as well as the rank three temperament melpomene tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.

As a 5-limit temperament, the patent val for 36edo is contorted, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a transversal for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.

Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.

Relation to 12edo

For people accustomed to 12edo, 36edo is one of the easiest (if not the easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which blue notes (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. Unlike 24edo, which has genuinely foreign intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals.

An easy way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, A is 33.333 cents above A and 33.333 cents below A. Or the colors could be written out (red A, blue C#, etc.) or abbreviated as rA, bC#, etc. This use of red and blue is consistent with color notation.

Because of the presence of blue notes, and the closeness with which intervals such as 4:7 are matched, 36edo is an ideal scale to use for African-American styles of music such as blues and jazz, in which chords containing the seventh harmonic are frequently used. The 5th and 11th harmonic fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 5:6 and flatly approximates 9:11, for instance, whereas the sharp 9:11 is 366.667 cents and the flat 5:6 is 300 cents. However, 10:11 and 11:15 each have a single (very close) approximation since they contain both the 5th and 11th harmonic.

36edo is fairly cosmopolitan because many other genres of world music can be played in it too. Because it contains 9edo as a subset, pelog (and mavila) easily adapt to it. Slendro can be approximated in several different ways. 36edo can function as a "bridge" between these genres and Western music. Arabic music does not adapt as well, however, since many versions contain quarter tones.

The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still pleasing). In contrast, the smallest interval in 24edo, which is 50 cents, sounds very bad to most ears.

People with perfect (absolute) pitch often have a harder time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding bad). This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes too.

"Quark"

In particle physics, baryons , which are the main building blocks of atomic nuclei, are always comprised of three quarks. One could draw an analogy between baryons and semitones (the main building block of 12edo); the baryon is comprised of three quarks and the semitone of three sixth-tones. The number of quarks in a colorless particle is always a multiple of three; similarly, the width of "colorless" intervals (the 12-edo intervals, which are neither red nor blue), expressed in terms of sixth-tones, is always a multiple of three. Because of this amusing coincidence, I (Mason Green) propose referring to the 33.333-cent sixth-tone interval as a "quark".

Approximations

3-limit (Pythagorean) approximations (same as 12edo):

3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.

4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents.

9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents.

16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents.

27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents.

32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents.

81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents.

128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.

7-limit approximations:

7 only:

7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.

8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.

49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents.

64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.

3 and 7:

7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.

12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.

9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents.

14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents.

28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents.

27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents.

21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents.

32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents.

49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents.

96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents.

49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents.

72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.

64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.

63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.

The following table gives an overview of all degrees of 36edo.

Degree	Size in cents	Approximate ratios of 2.3.7	Additional ratios of 2.3.7.13.17	ups and downs notation
0	0.00	1/1		P1	perfect unison	D
1	33.33	64/63, 49/48		^1	up unison	D^
2	66.67	28/27		vm2	downminor 2nd	Ebv
3	100.00	256/243	17/16, 18/17	m2	minor 2nd	Eb
4	133.33	243/224	14/13, 13/12	^m2	upminor 2nd	Eb^
5	166.67	54/49		vM2	downmajor 2nd	Ev
6	200.00	9/8		M2	major 2nd	E
7	233.33	8/7		^M2	upmajor 2nd	E^
8	266.67	7/6		vm3	downminor 3rd	Fv
9	300.00	32/27		m3	minor 3rd	F
10	333.33	98/81	17/14	^m3	upminor 3rd	F^
11	366.67	243/196	16/13, 26/21, 21/17	vM3	downmajor 3rd	F#v
12	400.00	81/64		M3	major 3rd	F#
13	433.33	9/7		^M3	upmajor 3rd	F#^
14	466.67	64/49, 21/16	17/13	v4	down 4th	Gv
15	500.00	4/3		P4	4th	G
16	533.33	49/36		^4	up 4th	G^
17	566.67		18/13	vA4	downaug 4th	G#v
18	600.00			A4, d5	aug 4th, dim 5th	G#, Ab
19	633.33		13/9	^d5	updim 5th	Ab^
20	666.67	72/49		v5	down 5th	Av
21	700.00	3/2		P5	5th	A
22	733.33	49/32, 32/21	26/17	^5	up fifth	A^
23	766.67	14/9		vm6	downminor 6th	Bbv
24	800.00	128/81		m6	minor 6th	Bb
25	833.33	392/243	13/8, 21/13, 34/21	^m6	upminor 6th	Bb^
26	866.67	81/49	28/17	vM6	downmajor 6th	Bv
27	900.00	27/16		M6	major 6th	B
28	933.33	12/7		^M6	upmajor 6th	B^
29	966.67	7/4		vm7	downminor 7th	Cv
30	1000.00	16/9		m7	minor 7th	C
31	1033.33	49/27		^m7	upminor 7th	C^
32	1066.67	448/243	13/7, 24/13	vM7	downmajor 7th	C#v
33	1100.00	243/128	32/17, 17/9	M7	major 7th	C#
34	1133.33	27/14		^M7	upmajor 7th	C#^
35	1166.67	63/32, 96/49		v8	down 8ve	Dv
36	1200.00	2/1		P8	8ve	D

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chord names in other EDOs.

Music

• Exponentially More Lost and Forgetful by Stephen Weigel played by flautists Orlando Cela and Wei Zhao
• Something by Herman Klein
• Hay by Joe Hayseed
• Boomers by Ivan Bratt
• Thoughts in Legolas Tuning by Chris Vaisvil