72edo

Theory

72-tone equal temperament, or 72-edo, divides the octave into 72 steps or moria. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of 24-tone equal temperament, a common and standard tuning of Arabic music, and has itself been used to tune Turkish music.

Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with 96-edo), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.

72-tone equal temperament approximates 11-limit just intonation exceptionally well, is consistent in the 17-limit, and is the ninth Zeta integral tuning. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other 5-limit major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.

72 is an excellent tuning for miracle temperament, especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.

Intervals

degrees	cents value	approximate ratios (17-limit)	ups and downs notation
0	0.000	1/1	P1	perfect unison	D
1	16.667	81/80	^1	up unison	^D
2	33.333	45/44	^^	double-up unison	^^D
3	50	33/32	^31, v3m2	triple-up unison, triple-down minor 2nd	^3D, v3Eb
4	66.667	25/24	vvm2	double-downminor 2nd	vvEb
5	83.333	21/20	vm2	downminor 2nd	vEb
6	100	35/33, 17/16, 18/17	m2	minor 2nd	Eb
7	116.667	15/14, 16/15	^m2	upminor 2nd	^Eb
8	133.333	27/25, 13/12, 14/13	v~2	downmid 2nd	^^Eb
9	150	12/11	~2	mid 2nd	v3E
10	166.667	11/10	^~2	upmid 2nd	vvE
11	183.333	10/9	vM2	downmajor 2nd	vE
12	200	9/8	M2	major 2nd	E
13	216.667	25/22, 17/15	^M2	upmajor 2nd	^E
14	233.333	8/7	^^M2	double-upmajor 2nd	^^E
15	250	81/70, 15/13	^3M2, v3m3	triple-up major 2nd, triple-down minor 3rd	^3E, v3F
16	266.667	7/6	vvm3	double-downminor 3rd	vvF
17	283.333	33/28, 13/11, 20/17	vm3	downminor 3rd	vF
18	300	25/21	m3	minor 3rd	F
19	316.667	6/5	^m3	upminor 3rd	^F
20	333.333	40/33, 17/14	v~3	downmid 3rd	^^F
21	350	11/9	~3	mid 3rd	^3F
22	366.667	99/80, 16/13, 21/17	^~3	upmid 3rd	vvF#
23	383.333	5/4	vM3	downmajor 3rd	vF#
24	400	44/35	M3	major 3rd	F#
25	416.667	14/11	^M3	upmajor 3rd	^F#
26	433.333	9/7	^^M3	double-upmajor 3rd	^^F#
27	450	35/27, 13/10	^3M3, v34	triple-up major 3rd, triple-down 4th	^3F#, v3G
28	466.667	21/16, 17/13	vv4	double-down 4th	vvG
29	483.333	33/25	v4	down 4th	vG
30	500	4/3	P4	perfect 4th	G
31	516.667	27/20	^4	up 4th	^G
32	533.333	15/11	v~4	downmid 4th	^^G
33	550	11/8	~4	mid 4th	^3G
34	566.667	25/18, 18/13	^~4	upmid 4th	vvG#
35	583.333	7/5	vA4, vd5	downaug 4th, updim 5th	vG#, vAb
36	600	99/70, 17/12	A4, d5	aug 4th, dim 5th	G#, Ab
37	616.667	10/7	^A4, ^d5	upaug 4th, downdim 5th	^G#, ^Ab
38	633.333	36/25, 13/9	v~5	downmid 5th	^^Ab
39	650	16/11	~5	mid 5th	v3A
40	666.667	22/15	^~5	upmid 5th	vvA
41	683.333	40/27	v5	down 5th	vA
42	700	3/2	P5	perfect 5th	A
43	716.667	50/33	^5	up 5th	^A
44	733.333	32/21	^^5	double-up 5th	^^A
45	750	54/35, 17/11	^35, v3m6	triple-up 5th, triple-down minor 6th	^3A, v3Bb
46	766.667	14/9	vvm6	double-downminor 6th	vvBb
47	783.333	11/7	vm6	downminor 6th	vBb
48	800	35/22	m6	minor 6th	Bb
49	816.667	8/5	^m6	upminor 6th	^Bb
50	833.333	81/50, 13/8	v~6	downmid 6th	^^Bb
51	850	18/11	~6	mid 6th	v3B
52	866.667	33/20, 28/17	^~6	upmid 6th	vvB
53	883.333	5/3	vM6	downmajor 6th	vB
54	900	27/16	M6	major 6th	B
55	916.667	56/33, 17/10	^M6	upmajor 6th	^B
56	933.333	12/7	^^M6	double-upmajor 6th	^^B
57	950	121/70	^3M6, v3m7	triple-up major 6th, triple-down minor 7th	^3B, v3C
58	966.667	7/4	vvm7	double-downminor 7th	vvC
59	983.333	44/25	vm7	downminor 7th	vC
60	1000	16/9	m7	minor 7th	C
61	1016.667	9/5	^m7	upminor 7th	^C
62	1033.333	20/11	v~7	downmid 7th	^^C
63	1050	11/6	~7	mid 7th	^3C
64	1066.667	50/27	^~7	upmid 7th	vvC#
65	1083.333	15/8	vM7	downmajor 7th	vC#
66	1100	66/35, 17/9	M7	major 7th	C#
67	1116.667	21/11	^M7	upmajor 7th	^C#
68	1133.333	27/14	^^M7	double-upmajor 7th	^^C#
69	1150	35/18	^3M7, v38	triple-up major 7th, triple-down octave	^3C#, v3D
70	1166.667	49/25	vv8	double-down octave	vvD
71	1183.333	99/50	v8	down octave	vD
72	1200	2/1	P8	perfect octave	D

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality	color	monzo format	examples
double-down minor	zo	{a, b, 0, 1}	7/6, 7/4
minor	fourthward wa	{a, b}, b < -1	32/27, 16/9
upminor	gu	{a, b, -1}	6/5, 9/5
mid	ilo	{a, b, 0, 0, 1}	11/9, 11/6
"	lu	{a, b, 0, 0, -1}	12/11, 18/11
downmajor	yo	{a, b, 1}	5/4, 5/3
major	fifthward wa	{a, b}, b > 1	9/8, 27/16
double-up major	ru	{a, b, 0, -1}	9/7, 12/7

All 72-edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:

color of the 3rd	JI chord	notes as edosteps	notes of C chord	written name	spoken name
zo	6:7:9	0-16-42	C vvEb G	Cvvm	C double-down minor
gu	10:12:15	0-19-42	C ^Eb G	C^m	C upminor
ilo	18:22:27	0-21-42	C v3E G	C~	C mid
yo	4:5:6	0-23-42	C vE G	Cv	C downmajor or C down
ru	14:18:27	0-26-42	C ^^E G	C^^	C double-upmajor or C double-up

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Approximations to prime harmonics

	prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19	prime 23	prime 29	prime 31
error	0.0¢	-1.955¢	-2.980¢	-2.159¢	-1.318¢	-7.194¢	-4.955¢	+2.487¢	+5.059¢	+3.755¢	-4.964¢

Commas

Commas tempered out by 72edo include...

3-limit
Pythagorean comma = 531441/524288 = |-19 12>

5-limit
kleisma = 15625/15552 = |-6 -5 6> ampersand = 34171875/33554432 = |-25 7 6> graviton = 129140163/128000000 = |-13 17 -6> ennealimma = 7629394531250/7625597484987 = |1 -27 18>

7-limit	11-limit	13-limit
............................... 225/224 1029/1024 2401/2400 4375/4374 16875/16807 19683/19600 420175/419904 250047/250000	....................... 243/242 385/384 441/440 540/539 1375/1372 3025/3024 4000/3993 6250/6237 9801/9800	....................... 169/168 325/324 351/350 364/363 625/624 676/675 729/728 1001/1000 1575/1573 1716/1715 2080/2079 6656/6655

Temperaments

It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit.

See also List of edo-distinct 72et rank two temperaments.

Scales

smithgw72a, smithgw72b, smithgw72c, smithgw72d, smithgw72e, smithgw72f, smithgw72g, smithgw72h, smithgw72i, smithgw72j

blackjack, miracle_8, miracle_10, miracle_12, miracle_12a, miracle_24hi, miracle_24lo

keenanmarvel, xenakis_chrome, xenakis_diat, xenakis_schrome

Euler(24255) genus in 72 equal

JuneGloom

Harmonic Scale

Mode 8 of the harmonic series -- overtones 8 through 16, octave repeating -- is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).

Overtones in "Mode 8":	8		9		10		11		12		13		14		15		16
...as JI Ratio from 1/1:	1/1		9/8		5/4		11/8		3/2		13/8		7/4		15/8		2/1
...in cents:	0		203.9		386.3		551.3		702.0		840.5		968.8		1088.3		1200.0
Nearest degree of 72edo:	0		12		23		33		42		50		58		65		72
...in cents:	0		200.0		383.3		550.0		700.0		833.3		966.7		1083.3		1200.0
Steps as Freq. Ratio:		9:8		10:9		11:10		12:11		13:12		14:13		15:14		16:15
...in cents:		203.9		182.4		165.0		150.6		138.6		128.3		119.4		111.7
Nearest degree of 72edo:		12		11		10		9		8		8		7		7
...in cents:		200.0		183.3		166.7		150.0		133.3		133.3		116.7		116.7

Linear temperaments

Periods per octave	Generator	Names
1	1\72	quincy
1	5\72	marvolo
1	7\72	miracle/benediction/manna
1	11\72
1	13\72
1	17\72	neominor
1	19\72	catakleismic
1	23\72
1	25\72	sqrtphi
1	29\72
1	31\72	marvo/zarvo
1	35\72	cotritone
2	1\72
2	5\72	harry
2	7\72
2	11\72	unidec/hendec
2	13\72	wizard/lizard/gizzard
2	17\72
3	1\72
3	5\72	tritikleismic
3	7\72
3	11\72	mirkat
4	1\72	quadritikleismic
4	5\72
4	7\72
6	1\72
6	5\72
8	1\72	octoid
8	2\72	octowerck
8	4\72
9	1\72
9	3\72	ennealimmal/ennealimmic
12	1\72	compton
18	1\72	hemiennealimmal
24	1\72	hours
36	1\72

Z function

72edo is the ninth zeta integral edo, as well as being a peak and gap edo, and the maximum value of the Z function in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.

Music

Kotekant play by Gene Ward Smith

Twinkle canon – 72 edo by Claudi Meneghin

Lazy Sunday by Jake Freivald in the lazysunday scale.

June Gloom #9 by Prent Rodgers

External links

• Wikipedia article on 72edo
• OrthodoxWiki Article on Byzantine chant, which uses 72edo
• Wikipedia article on Joe Maneri (1927-2009)
• Ekmelic Music Society/Gesellschaft für Ekmelische Musik, a group of composers and researchers dedicated to 72edo music
• Rick Tagawa's 72edo site, including theory and composers' list
• Danny Wier, composer and musician who specializes in 72-edo