72edo

← 71edo

72edo

73edo →

Prime factorization	23 × 32
Step size	16.6667 ¢
Fifth	42\72 (700 ¢) (→ 7\12)
Semitones (A1:m2)	6:6 (100 ¢ : 100 ¢)
Consistency limit	17
Distinct consistency limit	11

English Wikipedia has an article on:

72 equal temperament

The 72 equal divisions of the octave (72edo), or 72-tone equal temperament (72tet, 72et) when viewed from a regular temperament perspective, 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 12edo. 72edo is also a superset of 24edo, a common and standard tuning of Arabic music, and has itself been used to tune Turkish music.

Composers that used 72edo include Ivan Wyschnegradsky, Julián Carrillo (who is better associated with 96edo), Ezra Sims, James Tenney, Georg Friedrich Haas and the jazz musician Joe Maneri.

Theory

72edo 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 12edo, 72, 42 and 30 steps respectively, but the classic major third (5/4) measures 23 steps, not 24, and other 5-limit major intervals are one step flat of 12edo 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.

72edo is an excellent tuning for the miracle temperament, especially the 11-limit version, and the related rank-3 temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.

The 13th harmonic (octave reduced) is so closely mapped on acoustic phi that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.

72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo circulating temperament.

Prime harmonics

Approximation of prime harmonics in 72edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-1.96	-2.98	-2.16	-1.32	-7.19	-4.96	+2.49	+5.06	+3.76	+4.96
	Relative (%)	+0.0	-11.7	-17.9	-13.0	-7.9	-43.2	-29.7	+14.9	+30.4	+22.5	+29.8
Steps (reduced)		72 (0)	114 (42)	167 (23)	202 (58)	249 (33)	266 (50)	294 (6)	306 (18)	326 (38)	350 (62)	357 (69)

Subsets and supersets

Since 72 factors into 2³ × 3², 72edo has subset edos 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36. 144edo, which doubles it, provides a possible correction to its approximate harmonic 13.

Intervals

Degrees	Cents	Approximate Ratios *	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, 64/63	^^	dup unison	^^D
3	50.000	33/32	^31, v3m2	trup unison, trudminor 2nd	^3D, v3Eb
4	66.667	25/24	vvm2	dudminor 2nd	vvEb
5	83.333	21/20	vm2	downminor 2nd	vEb
6	100.000	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	150.000	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	200.000	9/8	M2	major 2nd	E
13	216.667	25/22, 17/15	^M2	upmajor 2nd	^E
14	233.333	8/7	^^M2	dupmajor 2nd	^^E
15	250.000	81/70, 15/13	^3M2, v3m3	trupmajor 2nd, trudminor 3rd	^3E, v3F
16	266.667	7/6	vvm3	dudminor 3rd	vvF
17	283.333	33/28, 13/11, 20/17	vm3	downminor 3rd	vF
18	300.000	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	350.000	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	400.000	44/35	M3	major 3rd	F#
25	416.667	14/11	^M3	upmajor 3rd	^F#
26	433.333	9/7	^^M3	dupmajor 3rd	^^F#
27	450.000	35/27, 13/10	^3M3, v34	trupmajor 3rd, trud 4th	^3F#, v3G
28	466.667	21/16, 17/13	vv4	dud 4th	vvG
29	483.333	33/25	v4	down 4th	vG
30	500.000	4/3	P4	perfect 4th	G
31	516.667	27/20	^4	up 4th	^G
32	533.333	15/11	^^4, v~4	dup 4th, downmid 4th	^^G
33	550.000	11/8	~4	mid 4th	^3G
34	566.667	25/18, 18/13	^~4, vvA4	upmid 4th, dudaug 4th	vvG#
35	583.333	7/5	vA4, vd5	downaug 4th, downdim 5th	vG#, vAb
36	600.000	99/70, 17/12	A4, d5	aug 4th, dim 5th	G#, Ab
37	616.667	10/7	^A4, ^d5	upaug 4th, updim 5th	^G#, ^Ab
38	633.333	36/25, 13/9	v~5, ^^d5	downmid 5th, dupdim 5th	^^Ab
39	650.000	16/11	~5	mid 5th	v3A
40	666.667	22/15	vv5, ^~5	dud 5th, upmid 5th	vvA
41	683.333	40/27	v5	down 5th	vA
42	700.000	3/2	P5	perfect 5th	A
43	716.667	50/33	^5	up 5th	^A
44	733.333	32/21	^^5	dup 5th	^^A
45	750.000	54/35, 17/11	^35, v3m6	trup 5th, trudminor 6th	^3A, v3Bb
46	766.667	14/9	vvm6	dudminor 6th	vvBb
47	783.333	11/7	vm6	downminor 6th	vBb
48	800.000	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	850.000	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	900.000	27/16	M6	major 6th	B
55	916.667	56/33, 17/10	^M6	upmajor 6th	^B
56	933.333	12/7	^^M6	dupmajor 6th	^^B
57	950.000	121/70	^3M6, v3m7	trupmajor 6th, trudminor 7th	^3B, v3C
58	966.667	7/4	vvm7	dudminor 7th	vvC
59	983.333	44/25	vm7	downminor 7th	vC
60	1000.000	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	1050.000	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	1100.000	66/35, 17/9	M7	major 7th	C#
67	1116.667	21/11	^M7	upmajor 7th	^C#
68	1133.333	27/14	^^M7	dupmajor 7th	^^C#
69	1150.000	35/18	^3M7, v38	trupmajor 7th, trud octave	^3C#, v3D
70	1166.667	49/25	vv8	dud octave	vvD
71	1183.333	99/50	v8	down octave	vD
72	1200.000	2/1	P8	perfect octave	D

* based on treating 72edo as a 17-limit temperament. For lower limits see Table of 72edo intervals.

Interval quality and chord names in color notation

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

Quality	Color	Monzo Format	Examples
dudminor	zo	(a b 0 1)	7/6, 7/40
minor	fourthward wa	(a b), b < -1	32/27, 16/9
upminor	gu	(a b -1)	6/5, 9/5
downmid	luyo	(a b 1 0 -1)	15/11
	tho	(a b 0 0 0 1)	13/8, 13/9
mid	ilo	(a, b, 0, 0, 1)	11/9, 11/6
	lu	(a, b, 0, 0, -1)	12/11, 18/11
upmid	logu	(a b -1 0 1)	11/10
	thu	(a b 0 0 0 -1)	16/13, 18/13
downmajor	yo	(a b 1)	5/4, 5/3
major	fifthward wa	(a b), b > 1	9/8, 27/16
dupmajor	ru	(a, b, 0, -1)	9/7, 12/7

All 72edo 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 dudminor
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 dupmajor or C dup

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

Remembering the pitch structure

The pitch structure is very easy to remember. In 72tet, 12edo is the Pythagorian ring; id est, every 6 degrees is the 3-limit.

Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:

• -1 degree corrects 5/4 (80/81)
• -2 degrees corrects 7/4 (63/64)
• +3 degrees corrects 11/8 (33/32)
• +2 degrees corrects 13/8 (1053/1024)
• 0 degree corrects 17/16 and 19/16 (4131/4096 and 513/512)

Notations

Sagittal

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 72edo in the Revo flavor of Sagittal:

JI approximation

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.

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 72edo. Prime harmonics are in bold. As 72edo is consistent in the 15-odd-limit, the results by direct approximation and patent val mapping are the same.

The following table shows how 15-odd-limit intervals are represented in 72edo. Prime harmonics are in bold.

As 72edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 72edo

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/6, 12/7	0.204	1.2
11/6, 12/11	0.637	3.8
7/5, 10/7	0.821	4.9
11/7, 14/11	0.841	5.0
9/5, 10/9	0.930	5.6
5/3, 6/5	1.025	6.2
11/8, 16/11	1.318	7.9
11/10, 20/11	1.662	10.0
9/7, 14/9	1.751	10.5
3/2, 4/3	1.955	11.7
7/4, 8/7	2.159	13.0
15/13, 26/15	2.259	13.6
11/9, 18/11	2.592	15.6
15/14, 28/15	2.776	16.7
5/4, 8/5	2.980	17.9
13/9, 18/13	3.284	19.7
15/11, 22/15	3.617	21.7
9/8, 16/9	3.910	23.5
13/10, 20/13	4.214	25.3
15/8, 16/15	4.935	29.6
13/7, 14/13	5.035	30.2
13/12, 24/13	5.239	31.4
13/11, 22/13	5.876	35.3
13/8, 16/13	7.194	43.2

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3.5	15625/15552, 531441/524288	[⟨72 114 167]]	+0.839	0.594	3.56
2.3.5.7	225/224, 1029/1024, 4375/4374	[⟨72 114 167 202]]	+0.822	0.515	3.09
2.3.5.7.11	225/224, 243/242, 385/384, 4000/3993	[⟨72 114 167 202 249]]	+0.734	0.493	2.96
2.3.5.7.11.13	169/168, 225/224, 243/242, 325/324, 385/384	[⟨72 114 167 202 266 249]]	+0.936	0.638	3.82
2.3.5.7.11.13.17	169/168, 221/220, 225/224, 243/242, 273/272, 325/324	[⟨72 114 167 202 249 266 294]]	+0.975	0.599	3.59

• 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are 99, 270, 224, 494, and 217, respectively.

Commas

Commas tempered out by 72edo include…

Prime Limit	Ratio	Monzo	Cents	Name(s)
3	(12 digits)	[-19 12⟩	23.46	Pythagorean comma
5	15625/15552	[-6 -5 6⟩	8.11	Kleisma
5	(16 digits)	[-25 7 6⟩	31.57	Ampersand
5	(18 digits)	[-13 17 -6⟩	15.35	Graviton
5	(26 digits)	[1 -27 18⟩	0.86	Ennealimma
7	225/224	[-5 2 2 -1⟩	7.71	Marvel comma
7	1029/1024	[-10 1 0 3⟩	8.43	Gamelisma
7	2401/2400	[-5 -1 -2 4⟩	0.72	Breedsma
7	4375/4374	[-1 -7 4 1⟩	0.40	Ragisma
7	16875/16807	[0 3 4 -5⟩	6.99	Mirkwai
7	19683/19600	[-4 9 -2 -2⟩	7.32	Cataharry
7	(12 digits)	[-6 -8 2 5⟩	1.12	Wizma
7	(12 digits)	[-4 6 -6 3⟩	0.33	Landscape comma
11	243/242	[-1 5 0 0 -2⟩	7.14	Rastma
11	385/384	[-7 -1 1 1 1⟩	4.50	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Werckisma
11	540/539	[2 3 1 -2 -1⟩	3.21	Swetisma
11	1375/1372	[-2 0 3 -3 1⟩	3.78	Moctdel
11	3025/3024	[-4 -3 2 -1 2⟩	0.57	Lehmerisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Wizardharry
11	6250/6237	[1 -4 5 -1 -1⟩	3.60	Liganellus comma
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Kalisma
11	(14 digits)	[16 -3 0 0 6⟩	2.04	Nexus comma
13	169/168	[-3 -1 0 -1 0 2⟩	10.27	Buzurgisma
13	325/324	[-2 -4 2 0 0 1⟩	5.34	Marveltwin comma
13	351/350	[-1 3 -2 -1 0 1⟩	4.94	Ratwolfsma
13	364/363	[2 -1 0 1 -2 1⟩	4.76	Gentle comma
13	625/624	[-4 -1 4 0 0 -1⟩	2.77	Tunbarsma
13	676/675	[2 -3 -2 0 0 2⟩	2.56	Island comma
13	729/728	[-3 6 0 -1 0 -1⟩	2.38	Squbema
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Sinbadma
13	1575/1573	[2 2 1 -2 -1⟩	2.20	Nicola
13	1716/1715	[2 1 -1 -3 1 1⟩	1.01	Lummic comma
13	2080/2079	[5 -3 1 -1 -1 1⟩	0.83	Ibnsinma
13	6656/6655	[9 0 -1 0 -3 1⟩	0.26012	Jacobin comma

Rank-2 temperaments

• List of edo-distinct 72et rank two temperaments

72edo 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.

Periods per 8ve	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	Semimiracle
2	11\72	Unidec / hendec
2	13\72	Wizard / lizard / gizzard
2	19\72	Bikleismic
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 / ennealimnic
12	1\72	Compton
18	1\72	Hemiennealimmal
24	1\72	Hours
36	1\72

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
• Aurora, Blackened skies, Lost spirit
• Harry Partch's 43-tone scale: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1

Harmonic scale

Mode 8 of the harmonic series – harmonics 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).

Harmonics 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

Music

Ambient Esoterica

• Goetic Synchronities (2023)

Jake Freivald

• Lazy Sunday^{[dead link]} in the lazysunday scale

English Wikipedia has an article on:

In vain (Haas)

Georg Friedrich Haas

• Blumenstück (2000)
• in vain (2000) (score)

Claudi Meneghin

• Twinkle canon – 72 edo^{[dead link]}
• The Miracle Canon, 3-in-1 on a Ground
• Sicilienne in Miracle

Prent Rodgers

• June Gloom #9^{[dead link]}

Gene Ward Smith

• Kotekant play (2010)

James Tenney

• Changes for Six Harps

Xeno Ov Eleas

• Χenomorphic Ghost Storm (2022)

External links

• OrthodoxWiki Article on Byzantine chant, which uses 72edo
• 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