72edo

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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 96edo), 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 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 50.000 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
6 100.000 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 double-upmajor 2nd ^^E
15 250.000 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
18 300.000 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 double-upmajor 3rd ^^F#
27 450.000 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
30 500.000 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
33 550.000 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
36 600.000 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
39 650.000 16/11 ~5 mid 5th v3A
40 666.667 22/15 ^~5 upmid 5th vvA
41 683.333 40/27 v5 down 5th vA
42 700.000 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
45 750.000 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
48 800.000 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 double-upmajor 6th ^^B
57 950.000 121/70 ^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 1000.000 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 double-upmajor 7th ^^C#
69 1150.000 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 1200.000 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.

Just approximation

Selected just intervals

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23 prime 29 prime 31
Error absolute (¢) 0.000 -1.955 -2.980 -2.159 -1.318 -7.194 -4.955 +2.487 +5.059 +3.756 +4.964
relative (%) 0.0 -11.7 -17.9 -13.0 -7.9 -43.2 -29.7 +14.9 +30.4 +22.5 +29.8

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 72et.

3-limit 5-limit 7-limit 11-limit 13-limit 17-limit 19-limit
Octave stretch (¢) +0.617 +0.839 +0.822 +0.734 +0.936 +0.975 +0.780
Error absolute (¢) 0.617 0.594 0.515 0.493 0.638 0.599 0.762
relative (%) 3.70 3.56 3.09 2.96 3.82 3.59 4.57
  • 72et has a lower relative error than any previous ETs in the 7-, 11-, 13-, 17-, and 19-limit. The next ET that does better in these subgroups is 99, 270, 224, 494, and 217, respectively.

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.

plot72.png

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

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 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

Scales

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

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