# 72edo

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 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.

# 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.

# 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

# Intervals

 degrees cents value pions 7mus approximate ratios (11-limit) ups and downs notation 0 1/1 P1 perfect unison D 1 16.667 17.667 21.333 (15.55516) 81/80 ^1 up unison D^ 2 33.333 35.333 42.667 (2A.AAB16) 45/44 ^^ double-up unison D^^ 3 50 53 64 (4016) 33/32 ^31, v3m2 triple-up unison, triple-down minor 2nd D^3, Ebv3 4 66.667 70.667 85.333 (55.55516) 25/24 vvm2 double-downminor 2nd Ebvv 5 83.333 88.333 106.667 (6A.AAB16) 21/20 vm2 downminor 2nd Ebv 6 100 106 128 (8016) 35/33 m2 minor 2nd Eb 7 116.667 123.667 149.333 (95.55516) 15/14 ^m2 upminor 2nd Eb^ 8 133.333 141.333 170.667 (AA.AAB16) 27/25 v~2 downmid 2nd Eb^^ 9 150 159 192 (C016) 12/11 ~2 mid 2nd Ev3 10 166.667 176.667 213.333 (D5.55516) 11/10 ^~2 upmid 2nd Evv 11 183.333 194.333 234.667 (EA.AAB16) 10/9 vM2 downmajor 2nd Ev 12 200 212 256 (10016) 9/8 M2 major 2nd E 13 216.667 229.667 277.333 (115.55516) 25/22 ^M2 upmajor 2nd E^ 14 233.333 247.333 298.667 (12A.AAB16) 8/7 ^^M2 double-upmajor 2nd E^^ 15 250 265 320 (14016) 81/70 ^3M2, v3m3 triple-up major 2nd, triple-down minor 3rd E^3, Fv3 16 266.667 282.333 341.333 (155.55516) 7/6 vvm3 double-downminor 3rd Fvv 17 283.333 300.333 362.667 (16A.AAB16) 33/28 vm3 downminor 3rd Fv 18 300 318 384 (18016) 25/21 m3 minor 3rd F 19 316.667 335.667 405.333 (195.55516) 6/5 ^m3 upminor 3rd F^ 20 333.333 353.333 426.667 (1AA.AAB16) 40/33 v~3 downmid 3rd F^^ 21 350 371 448 (1C016) 11/9 ~3 mid 3rd F^3 22 366.667 388.667 469.333 (1D5.55516) 99/80 ^~3 upmid 3rd F#vv 23 383.333 406.333 490.667 (1EA.AAB16) 5/4 vM3 downmajor 3rd F#v 24 400 424 512 (20016) 44/35 M3 major 3rd F# 25 416.667 441.667 533.333 (215.55516) 14/11 ^M3 upmajor 3rd F#^ 26 433.333 459.333 554.667 (22A.AAB16) 9/7 ^^M3 double-upmajor 3rd F#^^ 27 450 477 576 (24016) 35/27 ^3M3, v34 triple-up major 3rd, triple-down 4th F#^3, Gv3 28 466.667 494.667 597.333 (255.55516) 21/16 vv4 double-down 4th Gvv 29 483.333 512.333 618.667 (26A.AAB16) 33/25 v4 down 4th Gv 30 500 530 640 (28016) 4/3 P4 perfect 4th G 31 516.667 547.667 661.333 (295.55516) 27/20 ^4 up 4th G^ 32 533.333 565.333 682.667 (2AA.AAB16) 15/11 ^^4 double-up 4th G^^ 33 550 583 704 (2C016) 11/8 ^34 triple-up 4th G^3 34 566.667 600.667 725.333 (2D5.55516) 25/18 vvA4 double-down aug 4th G#vv 35 583.333 618.333 746.667 (2EA.AAB16) 7/5 vA4, vd5 downaug 4th, updim 5th G#v, Abv 36 600 636 768 (30016) 99/70 A4, d5 aug 4th, dim 5th G#, Ab 37 616.667 653.667 789.333 (315.55516) 10/7 ^A4, ^d5 upaug 4th, downdim 5th G#^, Ab^ 38 633.333 671.333 810.667 (32A.AAB16) 36/25 ^^d5 double-updim 5th Ab^^ 39 650 689 832 (34016) 16/11 v35 triple-down 5th Av3 40 666.667 706.667 853.333 (355.55516) 22/15 vv5 double-down 5th Avv 41 683.333 724.333 874.667 (36A.AAB16) 40/27 v5 down 5th Av 42 700 742 896 (38016) 3/2 P5 perfect 5th A 43 716.667 759.667 917.333 (395.55516) 50/33 ^5 up 5th A^ 44 733.333 777.333 948.667 (3AA.AAB16) 32/21 ^^5 double-up 5th A^^ 45 750 795 960 (3C016) 54/35 ^35, v3m6 triple-up 5th, triple-down minor 6th A^3, Bbv3 46 766.667 812.667 981.333 (3D5.55516) 14/9 vvm6 double-downminor 6th Bbvv 47 783.333 830.333 1002.667 (2EA.AAB16) 11/7 vm6 downminor 6th Bbv 48 800 848 1024 (40016) 35/22 m6 minor 6th Bb 49 816.667 865.667 1045.333 (415.55516) 8/5 ^m6 upminor 6th Bb^ 50 833.333 883.333 1066.667 (42A.AAB16) 81/50 v~6 downmid 6th Bb^^ 51 850 901 1088 (44016) 18/11 ~6 mid 6th Bv3 52 866.667 918.667 1109.333 (455.55516) 33/20 ^~6 upmid 6th Bvv 53 883.333 936.333 1130.667 (46A.AAB16) 5/3 vM6 downmajor 6th Bv 54 900 954 1152 (48016) 27/16 M6 major 6th B 55 916.667 971.667 1173.333 (495.55516) 56/33 ^M6 upmajor 6th B^ 56 933.333 989.333 1194.667 (4AA.AAB16) 12/7 ^^M6 double-upmajor 6th B^^ 57 950 1007 1216 (4C016) 121/70 ^3M6, v3m7 triple-up major 6th, triple-down minor 7th B^3, Cv3 58 966.667 1024.667 1237.333 (4D5.55516) 7/4 vvm7 double-downminor 7th Cvv 59 983.333 1042.333 1258.667 (4EA.AAB16) 44/25 vm7 downminor 7th Cv 60 1000 1060 1280 (50016) 16/9 m7 minor 7th C 61 1016.667 1077.667 1301.333 (515.55516) 9/5 ^m7 upminor 7th C^ 62 1033.333 1095.333 1322.667 (52A.AAB16) 20/11 v~7 downmid 7th C^^ 63 1050 1113 1344 (54016) 11/6 ~7 mid 7th C^3 64 1066.667 1130.667 1365.333 (555.55516) 50/27 ^~7 upmid 7th C#vv 65 1083.333 1148.333 1386.667 (56A.AAB16) 15/8 vM7 downmajor 7th C#v 66 1100 1166 1408 (58016) 66/35 M7 major 7th C# 67 1116.667 1183.667 1429.333 (595.55516) 21/11 ^M7 upmajor 7th C#^ 68 1133.333 1201.333 1450.667 (5AA.AAB16) 27/14 ^^M7 double-upmajor 7th C#^^ 69 1150 1219 1472 (5C016) 35/18 ^3M7, v38 triple-up major 7th, triple-down octave C#^3, Dv3 70 1166.667 1236.667 1493.333 (5D5.55516) 49/25 vv8 double-down octave Dvv 71 1183.333 1254.333 1514.667 (5EA.AAB16) 99/50 v8 down octave Dv 72 1200 1272 1536 (60016) 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 72edo chords can be named using ups and downs. 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 Ebvv G C.vvm 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 Ev3 G C~ C mid
yo 4:5:6 0-23-42 C Ev G C.v C downmajor or C dot down
ru 14:18:27 0-26-42 C E^^ G C.^^ C double-upmajor or C dot double-up

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

# 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 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

Lazy Sunday by Jake Freivald in the lazysunday scale.

June Gloom #9 by Prent Rodgers