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Commas and Temperaments | Harmonic Scale | Intervals | Linear temperaments | Z function | Music | Scales | External links
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 and Temperaments


72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and
the kleisma, 15625/15552,
the ampersand, 34171875/33554432 = |-25 7 6>,
the graviton, 129140163/128000000 = |-13 17 -6>, and
the ennealimma, 7629394531250/7625597484987 = |1 -27 18> in the 5-limit.

The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.

72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800.

For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.


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
approximate ratios (11-limit)
ups and downs notation
0
0
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
D^3, Ebv3
4
66.667
25/24
vvm2
double-downminor 2nd
Ebvv
5
83.333
21/20
vm2
downminor 2nd
Ebv
6
100
35/33
m2
minor 2nd
Eb
7
116.667
15/14
^m2
upminor 2nd
Eb^
8
133.333
27/25
v~2
downmid 2nd
Eb^^
9
150
12/11
~2
mid 2nd
Ev3
10
166.667
11/10
^~2
upmid 2nd
Evv
11
183.333
10/9
vM2
downmajor 2nd
Ev
12
200
9/8
M2
major 2nd
E
13
216.667
25/22
^M2
upmajor 2nd
E^
14
233.333
8/7
^^M2
double-upmajor 2nd
E^^
15
250
81/70
^3M2, v3m3
triple-up major 2nd,
triple-down minor 3rd
E^3, Fv3
16
266.667
7/6
vvm3
double-downminor 3rd
Fvv
17
283.333
33/28
vm3
downminor 3rd
Fv
18
300
25/21
m3
minor 3rd
F
19
316.667
6/5
^m3
upminor 3rd
F^
20
333.333
40/33
v~3
downmid 3rd
F^^
21
350
11/9
~3
mid 3rd
F^3
22
366.667
99/80
^~3
upmid 3rd
F#vv
23
383.333
5/4
vM3
downmajor 3rd
F#v
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
^3M3, v34
triple-up major 3rd,
triple-down 4th
F#^3, Gv3
28
466.667
21/16
vv4
double-down 4th
Gvv
29
483.333
33/25
v4
down 4th
Gv
30
500
4/3
P4
perfect 4th
G
31
516.667
27/20
^4
up 4th
G^
32
533.333
15/11
^^4
double-up 4th
G^^
33
550
11/8
^34
triple-up 4th
G^3
34
566.667
25/18
vvA4
double-down aug 4th
G#vv
35
583.333
7/5
vA4, vd5
downaug 4th, updim 5th
G#v, Abv
36
600
99/70
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
^^d5
double-updim 5th
Ab^^
39
650
16/11
v35
triple-down 5th
Av3
40
666.667
22/15
vv5
double-down 5th
Avv
41
683.333
40/27
v5
down 5th
Av
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
^35, v3m6
triple-up 5th,
triple-down minor 6th
A^3, Bbv3
46
766.667
14/9
vvm6
double-downminor 6th
Bbvv
47
783.333
11/7
vm6
downminor 6th
Bbv
48
800
35/22
m6
minor 6th
Bb
49
816.667
8/5
^m6
upminor 6th
Bb^
50
833.333
81/50
v~6
downmid 6th
Bb^^
51
850
18/11
~6
mid 6th
Bv3
52
866.667
33/20
^~6
upmid 6th
Bvv
53
883.333
5/3
vM6
downmajor 6th
Bv
54
900
27/16
M6
major 6th
B
55
916.667
56/33
^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
B^3, Cv3
58
966.667
7/4
vvm7
double-downminor 7th
Cvv
59
983.333
44/25
vm7
downminor 7th
Cv
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
C^3
64
1066.667
50/27
^~7
upmin 7th
C#vv
65
1083.333
15/8
vM7
downmajor 7th
C#v
66
1100
66/35
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
C#^3, Dv3
70
1166.667
49/25
vv8
double-down octave
Dvv
71
1183.333
99/50
v8
down octave
Dv
72
1200
2/1
P8
perfect octave
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.

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.

plot72.png

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

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

External links