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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...

ampersand = 34171875/33554432 = |-25 7 6>

graviton = 129140163/128000000 = |-13 17 -6>

ennealimma = 7629394531250/7625597484987 = |1 -27 18>

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.

## 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).## Intervals

triple-down minor 2nd

triple-down minor 3rd

triple-down 4th

triple-down minor 6th

triple-down minor 7th

triple-down octave

## Linear temperaments

## 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

Kotekantplayby Gene Ward SmithTwinkle canon – 72 edoby Claudi MeneghinLazy Sundayby Jake Freivald in the lazysunday scale.June Gloom #9by Prent Rodgers## Scales

smithgw72a, smithgw72b, smithgw72c, smithgw72d, smithgw72e, smithgw72f, smithgw72g, smithgw72h, smithgw72i, smithgw72jblackjack, 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