12edt

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← 11edt12edt13edt →
Prime factorization 22 × 3
Step size 158.496¢ 
Octave 8\12edt (1267.97¢) (→2\3edt)
Consistency limit 2
Distinct consistency limit 2
Special properties

12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for hemikleismic temperament. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.

Interval table

Steps Cents Approximate Ratios
0 0 1/1
1 158.496 21/19, 23/21
2 316.993 6/5, 13/11, 17/14, 23/19
3 475.489 17/13
4 633.985 13/9, 19/13
5 792.481 11/7, 14/9
6 950.978 19/11
7 1109.474 17/9, 21/11
8 1267.97 19/9, 23/11
9 1426.466
10 1584.963 5/2
11 1743.459 19/7
12 1901.955 3/1

Prime harmonics

Approximation of prime harmonics in 12edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +68.0 +0.0 +66.6 -40.4 -30.4 -2.6 +8.4 -25.6 -39.4 +34.8 +77.8
Relative (%) +42.9 +0.0 +42.0 -25.5 -19.2 -1.7 +5.3 -16.2 -24.9 +21.9 +49.1
Steps
(reduced)
8
(8)
12
(0)
18
(6)
21
(9)
26
(2)
28
(4)
31
(7)
32
(8)
34
(10)
37
(1)
38
(2)

Scala file

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

Theory

In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament.

Macrodiatonic and macromeantone

12edt can be viewed as a version of 12edo with octaves stretched out to tritaves, so it contains an extremely stretched diatonic scale or macrodiatonic scale (5L 2s<3/1>). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable-for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by 17/9 and the stretched major third by 13/9. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 17/9 to 13/9 tritave-reduced, rather than 4 3/2 to 5/4 octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like meansquared which repeats at 4/1).

Another example of a macrodiatonic scale is hyperpyth which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

Compositions

Instant Gamelan by Carlo Serafini

Tritave in 12 by Chris Vaisvil