12edt

← 11edt

12edt

13edt →

Prime factorization	22 × 3 (highly composite)
Step size	158.496 ¢
Octave	8\12edt (1267.97 ¢) (→ 2\3edt)
Consistency limit	3
Distinct consistency limit	3

12 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 12edt or 12ed3), is a nonoctave tuning system that divides the interval of 3/1 into 12 equal parts of about 158 ¢ each. Each step represents a frequency ratio of 3^1/12, or the 12th root of 3.

12edt corresponds 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.

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)

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 5L 2s⟨3/1⟩ scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the generator is now the size of a major seventh instead of a perfect fifth. 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.

Interval table

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	158.5	108.3	21/19, 23/21
2	317	216.7	6/5, 13/11, 17/14, 23/19
3	475.5	325	17/13
4	634	433.3	13/9, 19/13
5	792.5	541.7	11/7, 14/9
6	951	650	19/11
7	1109.5	758.3	17/9, 21/11
8	1268	866.7	19/9, 23/11
9	1426.5	975
10	1585	1083.3	5/2
11	1743.5	1191.7	19/7
12	1902	1300	3/1

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

Compositions

Instant Gamelan by Carlo Serafini

Tritave in 12 by Chris Vaisvil