12edt

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.

← 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

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

Prime harmonics

Approximation of harmonics in 12edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+68.0	+0.0	-22.6	+66.6	+68.0	-40.4	+45.4	+0.0	-23.9	-30.4	-22.6
	Relative (%)	+42.9	+0.0	-14.2	+42.0	+42.9	-25.5	+28.7	+0.0	-15.1	-19.2	-14.2
Steps (reduced)		8 (8)	12 (0)	15 (3)	18 (6)	20 (8)	21 (9)	23 (11)	24 (0)	25 (1)	26 (2)	27 (3)

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

Exactly analogous to meantone

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. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is hyperpyth which is found in 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