# 15edt

 ← 14edt 15edt 16edt →
Prime factorization 3 × 5
Step size 126.797¢
Octave 9\15edt (1141.17¢) (→3\5edt)
Fifth 6\15edt (760.782¢) (→2\5edt)
Semitones (A1:m2) 6:-3 (760.8¢ : -380.4¢)
Dual sharp fifth 6\15edt (760.782¢) (→2\5edt)
Dual flat fifth 5\15edt (633.985¢) (→1\3edt)
Dual major 2nd 2\15edt (253.594¢)
Consistency limit 2
Distinct consistency limit 2

15EDT is the equal division of the third harmonic into 15 parts of 126.7970 cents each, corresponding to 9.4639 edo.

## Properties

15EDT has harmonics 5 and 13 closely in tune, but does not do so well for 7 and 11, which are quite sharp. It tempers out the mowgli comma, |0 22 -15> in the 5-limit, which is tempered out by 19edo but has an optimal patent val of 303edo. As a 3.5.13 subgroup system, it tempers out 2197/2187 and 3159/3125. Using the patent val, it tempers out 375/343 and 6561/6125 in the 7-limit; 81/77, 125/121, and 363/343 in the 11-limit; 65/63, 169/165, 585/539, and 1287/1225 in the 13-limit; 51/49, 121/119, 125/119, 189/187, and 195/187 in the 17-limit (no-twos subgroup). 15EDT is related to the 2.3.5.13 subgroup temperament 19&123, which has a mapping [<1 0 0 0|, <0 15 22 35|], where the generator, an approximate 27/25, has a POTE tuning of 126.773, very close to 15EDT.

With the patent 4, it tempers out 36/35, 64/63, and 375/343 in the 3.4.5.7 subgroup; 45/44, 80/77, 81/77, and 363/343 in the 3.4.5.7.11 subgroup; 52/49, 65/63, 65/64, 143/140, and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, 52/51, 85/84, and 121/119 in the 3.4.5.7.11.13.17 subgroup (as well as 19ED4).

## Intervals

Degrees Cents Hekts Approximate Ratios Polaris nonatonic notation
0 1/1 H
1 126.797 86.667 14/13, 15/14, 16/15, 29/27 Ib
2 253.594 173.333 15/13 vH#, ^Ib
3 380.391 260 5/4 H#
4 507.188 346.667 4/3 I
5 633.985 433.333 13/9 J
6 760.782 520 14/9 K
7 887.579 606.667 5/3 L
8 1014.376 793.333 9/5 Mb
9 1141.173 780 27/14 vL#, ^Mb
10 1267.97 866.667 [[27/26|27/13] L#
11 1394.767 953.333 9/4 (9/8 plus an octave) M
12 1521.564 1040 12/5 (6/5 plus an octave) N
13 1648.361 1126.667 13/5 (13/10 plus an octave) O
14 1775.158 1213.333 14/5 (7/5 plus an octave) P
15 1901.955 1300 3/1 H

15edt contains 4 intervals from 5edt and 2 intervals from 3edt, meaning that it contains 6 redundant intervals and 8 new intervals. The new intervals introduced include good approximations to 15/14, 15/13, 4/3, 5/3 and their tritave inverses. This allows for new chord possibilities such as 1:3:4:5:9:12:13:14:15:16...

15edt also contains a 5L5s MOS similar to Blackwood Decatonic, which I call Ebony. This MOS has a period of 1/5 of the tritave and the generator is a single step. The major scale is sLsLsLsLsL, and the minor scale is LsLsLsLsLs.

15edt approximates the 5th and 13th harmonics (and 29th) very well. Taking these as consonances one obtains an 3L+3s MOS "augmented scale", in which three 13/9 intervals close to a tritave, and another three are set 5/3 away.

## Z function

Below is a plot of the no-twos Z function in the vicinity of 15edt:

Music: