Penslen

This is a problematic page or section. It lacks sufficient justification, content, or organization, and is subject to future overhaul or deletion.

Penslen is an aberrismic ternary scale with structure 5L5m6s and the sole MOS substitution scale of type 6s(5L5m).

11-limit penslen

Portent-tempered penslen is one of the best scale structures with which to access the 7-limit by tempering out 1029/1024 (equating (8/7)^3 to 3/2), and 11-limit by tempering out 385/384 (equating 6/5 * 8/7 to 11/8). These commas are both small and low-complexity.

The simplest tuning is 31edo, which tempers out both of the relevant commas.

Tuning shown is just 2.3.5, Slendric 7, Portent (Slendric + Keenanismic) 11, i.e.

$T(7)=(2^{10}/3{)}^{\frac{1}{3}}$

$T(11)=\frac{384}{5\cdot T(7)}$

T() denotes applying the tuning map.

Mode A3 (msLsmLsmLsmsLmsL) Just-2.3.5 Tuning

	0	1
3	702.0	51.6
2	468.0	1017.6
1	234.0	783.6
0	0.0	549.6
-1	966.0	315.6
-2	732.0	81.7
-3	498.0	1047.7
-4	264.1	813.7

Example Untempered Intervals

36/35	48.8
21/20	84.5
8/7	231.2
7/6	266.9
6/5	315.6
21/16	470.8
4/3	498.0
48/35	546.8
3/2	702.0
32/21	729.2
63/40	786.4
8/5	813.7
7/4	968.8
9/5	1017.6
64/35	1044.9
2/1	1200.0

23/16-offset penslen

In this version of penslen, the 8-step offset is ~23/16 tempered together with 10/7 by tempering out 161/160. You can also add prime 17 by tempering out 8211/8192 = (3*7*17*23)/2^13.

Tuning shown is just 2.3.23, and

$T(7)=(2^{10}/3{)}^{\frac{1}{3}},$

$T(5)=\frac{23\cdot T(7)}{2^5}.$

Mode G6 (LsmsLmsLmsLsmLsm) Just-2.3.23 Tuning

	0	1
3	702.0	130.2
2	468.0	1096.2
1	234.0	862.3
0	0.0	628.3
-1	966.0	394.3
-2	732.0	160.3
-3	498.0	1126.3
-4	264.1	892.3

Penslen in this temperament is represented by 46edo's 5:3:1 tuning.