Penslen

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

Interval Table (just-2.3.5; mode A3, msLsmLsmLsmsLmsL; 2.3.5 and 2.3.7 triads highlighted)

A3	G3	G8	A2	A7	G7	A1	A6	G6	A0	A5	G5	G10	A4	G4	G9
1200	1148	1118	966	936	884	732	702	650	498	468	416	386	234	182	152
1048	996	966	814	784	732	580	550	498	346	316	264	234	82	30	0
1018	966	936	784	754	702	550	520	468	316	286	234	204	52	0	1170
966	914	884	732	702	650	498	468	416	264	234	182	152	0	1148	1118
814	762	732	580	550	498	346	316	264	112	82	30	0	1048	996	966
784	732	702	550	520	468	316	286	234	82	52	0	1170	1018	966	936
732	680	650	498	468	416	264	234	182	30	0	1148	1118	966	914	884
702	650	620	468	438	386	234	204	152	0	1170	1118	1088	936	884	854
550	498	468	316	286	234	82	52	0	1048	1018	966	936	784	732	702
498	446	416	264	234	182	30	0	1148	996	966	914	884	732	680	650
468	416	386	234	204	152	0	1170	1118	966	936	884	854	702	650	620
316	264	234	82	52	0	1048	1018	966	814	784	732	702	550	498	468
264	212	182	30	0	1148	996	966	914	762	732	680	650	498	446	416
234	182	152	0	1170	1118	966	936	884	732	702	650	620	468	416	386
82	30	0	1048	1018	966	814	784	732	580	550	498	468	316	264	234
52	0	1170	1018	988	936	784	754	702	550	520	468	438	286	234	204
0	1148	1118	966	936	884	732	702	650	498	468	416	386	234	182	152

11-limit penslen

Portent (Slendric + Keenanismic)-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. (Together with Aberschismic, Portent implies 11-limit Rodan, making Rodan uniquely useful in the aberrismic theory of edos 41 and larger.)

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

Tuning shown is just 2.3.5, Slendric 7, 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.