Penslen: Difference between revisions
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'''Penslen''' is an [[aberrisma|aberrismic]] ternary scale with structure 5L5m6s and the sole [[MOS substitution]] scale of type 6s(5L5m). | '''Penslen''' (pattern msLsmLsmLsmsLmsL) is an [[aberrisma|aberrismic]] ternary scale with structure 5L5m6s and the sole [[MOS substitution]] scale of type 6s(5L5m). | ||
== 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. | The simplest tuning is 31edo, which tempers out both of the relevant commas. | ||
Tuning shown is just 2.3.5, Slendric 7, | Tuning shown is just 2.3.5, Slendric 7, Keenanismic 11, i.e. | ||
<math> | <math> | ||
| Line 14: | Line 16: | ||
T(11) = \frac{384}{5 \cdot T(7)} | T(11) = \frac{384}{5 \cdot T(7)} | ||
</math> | </math> | ||
T() denotes applying the tuning map. | |||
{| class="wikitable" | {| class="wikitable" | ||
| Line 103: | Line 107: | ||
|1200.0 | |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 | |||
<math>T(7) = (2^{10}/3)^{\frac{1}{3}},</math> | |||
<math>T(5) = \frac{23 \cdot T(7)}{2^5}.</math> | |||
{| class="wikitable" | |||
|+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. | |||
{{Cat|Ternary scales | {{Cat|Ternary scales | ||
MOS substitution scales | MOS substitution scales | ||
}} | }} | ||
Latest revision as of 09:06, 5 April 2026
Penslen (pattern msLsmLsmLsmsLmsL) is an aberrismic ternary scale with structure 5L5m6s and the sole MOS substitution scale of type 6s(5L5m).
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() denotes applying the tuning map.
| 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 |
| 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
| 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.
