5L 5s: Difference between revisions
Uncomfy with blackwoodian |
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| Pattern = LsLsLsLsLs | | Pattern = LsLsLsLsLs | ||
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== Name == | |||
The [[TAMNAMS]] temperament-agnostic name is '''pentasymmetric'''. The name "blackwood" is also used, although this is a misnomer as [[blackwood]] is an abstract temperament, not a concrete scale (a more proper name is "blackwood[10]"). | |||
== As a temperament == | |||
There is only one significant harmonic entropy minimum with this MOS pattern: [[Archytas_clan|blackwood]], in which intervals of the prime numbers 3 and 7 are all represented using steps of [[5edo|5edo]], and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5. | There is only one significant harmonic entropy minimum with this MOS pattern: [[Archytas_clan|blackwood]], in which intervals of the prime numbers 3 and 7 are all represented using steps of [[5edo|5edo]], and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5. | ||
The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being [[Rothenberg_propriety|proper]] (because 1\15 is in the middle of the range of good blackwood generators). | The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being [[Rothenberg_propriety|proper]] (because 1\15 is in the middle of the range of good blackwood generators). | ||
== Scale tree == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
Revision as of 18:53, 15 April 2021
| ↖ 4L 4s | ↑ 5L 4s | 6L 4s ↗ |
| ← 4L 5s | 5L 5s | 6L 5s → |
| ↙ 4L 6s | ↓ 5L 6s | 6L 6s ↘ |
┌╥┬╥┬╥┬╥┬╥┬┐ │║│║│║│║│║││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sLsLsLsLsL
Name
The TAMNAMS temperament-agnostic name is pentasymmetric. The name "blackwood" is also used, although this is a misnomer as blackwood is an abstract temperament, not a concrete scale (a more proper name is "blackwood[10]").
As a temperament
There is only one significant harmonic entropy minimum with this MOS pattern: blackwood, in which intervals of the prime numbers 3 and 7 are all represented using steps of 5edo, and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5.
The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being proper (because 1\15 is in the middle of the range of good blackwood generators).
Scale tree
| Generator | Cents | Comments | ||||
|---|---|---|---|---|---|---|
| 0\5 | 0 | |||||
| 1\30 | 40 | |||||
| 1\25 | 48 | |||||
| 240/(1+pi) | ||||||
| 1\20 | 60 | |||||
| 240/(1+e) | ||||||
| 2\35 | 68.57 | |||||
| 3\50 | 72 | |||||
| 1\15 | 80 | Blackwood is around here
Optimum rank range (L/s=2/1) for MOS | ||||
| 240/(1+sqrt(3)) | ||||||
| 3\40 | 90 | |||||
| 5\65 | 92.31 | Golden blackwood | ||||
| 240/(1+pi/2) | ||||||
| 2\25 | 96 | |||||
| 3\35 | 102.86 | |||||
| 4\45 | 103.33 | |||||
| 1\10 | 120 | |||||