5L 5s: Difference between revisions
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{{MOS intro}} | {{MOS intro}} | ||
== As a temperament == | == As a temperament == | ||
There is only one significant [[harmonic entropy]] minimum with this MOS pattern: [[limmic temperaments# | There is only one significant [[harmonic entropy]] minimum with this MOS pattern: [[limmic temperaments#5-limit_.28blackwood.29|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). | ||
Revision as of 09:02, 4 January 2024
| ↖ 4L 4s | ↑ 5L 4s | 6L 4s ↗ |
| ← 4L 5s | 5L 5s | 6L 5s → |
| ↙ 4L 6s | ↓ 5L 6s | 6L 6s ↘ |
┌╥┬╥┬╥┬╥┬╥┬┐ │║│║│║│║│║││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sLsLsLsLsL
5L 5s, named pentawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 5 small steps, with a period of 1 large step and 1 small step that repeats every 240.0 ¢, or 5 times every octave. Generators that produce this scale range from 120 ¢ to 240 ¢, or from 0 ¢ to 120 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.
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 | L | s | L/s | Comments | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Chroma-positive | Chroma-negative | ||||||||||
| 1\10 | 120.000 | 120.000 | 1 | 1 | 1.000 | ||||||
| 6\55 | 130.909 | 109.091 | 6 | 5 | 1.200 | Qintosec↑ | |||||
| 5\45 | 133.333 | 106.667 | 5 | 4 | 1.250 | ||||||
| 9\80 | 135.000 | 105.000 | 9 | 7 | 1.286 | ||||||
| 4\35 | 137.143 | 102.857 | 4 | 3 | 1.333 | ||||||
| 11\95 | 138.947 | 101.053 | 11 | 8 | 1.375 | ||||||
| 7\60 | 140.000 | 100.000 | 7 | 5 | 1.400 | Warlock | |||||
| 10\85 | 141.176 | 98.824 | 10 | 7 | 1.428 | ||||||
| 3\25 | 144.000 | 96.000 | 3 | 2 | 1.500 | L/s = 3/2 | |||||
| 11\90 | 146.667 | 93.333 | 11 | 7 | 1.571 | ||||||
| 8\65 | 147.692 | 92.308 | 8 | 5 | 1.600 | ||||||
| 13\105 | 148.571 | 91.429 | 13 | 8 | 1.625 | Unnamed golden tuning | |||||
| 5\40 | 150.000 | 90.000 | 5 | 3 | 1.667 | ||||||
| 12\95 | 151.579 | 88.421 | 12 | 7 | 1.714 | ||||||
| 7\55 | 152.727 | 87.273 | 7 | 4 | 1.750 | Quinkee | |||||
| 9\70 | 154.286 | 85.714 | 9 | 5 | 1.800 | ||||||
| 2\15 | 160.000 | 80.000 | 2 | 1 | 2.000 | Basic pentawood Blacksmith is optimal around here | |||||
| 9\65 | 166.154 | 73.846 | 9 | 4 | 2.250 | Trisedodge | |||||
| 7\50 | 168.000 | 72.000 | 7 | 3 | 2.333 | ||||||
| 12\85 | 169.412 | 70.588 | 12 | 5 | 2.400 | ||||||
| 5\35 | 171.429 | 68.571 | 5 | 2 | 2.500 | ||||||
| 13\90 | 173.333 | 66.667 | 13 | 5 | 2.600 | Unnamed golden tuning | |||||
| 8\55 | 174.545 | 65.455 | 8 | 3 | 2.667 | ||||||
| 11\75 | 176.000 | 64.000 | 11 | 4 | 2.750 | ||||||
| 3\20 | 180.000 | 60.000 | 3 | 1 | 3.000 | L/s = 3/1 | |||||
| 10\65 | 184.615 | 55.385 | 10 | 3 | 3.333 | ||||||
| 7\45 | 186.667 | 53.333 | 7 | 2 | 3.500 | ||||||
| 11\70 | 188.571 | 51.429 | 11 | 3 | 3.667 | ||||||
| 4\25 | 192.000 | 48.000 | 4 | 1 | 4.000 | ||||||
| 9\55 | 196.364 | 43.636 | 9 | 2 | 4.500 | ||||||
| 5\30 | 200.000 | 40.000 | 5 | 1 | 5.000 | ||||||
| 6\35 | 205.714 | 34.286 | 6 | 1 | 6.000 | Cloudtone↓ | |||||
| 1\5 | 240.000 | 0.000 | 1 | 0 | → inf | ||||||