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| '''5L 5s''' refers to the structure of octave-equivalent [[MOS]] scales with period 1\5 (one degree of [[5edo]] = 240¢) and generators ranging from 1\10 (one degree of [[10edo]] = 120¢) to 1\5 (240¢). In the case of 10edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). 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. Hence from a regular temperament perspective, this MOS pattern is essentially synonymous with blackwood.
| | {{Infobox MOS |
| | | Name = pentawood |
| | | Periods = 5 |
| | | nLargeSteps = 5 |
| | | nSmallSteps = 5 |
| | | Equalized = 1 |
| | | Collapsed = 0 |
| | | Pattern = LsLsLsLsLs |
| | }} |
| | {{MOS intro}} |
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| 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).
| | 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 reaches intervals of 5 like 6/5, 5/4, or 7/5. |
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| {| class="wikitable"
| | In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢. |
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| ! colspan="5" | Generator
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| ! | Cents
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| ! | Comments
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| |-
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| | | 0\5
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| | | 0
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| | style="text-align:center;" |
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| |-
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| | | 1\30
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| | | 40
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| |-
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| | | 1\25
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| | | 48
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| |-
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| | | 240/(1+pi)
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| |-
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| | | 1\20
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| | | 60
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| | style="text-align:center;" |
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| |-
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| | | 240/(1+e)
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| |-
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| | | 2\35
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| | | 68.57
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| |-
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| | | 3\50
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| | | 72
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| |-
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| | | 1\15
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| | | 80
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| | style="text-align:center;" | Blackwood is around here
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| Optimum rank range (L/s=2/1) for MOS
| | == Intervals == |
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| | {{MOS intervals}} |
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| | | 240/(1+sq<span style="line-height: 1.5;">rt(3)</span>)
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| |-
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| | | 3\40
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| | | 90
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| | style="text-align:center;" |
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| |-
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| | | 5\65
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| | | 92.31
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| | style="text-align:center;" | Golden blackwood
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| |-
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| | | 240/(1+pi/2)
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| |-
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| | | 2\25
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| | | 96
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| | style="text-align:center;" |
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| |3\35
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| |102.86
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| |-
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| |4\45
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| |103.33
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| |-
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| | | 1\10
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| | | 120
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| | style="text-align:center;" |
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| |}
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| [[Category:Scales]] | | ==Modes== |
| [[Category:MOS scales]] | | {{MOS mode degrees}} |
| [[Category:Abstract MOS patterns]]
| | |
| | == Scale tree == |
| | {{MOS tuning spectrum |
| | | 6/5 = Qintosec ↑ |
| | | 7/5 = Warlock |
| | | 13/8 = Unnamed golden tuning |
| | | 7/4 = Quinkee |
| | | 2/1 = Blacksmith is optimal around here |
| | | 9/4 = Trisedodge |
| | | 13/5 = Unnamed golden tuning |
| | | 6/1 = Cloudtone ↓ |
| | }} |
| | |
| | [[Category:Pentawood| ]] |
| | [[Category:10-tone scales]] |
| | <!-- main article --> |