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{{MOS intro}}
{{MOS intro}}
This MOS resembles [[5L 2s]], but with one large step removed.
4L 2s can be seen as a [[Warped diatonic|warped diatonic scale]], where one large step of diatonic (5L 2s) is removed, or as the equal-tempered whole-tone scale ([[6edo]]), but with two "whole tones" that are smaller than the others.


Scales with the true MOS pattern are always [[Rothenberg propriety|proper]], because there is only one small step per period. In addition, there are near-MOS patterns, such as LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).
== Name==
[[TAMNAMS]] suggests the temperament-agnostic name '''citric''' for this scale.
==Theory==
===Temperament interpretations===
There are three scales with this [[MOSScales|MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.
There are three scales with this [[MOSScales|MOS]] pattern that are significant minima of harmonic entropy. The first is [[antikythera]], or no-3's [[Diaschismic_family|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.


The second is [[Dicot_family|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.
The second is [[Dicot_family|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic_clan|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5.


In addition to the true MOS with pattern LLsLLs, all these scales also come in a near-MOS version, LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The true MOS is always proper, but the near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).
==Modes==
 
== Modes ==
{{MOS modes}}
{{MOS modes}}
== Scale tree ==
==Scale tree==
{| class="wikitable"
{| class="wikitable"
|-
|-
! colspan="11" | Generator
! colspan="11" |Generator
! | Cents
! | Cents
! | Comments
! | Comments
|-
|-
| | 1\6
| |1\6
| |
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Line 38: Line 42:
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| | 200
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| style="text-align:center;" |  
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| |200
| style="text-align:center;" |
|-
|-
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| |
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| |6\34
| |  
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| | 6\34
| |
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| |211.76
| |
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| |  
| | 211.76
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
Line 60: Line 66:
| |  
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| | 5\28
| | 5\28
| |  
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| | 214.29
| | 214.29
| style="text-align:center;" | Antikythera is around here
| style="text-align:center;" |Antikythera is around here
|-
|-
| |  
| |
| |  
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| | 4\22
| |4\22
| |  
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| | 218.18
| |218.18
| style="text-align:center;" |  
| style="text-align:center;" |
|-
|-
| |  
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| |
| | 3\16
| |3\16
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| | 225
| |225
| style="text-align:center;" |  
| style="text-align:center;" |
|-
|-
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| | 227.56
| | 227.56
| |  
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|-
|-
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Line 116: Line 122:
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| | 8\42
| | 8\42
| |  
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| | 228.57
| |228.57
| style="text-align:center;" |  
| style="text-align:center;" |
|-
|-
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| | 600/(1+phi)
| |600/(1+phi)
| style="text-align:center;" | Golden lemba
| style="text-align:center;" | Golden lemba
|-
|-
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| | 13\68
| |13\68
| |  
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| | 229.41
| |229.41
| style="text-align:center;" |  
| style="text-align:center;" |
|-
|-
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| |5\26
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| | 5\26
| |230.77
| |
| style="text-align:center;" |
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| | 230.77
| style="text-align:center;" |  
|-
|-
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Line 183: Line 189:
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| | 7\36
| |7\36
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| | 233.33
| |233.33
| style="text-align:center;" | Lemba is around here
| style="text-align:center;" | Lemba is around here
|-
|-
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| | 2\10
| |2\10
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Line 204: Line 209:
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| | 240
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| style="text-align:center;" | Boundary of propriety for near-MOS
| |240
| style="text-align:center;" |Boundary of propriety for near-MOS


Optimum rank range (L/s=2/1) for MOS
Optimum rank range (L/s=2/1) for MOS
|-
|-
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| |5\24
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| | 5\24
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| | 250
| | 250
| style="text-align:center;" | Decimal is around here
| style="text-align:center;" |Decimal is around here
|-
|-
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Line 231: Line 237:
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| | 251.89
| |251.89
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|-
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| | 8\38
| |8\38
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Line 251: Line 257:
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| | 252.63
| | 252.63
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|-
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| |253.39
| |
| style="text-align:center;" |<span style="display: block; text-align: center;">L/s = e</span>
| |
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| | 253.39
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = e</span>
|-
|-
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| | 3\14
| |3\14
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Line 278: Line 281:
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| | 257.14
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| |257.14
| style="text-align:center;" | L/s = 3
| style="text-align:center;" | L/s = 3
|-
|-
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| | 258.81
| |258.81
| style="text-align:center;" | L/s = pi
| style="text-align:center;" | L/s = pi
|-
|-
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| |4\18
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| | 4\18
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| |266.67
| |  
| style="text-align:center;" |L/s = 4
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| | 266.67
| style="text-align:center;" | L/s = 4
|-
|-
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| |5\22
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| |272.73
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| style="text-align:center;" |
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| | 5\22
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| | 272.73
| style="text-align:center;" |  
|-
|-
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| |6\26
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| |276.92
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| style="text-align:center;" |Doublewide is around here
| |
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| | 6\26
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| | 276.92
| style="text-align:center;" | Doublewide is around here
|-
|-
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| | 7\30
| | 7\30
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| | 280
| | 280
| style="text-align:center;" |  
| style="text-align:center;" |
|-
|-
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Line 355: Line 361:
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| |8\34
| | 8\34
| |
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| |282.35
| | 282.35
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
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| |9\38
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| | 9\38
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| | 284.21
| | 284.21
| style="text-align:center;" |  
| style="text-align:center;" |
|-
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| | 10\42
| |10\42
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| | 285.71
| |285.71
| style="text-align:center;" |  
| style="text-align:center;" |
|-
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| | 11\46
| |11\46
| | 286.96
| |286.96
| style="text-align:center;" |  
| style="text-align:center;" |
|-
|-
| | 1\4
| |1\4
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| | 300
| |300
| style="text-align:center;" |  
| style="text-align:center;" |
|}
|}


[[Category:6-tone scales]]
[[Category:6-tone scales]]
Retrieved from "https://en.xen.wiki/w/4L_2s"