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2L 3s: Difference between revisions

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== Scale tree ==
== Scale tree ==
Generator ranges:
{{Todo| expand |inline=1|comment=Add back entries from original scale tree.}}
* Chroma-positive generator: 480 cents (2\5) to 600 cents (1\2)
{{Scale tree}}
* Chroma-negative generator: 600 cents (1\2) to 720 cents (3\5)
 
{{Todo| cleanup |inline=1|comment=Rework this scale tree.}}
 
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | s
! | L-s
! | |L-2s|
! | Scale steps
! | Trichord
! | Comments
|-
| | 2\5
| |
| |
| |
| |
| |
| | 480
| | 240
| | 0
| | 240
| | 1 1 1 1 1
| | 1 1
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 11\27
| | 488.89
| | 222.22
| | 44.44
| | 177.78
| | 6 5 5 6 5
| | 6 5
| style="text-align:center;" | Slendro (insofar as it resembles a MOS)
 
would be in this region
|-
| |
| |
| |
| |
| | 9\22
| |
| | 490.91
| | 218.18
| | 54.545
| | 163.64
| | 5 4 4 5 4
| | 5 4
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 16\39
| | 492.31
| | 215.38
| | 61.54
| | 153.85
| | 9 7 7 9 7
| | 9 7
| style="text-align:center;" | No-5's superpyth/dominant is around here
|-
| |
| |
| |
| | 7\17
| |
| |
| | 494.12
| | 211.76
| | 70.59
| | 141.18
| | 4 3 3 4 3
| | 4 3
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 19\46
| | 495.65
| | 208.7
| | 78.26
| | 130.435
| | 11 8 8 11 8
| | 11 8
| |
|-
| |
| |
| |
| |
| | 12\29
| |
| | 496.55
| | 206.9
| | 82.76
| | 124.14
| | 7 5 5 7 5
| | 7 5
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 17\41
| | 497.56
| | 204.88
| | 87.8
| | 117.07
| | 10 7 7 10 7
| | 10 7
| style="text-align:center;" | Pythagorean pentatonic is around here
|-
| |
| |
| | 5\12
| |
| |
| |
| | 500
| | 200
| | 100
| | 100
| | 3 2 2 3 2
| | 3 2
| style="text-align:center;" | Familiar 12-equal pentatonic
 
(also optimum rank range: L/s=3/2)
|-
| |
| |
| |
| |
| |
| |
| | 502.305
| | 195.39
| | 111.53
| | 83.86
| | pi 2 pi 2 2
| | pi 2
| |
|-
| |
| |
| |
| |
| |
| | 18\43
| | 502.33
| | 195.35
| | 111.63
| | 83.72
| | 11 7 7 11 7
| | 11 7
| |
|-
| |
| |
| |
| |
| | 13\31
| |
| | 503.23
| | 193.55
| | 116.13
| | 77.42
| | 8 5 5 8 5
| | 8 5
| style="text-align:center;" | Optimal meantone pentatonic
 
is around here
|-
| |
| |
| |
| |
| |
| |
| | 1200/(4-phi)
| | 192.43
| | 118.93
| | 73.50
| | phi 1 1 phi 1
| | phi 1
| style="text-align:center;" | Golden meantone
|-
| |
| |
| |
| |
| |
| | 21\50
| | 504
| | 192
| | 120
| | 72
| | 13 8 8 13 8
| | 13 8
| style="text-align:center;" |
|-
| |
| |
| |
| | 8\19
| |
| |
| | 505.26
| | 189.47
| | 126.32
| | 63.16
| | 5 3 3 5 3
| | 5 3
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 19\45
| | 506.67
| | 186.67
| | 133.33
| | 53.33
| | 12 7 7 12 7
| | 12 7
| |
|-
| |
| |
| |
| |
| |
| |
| | 507.18
| | 185.64
| | 135.9
| | 49.74
| | √3 1 √3 1 1
| | √3 1
| |
|-
| |
| |
| |
| |
| | 11\26
| |
| | 507.69
| | 184.615
| | 138.46
| | 46.15
| | 7 4 4 7 4
| | 7 4
| |
|-
| |
| |
| |
| |
| |
| | 14\33
| | 509.09
| | 181.82
| | 145.455
| | 36.36
| | 9 5 5 9 5
| | 9 5
| |
|-
| |
| | 3\7
| |
| |
| |
| |
| | 514.29
| | 171.43
| | 171.43
| | 0
| | 2 1 1 2 1
| | 2 1
| style="text-align:center;" | (Boundary of propriety: smaller
 
generators than this are strictly proper)
|-
| |
| |
| |
| |
| |
| | 13\30
| | 520
| | 160
| | 200
| | 40
| | 9 4 4 9 4
| | 9 4
| |
|-
| |
| |
| |
| |
| | 10\23
| |
| | 521.74
| | 156.52
| | 208.7
| | 52.17
| | 7 3 3 7 3
| | 7 3
| |
|-
| |
| |
| |
| |
| |
| | 17\39
| | 523.08
| | 153.84
| | 215.385
| | 61.54
| | 12 5 5 12 5
| | 12 5
| |
|-
| |
| |
| |
| | 7\16
| |
| |
| | 525
| | 150
| | 225
| | 75
| | 5 2 2 5 2
| | 5 2
| style="text-align:center;" | 5-note subset of pelog (insofar as it
 
resembles a MOS) would be in this region
|-
| |
| |
| |
| |
| |
| | 18\41
| | 526.83
| | 146.34
| | 234.15
| | 87.8
| | 13 5 5 13 5
| | 13 5
| |
|-
| |
| |
| |
| |
| |
| |
| | 600(25+√5)/31
| | 145.7
| | 235.75
| | 90.05
| | phi+1 1 1 phi+1 1
| | phi+1 1
| |
|-
| |
| |
| |
| |
| | 11\25
| |
| | 528
| | 144
| | 240
| | 96
| | 8 3 3 8 3
| | 8 3
| |
|-
| |
| |
| |
| |
| |
| |
| | 528.88
| | 142.24
| | 244.405
| | 102.17
| | e 1 e 1 1
| | e 1
| style="text-align:center;" | L/s = e
|-
| |
| |
| |
| |
| |
| | 15\34
| | 529.41
| | 141.18
| | 247.06
| | 105.88
| | 11 4 4 11 4
| | 11 4
| |
|-
| |
| |
| | 4\9
| |
| |
| |
| | 533.33
| | 133.33
| | 266.67
| | 133.33
| | 3 1 1 3 1
| | 3 1
| style="text-align:center;" | L/s = 3
|-
| |
| |
| |
| |
| |
| |
| | 535.36
| | 129.26
| | 276.835
| | 147.57
| | pi 1 pi 1 1
| | pi 1
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
| |
| |
| |
| |
| |
| | 13\29
| | 537.93
| | 124.14
| | 289.655
| | 165.52
| | 10 3 3 10 3
| | 10 3
| |
|-
| |
| |
| |
| |
| | 9\20
| |
| | 540
| | 120
| | 240
| | 180
| | 7 2 2 7 2
| | 7 2
| |
|-
| |
| |
| |
| |
| |
| | 14\31
| | 541.935
| | 116.13
| | 309.68
| | 193.55
| | 11 3 3 11 3
| | 11 3
| |
|-
| |
| |
| |
| | 5\11
| |
| |
| | 545.45
| | 109.09
| | 327.27
| | 218.18
| | 4 1 1 4 1
| | 4 1
| style="text-align:center;" | L/s = 4
|-
| |
| |
| |
| |
| |
| | 11\24
| | 550
| | 100
| | 350
| | 250
| | 9 2 2 9 2
| | 9 2
| |
|-
| |
| |
| |
| |
| | 6\13
| |
| | 553.85
| | 92.31
| | 369.23
| | 276.92
| | 5 1 1 5 1
| | 5 1
| |
|-
| |
| |
| |
| |
| |
| | 7\15
| | 560
| | 80
| | 480
| | 400
| | 6 1 1 6 1
| | 6 1
| |
|-
| | 1\2
| |
| |
| |
| |
| |
| | 600
| | 0
| | 600
| | 600
| | 1 0 0 1 0
| | 1 0
| style="text-align:center;" | a degenerated pentatonic scale with only 2 different steps
|}


From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
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