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=5L 2s - "diatonic"=
=5L 2s - "diatonic"=


One way of distinguishing the "diatonic" scale is by considering it a [[MOSScales|moment of symmetry]] scale produced by a chain of "fifths". This will include [[12edo|12edo]]'s diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".
One way of distinguishing the "diatonic" scale is by considering it a [[MOSScales|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths"). This will include [[12edo]]'s diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".


It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.
It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.


==substituting step sizes==
==Substituting step sizes==


The 5L 2s MOS scale has this generalized form.
The 5L 2s MOS scale has this generalized form.
Line 15: Line 15:
2 2 1 2 2 2 1
2 2 1 2 2 2 1


When L=3, s=1, you have [[17edo|17edo]]:
When L=3, s=1, you have [[17edo]]: 3 3 1 3 3 3 1


3 3 1 3 3 3 1
When L=3, s=2, you have [[19edo]]: 3 3 2 3 3 3 2


When L=3, s=2, you have [[19edo|19edo]]:
When L=4, s=1, you have [[22edo]]: 4 4 1 4 4 4 1


3 3 2 3 3 3 2
When L=4, s=3, you have [[26edo]]: 4 4 3 4 4 4 3


When L=4, s=1, you have [[22edo|22edo]]:
When L=5, s=1, you have [[27edo]]: 5 5 1 5 5 5 1


4 4 1 4 4 4 1
When L=5, s=2, you have [[29edo]]: 5 5 2 5 5 5 2


When L=4, s=3, you have [[26edo|26edo]]:
When L=5, s=3, you have [[31edo]]: 5 5 3 5 5 5 3


4 4 3 4 4 4 3
When L=5, s=4, you have [[33edo]]: 5 5 4 5 5 5 4


When L=5, s=1, you have [[27edo|27edo]]:
So you have scales where L and s are nearly equal, which approach [[7edo]]:
 
5 5 1 5 5 5 1
 
When L=5, s=2, you have [[29edo|29edo]]:
 
5 5 2 5 5 5 2
 
When L=5, s=3, you have [[31edo|31edo]]:
 
5 5 3 5 5 5 3
 
When L=5, s=4, you have [[33edo|33edo]]:
 
5 5 4 5 5 5 4
 
So you have scales where L and s are nearly equal, which approach [[7edo|7edo]]:


1 1 1 1 1 1 1
1 1 1 1 1 1 1


And you have scales where s becomes so small it approaches zero, which would give us [[5edo|5edo]]:
And you have scales where s becomes so small it approaches zero, which would give us [[5edo]]:


1 1 0 1 1 1 0 or 1 1 1 1 1
1 1 0 1 1 1 0 = 1 1 1 1 1


==a continuum of temperaments==
==A continuum of temperaments==


So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo:
So if 4\7 (three degrees of 7edo) is at one extreme and 3\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12edo:


{| class="wikitable"
{| class="wikitable"
|-
|-
| | 3\7
| | 4\7
| |  
| |  
|-
|-
| |  
| |  
| | 5\12
| | 7\12
|-
|-
| | 2\5
| | 3\5
| |  
| |  
|}
|}


If we carry this freshman-summing out a little further, new, larger [[EDO|edo]]s pop up in our continuum.
If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.


{| class="wikitable"
{| class="wikitable"
|-
! colspan="8" | generator
! colspan="6" | generator
! | cents
! |  
! | L
! | in cents
! | s
! | tetrachord
! | L/s
! |
! |
! |  
! |  
! | comments
! | comments
|-
|-
| | 3\7
| 4\7||||||||||||||||685.714||1||1||1.000||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 514.286
| style="text-align:center;" | 1 1 1
| | 239.2945
| | 274.991
| | 307.521
| | 378.193
| style="text-align:center;" |  
|-
|-
| | 59\138
| ||||||||||||||''63\110''||687.273||16||15||1.067||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 513.0435
| style="text-align:center;" | 20 20 19
| | 238.673
| | 274.370
| | 308.142
| | 378.8145
| |  
|-
|-
| | 56\131
| ||||||||||||||''59\103''||687.379||15||14||1.071||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.977
| style="text-align:center;" | 19 19 18
| | 238.640
| | 274.337
| | 308.175
| | 378.848
| |  
|-
|-
| | 53\124
| ||||||||||||||''55\96''||687.500||14||13||1.077||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.903
| style="text-align:center;" | 18 18 17
| | 238.603
| | 274.300
| | 308.212
| | 378.885
| |  
|-
|-
| | 50\117
| ||||||||||||||''51\89''||687.640||13||12||1.083||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.8205
| style="text-align:center;" | 17 17 16
| | 238.562
| | 274.259
| | 308.2535
| | 378.926
| |  
|-
|-
| | 47\110
| ||||||||||||||''47\82''||687.805||12||11||1.091||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.727
| style="text-align:center;" | 16 16 15
| | 238.515
| | 274.212
| | 308.300
| | 378.973
| |  
|-
|-
| | 44\103
| ||||||||||||||''43\75''||688.000||11||10||1.100||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.621
| style="text-align:center;" | 15 15 14
| | 238.462
| | 274.159
| | 308.353
| | 379.0255
| |  
|-
|-
| | 41\96
| ||||||||||||||''39\68''||688.235||10||9||1.111||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.500
| style="text-align:center;" | 14 14 13
| | 238.402
| | 274.098
| | 308.414
| | 379.086
| |  
|-
|-
| | 38\89
| ||||||||||||||''35\61''||688.525||9||8||1.125||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.360
| style="text-align:center;" | 13 13 12
| | 238.331
| | 274.028
| | 308.484
| | 379.156
| |  
|-
|-
| | 35\82
| ||||||||||||||31\54||688.889||8||7||1.143||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.195
| style="text-align:center;" | 12 12 11
| | 238.249
| | 273.946
| | 308.566
| | 379.239
| |  
|-
|-
| | 32\75
| ||||||||||||27\47||||689.362||7||6||1.167||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 512.000
| style="text-align:center;" | 11 11 10
| | 238.152
| | 273.848
| | 308.664
| | 379.336
| |  
|-
|-
| | 29\68
| ||||||||||23\40||||||690.000||6||5||1.200||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 511.765
| style="text-align:center;" | 10 10 9
| | 238.034
| | 273.731
| | 308.781
| | 379.454
| |  
|-
|-
| | 26\61
| ||||||||19\33||||||||690.909||5||4||1.250||
| colspan="2" |
| |
| |
| |
| |
| style="text-align:center;" | 511.475
| style="text-align:center;" | 9 9 8
| | 237.889
| | 273.586
| | 308.926
| | 379.5985
| |  
|-
| | 23\54
| colspan="2" |
| |
| |
| |
| |  
| style="text-align:center;" | 511.111
| style="text-align:center;" | 8 8 7
| | 237.707
| | 273.404
| | 309.108
| | 379.781
| |
|-
| | 20\47
| colspan="2" |
| |
| |
| |
| |
| style="text-align:center;" | 510.638
| style="text-align:center;" | 7 7 6
| | 237.471
| | 273.168
| | 309.345
| | 380.017
| |
|-
| | 17\40
| colspan="2" |
| |
| |
| style="text-align:center;" |
| |
| style="text-align:center;" | 510.000
| style="text-align:center;" | 6 6 5
| | 237.152
| | 272.848
| | 309.664
| | 380.336
| style="text-align:center;" |
|-
| | 14\33
| colspan="2" |
| |
| |
| style="text-align:center;" |
| |
| style="text-align:center;" | 509.091
| style="text-align:center;" | 5 5 4
| | 236.697
| | 272.394
| | 310.118
| | 380.791
| style="text-align:center;" |
|-
| |
| colspan="2" | 25\59
| |
| |
| style="text-align:center;" |
| |
| style="text-align:center;" | 508.475
| style="text-align:center;" | 9 9 7
| | 236.389
| | 272.086
| | 310.4265
| | 381.0985
| style="text-align:center;" |  
|-
|-
| | 11\26
| ||||||||||34\59||||||691.525||9||7||1.286||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 507.692
| style="text-align:center;" | 4 4 3
| | 235.998
| | 271.695
| | 310.817
| | 381.491
| style="text-align:center;" |  
|-
|-
| |  
| ||||||15\26||||||||||692.308||4||3||1.333||
| colspan="2" | 30\71
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 507.042
| style="text-align:center;" | 11 11 8
| | 235.672
| | 271.3695
| | 311.142
| | 381.846
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||41\71||||||692.958||11||8||1.375||
| colspan="2" | 19\45
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 506.667
| style="text-align:center;" | 7 7 5
| | 235.485
| | 271.182
| | 311.33
| | 382.003
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||26\45||||||||693.333||7||5||1.400||
| colspan="2" | 27\64
| |  
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" | 506.250
| style="text-align:center;" | 10 10 7
| | 235.277
| | 270.973
| | 311.539
| | 382.211
| style="text-align:center;" |  
|-
|-
| | 8\19
| ||||||||||37\64||||||693.750||10||7||1.429||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 505.263
| style="text-align:center;" | 3 3 2
| | 234.783
| | 270.480
| | 312.032
| | 382.705
| style="text-align:center;" | Optimum rank range (L/s=3/2) diatonic
|-
|-
| |  
| ||||11\19||||||||||||694.737||3||2||1.500||Optimum rank range (L/s=3/2) diatonic
| colspan="2" | 37\88
| |  
| |  
| |  
| |  
| style="text-align:center;" | 504.5455
| style="text-align:center;" | 14 14 9
| | 234.424
| | 270.121
| | 312.391
| | 383.0635
| style="text-align:center;" | LucyTuning
|-
|-
| |  
| ||||||||||||51\88||||695.455||14||9||1.556||
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 504.356
| style="text-align:center;" | <span style="display: block; text-align: center;">pi pi 2</span>
| | 234.329
| | 270.026
| | 312.486
| | 383.158
| |  
|-
|-
| |  
| ||||||||||||||||695.644||&pi;||2||1.571||LucyTuning
| colspan="2" | 29\69
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 504.348
| style="text-align:center;" | 11 11 7
| | 234.3255
| | 270.022
| | 312.490
| | 383.172
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||40\69||||||695.652||11||7||1.571||
| colspan="2" | 21\50
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 504.000
| style="text-align:center;" | 8 8 5
| | 234.152
| | 269.848
| | 312.664
| | 383.336
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||29\50||||||||696.000||8||5||1.600||
| colspan="2" |  
| | 55\131
| |  
| |
| |  
| style="text-align:center;" | 503.817
| style="text-align:center;" | 21 21 13
| | 234.060
| | 269.757
| | 312.755
| | 383.428
| |  
|-
|-
| |  
| ||||||||||||76\131||||696.183||21||13||1.615||
| colspan="2" |  
| |  
| |  
| | 144\343
| |  
| style="text-align:center;" | 503.790
| style="text-align:center;" | 55 55 34
| | 234.047
| | 269.743
| | 312.769
| | 383.441
| |  
|-
|-
| |  
| ||||||||||||||''199\343''||696.210||55||34||1.618||
| colspan="2" |  
| |  
| |  
| |  
| | 233\555
| style="text-align:center;" | 503.784
| style="text-align:center;" | 89 89 55
| | 234.0435
| | 269.740
| | 312.772
| | 383.444
| style="text-align:center;" | Golden meantone
|-
|-
| |  
| ||||||||||||||||696.215||&phi;||1||1.618||Golden meantone
| colspan="2" |  
| |  
| | 89\212
| |  
| |  
| style="text-align:center;" | 503.774
| style="text-align:center;" | 34 34 21
| | 234.038
| | 269.735
| | 312.777
| | 383.449
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||''322\555''||696.216||89||55||1.618||
| colspan="2" |  
| | 34\81
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 503.704
| style="text-align:center;" | 13 13 8
| | 234.003
| | 269.700
| | 312.811
| | 383.485
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||123\212||696.226||34||21||1.619||
| colspan="2" | 13\31
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 503.226
| style="text-align:center;" | 5 5 3
| | 233.7645
| | 269.461
| | 313.051
| | 383.723
| style="text-align:center;" | Meantone is in this region
|-
|-
| |  
| ||||||||||47\81||||||696.296||13||8||1.625||
| colspan="2" |  
| | 31\74
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 502.703
| style="text-align:center;" | 12 12 7
| | 233.503
| | 269.200
| | 313.312
| | 383.985
| style="text-align:center;" |  
|-
|-
| |  
| ||||||18\31||||||||||696.774||5||3||1.667||Meantone is in this region
| |  
| |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 502.5135
| style="text-align:center;" | <span style="background-color: #ffffff;">√3 √3 1</span>
| | 233.408
| | 269.105
| | 313.407
| | 384.079
| |  
|-
|-
| |  
| ||||||||||43\74||||||697.297||12||7||1.714||
| colspan="2" | 18\43
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 502.326
| style="text-align:center;" | 7 7 4
| | 233.314
| | 269.011
| | 313.501
| | 384.183
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||||697.487||√3||1||1.732||
| colspan="2" | 23\55
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 501.818
| style="text-align:center;" | 9 9 5
| | 233.061
| | 268.7575
| | 313.754
| | 384.428
| style="text-align:center;" |  
|-
|-
| | 5\12
| ||||||||25\43||||||||697.674||7||4||1.750||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" | 500.000
| style="text-align:center;" | 2 2 1
| | 232.152
| | 267.848
| | 314.664
| | 385.336
| style="text-align:center;" | Boundary of propriety
 
(generators larger than this are proper)
|-
|-
| |  
| ||||||||||32\55||||||698.182||9||5||1.800||
| colspan="2" | 42\101
| |  
| |  
| |  
| |  
| style="text-align:center;" | 499.010
| style="text-align:center;" | 17 17 8
| | 231.6565
| | 267.353
| | 315.159
| | 385.831
| |  
|-
|-
| |  
| ||||||||||||39\67||||698.507||11||6||1.833||
| colspan="2" | <span style="display: block; text-align: center;">37\89</span>
| |  
| |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 498.876
| style="text-align:center;" | 15 15 7
| | 231.590
| | 267.287
| | 315,226
| | 385.898
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||46\79||698.734||13||7||1.857||
| colspan="2" | <span style="display: block; text-align: center;">32\77</span>
| |  
| |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" | 498.701
| style="text-align:center;" | 13 13 6
| | 231.502
| | 267.199
| | 315.313
| | 385.986
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||''53\91''||698.901||15||8||1.875||
| colspan="2" | 27\65
| |  
| |  
| |  
| |  
| style="text-align:center;" | 498.4615
| style="text-align:center;" | 11 11 5
| | 231.382
| | 267.079
| | 315.433
| | 386.105
| |  
|-
|-
| |  
| ||||||||||||||''60\103''||699.029||17||9||1.889||
| colspan="2" | 22\53
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 498.113
| style="text-align:center;" | 9 9 4
| | 231.208
| | 266.905
| | 315.609
| | 386.278
| style="text-align:center;" | Pythagorean is around here
|-
|-
| |  
| ||7\12||||||||||||||700.000||2||1||2.000||Boundary of propriety (generators smaller than this are proper)
| colspan="2" | 17\41
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 497.591
| style="text-align:center;" | 7 7 3
| | 230.932
| | 266.629
| | 315.883
| | 386.556
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||''59\101''||700.990||17||8||2.125||
| colspan="2" | 29\70
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 497.143
| style="text-align:center;" | 12 12 5
| | 230.723
| | 266.420
| | 316.092
| | 386.765
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||''52\89''||701.124||15||7||2.143||
| colspan="2" | 12\29
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 496.552
| style="text-align:center;" | 5 5 2
| | 230.4275
| | 266.124
| | 316.388
| | 387.061
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||||45\77||701.299||13||6||2.167||
| colspan="2" |  
| | 31\75
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 496.000
| style="text-align:center;" | 13 13 5
| | 230.152
| | 265.848
| | 316.664
| | 387.336
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||38\65||||701.539||11||5||2.200||
| |  
| |  
| |  
| |  
| | 81\196
| |
| style="text-align:center;" | 495.918
| style="text-align:center;" | 34 34 13
| | 230.111
| | 265.808
| | 316.705
| | 387.377
| |  
|-
|-
| |  
| ||||||||||31\53||||||701.887||9||4||2.250||
| |  
| |  
| |  
| |  
| |
| | 131\317
| style="text-align:center;" | 495.899
| style="text-align:center;" | 55 55 21
| | 230.101
| | 265.798
| | 316.714
| | 387.387
| |  
|-
|-
| |  
| ||||||||||||||||701.955||||||2.260||Pythagorean (g = 3/2 ; L=9/8 ; s=256/243)
| |  
| |  
| |  
| | 50\121
| |  
| |  
| style="text-align:center;" | 495.868
| style="text-align:center;" | 21 21 8
| | 230.0855
| | 265.782
| | 316.73
| | 387.402
| |  
|-
|-
| |  
| ||||||||24\41||||||||702.409||7||3||2.333||
| colspan="2" | 19\46
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 495.652
| style="text-align:center;" | 8 8 3
| | 229.978
| | 265.6745
| | 316.837
| | 387.511
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||41\70||||||702.857||12||5||2.400||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 495.393
| style="text-align:center;" | <span style="display: block; text-align: center;">e e 1</span>
| | 229.848
| | 265.545
| | 316.967
| | 387.639
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = e</span>
|-
|-
| |  
| ||||||17\29||||||||||703.448||5||2||2.500||
| colspan="2" | 26\63
| |  
| |  
| |  
| |  
| style="text-align:center;" | <span style="display: block; text-align: center;">495.238</span>
| style="text-align:center;" | 11 11 4
| | 229.771
| | 265.4675
| | 317.045
| | 387.717
| style="text-align:center;" | <span style="display: block; text-align: center;">
 
</span>
|-
|-
| | 7\17
| ||||||||||44\75||||||704.000||13||5||2.600||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 494.118
| style="text-align:center;" | 3 3 1
| | 229.210
| | 264.907
| | 317.596
| | 388.286
| style="text-align:center;" | L/s = 3
|-
|-
| |  
| ||||||||||||||115\196||704.082||34||13||2.615||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 493.553
| style="text-align:center;" | pi pi 1
| | 228.928
| | 264.625
| | 317.887
| | 388.56
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
|-
| |  
| ||||||||||||||''186\317''||704.101||55||21||2.619||
| colspan="2" | 23\56
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 492.857
| style="text-align:center;" | 10 10 3
| | 228.580
| | 264.277
| | 318.235
| | 388.908
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||||||71\121||||704.132||21||8||2.625||
| colspan="2" | 16\39
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 492.308
| style="text-align:center;" | 7 7 2
| | 228.305
| | 264.002
| | 318.51
| | 389.182
| style="text-align:center;" |  
|-
|-
| |  
| ||||||||27\46||||||||704.348||8||3||2.667||
| colspan="2" | 25\61
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 491.803
| style="text-align:center;" | 11 11 3
| | 228.053
| | 263.750
| | 318.761
| | 389.436
| style="text-align:center;" |  
|-
|-
| | 9\22
| ||||||||||||||||704.607||e||1||2.718||
| colspan="2" |
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 490.909
| style="text-align:center;" | 4 4 1
| | 227.606
| | 263.303
| | 319.209
| | 389.882
| style="text-align:center;" | (No-5's) superpyth is in this region
 
L/s = 4
|-
|-
| |  
| ||||||||||37\63||||||704.762||11||4||2.750||
| colspan="2" | 20\49
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 489.796
| style="text-align:center;" | 9 9 2
| | 227.050
| | 262.746
| | 319.766
| | 390.438
| style="text-align:center;" |  
|-
|-
| | 11\27
| ||||||||||||47\80||||705.000||14||5||2.800||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 488.889
| style="text-align:center;" | 5 5 1
| | 226.596
| | 262.293
| | 320.219
| | 390.892
| style="text-align:center;" |  
|-
|-
| | 13\32
| ||||10\17||||||||||||705.882||3||1||3.000||
| colspan="2" |  
| |
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 487.500
| style="text-align:center;" | 6 6 1
| | 225.9015
| | 261.598
| | 320.914
| | 391.596
| style="text-align:center;" |  
|-
|-
| | 15\37
| ||||||||||||||||706.447||&pi;||1||3.142||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 486.4865
| style="text-align:center;" | 7 7 1
| | 225.395
| | 261.092
| | 321.4205
| | 392.093
| |  
|-
|-
| | 17\42
| ||||||||||33\56||||||707.143||10||3||3.333||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" | 485.714
| style="text-align:center;" | 8 8 1
| | 225.009
| | 260.7055
| | 321.807
| | 392.479
| style="text-align:center;" |  
|-
|-
| | 19\47
| ||||||||23\39||||||||707.692||7||2||3.500||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 485.106
| style="text-align:center;" | 9 9 1
| | 224.705
| | 260.402
| | 322.111
| | 392.783
| |  
|-
|-
| | 21\52
| ||||||||||36\61||||||708.197||11||3||3.667||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 484.615
| style="text-align:center;" | 10 10 1
| | 224.459
| | 260.156
| | 322.356
| | 393.0285
| |  
|-
|-
| | 23\57
| ||||||13\22||||||||||709.091||4||1||4.000||(No-5's) superpyth is in this region
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 484.2105
| style="text-align:center;" | 11 11 1
| | 224.257
| | 259.954
| | 322.5585
| | 393.231
| |  
|-
|-
| | 25\62
| ||||||||||29\49||||||710.204||9||2||4.500||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 483.871
| style="text-align:center;" | 12 12 1
| | 224.087
| | 259.784
| | 322.728
| | 393.401
| |  
|-
|-
| | 27\67
| ||||||||16\27||||||||711.111||5||1||5.000||
| colspan="2" |  
| |  
| |  
| |
| |  
| style="text-align:center;" | 483.582
| style="text-align:center;" | 13 13 1
| | 223.943
| | 259.6395
| | 322.873
| | 393.545
| |  
|-
|-
| | 29\72
| ||||||||||19\32||||||712.500||6||1||6.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 483.333
| style="text-align:center;" | 14 14 1
| | 223.818
| | 259.515
| | 322.997
| | 393.6695
| |  
|-
|-
| | 31\77
| ||||||||||||22\37||||713.514||7||1||7.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 483.117
| style="text-align:center;" | 15 15 1
| | 223.710
| | 259.407
| | 323.105
| | 393.778
| |  
|-
|-
| | 33\82
| ||||||||||||||25\42||714.286||8||1||8.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.927
| style="text-align:center;" | 16 16 1
| | 223.615
| | 259.312
| | 323.200
| | 393.873
| |  
|-
|-
| | 35\87
| ||||||||||||||''28\47''||714.894||9||1||9.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.759
| style="text-align:center;" | 17 17 1
| | 223.531
| | 259.228
| | 323.2845
| | 393.957
| |  
|-
|-
| | 37\92
| ||||||||||||||''31\52''||715.385||10||1||10.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.609
| style="text-align:center;" | 18 18 1
| | 223.456
| | 259.153
| | 323.539
| | 394.032
| |  
|-
|-
| | 39\97
| ||||||||||||||''34\57''||715.790||11||1||11.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.474
| style="text-align:center;" | 19 19 1
| | 223.389
| | 259.0855
| | 323.427
| | 394.099
| |  
|-
|-
| | 41\102
| ||||||||||||||''37\62''||716.129||12||1||12.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.353
| style="text-align:center;" | 20 20 1
| | 223.328
| | 259.025
| | 323.487
| | 394.160
| |  
|-
|-
| | 43\107
| ||||||||||||||''40\67''||716.418||13||1||13.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.243
| style="text-align:center;" | 21 21 1
| | 223.273
| | 258.970
| | 323.542
| | 394.215
| |  
|-
|-
| | 45\112
| ||||||||||||||''43\72''||716.667||14||1||14.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.143
| style="text-align:center;" | 22 22 1
| | 223.223
| | 258.920
| | 323.592
| | 394.265
| |  
|-
|-
| | 47\117
| ||||||||||||||''46\77''||716.883||15||1||15.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 482.051
| style="text-align:center;" | 23 23 1
| | 223.177
| | 258.874
| | 323.638
| | 394.311
| |  
|-
|-
| | 49\122
| ||||||||||||||''49\82''||717.073||16||1||16.000||
| colspan="2" |  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 481.967
| style="text-align:center;" | 24 24 1
| | 223.135
| | 258.832
| | 322.680
| | 394.353
| |  
|-
|-
| | 2\5
| 3\5||||||||||||||||720.000||1||0||-> inf||
| colspan="2" |  
| |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" | 480.000
| style="text-align:center;" | 1 1 0
| | 222.152
| | 257.848
| | 324.664
| | 395.336
| style="text-align:center;" |  
|}
|}


Temperaments above 5\12 on this chart are called "negative temperaments" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 8\19) and 1/4-comma (close to 13\31). As these tunings approach 3\7, the majors become flatter and the minors become sharper.
Temperaments above 7\12 on this chart are called "negative temperaments" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.


Temperaments below 5\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 22\53) as well as superpyth temperaments such as 7\17 and 9\22. As these tunings approach 2\5, the majors become sharper and the minors become flatter. Around 9\22, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
Temperaments below 7\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth temperaments such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.


[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]
[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]


5L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]].
5L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]].
Retrieved from "https://en.xen.wiki/w/5L_2s"