7L 5s: Difference between revisions

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== Scale tree ==
== Scale tree ==
{| class="wikitable"
{| class="wikitable center-all"
|-
! colspan="6" | Generator
! colspan="6" | Generator
! | in cents
! Cents
! | Comments
! L
|-
! s
| style="text-align:center;" | 3\7
! L/s
| |
! Comments
| |
| |
| |
| |
| | 514.286
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 20\47
| | 510.638
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 17\40
| |
| | 510
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 31\73
| | 509.589
| |
|-
| |
| |
| |
| style="text-align:center;" | 14\33
| |
| |
| | 509.091
| |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 39\92
| | 508.696
| |
|-
| |
| |
| |
| |
| style="text-align:center;" | 25\59
| |
| | 508.475
| |
|-
|-
| |
| 5\12 || || || || || || 500.000 || 1 || 1 || 1.000 ||  
| |
| |
| |
| |
| style="text-align:center;" | 36\85
| | 508.235
| |
|-
|-
| |  
| || || || || || 28\67 || 501.493 || 6 || 5 || 1.200 ||  
| |  
| |  
| |  
| |  
| |  
| | 507.934
| style="text-align:center;" | L/s = pi
|-
|-
| |
| || || || || 23\55 || || 501.818 || 5 || 4 || 1.250 ||  
| |
| style="text-align:center;" | 11\26
| |
| |
| |
| | 507.692
| style="text-align:center;" | Flattone
|-
|-
| |  
| || || || || || 41\98 || 502.041 || 9 || 7 || 1.286 ||  
| |  
| |  
| |  
| |  
| style="text-align:center;" | 41\97
| | 507.216
| style="text-align:center;" | Flattone
|-
|-
| |
| || || || 18\43 || || || 502.326 || 4 || 3 || 1.333 || Meantone/meridetone
| |
| |
| |
| |
| |
| | 507.151
| style="text-align:center;" | L/s = e
|-
|-
| |
| || || || || || 49\117 || 502.564 || 11 || 8 || 1.375 ||
| |
| |
| |
| style="text-align:center;" | 30\71
| |
| | 507.042
| style="text-align:center;" | Flattone
|-
|-
| |  
| || || || || 31\74 || || 502.703 || 7 || 5 || 1.400 || Meantone/huygnes
| |  
| |  
| |  
| |  
| |  
| | 506.9365
| style="text-align:center;" | L/s = phi+1
|-
|-
| |
| || || || || || 44\105 || 502.857 || 10 || 7 || 1.428 ||  
| |
| |
| |
| |
| style="text-align:center;" | 49\116
| | 506.897
| style="text-align:center;" | Flattone
|-
|-
| |
| || || 13\31 || || || || 503.226 || 3 || 2 || 1.500 || L/s = 3/2
| |
| |
| style="text-align:center;" | 19\45
| |
| |
| | 506.667
| style="text-align:center;" | Flattone
|-
|-
| |
| || || || || || 47\112 || 503.571 || 11 || 7 || 1.571 ||
| |
| |
| |
| |
| style="text-align:center;" | 46\109
| | 506.422
| style="text-align:center;" | Flattone
|-
|-
| |
| || || || || 34\81 || || 503.704 || 8 || 5 || 1.600 || Meantone
| |
| |
| |
| style="text-align:center;" | 27\64
| |
| | 506.25
| style="text-align:center;" | Flattone
|-
|-
| |
| || || || || || 55\131 || 503.817 || 13 || 8 || 1.625 || Golden meantone
| |
| |
| |
| |
| style="text-align:center;" | 35\83
| | 506.024
| |
|-
|-
| |
| || || || 21\50 || || || 504.000 || 5 || 3 || 1.667 || Meantone/meanpop
| style="text-align:center;" | 8\19
| |
| |
| |
| |
| | 505.263
| style="text-align:center;" | Boundary of propriety: generators smaller than this are proper.
 
Meanenneadecal; L/s = 2
|-
|-
| |
| || || || || || 50\119 || 504.202 || 12 || 7 || 1.714 ||
| |
| |
| |
| |
| style="text-align:center;" | 37\88
| | 504.5455
| |
|-
|-
| |
| || || || || 29\69 || || 504.348 || 7 || 4 || 1.750 ||
| |
| |
| |
| |
| |
| | 504.507
| style="text-align:center;" | [[Lucy Tuning]]
|-
|-
| |
| || || || || || 37\88 || 504.545 || 9 || 5 || 1.800 ||
| |
| |
| |
| style="text-align:center;" | 29\69
| |
| | 504.348
| |
|-
|-
| |  
| || 8\19 || || || || || 505.263 || 2 || 1 || 2.000 || Basic m-chromatic<br>(Generators smaller than this are proper)
| |  
| |  
| |  
| |  
| |  
| | 504.275
| style="text-align:center;" | L/s = sqrt(3)
|-
|-
| |
| || || || || || 35\83 || 506.024 || 9 || 4 || 2.250 ||
| |
| |
| |
| |
| style="text-align:center;" | 50\119
| | 504.202
| |
|-
|-
| |
| || || || || 27\64 || || 506.250 || 7 || 3 || 2.333 ||  
| |
| |
| style="text-align:center;" | 21\50
| |
| |
| | 504
| style="text-align:center;" | Meantone / Meanpop
|-
|-
| |
| || || || || || 46\109 || 506.422 || 12 || 5 || 2.400 ||
| |
| |
| |
| |
| style="text-align:center;" | 55\131
| | 503.817
| style="text-align:center;" | Golden meantone
|-
|-
| |  
| || || || 19\45 || || || 506.667 || 5 || 2 || 2.500 || Flattone
| |  
| |  
| |  
| |  
| |  
| | 503.7855
| style="text-align:center;" | L/s = phi
|-
|-
| |
| || || || || || 49\116 || 506.897 || 13 || 5 || 2.600 || Golden flattone
| |
| |
| |
| style="text-align:center;" | 34\81
| |
| | 503.704
| style="text-align:center;" | Optimal meantone is around here
|-
|-
| |
| || || || || 30\71 || || 507.042 || 8 || 3 || 2.667 ||
| |
| |
| |
| |
| style="text-align:center;" | 47\112
| | 503.571
| style="text-align:center;" | Meantone / Meanpop
|-
|-
| |  
| || || || || || 41\97 || 507.216 || 11 || 4 || 2.750 ||
| |  
| |  
| |  
| |  
| |  
| | 503.5685
| style="text-align:center;" | L/s = pi/2
|-
|-
| |
| || || 11\26 || || || || 507.692 || 3 || 1 || 3.000 || L/s = 3/1
| |
| style="text-align:center;" | 13\31
| |
| |
| |
| | 503.226
| style="text-align:center;" | Optimal rank range (L/s = 3/2)
|-
|-
| |
| || || || || || 36\85 || 508.235 || 10 || 3 || 3.333 ||
| |
| |
| |
| |
| style="text-align:center;" | 44\105
| | 502.857
| style="text-align:center;" | Meantone / Huygens
|-
|-
| |
| || || || || 25\59 || || 508.475 || 7 || 2 || 3.500 ||
| |
| |
| |
| style="text-align:center;" | 31\74
| |
| | 502.703
| style="text-align:center;" | Meantone / Huygens
|-
|-
| |
| || || || || || 39\92 || 508.696 || 11 || 3 || 3.667 ||
| |
| |
| |
| |
| style="text-align:center;" | 49\117
| | 502.564
| |
|-
|-
| |
| || || || 14\33 || || || 509.091 || 4 || 1 || 4.000 ||
| |
| |
| style="text-align:center;" | 18\43
| |
| |
| | 502.326
| style="text-align:center;" | Meantone / Meridetone
|-
|-
| |
| || || || || || 31\73 || 509.589 || 9 || 2 || 4.500 ||
| |
| |
| |
| |
| style="text-align:center;" | 41\98
| | 502.041
| |
|-
|-
| |
| || || || || 17\40 || || 510.000 || 5 || 1 || 5.000 ||
| |
| |
| |
| style="text-align:center;" | 23\55
| |
| | 501.818
| |
|-
|-
| |
| || || || || || 20\47 || 510.638 || 6 || 1 || 6.000 ||
| |
| |
| |
| |
| style="text-align:center;" | 28\67
| | 501.493
| |
|-
|-
| style="text-align:center;" | 5\12
| 3\7 || || || || || || 514.286 || 1 || 0 || → inf ||
| |
| |
| |
| |
| |
| | 500
| |
|}
|}


Revision as of 09:01, 11 April 2021

↖ 6L 4s ↑ 7L 4s 8L 4s ↗
← 6L 5s 7L 5s 8L 5s →
↙ 6L 6s ↓ 7L 6s 8L 6s ↘ 
┌╥╥┬╥┬╥╥┬╥┬╥┬┐
│║║│║│║║│║│║││
││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLsLLsLsLs
sLsLsLLsLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\12 to 3\7 (500.0 ¢ to 514.3 ¢)
Dark 4\7 to 7\12 (685.7 ¢ to 700.0 ¢)
TAMNAMS information
Related to 5L 2s (diatonic)
With tunings 1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent 5L 2s
Sister 5L 7s
Daughters 12L 7s, 7L 12s
Neutralized 2L 10s
2-Flought 19L 5s, 7L 17s
Equal tunings
Equalized (L:s = 1:1) 5\12 (500.0 ¢)
Supersoft (L:s = 4:3) 18\43 (502.3 ¢)
Soft (L:s = 3:2) 13\31 (503.2 ¢)
Semisoft (L:s = 5:3) 21\50 (504.0 ¢)
Basic (L:s = 2:1) 8\19 (505.3 ¢)
Semihard (L:s = 5:2) 19\45 (506.7 ¢)
Hard (L:s = 3:1) 11\26 (507.7 ¢)
Superhard (L:s = 4:1) 14\33 (509.1 ¢)
Collapsed (L:s = 1:0) 3\7 (514.3 ¢)

7L 5s is the MOS pattern of the meantone chromatic scale, which is the only notable harmonic entropy minimum in this range of possible scales. In meantone, "diatonic" semitones (~16/15), such as appear in the diatonic scale, for example B-C, are larger than "chromatic" semitones (~25/24), the intervals by which flats and sharps alter pitches.

The meantone chromatic scale is proper.

Scales

Scale tree

Generator Cents L s L/s Comments
5\12 500.000 1 1 1.000
28\67 501.493 6 5 1.200
23\55 501.818 5 4 1.250
41\98 502.041 9 7 1.286
18\43 502.326 4 3 1.333 Meantone/meridetone
49\117 502.564 11 8 1.375
31\74 502.703 7 5 1.400 Meantone/huygnes
44\105 502.857 10 7 1.428
13\31 503.226 3 2 1.500 L/s = 3/2
47\112 503.571 11 7 1.571
34\81 503.704 8 5 1.600 Meantone
55\131 503.817 13 8 1.625 Golden meantone
21\50 504.000 5 3 1.667 Meantone/meanpop
50\119 504.202 12 7 1.714
29\69 504.348 7 4 1.750
37\88 504.545 9 5 1.800
8\19 505.263 2 1 2.000 Basic m-chromatic
(Generators smaller than this are proper)
35\83 506.024 9 4 2.250
27\64 506.250 7 3 2.333
46\109 506.422 12 5 2.400
19\45 506.667 5 2 2.500 Flattone
49\116 506.897 13 5 2.600 Golden flattone
30\71 507.042 8 3 2.667
41\97 507.216 11 4 2.750
11\26 507.692 3 1 3.000 L/s = 3/1
36\85 508.235 10 3 3.333
25\59 508.475 7 2 3.500
39\92 508.696 11 3 3.667
14\33 509.091 4 1 4.000
31\73 509.589 9 2 4.500
17\40 510.000 5 1 5.000
20\47 510.638 6 1 6.000
3\7 514.286 1 0 → inf
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