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7L 3s: Difference between revisions

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Undo revision 77343 by Moremajorthanmajor (talk) Yet another baffling readability reducing column addition using measurement intervals that appear nowhere else on the wiki.
Tag: Undo
→Scale tree: improvement
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TODO: add interval table
TODO: add interval table


==Scale tree==
== Scale tree ==
The generator range reflects two extremes: one where L=s (3\10), and another where s=0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 -- five degrees of [[17edo|17edo]]:
The generator range reflects two extremes: one where L = s (3\10), and another where s = 0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 – five degrees of [[17edo]]:


{| class="wikitable"
{| class="wikitable center-all"
! colspan="6" rowspan="2" | Generator
! colspan="2" | Cents
! rowspan="2" | L
! rowspan="2" | s
! rowspan="2" | L/s
! rowspan="2" | Comments
|-
|-
! colspan="6" | Generator (fraction of octave)
! Chroma-positive
! | Generator (cents)
! Chroma-negative
! | Comments
|-
|-
| | 3\10
| 7\10 || || || || || || 840.000 || 360.000 || 1 || 1 || 1.000 ||  
| |
| |
| |
| |
| |
| | 360
| style="text-align:center;" |
|-
|-
| |
| || || || || || 40\57 || 842.105 || 357.895 || 6 || 5 || 1.200 || Restles↑
| |
| |
| |
| |
| | 17\57
| | 357.895
| |
|-
|-
| |
| || || || || 33\47 || || 842.553 || 357.447 || 5 || 4 || 1.250 ||  
| |
| |
| |
| | 14\47
| |
| | 357.45
| |
|-
|-
| |
| || || || || || 59\84 || 842.857 || 357.143 || 9 || 7 || 1.286 ||  
| |
| |
| | 11\37
| |
| |
| | 356.76
| |
|-
|-
| |
| || || || 26\37 || || || 843.243 || 356.757 || 4 || 3 || 1.333 ||
| |
| | 8\27
| |
| |
| |
| | 355.56
| style="text-align:center;" | Optimum rank range (L/s=3/2)
|-
|-
| |
| || || || || || 71\101 || 843.564 || 356.436 || 11 || 8 || 1.375 ||  
| |
| |
| |
| |
| |
| | 355.11
| |
|-
|-
| |
| || || || || 45\64 || || 843.750 || 356.250 || 7 || 5 || 1.400 || Beatles
| |
| |
| |
| | 21\71
| |
| | 354.93
| style="text-align:center;" |
|-
|-
| |
| || || || || || 64\91 || 843.956 || 356.044 || 10 || 7 || 1.428 ||  
| |
| |
| |
| |
| | 34\115
| | 354.78
| style="text-align:center;" | Golden neutral thirds
|-
|-
| |
| || || 19\27 || || || || 844.444 || 355.555 || 3 || 2 || 1.500 || L/s = 3/2, ringo
| |
| |
| | 13\44
| |
| |
| | 354.55
| style="text-align:center;" |
|-
|-
| |
| || || || || || 69\98 || 844.698 || 355.102 || 11 || 7 || 1.571 ||  
| |
| |
| |
| |
| |
| | 354.19
| |
|-
|-
| |
| || || || || 50\71 || || 845.070 || 354.930 || 8 || 5 || 1.600 ||  
| | 5\17
| |
| |
| |
| |
| | 352.94
| style="text-align:center;" | Boundary of propriety: generators larger than this are proper
|-
|-
| |
| || || || || || 81\115 || 845.217 || 354.783 || 13 || 8 || 1.625 || Unnamed golden tuning
| |
| |
| | 12\41
| |
| |
| | 351.22
| style="text-align:center;" |
|-
|-
| |
| || || || 31\44 || || || 845.455 || 354.545 || 5 || 3 || 1.667 ||
| |
| |
| |
| |
| | 31\106
| | 350.94
| style="text-align:center;" | No-5's neutral thirds (observing 81/80) is around here
|-
|-
| |
| || || || || || 74\105 || 845.714 || 354.286 || 12 || 7 || 1.714 ||  
| |
| |
| |
| | 19\65
| |
| | 350.77
| style="text-align:center;" |
|-
|-
| |
| || || || || 43\61 || || 845.902 || 354.098 || 7 || 4 || 1.750 ||
| |
| |
| |
| |
| |
| | 350.64
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = e</span>
|-
|-
| |
| || || || || || 55\78 || 846.154 || 353.846 || 9 || 5 || 1.800 ||
| |
| | 7\24
| |
| |
| |
| | 350
| style="text-align:center;" |
|-
|-
| |
| || 12\17 || || || || || 847.059 || 352.941 || 2 || 1 || 2.000 || Basic dicotonic<br>(Generators smaller than this are proper)
| |
| |
| |
| |
| |
| | 349.72
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
|-
| |
| || || || || || 53\75 || 848.000 || 352.000 || 9 || 4 || 2.250 ||
| |
| |
| |
| | 16\55
| |
| | 349.09
| |
|-
|-
| |
| || || || || 41\58 || || 848.273 || 351.724 || 7 || 3 || 2.333 ||  
| |
| |
| | 9\31
| |
| |
| | 348.39
| style="text-align:center;" | Mohajira (tempering out 81/80) is around here
|-
|-
| |
| || || || || || 70\99 || 848.485 || 351.515 || 12 || 5 || 2.400 || Hemififths
| |
| |
| |
| | 11\38
| |
| | 347.37
| |
|-
|-
| |
| || || || 29\41 || || || 848.780 || 351.220 || 5 || 2 || 2.500 || Neutrominant
| |
| |
| |
| |
| | 13\45
| | 346.67
| |
|-
|-
| | 2\7
| || || || || || 75\106 || 849.507 || 350.943 || 13 || 5 || 2.600 || Unnamed golden tuning
| |
|-
| |
| || || || || 46\65 || || 849.231 || 350.769 || 8 || 3 || 2.667 || Mohamaq
| |
|-
| |
| || || || || || 63\89 || 849.438 || 350.562 || 11 || 4 || 2.750 ||
| |
|-
| | 342.86
| || || 17\24 || || || || 850.000 || 350.000 || 3 || 1 || 3.000 || L/s = 3/1
| style="text-align:center;" |
|-
| || || || || || 56\79 || 850.633 || 349.367 || 10 || 3 || 3.333 ||
|-
| || || || || 39\55 || || 850.909 || 349.091 || 7 || 2 || 3.500 ||
|-
| || || || || || 61\86 || 851.613 || 358.837 || 11 || 3 || 3.667 ||
|-
| || || || 22\31 || || || 851.613 || 348.387 || 4 || 1 || 4.000 || Migration/mohajira
|-
| || || || || || 49\69 || 852.174 || 347.826 || 9 || 2 || 4.500 ||  
|-
| || || || || 27\38 || || 852.632|| 347.368 || 5 || 1 || 5.000 ||
|-
| || || || || || 32\45 || 853.333 || 346.667 || 6 || 1 || 6.000 || Ptolemy
|-
| 5\7 || || || || || || 857.143 || 342.867 || 1 || 0 || → inf ||
|}
|}


The scale produced by stacks of 5\17 is the [[17edo_neutral_scale|17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic_pairs|chromatic pair]].
The scale produced by stacks of 5\17 is the [[17edo neutral scale]]. Between 11/38 and 16/55, with 9/31 in between, is the mohajira/mohaha/mohoho range, where mohaha and mohoho use the MOS as the chromatic scale of a [[Chromatic pairs|chromatic pair]].
 
Other compatible edos include: [[37edo]], [[27edo]], [[44edo]], [[41edo]], [[24edo]], [[31edo]].


Other compatible edos include: [[37edo|37edo]], [[27edo|27edo]], [[44edo|44edo]], [[41edo|41edo]], [[24edo|24edo]], [[31edo|31edo]].
You can also build this scale by stacking neutral thirds that are not members of edos – for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 – or the square root of 3:2 (a bisected just perfect fifth).


You can also build this scale by stacking neutral thirds that are not members of edos -- for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 -- or the square root of 3:2 (a bisected just perfect fifth).
== Rank-2 temperaments ==
== Rank-2 temperaments ==
==7-note subsets==
==7-note subsets==
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