7L 3s: Difference between revisions

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! colspan="6" rowspan="2" | Generator
! colspan="6" rowspan="2" | Generator
! colspan="2" | Cents
! colspan="2" | Cents
! colspan="2" |Iotas
! rowspan="2" | L
! rowspan="2" | L
! rowspan="2" | s
! rowspan="2" | s
Line 47: Line 48:
! Chroma-positive
! Chroma-positive
! Chroma-negative
! Chroma-negative
!Chroma-positive
!Chroma-negative
|-
|-
| 7\10 || || || || || || 840.000 || 360.000 || 1 || 1 || 1.000 ||  
| 7\10 || || || || || || 840.000 || 360.000  
|1190
|510|| 1 || 1 || 1.000 ||  
|-
|-
| || || || || || 40\57 || 842.105 || 357.895 || 6 || 5 || 1.200 || Restles↑
| || || || || || 40\57 || 842.105 || 357.895  
|1192.982
|507.018|| 6 || 5 || 1.200 || Restles↑
|-
|-
| || || || || 33\47 || || 842.553 || 357.447 || 5 || 4 || 1.250 ||  
| || || || || 33\47 || || 842.553 || 357.447  
|1193.671
|506.329|| 5 || 4 || 1.250 ||  
|-
|-
| || || || || || 59\84 || 842.857 || 357.143 || 9 || 7 || 1.286 ||  
| || || || || || 59\84 || 842.857 || 357.143  
|1194.048
|505.952|| 9 || 7 || 1.286 ||  
|-
|-
| || || || 26\37 || || || 843.243 || 356.757 || 4 || 3 || 1.333 ||  
| || || || 26\37 || || || 843.243 || 356.757  
|1194.595
|505.405|| 4 || 3 || 1.333 ||  
|-
|-
| || || || || || 71\101 || 843.564 || 356.436 || 11 || 8 || 1.375 ||  
| || || || || || 71\101 || 843.564 || 356.436  
|1195.05
|504.95|| 11 || 8 || 1.375 ||  
|-
|-
| || || || || 45\64 || || 843.750 || 356.250 || 7 || 5 || 1.400 || Beatles
| || || || || 45\64 || || 843.750 || 356.250  
|1195.3125
|504.6875|| 7 || 5 || 1.400 || Beatles
|-
|-
| || || || || || 64\91 || 843.956 || 356.044 || 10 || 7 || 1.428 ||  
| || || || || || 64\91 || 843.956 || 356.044  
|1195.604
|504.396|| 10 || 7 || 1.428 ||  
|-
|-
| || || 19\27 || || || || 844.444 || 355.555 || 3 || 2 || 1.500 || L/s = 3/2, suhajira/ringo
| || || 19\27 || || || || 844.444 || 355.556
|1196.296
|503.704|| 3 || 2 || 1.500 || L/s = 3/2, suhajira/ringo
|-
|-
| || || || || || 69\98 || 844.698 || 355.102 || 11 || 7 || 1.571 ||  
| || || || || || 69\98 || 844.698 || 355.102  
|1196.939
|503.061|| 11 || 7 || 1.571 ||  
|-
|-
| || || || || 50\71 || || 845.070 || 354.930 || 8 || 5 || 1.600 ||  
| || || || || 50\71 || || 845.070 || 354.930  
|1197.183
|502.817|| 8 || 5 || 1.600 ||  
|-
|-
| || || || || || 81\115 || 845.217 || 354.783 || 13 || 8 || 1.625 || Golden suhajira
| || || || || || 81\115 || 845.217 || 354.783  
|1197.391
|502.609|| 13 || 8 || 1.625 || Golden suhajira
|-
|-
| || || || 31\44 || || || 845.455 || 354.545 || 5 || 3 || 1.667 ||  
| || || || 31\44 || || || 845.455 || 354.545  
|1197.773
|502.227|| 5 || 3 || 1.667 ||  
|-
|-
| || || || || || 74\105 || 845.714 || 354.286 || 12 || 7 || 1.714 ||  
| || || || || || 74\105 || 845.714 || 354.286  
|1198.095
|501.905|| 12 || 7 || 1.714 ||  
|-
|-
| || || || || 43\61 || || 845.902 || 354.098 || 7 || 4 || 1.750 ||
| || || || || 43\61 || || 845.902 || 354.098  
|1198.361
|501.639|| 7 || 4 || 1.750 ||
|-
|-
| || || || || || 55\78 || 846.154 || 353.846 || 9 || 5 || 1.800 ||
| || || || || || 55\78 || 846.154 || 353.846  
|1198.717
|501.283|| 9 || 5 || 1.800 ||
|-
|-
| || 12\17 || || || || || 847.059 || 352.941 || 2 || 1 || 2.000 || Basic dicotonic<br>(Generators smaller than this are proper)
| || 12\17 || || || || || 847.059 || 352.941  
|1200
|500|| 2 || 1 || 2.000 || Basic dicotonic<br>(Generators smaller than this are proper)
|-
|-
| || || || || || 53\75 || 848.000 || 352.000 || 9 || 4 || 2.250 ||
| || || || || || 53\75 || 848.000 || 352.000  
|1201.333
|498.667|| 9 || 4 || 2.250 ||
|-
|-
| || || || || 41\58 || || 848.273 || 351.724 || 7 || 3 || 2.333 ||  
| || || || || 41\58 || || 848.273 || 351.724  
|1201.724
|498.276|| 7 || 3 || 2.333 ||  
|-
|-
| || || || || || 70\99 || 848.485 || 351.515 || 12 || 5 || 2.400 || Hemif/hemififths
| || || || || || 70\99 || 848.485 || 351.515  
|1202.02
|497.98|| 12 || 5 || 2.400 || Hemif/hemififths
|-
|-
| || || || 29\41 || || || 848.780 || 351.220 || 5 || 2 || 2.500 || Mohaha/neutrominant
| || || || 29\41 || || || 848.780 || 351.220  
|1202.439
|497.561|| 5 || 2 || 2.500 || Mohaha/neutrominant
|-
|-
| || || || || || 75\106 || 849.057 || 350.943 || 13 || 5 || 2.600 || Hemif/salsa/karadeniz
| || || || || || 75\106 || 849.057 || 350.943  
|1202.83
|497.17|| 13 || 5 || 2.600 || Hemif/salsa/karadeniz
|-
|-
| || || || || 46\65 || || 849.231 || 350.769 || 8 || 3 || 2.667 || Mohaha/mohamaq
| || || || || 46\65 || || 849.231 || 350.769  
|1203.077
|496.923|| 8 || 3 || 2.667 || Mohaha/mohamaq
|-
|-
| || || || || || 63\89 || 849.438 || 350.562 || 11 || 4 || 2.750 ||
| || || || || || 63\89 || 849.438 || 350.562  
|1203.371
|496.629|| 11 || 4 || 2.750 ||
|-
|-
| || || 17\24 || || || || 850.000 || 350.000 || 3 || 1 || 3.000 || L/s = 3/1
| || || 17\24 || || || || 850.000 || 350.000  
|1204.167
|495.833|| 3 || 1 || 3.000 || L/s = 3/1
|-
|-
| || || || || || 56\79 || 850.633 || 349.367 || 10 || 3 || 3.333 ||
| || || || || || 56\79 || 850.633 || 349.367  
|1205.063
|494.937|| 10 || 3 || 3.333 ||
|-
|-
| || || || || 39\55 || || 850.909 || 349.091 || 7 || 2 || 3.500 ||  
| || || || || 39\55 || || 850.909 || 349.091  
|1205.455
|494.545|| 7 || 2 || 3.500 ||  
|-
|-
| || || || || || 61\86 || 851.613 || 358.837 || 11 || 3 || 3.667 ||  
| || || || || || 61\86 || 851.613 || 358.837  
|1205.814
|494.186|| 11 || 3 || 3.667 ||  
|-
|-
| || || || 22\31 || || || 851.613 || 348.387 || 4 || 1 || 4.000 || Mohaha/migration/mohajira
| || || || 22\31 || || || 851.613 || 348.387  
|1206.452
|493.548|| 4 || 1 || 4.000 || Mohaha/migration/mohajira
|-
|-
| || || || || || 49\69 || 852.174 || 347.826 || 9 || 2 || 4.500 ||  
| || || || || || 49\69 || 852.174 || 347.826  
|1207.246
|492.754|| 9 || 2 || 4.500 ||  
|-
|-
| || || || || 27\38 || || 852.632|| 347.368 || 5 || 1 || 5.000 ||  
| || || || || 27\38 || || 852.632|| 347.368  
|1207.895
|492.105|| 5 || 1 || 5.000 ||  
|-
|-
| || || || || || 32\45 || 853.333 || 346.667 || 6 || 1 || 6.000 || Mohaha/ptolemy
| || || || || || 32\45 || 853.333 || 346.667  
|1208.889
|491.111|| 6 || 1 || 6.000 || Mohaha/ptolemy
|-
|-
| 5\7 || || || || || || 857.143 || 342.867 || 1 || 0 || → inf ||
| 5\7 || || || || || || 857.143 || 342.867  
|1214.286
|485.714|| 1 || 0 || → inf ||
|}
|}


Revision as of 23:16, 7 September 2021

↖ 6L 2s ↑ 7L 2s 8L 2s ↗
← 6L 3s 7L 3s 8L 3s →
↙ 6L 4s ↓ 7L 4s 8L 4s ↘ 
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\10 to 5\7 (840.0 ¢ to 857.1 ¢)
Dark 2\7 to 3\10 (342.9 ¢ to 360.0 ¢)
TAMNAMS information
Name dicoid
Prefix dico-
Abbrev. di
Related MOS scales
Parent 3L 4s
Sister 3L 7s
Daughters 10L 7s, 7L 10s
Neutralized 4L 6s
2-Flought 17L 3s, 7L 13s
Equal tunings
Equalized (L:s = 1:1) 7\10 (840.0 ¢)
Supersoft (L:s = 4:3) 26\37 (843.2 ¢)
Soft (L:s = 3:2) 19\27 (844.4 ¢)
Semisoft (L:s = 5:3) 31\44 (845.5 ¢)
Basic (L:s = 2:1) 12\17 (847.1 ¢)
Semihard (L:s = 5:2) 29\41 (848.8 ¢)
Hard (L:s = 3:1) 17\24 (850.0 ¢)
Superhard (L:s = 4:1) 22\31 (851.6 ¢)
Collapsed (L:s = 1:0) 5\7 (857.1 ¢)

7L 3s refers to the structure of moment of symmetry scales built from a 10-tone chain of neutral thirds (assuming a period of an octave):

L s L L L s L L s L

Graham Breed has a page on his website dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:

t q t t t q t t q t

Names

This MOS is called dicotonic (named after the abstract temperaments dicot and more specifically 11-limit dichotic) in TAMNAMS.

Intervals

The generator (g) will fall between 343 cents (2\7 - two degrees of 7edo and 360 cents (3\10 - three degrees of 10edo), hence a neutral third.

2g, then, will fall between 686 cents (4\7) and 720 cents (3\5), the range of diatonic fifths.

The "large step" will fall between 171 cents (1\7) and 120 cents (1\10), ranging from a submajor second to a sinaic.

The "small step" will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.

The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.

Note: In TAMNAMS, a k-step interval class in dicotonic may be called a "k-step", "k-mosstep", or "k-dicostep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

TODO: add interval table

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:

Generator Cents Iotas L s L/s Comments
Chroma-positive Chroma-negative Chroma-positive Chroma-negative
7\10 840.000 360.000 1190 510 1 1 1.000
40\57 842.105 357.895 1192.982 507.018 6 5 1.200 Restles↑
33\47 842.553 357.447 1193.671 506.329 5 4 1.250
59\84 842.857 357.143 1194.048 505.952 9 7 1.286
26\37 843.243 356.757 1194.595 505.405 4 3 1.333
71\101 843.564 356.436 1195.05 504.95 11 8 1.375
45\64 843.750 356.250 1195.3125 504.6875 7 5 1.400 Beatles
64\91 843.956 356.044 1195.604 504.396 10 7 1.428
19\27 844.444 355.556 1196.296 503.704 3 2 1.500 L/s = 3/2, suhajira/ringo
69\98 844.698 355.102 1196.939 503.061 11 7 1.571
50\71 845.070 354.930 1197.183 502.817 8 5 1.600
81\115 845.217 354.783 1197.391 502.609 13 8 1.625 Golden suhajira
31\44 845.455 354.545 1197.773 502.227 5 3 1.667
74\105 845.714 354.286 1198.095 501.905 12 7 1.714
43\61 845.902 354.098 1198.361 501.639 7 4 1.750
55\78 846.154 353.846 1198.717 501.283 9 5 1.800
12\17 847.059 352.941 1200 500 2 1 2.000 Basic dicotonic
(Generators smaller than this are proper)
53\75 848.000 352.000 1201.333 498.667 9 4 2.250
41\58 848.273 351.724 1201.724 498.276 7 3 2.333
70\99 848.485 351.515 1202.02 497.98 12 5 2.400 Hemif/hemififths
29\41 848.780 351.220 1202.439 497.561 5 2 2.500 Mohaha/neutrominant
75\106 849.057 350.943 1202.83 497.17 13 5 2.600 Hemif/salsa/karadeniz
46\65 849.231 350.769 1203.077 496.923 8 3 2.667 Mohaha/mohamaq
63\89 849.438 350.562 1203.371 496.629 11 4 2.750
17\24 850.000 350.000 1204.167 495.833 3 1 3.000 L/s = 3/1
56\79 850.633 349.367 1205.063 494.937 10 3 3.333
39\55 850.909 349.091 1205.455 494.545 7 2 3.500
61\86 851.613 358.837 1205.814 494.186 11 3 3.667
22\31 851.613 348.387 1206.452 493.548 4 1 4.000 Mohaha/migration/mohajira
49\69 852.174 347.826 1207.246 492.754 9 2 4.500
27\38 852.632 347.368 1207.895 492.105 5 1 5.000
32\45 853.333 346.667 1208.889 491.111 6 1 6.000 Mohaha/ptolemy
5\7 857.143 342.867 1214.286 485.714 1 0 → inf

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 pair.

Other compatible edos include: 37edo, 27edo, 44edo, 41edo, 24edo, 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).

Rank-2 temperaments

7-note subsets

If you stop the chain at 7 tones, you have a heptatonic scale of the form 3L 4s:

L s s L s L s

The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for "tone," uppercase because it is a wider tone than t. Thus, we have:

T t t T t T t

This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:

T t t T t t T

which is not a complete moment of symmetry scale in itself, but a subset of one.

Tetrachordal structure

Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a tetrachordal scale. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.

I ( - Andrew Heathwaite) offer "a" to refer to a step of 2t (for "augmented second")

Thus, the possible tetrachords are:

T t t

t T t

t t T

a q t

a t q

t a q

t q a

q a t

q t a

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