7L 3s

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7L 3s
Brightest mode LLLsLLsLLs
Period 2/1
Range for bright generator 7\10 (840.0¢) to 5\7 (857.1¢)
Parent MOS 3L 4s
Daughter MOSes 10L 7s, 7L 10s
Sister MOS 3L 7s
TAMNAMS name neutertonic
Equal tunings
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¢)

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 neutertonic (from neutral and tertial) 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 neutertonic 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.

# generators up Notation (1/1 = 0) name In L's and s's # generators up Notation of 2/1 inverse name In L's and s's
The 10-note MOS has the following intervals (from some root):
0 0 perfect unison 0 0 0 perfect 10-step 7L+3s
1 7 perfect 7-step 5L+2s -1 3 perfect 3-step 2L+1s
2 4 major 4-step 3L+1s -2 6 minor 6-step 4L+2s
3 1 major (1-)step 1L -3 9v minor 9-step 6L+3s
4 8 major 8-step 6L+2s -4 2v minor 2-step 1L+1s
5 5 major 5-step 4L+1s -5 5v minor 5-step 3L+2s
6 2 major 2-step 2L -6 8v minor 8-step 5L+3s
7 9 major 9-step 7L+2s -7 1v minor (1-)step 1s
8 6^ major 6-step 5L+1s -8 4v minor 4-step 2L+2s
9 3^ augmented 3-step 3L -9 7v diminished 7-step 4L+3s
10 0^ augmented unison 1L-1s -10 0v diminished 10-step 6L+4s
11 7^ augmented 7-step 6L+1s -11 3v diminished 3-step 1L+2s
The chromatic 17-note MOS (either 7L 10s, 10L 7s, or 17edo) also has the following intervals (from some root):
12 4^ augmented 4-step 4L -12 6v diminished 6-step 3L+3s
13 1^ augmented (1-)step 2L-1s -13 9w diminished 9-step 5L+4s
14 8^ augmented 8-step 8L+1s -14 2w diminished 2-step 2s
15 5^ augmented 5-step 5L -15 5w diminished 5-step 2L+3s
16 2^ augmented 2-step 3L-1s -16 8w diminished 8-step 4L+4s

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 L s L/s Comments
Chroma-positive Chroma-negative
7\10 840.000 360.000 1 1 1.000
40\57 842.105 357.895 6 5 1.200 Restles↑
33\47 842.553 357.447 5 4 1.250
59\84 842.857 357.143 9 7 1.286
26\37 843.243 356.757 4 3 1.333
71\101 843.564 356.436 11 8 1.375
45\64 843.750 356.250 7 5 1.400 Beatles
64\91 843.956 356.044 10 7 1.428
19\27 844.444 355.556 3 2 1.500 L/s = 3/2, suhajira/ringo
69\98 844.698 355.102 11 7 1.571
50\71 845.070 354.930 8 5 1.600
81\115 845.217 354.783 13 8 1.625 Golden suhajira
31\44 845.455 354.545 5 3 1.667
74\105 845.714 354.286 12 7 1.714
43\61 845.902 354.098 7 4 1.750
55\78 846.154 353.846 9 5 1.800
12\17 847.059 352.941 2 1 2.000 Basic neutertonic
(Generators smaller than this are proper)
53\75 848.000 352.000 9 4 2.250
41\58 848.273 351.724 7 3 2.333
70\99 848.485 351.515 12 5 2.400 Hemif/hemififths
29\41 848.780 351.220 5 2 2.500 Mohaha/neutrominant
75\106 849.057 350.943 13 5 2.600 Hemif/salsa/karadeniz
46\65 849.231 350.769 8 3 2.667 Mohaha/mohamaq
63\89 849.438 350.562 11 4 2.750
17\24 850.000 350.000 3 1 3.000 L/s = 3/1
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 Mohaha/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 Mohaha/ptolemy
5\7 857.143 342.867 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 (External-6855b5f4f272812f2538853afd1c4157-withext.jpg - 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