User:Moremajorthanmajor/Greater dicoid

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7L 3s refers to the structure of moment of symmetry scales built from a 10-tone chain of neutral or major thirds (assuming a period of an octave up to 10/9edo):

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 Greater dicoid (from dicot, an exotemperament) in TAMNAMS.

Intervals

The generator (g) will fall between 343 cents (2\7 - two degrees of 7edo and 400 cents (3\9 - three degrees of 9edo), hence a neutral or major third.

2g, then, will fall between 686 cents (4\7) and 800 cents (2\3), the range of diatonic fifths and subminor sixths.

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 dicoid 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 15/7 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 6v 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 17edIX) 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 edIX would be (3+2)\(10+7) = 5\17 – five degrees of 17edIX:

Generator Normalized Cents L s L/s Comments
Bright Dark
7\10 933.333 400.000 1 1 1.000
40\57 923.077 392.307 6 5 1.200 Restles↑
73\104 922.105 391.579 11 9 1.222
106\151 921.739 391.304 16 13 1.231
33\47 920.930 390.698 5 4 1.250
92\131 920.000 390.000 13 11 1.273
59\84 919.481 389.610 9 7 1.286
85\121 918.919 389.189 13 10 1.000
111\158 918.621 388.966 17 13 1.308
26\37 917.647 388.235 4 3 1.333
97\138 916.535 387.401 15 11 1.364
71\101 916.129 387.097 11 8 1.375
45\64 915.254 386.441 7 5 1.400 Beatles
64\91 914.286 385.714 10 7 1.428
83\118 913.716 385.321 13 9 1.444
19\27 912.000 384.000 3 2 1.500 L/s = 3/2, suhajira/ringo
164\233 911.111 383.333 26 17 1.529
145\206 910.995 383.246 23 15 1.533
126\179 910.843 383.133 20 13 1.538
107\152 910.638 382.988 17 11 1.545
88\125 910.345 382.759 14 9 1.556
69\98 909.890 382.418 11 7 1.571
50\71 909.091 381.818 8 5 1.600
81\115 908.411 381.308 13 8 1.625 Golden suhajira
31\44 907.317 380.489 5 3 1.667
74\105 906.122 379.592 12 7 1.714
43\61 905.263 378.947 7 4 1.750
55\78 904.110 378.082 9 5 1.800
67\95 903.371 377.528 11 6 1.833
79\112 902.857 377.143 13 7 1.857
91\129 902.479 376.860 15 8 1.875
103\146 902.190 376.642 17 9 1.889
115\163 901.961 376.471 19 10 1.900
12\17 900.000 375.000 2 1 2.000 Basic Greater dicoid
(Generators smaller than this are proper)
173\245 898.701 374.026 29 14 2.071
161\228 898.605 373.953 27 13 2.077
149\211 898.492 373.869 25 12 2.083
137\194 898.361 373.770 23 11 2.091
125\177 898.204 373.653 21 10 2.100
113\160 898.013 373.510 19 9 2.111
101\143 897.778 373.333 17 8 2.125
89\126 897.479 373.109 15 7 2.143
77\109 897.087 372.816 13 6 2.167
65\92 896.552 372.414 11 5 2.200
53\75 895.775 371.831 9 4 2.250
41\58 894.545 370.909 7 3 2.333
70\99 893.617 370.213 12 5 2.400 Hemif/hemififths
29\41 892.307 369.231 5 2 2.500 Mohaha/neutrominant
75\106 891.089 368.317 13 5 2.600 Hemif/salsa/karadeniz
46\65 890.322 367.742 8 3 2.667 Mohaha/mohamaq
63\89 889.412 367.059 11 4 2.750
80\113 888.889 366.667 14 5 2.800
97\137 888.550 366.412 17 6 2.833
114\161 888.312 366.234 20 7 2.857
131\185 888.136 366.102 23 8 2.875
148\209 888.000 366.000 26 9 2.889
165\233 887.892 365.919 29 10 2.900
17\24 886.957 365.213 3 1 3.000 L/s = 3/1
90\127 885.246 363.934 16 5 3.200
73\103 884.848 363.636 13 4 3.250
56\79 884.210 363.158 10 3 3.333
39\55 883.018 362.264 7 2 3.500
61\86 881.928 361.446 11 3 3.667
22\31 880.000 360.000 4 1 4.000 Mohaha/migration/mohajira
137\193 879.144 359.358 25 6 4.167
115\162 878.980 359.236 21 5 4.200
93\131 878.740 359.055 17 4 4.250
71\100 878.351 358.762 13 3 4.333
49\69 877.612 358.209 9 2 4.500
76\107 876.923 357.692 14 3 4.667
27\38 875.676 356.757 5 1 5.000
59\83 874.074 355.556 11 2 5.500
91\128 873.600 355.200 17 3 5.667
32\45 872.727 354.545 6 1 6.000 Mohaha/ptolemy
5\7 857.143 342.857 1 0 → inf

The scale produced by stacks of 5\17 is the 17edIX 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 edIXs include: 37edIX, 27edIX, 44edIX, 41edIX, 24edIX, 31edIX.

You can also build this scale by stacking neutral thirds that are not members of edIXs – for instance, frequency ratios 11:9, 5:4, 21:17, 16:13 – or the square root of 3:2 or 11:7 (a bisected just perfect fifth or undecimal subminor sixth).

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

See also

7L 3s (33/16-equivalent) - harmonic subminor ninth tuning

7L 3s (44/21-equivalent) - Neogothic minor ninth tuning

7L 3s (21/10-equivalent) - septimal chromatic minor ninth tuning

7L 3s (15/7-equivalent) - septimal diatonic minor ninth tuning