7L 3s
| ↖ 6L 2s | ↑ 7L 2s | 8L 2s ↗ |
| ← 6L 3s | 7L 3s | 8L 3s → |
| ↙ 6L 4s | ↓ 7L 4s | 8L 4s ↘ |
┌╥╥╥┬╥╥┬╥╥┬┐ │║║║│║║│║║││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sLLsLLsLLL
| ↖ 6L 2s | ↑ 7L 2s | 8L 2s ↗ |
| ← 6L 3s | 7L 3s | 8L 3s → |
| ↙ 6L 4s | ↓ 7L 4s | 8L 4s ↘ |
┌╥╥╥┬╥╥┬╥╥┬┐ │║║║│║║│║║││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sLLsLLsLLL
7L 3s(<15/7>) refers to the structure of moment of symmetry scales built from a 10-tone chain of (neutral) thirds (assuming a period of an octave or a minor ninth):
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.
| # 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 | ed17\16 | L | s | L/s | Comments | ||||
|---|---|---|---|---|---|---|---|---|---|---|
| edo | Neapolitan (normalized) | |||||||||
| 7\10 | 840.000 | 933.333 (1171.242 śata) | 1190.000 | 1 | 1 | 1.000 | ||||
| 40\57 | 842.105 | 923.077 (1158.371 śata) | 1192.9825 | 6 | 5 | 1.200 | (Neapolitan-)Restles↑ | |||
| 73\104 | 842.308 | 922.105 (1157.152 śata) | 1193.269 | 11 | 9 | 1.222 | ||||
| 106\151 | 842.384 | 921.739 (1156.692 śata) | 1193.3775 | 16 | 13 | 1.231 | ||||
| 33\47 | 842.553 | 920.930 (1155.677 śata) | 1193.617 | 5 | 4 | 1.250 | ||||
| 92\131 | 842.748 | 920.000 (1154.510 śata) | 1193.893 | 14 | 11 | 1.273 | ||||
| 59\84 | 842.857 | 919.4805 (1153.858 śata) | 1194.048 | 9 | 7 | 1.286 | ||||
| 85\121 | 842.975 | 918.919 (1153.153 śata) | 1194.215 | 13 | 10 | 1.300 | ||||
| 111\158 | 843.038 | 918.621 (1152.779 śata) | 1194.304 | 17 | 13 | 1.308 | ||||
| 26\37 | 843.243 | 917.647 (1151.557 śata) | 1194.595 | 4 | 3 | 1.333 | ||||
| 97\138 | 843.478 | 916.535 (1150.162 śata) | 1194.9275 | 15 | 11 | 1.364 | ||||
| 71\101 | 843.564 | 916.129 (1149.652 śata) | 1195.0495 | 11 | 8 | 1.375 | ||||
| 45\64 | 843.750 | 915.254 (1148.554 śata) | 1195.3125 | 7 | 5 | 1.400 | (Neapolitan-)Beatles | |||
| 64\91 | 843.956 | 914.286 (1147.339 śata) | 1195.604 | 10 | 7 | 1.428 | ||||
| 83\118 | 844.068 | 913.7615 (1146.681 śata) | 1195.763 | 13 | 9 | 1.444 | ||||
| 102\145 | 844.138 | 913.433 (1146.269 śata) | 1195.862 | 16 | 11 | 1.4545 | ||||
| 121\172 | 844.186 | 913.2075 (1145.986 śata) | 1195.930 | 19 | 13 | 1.4615 | ||||
| 19\27 | 844.444 | 912.000 (1144.471 śata) | 1196.296 | 3 | 2 | 1.500 | L/s = 3/2, (Neapolitan-)suhajira/ringo | |||
| 107\152 | 844.736 | 910.638 (1142.762 śata) | 1196.7105 | 17 | 11 | 1.5455 | ||||
| 88\125 | 844.8 | 910.345 (1142.3935 śata) | 1196.800 | 14 | 9 | 1.556 | ||||
| 69\98 | 844.898 | 909.890 (1141.823 śata) | 1196.939 | 11 | 7 | 1.571 | ||||
| 50\71 | 845.070 | 909.091 (1140.820 śata) | 1197.183 | 8 | 5 | 1.600 | ||||
| 81\115 | 845.217 | 908.411 (1139.967 śata) | 1197.391 | 13 | 8 | 1.625 | Golden (Neapolitan-)suhajira | |||
| 31\44 | 845.455 | 907.317 (1138.594 śata) | 1197.727 | 5 | 3 | 1.667 | ||||
| 74\105 | 845.714 | 906.122 (1137.095 śata) | 1198.095 | 12 | 7 | 1.714 | ||||
| 43\61 | 845.902 | 905.263 (1136.0165 śata) | 1198.361 | 7 | 4 | 1.750 | ||||
| 98\139 | 846.043 | 904.615 (1135.204 śata) | 1198.561 | 16 | 9 | 1.778 | ||||
| 55\78 | 846.154 | 904.110 (1134.569 śata) | 1198.718 | 9 | 5 | 1.800 | ||||
| 67\95 | 846.316 | 903.371 (1133.642 śata) | 1198.947 | 11 | 6 | 1.833 | ||||
| 79\112 | 846.429 | 902.857 (1132.997 śata) | 1199.107 | 13 | 7 | 1.857 | ||||
| 12\17 | 847.059 | 900.000 (1129.412 śata) | 1200.000 | 2 | 1 | 2.000 | Basic (Neapolitan-)dicotonic (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 | (Neapolitan-)Hemif/hemififths | ||||
| 29\41 | 848.780 | 351.220 | 5 | 2 | 2.500 | (Neapolitan-)Mohaha/neutrominant | ||||
| 75\106 | 849.057 | 350.943 | 13 | 5 | 2.600 | (Neapolitan-)Hemif/salsa/karadeniz | ||||
| 46\65 | 849.231 | 350.769 | 8 | 3 | 2.667 | (Neapolitan-)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 | (Neapolitan-)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 | (Neapolitan-)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 (
- 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