User:Moremajorthanmajor/7L 3s (15/7-equivalent)
| ↖ 6L 2s⟨15/7⟩ | ↑ 7L 2s⟨15/7⟩ | 8L 2s⟨15/7⟩ ↗ |
| ← 6L 3s⟨15/7⟩ | 7L 3s<15/7> | 8L 3s⟨15/7⟩ → |
| ↙ 6L 4s⟨15/7⟩ | ↓ 7L 4s⟨15/7⟩ | 8L 4s⟨15/7⟩ ↘ |
┌╥╥╥┬╥╥┬╥╥┬┐ │║║║│║║│║║││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sLLsLLsLLL
7L 3s⟨15/7⟩ is a 15/7-equivalent (non-octave) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every interval of 15/7 (1319.4¢). Generators that produce this scale range from 923.6¢ to 942.5¢, or from 377¢ to 395.8¢.
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 377 cents (2\7 - two degrees of 7ed15/7) and 396 cents (3\10 - three degrees of 10ed15/7), hence a major third.
2g, then, will fall between 754 cents (4\7) and 792 cents (3\5), the range of diatonic subminor sixths.
The "large step" will fall between 188.5 cents (1\7) and 131.9 cents (1\10), ranging from a small major second to a sinaic.
The "small step" will fall between 0 cents and 131.9 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.
The most frequent interval, then is the major third (and its inversion, the diminished seventh), followed by the superfourth and subminor sixth.
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 17ed15/7) 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 17ed15/7:
| Generator(ed15/7) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 7\10 | 923.610 | 395.833 | 1:1 | 1.000 | Equalized 7L 3s⟨15/7⟩ | |||||
| 40\57 | 925.925 | 393.518 | 6:5 | 1.200 | ||||||
| 33\47 | 926.417 | 393.026 | 5:4 | 1.250 | ||||||
| 59\84 | 926.751 | 392.691 | 9:7 | 1.286 | ||||||
| 26\37 | 927.176 | 392.267 | 4:3 | 1.333 | Supersoft 7L 3s⟨15/7⟩ | |||||
| 71\101 | 927.529 | 391.914 | 11:8 | 1.375 | ||||||
| 45\64 | 927.733 | 391.710 | 7:5 | 1.400 | ||||||
| 64\91 | 927.960 | 391.483 | 10:7 | 1.429 | ||||||
| 19\27 | 928.497 | 390.946 | 3:2 | 1.500 | Soft 7L 3s⟨15/7⟩ | |||||
| 69\98 | 928.995 | 390.447 | 11:7 | 1.571 | ||||||
| 50\71 | 929.185 | 390.258 | 8:5 | 1.600 | ||||||
| 81\115 | 929.347 | 390.096 | 13:8 | 1.625 | ||||||
| 31\44 | 929.607 | 389.835 | 5:3 | 1.667 | Semisoft 7L 3s⟨15/7⟩ | |||||
| 74\105 | 929.893 | 389.550 | 12:7 | 1.714 | ||||||
| 43\61 | 930.099 | 389.344 | 7:4 | 1.750 | ||||||
| 55\78 | 930.376 | 389.066 | 9:5 | 1.800 | ||||||
| 12\17 | 931.371 | 388.071 | 2:1 | 2.000 | Basic 7L 3s⟨15/7⟩ Scales with tunings softer than this are proper | |||||
| 53\75 | 932.406 | 387.037 | 9:4 | 2.250 | ||||||
| 41\58 | 932.710 | 386.733 | 7:3 | 2.333 | ||||||
| 70\99 | 932.939 | 386.503 | 12:5 | 2.400 | ||||||
| 29\41 | 933.264 | 386.178 | 5:2 | 2.500 | Semihard 7L 3s⟨15/7⟩ | |||||
| 75\106 | 933.568 | 385.875 | 13:5 | 2.600 | ||||||
| 46\65 | 933.760 | 385.683 | 8:3 | 2.667 | ||||||
| 63\89 | 933.988 | 385.455 | 11:4 | 2.750 | ||||||
| 17\24 | 934.605 | 384.837 | 3:1 | 3.000 | Hard 7L 3s⟨15/7⟩ | |||||
| 56\79 | 935.301 | 384.142 | 10:3 | 3.333 | ||||||
| 39\55 | 935.605 | 383.838 | 7:2 | 3.500 | ||||||
| 61\86 | 935.884 | 383.559 | 11:3 | 3.667 | ||||||
| 22\31 | 936.379 | 383.064 | 4:1 | 4.000 | Superhard 7L 3s⟨15/7⟩ | |||||
| 49\69 | 936.996 | 382.447 | 9:2 | 4.500 | ||||||
| 27\38 | 937.499 | 381.944 | 5:1 | 5.000 | ||||||
| 32\45 | 938.270 | 381.172 | 6:1 | 6.000 | ||||||
| 5\7 | 942.459 | 376.984 | 1:0 | → ∞ | Collapsed 7L 3s⟨15/7⟩ | |||||
The scale produced by stacks of 5\17 is the 17ed15/7 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 ed15/7s include: 37ed15/7, 27ed15/7, 44ed15/7, 41ed15/7, 24ed15/7, 31ed15/7.
You can also build this scale by stacking neutral thirds that are not members of ed15/7s – for instance, the frequency ratio 5:4 – or the square root of 11:7 (a bisected 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 (
- 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