User:Moremajorthanmajor/7L 3s (15/7-equivalent)

From Xenharmonic Wiki
Jump to navigation Jump to search
↖ 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⟩ ↘ 
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 15/7 (1319.4¢)
Period 15/7 (1319.4¢)
Generator size(ed15/7)
Bright 7\10 to 5\7 (923.6¢ to 942.5¢)
Dark 2\7 to 3\10 (377.0¢ to 395.8¢)
Related MOS scales
Parent 3L 4s⟨15/7⟩
Sister 3L 7s⟨15/7⟩
Daughters 10L 7s⟨15/7⟩, 7L 10s⟨15/7⟩
Neutralized 4L 6s⟨15/7⟩
2-Flought 17L 3s⟨15/7⟩, 7L 13s⟨15/7⟩
Equal tunings(ed15/7)
Equalized (L:s = 1:1) 7\10 (923.6¢)
Supersoft (L:s = 4:3) 26\37 (927.2¢)
Soft (L:s = 3:2) 19\27 (928.5¢)
Semisoft (L:s = 5:3) 31\44 (929.6¢)
Basic (L:s = 2:1) 12\17 (931.4¢)
Semihard (L:s = 5:2) 29\41 (933.3¢)
Hard (L:s = 3:1) 17\24 (934.6¢)
Superhard (L:s = 4:1) 22\31 (936.4¢)
Collapsed (L:s = 1:0) 5\7 (942.5¢)

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:


Scale Tree and Tuning Spectrum of 7L 3s⟨15/7⟩
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 (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

Retrieved from "https://en.xen.wiki/index.php?title=User:Moremajorthanmajor/7L_3s_(15/7-equivalent)&oldid=148055"