User:Moremajorthanmajor/9L 4s (7/2-equivalent)

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↖8L 3s⟨7/2⟩ ↑9L 3s⟨7/2⟩ 10L 3s⟨7/2⟩↗
←8L 4s⟨7/2⟩9L 4s<7/2> 10L 4s⟨7/2⟩→
↙8L 5s⟨7/2⟩ ↓9L 5s⟨7/2⟩ 10L 5s⟨7/2⟩↘
 
┌╥╥╥┬╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║│║║││
│││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLsLLs
sLLsLLsLLsLLL
Equave 7/2 (2168.8¢)
Period 7/2 (2168.8¢)
Generator size(ed7/2)
Bright 10\13 to 7\9 (1668.3¢ to 1686.9¢)
Dark 2\9 to 3\13 (482.0¢ to 500.5¢)
Related MOS scales
Parent 4L 5s⟨7/2⟩
Sister 4L 9s⟨7/2⟩
Daughters 13L 9s⟨7/2⟩
9L 13s⟨7/2⟩
Equal tunings(ed7/2)
Equalized (L:s = 1:1) 10\13 (1668.3¢)
Supersoft (L:s = 4:3) 37\48 (1671.8¢)
Soft (L:s = 3:2) 27\35 (1673.1¢)
Semisoft (L:s = 5:3) 44\57 (1674.2¢)
Basic (L:s = 2:1) 17\22 (1675.9¢)
Semihard (L:s = 5:2) 41\53 (1677.8¢)
Hard (L:s = 3:1) 24\31 (1679.1¢)
Superhard (L:s = 4:1) 31\40 (1680.8¢)
Collapsed (L:s = 1:0) 7\9 (1686.9¢)

9L 4s⟨7/2⟩ is a 7/2-equivalent (non-octave) moment of symmetry scale containing 9 large steps and 4 small steps, repeating every interval of 7/2 (2168.8¢). Generators that produce this scale range from 1668.3¢ to 1686.9¢, or from 482¢ to 500.5¢.


Shōsūshī - so named because of the four small steps in its diatonic MOS - is an excellent 8*-limit temperament tempering out 64/63.

Properties

In shōsūshī, the just major 21st (15/2) is divided into 7 equal steps. These steps form quartal harmonies (1-4/3-7/4-7/3…). Alternately, the ‘perfect eleventh’ 8/3 also makes a good shōsūshī generator. Compatible equal temperaments include 22ed7/2, 31ed7/2, 40ed7/2 and 49ed7/2.

Notation

Because shōsūshī is a Mahjong hand, its MOS has 13 tones, as in the 13 tiles in a Mahjong hand and its “13 han” Japanese value. Thus the nominals are given as Japanese kana, and to avoid confusion with traditional Japanese notations, hiragana are used.

The notation used in this article is くすつぬふむゆきしちにひみ (ku-su-tsu-nu-fu-mu-yu-ki-shi-chi-ni-hi-mi) = LLsLLLsLLsLLs (Ionian b14), #/b mostly two chromas up/down.

Thus the 22ed7/2 gamut is as follows:

く/み# か/そ す さ/と つ/の ぬ/た な/ほ ふ は/も む ま/よ ゆ/け き/や き#/せ し し#/て ち/ね ち#/に に#/へ ひ ひ#/め み/こ

The 31ed7/2 gamut is notated as follows:

く み#/そ か す と さ/の つ ぬ た/ほ な ふ も は む よ ま/け ゆ き や/せ き# し て し#/ね ち に ち#/へ に# ひ め ひ#/け こ

The 35ed7/2 gamut:

く か/そb そ/か# す さ/とb と/さ# つ/のb た/の ぬ/た# な/ほb ほ/な# ふ は/もb も/は# む よ/けb け/よ# ゆ/やb や/ま き/ま# き#/せb せ し し#/てb て ち/ねb ち#/ね に に#/へb へ ひ ひ#/めb め み/こb み#/こ

Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
か (く#) 1\48, 45.18 1\35, 61.97 2\57, 76.10 1\22, 98.58 3\53, 122.76 2\31, 139.92 3\40, 162.66
そ (すb) 3\48, 135.55 2\35, 123.93 3\57, 114.15 2\53, 81.84 1\31, 69.96 1\40, 54.22
す 4\48, 180.735 3\35, 185.90 5\57, 190.25 2\22, 197.17 5\53, 204.61 3\31, 209.89 4\40, 218.88
さ (す#) 5\48, 225.92 4\35, 247.87 7\57, 266.35 3\22, 295.75 8\53, 327.37 5\31, 349.81 7\40, 379.545
と (つb) 7\48, 316.29 5\35, 309.83 8\57, 304.40 7\53, 286.45 4\31, 279.85 5\40, 271.10
つ 8\48, 361.47 6\35, 371.80 10\57, 380.495 4\22, 394.33 10\53, 409.21 6\31, 419.77 8\40, 433.765
た (つ#) 9\48, 406.655 7\35, 433.765 12\57, 456.595 5\22, 492.915 13\53, 531.98 8\31, 559.70 11\40, 596.43
の (ぬb) 10\48, 451.83 11\57, 418.545 4\22, 394.33 9\53, 368.29 5\31, 349.81 6\40, 325.32
ぬ 11\48, 497.02 8\35, 495.73 13\57, 494.645 5\22, 492.915 12\53, 491.055 7\31, 489.735 9\40, 487.99
な (ぬ#) 12\48, 542.21 9\35, 557.70 15\57, 570.74 6\22, 591.50 15\53, 613.82 9\31, 629.66 12\40, 650.65
ほ (ふb) 14\48, 632.57 10\35, 619.665 16\57, 608.79 14\53, 572.90 8\31, 559.70 10\40, 542.21
ふ 15\48, 677.76 11\35, 681.63 18\57, 684.89 7\22, 690.08 17\53, 695.66 10\31, 699.62 13\40, 704.87
は (ふ#) 16\48, 722.94 12\35, 743.60 20\57, 760.99 8\22, 788.66 20\53, 818.425 12\31, 839.55 16\40, 867.53
も (むb) 18\48, 813.31 13\35, 805.56 21\57, 799.04 19\53, 777.50 11\31, 769.58 14\40, 759.09
む 19\48, 850.49 14\35, 867.53 23\57, 875.14 9\22, 887.25 22\53, 900.27 13\31, 909.51 17\40, 921.75
よ (む#) 20\48, 903.68 15\35, 929.50 25\57, 951.24 10\22, 985.83 25\53, 1023.03 15\31, 1049.43 20\40, 1084.41
け (ゆb) 22\48, 994.045 16\35, 991.46 26\57, 989.29 24\53, 982.11 14\31, 979.47 18\40, 975.97
ゆ 23\48, 1039.23 17\35, 1053.43 28\57, 1065.39 11\22, 1084.41 27\53, 1104.87 16\31, 1119.39 21\40, 1138.63
や (ゆ#) 24\48, 1084.41 18\35, 1115.40 30\57, 1141.49 12\22, 1183.00 30\29, 1227.63 15\31, 1049.43 24\40, 1301.30
ま (きb) 25\48, 1129.60 29\57, 1103.44 11\22, 1084.41 26\53, 1063.95 18\31, 1259.32 19\40, 1030.19
き 26\48, 1174.78 19\35, 1177.36 31\57, 1179.54 12\22, 1183.00 29\53, 1186.72 17\31, 1189.36 22\40, 1192.85
き# 27\48, 1219.965 20\35, 1239.32 33\57, 1255.64 13\22, 1281.58 32\53, 1309.48 18\31, 1259.32 25\40, 1355.52
せ (しb) 29\48, 1310.33 21\35, 1301.29 34\57, 1293.69 31\53, 1268.56 19\31, 1329.28 23\40, 1247.075
し 30\48, 1355.52 22\35, 1363.26 36\57, 1369.785 14\22, 1380.16 34\53, 1391.32 20\31, 1399.24 26\40, 1409.74
し# 31\48, 1400.70 23\35, 1425.23 38\57, 1445.88 15\22, 1478.745 37\53, 1514.09 22\31, 1539.17 29\40, 1572.40
て (ちb) 33\48, 1491.07 24\35, 1487.195 39\57, 1483.93 36\53, 1473.165 21\31, 1469.205 27\40, 1463.96
ち 34\48, 1536.25 25\35, 1549.16 41\57, 1560.03 16\22, 1577.33 39\53, 1595.93 23\31, 1609.13 30\40, 1626.62
ち# 35\48, 1581.44 26\35, 1611.13 43\57, 1636.13 17\22, 1675.91 42\53, 1718.69 25\31, 1749.05 33\40, 1789.28
ね (にb) 36\48, 1626.62 42\57, 1598.08 16\22, 1577.33 38\53, 1555.01 22\31, 1539.17 28\40, 1518.18
に 37\48, 1671.80 27\35, 1673.095 44\57, 1674.18 17\22, 1675.91 41\53, 1677.77 24\31, 1679.09 31\40, 1680.84
に# 38\48, 1716.99 28\35, 1735.06 46\57, 1750.28 18\22, 1774.49 44\53, 1800.535 26\31, 1819.015 34\40, 1843.50
へ (ひb) 40\48, 1807.355 29\35, 1797.03 47\57, 1788.33 43\53, 1759.61 25\31, 1749.05 32\40, 1735.06
ひ 41\48, 1852.54 30\35, 1858.99 49\57, 1864.43 19\22, 1873.08 46\53, 1882.37 27\31, 1888.98 35\40, 1897.72
ひ# 42\48, 1897.72 31\35, 1920.96 51\57, 1940.53 20\22, 1971.66 49\53, 2005.14 29\31, 2028.90 38\40, 2060.385
め (みb) 44\48, 1988.09 32\35, 1982.93 52\57, 1978.58 48\53, 1964.22 28\31, 1958.94 36\40, 1951.94
み 45\48, 2033.27 33\35, 2044.89 54\57, 2054.68 21\22, 2070.24 51\53, 2086.98 30\31, 2098.86 39\40, 2114.605
み# 46\48, 2078.46 34\35, 2106.86 56\57, 2130.78 22\22, 2168.83 54\53, 2209.75 32\31, 2238.79 42\40, 2223.05
こ (くb) 47\48, 2123.64 55\57, 2092.73 21\22, 2070.24 50\53, 2046.06 29\31, 2028.90 37\40, 2006.16
く 48\48, 2168.83 35\35, 2168.83 57\57, 2168.83 22\22, 2168.83 53\53, 2168.83 31\31, 2168.83 40\40, 2168.83

Intervals

Generators Fourteenth notation Interval category name Generators Notation of 7/2 inverse Interval category name
The 13-note MOS has the following intervals (from some root):
0 く perfect unison 0 く “perfect” minor fourteenth
1 に natural eleventh -1 ぬ natural fourth
2 き perfect octave -2 け (ゆb) minor seventh
3 ふ natural fifth -3 て (ちb) minor tenth
4 す major second -4 め (みb) minor thirteenth
5 ひ natural twelfth (tritave) -5 と (つb) minor third
6 し major ninth -6 も (むb) minor sixth
7 む major sixth -7 せ (しb) minor ninth
8 つ major third -8 へ (ひb) diminished twelfth
9 み major thirteenth -9 そ (すb) minor second
10 ち major tenth -10 ほ (ふb) diminished fifth
11 ゆ major seventh -11 ま (きb) diminished octave
12 な (ぬ#) augmented fourth -12 ね (にb) diminished eleventh
The chromatic 22-note MOS (either 13L 9s, 9L 13s or 22ed7/2) also has these intervals
13 か (く#) augmented unison -13 こ (くb) diminished fourteenth
14 に# augmented eleventh -14 の (ぬb) diminished fourth
15 き# augmented octave -15 けb (ゆbb) diminished seventh
16 は (ふ#) augmented fifth -16 てb (ちbb) diminished tenth
17 さ (す#) augmented second -17 めb (みbb) diminished thirteenth
18 に# augmented twelfth -18 とb (つbb) diminished third
19 し# augmented ninth -19 もb (むbb) diminished sixth
20 よ (む#) augmented sixth -20 せb (しbb) diminished ninth
21 た (つ#) augmented third -21 へb (ひbb) doubly diminished twelfth

Genchain

The generator chain for this scale is as follows:

へb (ひbb) せb (しbb) もb (むbb) とb (つbb) めb (みbb) てb (ちbb) けb (ゆbb) →
dd12 d9 d6 d3 d13 d10 d7
の (ぬb) こ (くb) ね (にb) ま (きb) ほ (ふb) そ (すb) へ (ひb) せ (しb) も (むb) と (つb) め (みb) て (ちb) け (ゆb) →
d4 d14 d11 d8 d5 m2 d12 m9 m6 m3 m13 m10 m7
ぬ く に き ふ す ひ し む つ み ち ゆ →
N4 P1 N11 P8 N5 M2 N12 M9 M6 M3 M13 M10 M7
な (ぬ#) か (く#) に# き# は (ふ#) さ (す#) に# し# よ (む#) た (つ#)
A4 A1 A11 A8 A5 A2 A12 A9 A6 A3

Modes

The mode names are based on the diatonic scale:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Lydian Dominant ♮ 11 LLLsLLsLLsLLs 12|0 M M A P M M P M M P P M
Major b14 LLsLLLsLLsLLs 11|1 M M P P M M P M M P P M
Mixolydian LLsLLsLLLsLLs 10|2 M M P P M m P M M P P M
Mixolydian b10 (Persian) LLsLLsLLsLLLs 9|3 M M P P M m P M m P P M
Aeolian Dominant ♮6 LLsLLsLLsLLsL 8|4 M M P P M m P M m P P m
Dorian b13 LsLLLsLLsLLsL 7|5 M m P P M m P M m P P m
Minor LsLLsLLLsLLsL 6|6 M m P P m m P M m P P m
Phrygian ♮2 LsLLsLLsLLLsL 5|7 M m P P m m P m m P P m
Locrian ♮2 ♮5 LsLLsLLsLLsLL 4|8 M m P P m m P m m P d m
Locrian ♮5 sLLLsLLsLLsLL 3|9 m m P P m m P m m P d m
Locrian sLLsLLLsLLsLL 2|10 m m P d m m P m m P d m
Locrian b8 sLLsLLsLLLsLL 1|11 m m P d m m d m m P d m
Locrian b8 b11 sLLsLLsLLsLLL 0|12 m m P d m m d m m d d m

Cyclic Permutation Order

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Mixolydian b10 (Persian) LLsLLsLLsLLLs 9|3 M M P P M m P M m P P M
Phrygian ♮2 LsLLsLLsLLLsL 5|7 M m P P m m P m m P P m
Locrian b8 sLLsLLsLLLsLL 1|11 m m P d m m d m m P d m
Mixolydian LLsLLsLLLsLLs 10|2 M M P P M m P M M P P M
Minor LsLLsLLLsLLsL 6|6 M m P P m m P M m P P m
Locrian sLLsLLLsLLsLL 2|10 m m P d m m P m m P d m
Major b14 LLsLLLsLLsLLs 11|1 M M P P M M P M M P P M
Dorian b13 LsLLLsLLsLLsL 7|5 M m P P M m P M m P P m
Locrian ♮5 sLLLsLLsLLsLL 3|9 m m P P m m P m m P d m
Lydian Dominant ♮11 LLLsLLsLLsLLs 12|0 M M A P M M P M M P P M
Aeolian Dominant ♮6 LLsLLsLLsLLsL 8|4 M M P P M m P M m P P m
Locrian ♮2 ♮5 LsLLsLLsLLsLL 4|8 M m P P m m P m m P d m
Locrian b8 b11 sLLsLLsLLsLLL 0|12 m m P d m m d m m d d m

Temperaments

The Shōsūshī temperament spells its major hexad 84:108:126:162:189:224 root-2(4g-3p)-(3g-2p)-(11g-8p)-2(3g-2p)-(1g) (p = 7/2, g = 8/3) and its minor hexad 12:14:18:21:27:32 or 60:72:90:108:135:160 root-(-5g-p)-(3g-2p)-(-2g)-2(3g-2p)-(1g) (p = 7/2, g = 8/3).

Shōsūshī-Superpyth

Subgroup: 7/2.2.3

Comma list: 64/63

POL2 generator: ~4/3 = 486.8323¢

Mapping: [⟨1 1 2], ⟨0 -2 -5]]

Vals: 22ed7/2, 31ed7/2, 40ed7/2, 49ed7/2

Scale tree

The spectrum looks like this:

Scale Tree and Tuning Spectrum of 9L 4s⟨7/2⟩
Generator(ed7/2) Cents Step Ratio Comments
Bright Dark L:s Hardness
10\13 1668.328 500.498 1:1 1.000 Equalized 9L 4s⟨7/2⟩
57\74 1670.582 498.244 6:5 1.200
47\61 1671.063 497.763 5:4 1.250
84\109 1671.389 497.437 9:7 1.286
37\48 1671.803 497.023 4:3 1.333 Supersoft 9L 4s⟨7/2⟩
101\131 1672.148 496.678 11:8 1.375
64\83 1672.348 496.478 7:5 1.400
91\118 1672.569 496.257 10:7 1.429
27\35 1673.094 495.732 3:2 1.500 Soft 9L 4s⟨7/2⟩
98\127 1673.582 495.244 11:7 1.571
71\92 1673.768 495.058 8:5 1.600
115\149 1673.926 494.900 13:8 1.625
44\57 1674.181 494.645 5:3 1.667 Semisoft 9L 4s⟨7/2⟩
105\136 1674.461 494.365 12:7 1.714
61\79 1674.663 494.163 7:4 1.750
78\101 1674.935 493.891 9:5 1.800
17\22 1675.911 492.915 2:1 2.000 Basic 9L 4s⟨7/2⟩
Scales with tunings softer than this are proper
75\97 1676.927 491.899 9:4 2.250
58\75 1677.225 491.601 7:3 2.333
99\128 1677.451 491.375 12:5 2.400
41\53 1677.771 491.055 5:2 2.500 Semihard 9L 4s⟨7/2⟩
106\137 1678.070 490.756 13:5 2.600
65\84 1678.258 490.568 8:3 2.667
89\115 1678.483 490.343 11:4 2.750
24\31 1679.091 489.735 3:1 3.000 Hard 9L 4s⟨7/2⟩
79\102 1679.777 489.049 10:3 3.333
55\71 1680.076 488.750 7:2 3.500
86\111 1680.352 488.474 11:3 3.667
31\40 1680.840 487.986 4:1 4.000 Superhard 9L 4s⟨7/2⟩
69\89 1681.449 487.377 9:2 4.500
38\49 1681.947 486.879 5:1 5.000
45\58 1682.710 486.116 6:1 6.000
7\9 1686.865 481.961 1:0 → ∞ Collapsed 9L 4s⟨7/2⟩
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