User:Moremajorthanmajor/9L 4s (7/2-equivalent)
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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⟩
Neutralized
5L 8s⟨7/2⟩
2-Flought
22L 4s⟨7/2⟩, 9L 17s⟨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¢)
| ↖ 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⟩ ↘ |
┌╥╥╥┬╥╥┬╥╥┬╥╥┬┐ │║║║│║║│║║│║║││ │││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┘
sLLsLLsLLsLLL
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
| 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
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:
| 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⟩ | |||||