User:Moremajorthanmajor/9L 4s (minor fourteenth-equivalent)

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Shōsūshī - so named because of the four small steps in its diatonic MOS - is an excellent 8*-limit temperament tempering out 64/63 or an excellent 9 (18)-limit temperament tempering out 81/80.

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 22edXIV; 31edXIV, 40edXIV and 49edXIV (7-ward) and 35edXIV, 48edXIV and 61edXIV (9-ward).

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 22edXIV 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 31edXIV 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 35edXIV 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
Shōsūshī Petrushka
か (く#) А# 1\48, 46.154 1\35, 63.158 2\57, 77.419 1\22, 100 3\53, 124.138 2\31, 141.176 3\40, 163.63
そ (すb) Бь 3\48, 138.462 2\35, 126.316 3\57, 116.129 2\53, 82.759 1\31, 70.588 1\40, 54.54
す Б 4\48, 184.615 3\35, 189.454 5\57, 193.548 2\22, 200 5\53, 206.897 3\31, 211.765 4\40, 218.18
さ (す#) Б# 5\48, 230.769 4\35, 252.632 7\57, 270.968 3\22, 300 8\53, 331.034 5\31, 352.941 7\40, 381.81
と (つb) Вь 7\48, 323.077 5\35, 315.789 8\57, 309.677 7\53, 289.655 4\31, 282.353 5\40, 272.72
つ В 8\48, 369.231 6\35, 378.947 10\57, 387.097 4\22, 400 10\53, 413.739 6\31, 423.529 8\40, 436.36
た (つ#) В# 9\48, 415.385 7\35, 442.105 12\57, 464.516 5\22, 500 13\53, 537.069 8\31, 564.706 11\40, 600
の (ぬb) Гь 10\48. 461.538 11\57, 425.806 4\22, 400 9\53, 372.414 5\31, 352.941 6\40, 327.27
ぬ Г 11\48, 507.692 8\35, 505.263 13\57, 503.226 5\22, 500 12\53, 496.552 7\31, 494.118 9\40, 490.90
な (ぬ#) Г# 12\48, 553.846 9\35, 568.421 15\57, 580.645 6\22, 600 15\53, 620.690 9\31, 635.294 12\40, 654.54
ほ (ふb) Дь 14\48, 646.154 10\35, 631.579 16\57, 619.355 14\53, 579.310 8\31, 564.706 10\40, 545.45
ふ Д 15\48, 692.308 11\35, 694.737 18\57, 696.774 7\22, 700 17\53, 703.448 10\31, 705.882 13\40, 709.09
は (ふ#) Д# 16\48, 738.462 12\35, 757.895 20\57, 774.194 8\22, 800 20\53, 827.586 12\31, 847.059 16\40, 872.72
も (むb) Еь 18\48, 830.769 13\35, 821.053 21\57, 812.903 19\53, 786.261 11\31, 776.471 14\40, 763.63
む Е 19\48, 876.923 14\35, 884.211 23\57, 890.323 9\22, 900 22\53, 910.345 13\31, 917.647 17\40, 927.27
よ (む#) Е# 20\48, 923.077 15\35, 947.638 25\57, 967.742 10\22, 1000 25\53, 1034.483 15\31, 1058.824 20\40, 1090.90
け (ゆb) Ёь 22\48, 1015.385 16\35, 1010.526 26\57, 1006.452 24\53, 993.103 14\31, 988.235 18\40, 981.81
ゆ Ё 23\48, 1061.538 17\35, 1073.684 28\57, 1083.871 11\22, 1100 27\53, 1117.241 16\31, 1129.412 21\40, 1145.45
や (ゆ#) Ё# 24\48, 1107.692 18\35, 1136.842 30\57, 1161.290 12\22, 1200 30\29, 1241.176 15\31, 1058.824 24\40, 1309.09
ま (きb) Жь 25\48, 1153.846 29\57, 1122.581 11\22, 1100 26\53, 1075.862 18\31, 1270.588 19\40, 1036.36
き Ж 26\48, 1200 19\35, 1200 31\57, 1200 12\22, 1200 29\53, 1200 17\31, 1200 22\40, 1200
き# Ж# 27\48, 1246.154 20\35, 1263.158 33\57, 1277.419 13\22, 1300 32\53, 1324.138 18\31, 1270.588 25\40, 1363.63
せ (しb) Зь 29\48, 1338.462 21\35, 1326.316 34\57, 1316.129 31\53, 1282.759 19\31, 1341.176 23\40, 1254.54
し З 30\48, 1384.615 22\35, 1389.474 36\57, 1393.548 14\22, 1400 34\53, 1406.897 20\31, 1411.765 26\40, 1418.18
し# З# 31\48, 1430.769 23\35, 1452.632 38\57, 1470.968 15\22, 1500 37\53, 1531.034 22\31, 1552.941 29\40, 1581.81
て (ちb) Иь 33\48, 1523.077 24\35, 1515.789 39\57, 1509.677 36\53, 1489.655 21\31, 1482.353 27\40, 1472.72
ち И 34\48, 1569.231 25\35, 1578.947 41\57, 1587.097 16\22, 1600 39\53, 1613.793 23\31, 1623.529 30\40, 1636.36
ち# И# 35\48, 1615.385 26\35, 1642.105 43\57, 1664.516 17\22, 1700 42\53, 1737.069 25\31, 1764.706 33\40, 1800
ね (にb) Йь 36\48, 1661.538 42\57, 1625.806 16\22, 1600 38\53, 1572.414 22\31, 1552.941 28\40, 1527.27
に Й 37\48, 1707.692 27\35, 1705.263 44\57, 1703.226 17\22, 1700 41\53, 1696.552 24\31, 1694.118 31\40, 1690.90
に# Й# 38\48, 1753.846 28\35, 1768.421 46\57, 1780.645 18\22, 1800 44\53, 1820.690 26\31, 1835.294 34\40, 1854.54
へ (ひb) Кь 40\48, 1846.158 29\35, 1831.579 47\57, 1819.355 43\53, 1779.310 25\31, 1764.706 32\40, 1745.45
ひ К 41\48, 1892.308 30\35, 1894.737 49\57, 1896.774 19\22, 1900 46\53, 1903.448 27\31, 1905.882 35\40, 1909.09
ひ# К# 42\48, 1938.462 31\35, 1957.895 51\57, 1974.194 20\22, 2000 49\53, 2027.586 29\31, 2047.059 38\40, 2072.72
め (みb) Ль 44\48, 2030.769 32\35, 2021.053 52\57, 2012.903 48\53, 1986.207 28\31, 1976.471 36\40, 1963.63
み Л 45\48, 2076.923 33\35, 2084.211 54\57, 2090.323 21\22, 2100 51\53, 2110.345 30\31, 2117.647 39\40, 2127.27
み# Л# 46\48, 2123.077 34\35, 2147.368 56\57, 2167.742 22\22, 2200 54\53, 2234.483 32\31, 2258.824 42\40, 2290.90
こ (くb) Мь 47\48, 2169.231 55\57, 2129.032 21\22, 2100 50\53, 2068.966 29\31, 2047.059 37\40, 2018.18
く М 48\48, 2215.385 35\35, 2210.526 57\57, 2206.452 22\22, 2200 53\53, 2193.103 31\31, 2188.235 40\40, 2181.81
か (く#) М# 49\48, 2261.538 36\35, 2273.684 59\57, 2283.871 23\22, 2300 56\53, 2317.241 33\31, 2329.418 43\40, 2345.45
そ (すb) Нь 51\48, 2353.846 37\35, 2336.842 60\57, 2322.581 55\53, 2275.862 32\31, 2258.854 41\40, 2236.36
す Н 52\48, 2400 38\35, 2400 62\57, 2400 24\22, 2400 58\53, 2400 34\31, 2400 44\40, 2400
さ (す#) Н# 53\48, 2446.154 39\35, 2463.158 64\57, 2477.419 25\22, 2500 61\53, 2524.138 36\31, 2541.176 47\40, 2563.63
と (つb) Оь 55\48, 2538.462 40\35, 2526.316 65\57, 2516.129 60\53, 2482.759 35\31, 2570.588 45\40, 2454.54
つ О 56\48, 2584.615 41\35, 2589.474 67\57, 2593.548 26\22, 2600 63\53, 2606.897 37\31, 2611.765 48\40, 2618.18
た (つ#) О# 57\48, 2630.792 42\35, 2652.632 69\57, 2670.968 27\22, 2700 66\53, 2731.034 39\31, 2752.941 51\40, 2781.81
の (ぬb) Пь 58\48, 2676.923 68\57, 2632.258 26\22, 2600 62\53, 2565.517 36\31, 2541.176 46\40, 2509.09
ぬ П 59\48, 2723.077 43\35, 2715.789 70\57, 2709.677 27\22, 2700 65\53, 2689.655 38\31, 2682.353 49\40, 2672.72
な (ぬ#) П# 60\48, 2769.321 44\35, 2778.947 72\57, 2787.097 28\22, 2800 68\53, 2813.739 40\31, 2823.529 52\40, 2836.36
ほ (ふb) Рь 62\48, 2861.538 45\35, 2842.105 73\57, 2825.806 67\53, 2772.414 39\31, 2752.941 51\40, 2781.81
ふ Р 63\48, 2907.692 46\35, 2905.263 75\57, 2903.226 29\22, 2900 70\53, 2896.552 41\31, 2894.118 53\40, 2890.90
は (ふ#) Р# 64\48, 2953.846 47\35, 2968.421 77\57, 2980.645 30\22, 3000 73\53, 3020.690 43\31, 3035.294 56\40, 3054.54
も (むb) Сь 66\48, 3046.154 48\35, 3031.579 78\57, 3019.355 72\53, 2979.310 42\31, 2964.706 54\40, 2945.45
む С 67\48, 3092.308 49\35, 3094.737 80\57, 3096.774 31\22, 3100 75\53, 3103.448 44\31, 3105.882 57\40, 3109.09
よ (む#) С# 68\48, 3138.462 50\35, 3157.895 82\57, 3174.194 32\22, 3200 78\53, 3227.586 46\31, 3247.059 60\40, 3272.72
け (ゆb) Ть 70\48, 3230.769 51\35, 3221.053 83\57, 3212.903 77\53, 3186.207 45\31, 3176.471 59\40, 3163.63
ゆ Т 71\48, 3276.923 52\35, 3284.211 85\57, 3290.323 33\22, 3300 80\53, 3310.345 47\31, 3317.647 61\40, 3327.27
や (ゆ#) Т# 72\48, 3323.077 53\35, 3347.368 86\57, 3329.032 34\22, 3400 83\53, 3434.483 49\31, 3458.824 64\40, 3490.90
ま (きb) Уь 73\48, 3369.231 87\57, 3367.742 33\22, 3300 79\53, 3268.966 46\31, 3247.059 59\40, 3218.18
き У 74\48, 3415.385 54\35, 3410.526 88\57, 3406.452 34\22, 3400 82\53, 3393.103 48\31, 3388.235 62\40, 3381.81
き# У# 75\48, 3461.538 55\35, 3473.684 90\57, 3483.871 35\22, 3500 85\53, 3517.241 50\31, 3529.421 65\40, 3545.45
せ (しb) Фь 77\48, 3553.846 56\35, 3536.842 92\57, 3522.581 84\53, 3475.862 49\31, 3458.824 63\40, 3436.36
し Ф 78\48, 3600 57\35, 3600 93\57, 3600 36\22, 3600 87\53, 3600 51\31, 3600 66\40, 3600
し# Ф# 79\48, 3646.154 58\35, 3663.158 95\57, 3677.419 37\22, 3700 90\53, 3724.138 53\31, 3741.176 69\40, 3763.63
て (ちb) Хь 81\48, 3738.462 59\35, 3726.316 96\57, 3716.129 89\53, 3682.759 52\31, 3670.588 67\40, 3654.54
ち Х 82\48, 3784.615 60\35, 3789.474 98\57, 3793.458 38\22, 3800 92\53, 3806.897 54\31, 3811.765 70\40, 3818.18
ち# Х# 83\48, 3830.077 61\35, 3852.632 100\57, 3870.968 39\22, 3900 95\53, 3931.034 56\31, 3952.941 73\40, 3981.81
ね (にb) Ць 84\48, 3876.923 99\57, 3832.258 38\22, 3800 91\53, 3765.517 53\31, 3741.176 68\40, 3709.09
に Ц 85\48, 3923.077 62\35, 3915.789 101\57, 3909.677 39\22, 3900 94\53, 3889.655 55\31, 3882.353 71\40, 3872.72
に# Ц# 86\48, 3969.231 63\35, 3978.947 103\57, 3987.097 40\22, 4000 97\53, 4013.793 57\31, 4023.529 74\40, 4036.36
へ (ひb) Чь 88\48, 4061.538 64\35, 4042.105 104\57, 4025.846 96\53, 3972.414 56\31, 3952.941 69\40, 3763.63
ひ Ч 89\48, 4107.692 65\35, 4105.263 106\57, 4103.226 41\22, 4100 99\53, 4096.552 58\31, 4094.118 75\40, 4090.90
ひ# Ч# 90\48, 4153.846 66\35, 4168.421 108\57, 4180.645 42\22, 4200 102\53, 4220.690 60\31, 4235.294 78\40, 4254.54
め (みb) Шь 92\48, 4246.154 67\35, 4231.579 110\57, 4219.355 101\53, 4179.310 59\31, 4164.706 76\40, 4145.45
み Ш 93\48, 4292.308 68\35, 4294.737 111\57, 4296.774 43\22, 4300 104\53, 4303.448 61\31, 4305.882 79\40, 4309.09
み# Ш# 94\48, 4338.462 69\35, 4357.895 113\57, 4374.194 44\22, 4400 107\53, 4427.586 63\31, 4447.059 82\40, 4472.72
こ (くb) Аь 95\48, 4384.615 112\57, 4335.484 43\22, 4300 103\53, 4262.931 60\31, 4235.294 77\40, 4200
く А 96\48, 4430.769 70\35, 4421.053 114\57, 4412.903 44\22, 4400 106\53, 4386.207 62\31, 4376.471 80\40, 4363.63

Intervals

Generators Fourteenth notation Interval category name Generators Notation of 7/2ish 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 22edXIV) 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 ♮11 b14 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
Hindu ♮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 P 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

Natural Dorian and Phrygian modes are shōsūshī MODMOS by one alteration, as are the close approximation to Lydian and the dominant version of Mixolydian and the analog to Harmonic Minor only has two step sizes:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Dorian LsLLLsLLsLLLs 7|5 #13 M m P P M m P M m P P M
Mixolydian Dominant LLsLLsLLLsLsL 6|6 #3 #10 M M P P m m P M M P P m
Persian ♮7 LLsLLLsLsLLLs 5|7 #3 #6 #7 M M P P M M P M m P P M
Lydian Dominant ♮11 LLLsLsLLLsLLs 4|8 #3 #4 #5 #9 #10 #12 #13 M M A P M M P M M P P M
Locrian b4 b8 b11 sLsLLLsLLsLLL 3|9 b4 b5 b8 b11 m m d d m m d m m d d m
Lydian b14 LLLsLLsLLLsLs 2|10 #5 #6 #7 #9 #10 #12 #13 M M P P M M P M M A P M
Locrian b11 sLLsLLLsLsLLL 1|11 #8 b11 m m P d m m P m m d d m
Phrygian b8 sLLLsLsLLLsLL 0|12 #5 #11 #12 m m P P m m d m m P P m
Locrian ♮2 LsLsLLLsLLsLL 12|0 b3 b5 b6 b7 b9 b10 b12 b13 M m P d m m P m m P P M
Phrygian sLLLsLLsLLLsL 11|1 b2 b3 b6 b7 m m P P m m P m m P P m
Minor b12 LsLLsLLLsLsLL 10|2 b3 b6 b10 b12 b13 M m P P m m P M m P d m
Phrygian ♮2 ♮6 LsLLLsLsLLLsL 9|3 b3 b9 b13 M m P P M m P m m P P m
Hindu LLsLsLLLsLLsL 8|4 b6 M M P P m m P M m P P m
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Harmonic Minor LsLLsLLLsLLLs 6|6 #13 M m P P m m P M m P P M
Dorian ♮10 b13 LsLLLsLLLsLsL 5|7 #6 #10 M m P P M m P M M P P M
Ionian Dominant LLsLLLsLLsLsL 4|8 #3 #4 #5 #6 #7 #9 #10 #12 M M P P M M P M M P P m
Lydian b10 ♮11 b14 LLLsLLsLsLLLs 3|9 #2 #3 #4 #6 #7 #9 #10 #11 #12 #13 M M A P M M P M m P P M
Locrian bb7 b8 sLLsLsLLLsLLL 2|10 bb7 b8 m m P d m d d m m P d m
Locrian b4 b8 b11 sLsLLLsLLLsLL 1|11 b4 b11 m m d d m m d m m d d m
Lydian #8 b14 LLLsLLLsLLsLs 0|12 #2 #3 #4 #5 #6 #7 x8 #9 #10 x11 #12 #13 M M P P M M A M M A P M
Locrian ♮5 b11 sLLLsLLsLsLLL 12|0 b2 b3 b6 b7 b9 b10 b11 b12 b13 m m P P m m P m m d d m
Minor b8 b9 b12 LsLLsLsLLLsLL 11|1 b3 b6 b7 b8 b9 b12 M m P P m m d m m P d m
Minor b5 b9 LsLsLLLsLLLsL 10|2 b3 b5 b6 b7 b9 M m P d M m P m m P d m
Phrygian ♮9 sLLLsLLLsLLsL 9|3 b2 b3 b6 b13 m m P P m m P M m P P m
Dorian b12 b13 LsLLLsLLsLsLL 8|4 b2 b3 b6 b13 M m P P M m P M m P m m
Persian Dominant b9 LLsLLsLsLLLsL 7|5 #3 b9 b10 M m P P M m P m m P P M
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Phrygian sLLLsLLsLLLsL 5|7 b2 m m P P m m P m m P P m
Minor b12 LsLLsLLLsLsLL 4|8 #9 b12 M m P P m m P M m P d m
Phrygian ♮2 ♮6 LsLLLsLsLLLsL 3|9 #2 #6 M m P P M m P m m P P m
Hindu LLsLsLLLsLLsL 2|10 #2 #5 #9 #12 M M P P m m P M m P P m
Dorian LsLLLsLLsLLLs 1|11 #2 #5 #8 #9 #12 M m P P M m P M m P P M
Mixolydian Dominant LLsLLsLLLsLsL 0|12 #2 #3 #5 #8 #9 #10 #11 #12 M M P P m m P M m P P m
Persian ♮7 LLsLLLsLsLLLs 12|0 b4 b10 M M P P M M P M m P P M
Lydian Dominant ♮11 LLLsLsLLLsLLs 11|1 #4 b7 M M A P M M P M M P P M
Locrian b4 b8 b11 sLsLLLsLLsLLL 10|2 b2 b3 b4 b5 b6 b8 b9 b10 b11 b12 b13 m m d d m m d m m d d m
Lydian b14 LLLsLLsLLLsLs 9|3 #4 #7 M M P P M M P M M A P M
Locrian b11 sLLsLLLsLsLLL 8|4 b2 b3 b5 b6 b8 b9 b11 b12 b13 m m P d m m P m m d d m
Phrygian b8 sLLLsLsLLLsLL 7|5 b2 b5 b6 b8 b9 b11 b12 b13 m m P P m m d m m P P m
Minor b5 b9 b12 LsLsLLLsLLsLL 6|6 b5 b9 b12 M m P P m m P m m P P M
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Lydian b14 LLLsLLsLLLsLs 12|0 #11 M M P P M M P M M A P M
Locrian b11 sLLsLLLsLsLLL 11|1 b2 b3 b5 b6 b7 b9 b10 b12 b13 m m P d m m P m m d d m
Phrygian b8 sLLLsLsLLLsLL 10|2 b2 b3 b4 b8 b9 b10 b12 b13 m m P d m m P m m d d m
Minor b5 b9 b12 LsLsLLLsLLsLL 9|3 b3 b5 b6 b9 b12 b13 m m P P m m d m m P P m
Phrygian sLLLsLLsLLLsL 8|4 b6 b9 m m P P m m P m m P P M
Minor b12 LsLLsLLLsLsLL 7|5 b6 b12 m m P P m m P M m P d m
Minor b9 LsLLLsLsLLLsL 6|6 b9 M m P P m m P m m P P m
Hindu LLsLsLLLsLLsL 5|7 #3 #9 M M P P M m P M m P P m
Dorian LsLLLsLLsLLLs 4|8 #6 #9 #10 M m P P M m P M m P P m
Mixolydian Dominant LLsLLsLLLsLsL 3|9 #2 #3 #6 #9 #10 #12 M m P P M m P M M P P M
Persian ♮7 LLsLLLsLsLLLs 2|10 #2 #3 #5 #6 #7 #9 #12 #13 M M P P M M P M m P P M
Lydian Dominant ♮11 LLLsLsLLLsLLs 1|11 #2 #3 #4 #5 #6 #8 #9 #10 #12 M M A P M M P M M P P M
Locrian b4 b8 b11 sLsLLLsLLsLLL 0|12 b4 m m d d m m d m m d d m
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Lydian b14 LLLsLLsLLLsLs 12|0 #11 M M P P M M P M M A P M
Locrian b11 sLLsLLLsLsLLL 11|1 b2 b3 b5 b6 b7 b9 b10 b12 b13 m m P d m m P m m d d m
Phrygian b8 sLLLsLsLLLsLL 10|2 b2 b3 b4 b8 b9 b10 b12 b13 m m P d m m P m m d d m
Minor b5 b9 b12 LsLsLLLsLLsLL 9|3 b3 b5 b6 b9 b12 b13 m m P P m m d m m P P m
Phrygian sLLLsLLsLLLsL 8|4 b6 b9 m m P P m m P m m P P M
Minor b12 LsLLsLLLsLsLL 7|5 b6 b12 m m P P m m P M m P d m
Minor b9 LsLLLsLsLLLsL 6|6 b9 M m P P m m P m m P P m
Hindu LLsLsLLLsLLsL 5|7 #3 #9 M M P P M m P M m P P m
Dorian LsLLLsLLsLLLs 4|8 #6 #9 #10 M m P P M m P M m P P m
Mixolydian Dominant LLsLLsLLLsLsL 3|9 #2 #3 #6 #9 #10 #12 M m P P M m P M M P P M
Persian ♮7 LLsLLLsLsLLLs 2|10 #2 #3 #5 #6 #7 #9 #12 #13 M M P P M M P M m P P M
Lydian Dominant ♮11 LLLsLsLLLsLLs 1|11 #2 #3 #4 #5 #6 #8 #9 #10 #12 M M A P M M P M M P P M
Locrian b4 b8 b11 sLsLLLsLLsLLL 0|12 b4 m m d d m m d m m d d m
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Mixolydian Dominant LLsLLsLLLsLsL 10|2 b13 M M P P m m P M m P P m
Persian ♮7 LLsLLLsLsLLLs 9|3 #7 M M P P M M P M m P P M
Major #4 b14 LLLsLsLLLsLLs 8|4 #3 #4 #7 #10 #13 M M A P M M P M m P P M
Locrian b4 b8 b11 sLsLLLsLLsLLL 7|5 b2 b4 b5 b6 b8 b9 b10 b11 b12 m m d d m m d m m d d m
Lydian b14 LLLsLLsLLLsLs 6|6 #3 #4 #6 #7 #10 #11 #13 M M A P M M P M M A P M
Locrian b11 sLLsLLLsLsLLL 5|7 b2 b5 b6 b9 b11 m m P d m m P m m d d m
Phrygian b8 sLLLsLsLLLsLL 4|8 b8 #12 m m P P m m d m m P P m
Locrian ♮2 LsLsLLLsLLsLL 3|9 #5 #6 #7 #8 #9 #10 #12 M m P d M m P M m P P m
Phrygian sLLLsLLsLLLsL 2|10 #5 #12 m m P P m m P m m P P m
Minor b12 LsLLsLLLsLsLL 1|11 #2 #5 #6 #8 #9 #12 M m P P m m P M m P d m
Phrygian ♮2 ♮6 LsLLLsLsLLLsL 0|12 #2 #5 #8 #11 #12 M m P P M m P m m P P m
Hindu LLsLsLLLsLLsL 12|0 b3 b4 b7 b10 M M P P m m P M m P P m
Dorian LsLLLsLLsLLLs 11|1 b3 b7 b10 M m P P M m P M m P P 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♮11 b14 LLLsLLsLLsLLs 12|0 M M A P M M P M M P P M
Hindu ♮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
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Persian ♮7 LLsLLLsLsLLLs 9|3 #7 M M P P M M P M m P P M
Phrygian ♮2 ♮6 LsLLLsLsLLLsL 5|7 #6 M m P P M m P m m P P m
Phrygian sLLLsLsLLLsLL 5|7 b2 m m P P m m P m m P P m
Major #4 b14 LLLsLsLLLsLLs 11|1 #4 M M A P M M P M M P P M
Hindu LLsLsLLLsLLsL 8|4 b6 M M P P M m P M m P P m
Locrian ♮2 LsLsLLLsLLsLL 2|10 #2 M m P d m m P m m P d m
Locrian b4 b8 b11 sLsLLLsLLsLLL 0|12 b4 m m d d m m d m m d d m
Dorian LsLLLsLLsLLLs 7|5 #13 M m P P M m P M m P P M
Locrian b11 sLLsLLLsLsLLL 2|10 b11 m m P d m m P m m d d m
Lydian b14 LLLsLLsLLLsLs 12|0 #11 M M A P M M P M M A P M
Mixolydian Dominant LLsLLsLLLsLsL 10|2 b13 M M P P M m P M M P P m
Minor b12 LsLLsLLLsLsLL 6|6 b12 m m P P m m P m m P d m
Locrian b11 sLLsLLLsLsLLL 2|10 b11 m m P d m m P m m d d m
Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th
Persian #4 LLLsLsLLsLLLs 9|3 #4 M M A P M m P M m P P M
Hindu b9 LLsLsLLsLLLsL 8|4 b6 b9, 5|7 #3 M M P P m m P m m P P m
Locrian ♮2 LsLsLLsLLLsLL 2|10 #2 M m P d m m d m m P d m
Locrian bb7 b8 sLsLLsLLLsLLL 1|11 bb7 m m P d m d d m m P d m
Harmonic Minor LsLLsLLLsLLLs 6|6 #13 M m P P m m P M m P P M
Locrian ♮12 sLLsLLLsLLLsL 2|10 #12 m m P d m m P m m P M m
Major #11 b14 LLsLLLsLLLsLs 11|1 #11 M M P P M M P M M A P M
Dorian ♮10 b13 LsLLLsLLLsLsL 7|5 #10 M m P P M m P M M P P m
Locrian ♮5 ♮9 sLLLsLLLsLsLL 3|9 #9 m m P P m m P M m P d m
Lydian #8♮11 b14 LLLsLLLsLsLLs 12|0 #8 M M A P M M P M M P P M
Ionian Dominant LLsLLLsLsLLsL 11|1 b13, 8|4 #7 M M P P M M P M M P P m
Locrian ♮2 ♮5 ♮6 LsLLLsLsLLsLL 4|8 #6 M m P P M m P m m P d m
Locrian ♮5 b8 b11 sLLLsLsLLsLLL 0|12 #5 m m P P m m d m m d d m

Temperaments

The Shōsūshī temperament spells its major hexad 24:30:36:45:54:64 or 84:108:126:162:189:224 root-2(4g-3p)-(3g-2p)-(11g-8p)-2(3g-2p)-(1g) (p = 18/5 or 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 = 18/5 or 7/2, g = 8/3).

Shōsūshī-Meantone

Subgroup: 18/5.2.3

Comma list: 81/80

POL2 generator: ~4/3 = 506.8820¢

Mapping: [⟨1 1 2], ⟨0 -2 -5]]

Vals: 22ed18/5, 35ed18/5, 48ed18/5, 61ed18/5

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:

Generator Normalized L s L/s Comments
3\13 514.286 1 1 1.000 Equalised
17\74 510.000 6 5 1.200
14\61 509.09 5 4 1.250
25\109 508.474 9 7 1.286
11\48 507.692 4 3 1.333
19\83 506.6 7 5 1.400
8\35 505.263 3 2 1.500 Shōsūshī-Meantone starts here
21\92 504.000 8 5 1.600
13\57 503.226 5 3 1.667
18\79 502.326 7 4 1.750
23\101 501.81 9 5 1.800
5\22 500.000 2 1 2.000 Shōsūshī-Meantone ends, Shōsūshī-Pythagorean begins
42\185 499.0099 17 8 2.125
37\163 498.876 15 7 2.143
32\141 498.701 13 6 2.167
27\119 498.462 11 5 2.200
22\97 498.113 9 4 2.250
17\75 497.561 7 3 2.333
12\53 496.552 5 2 2.500 Shōsūshī-Neogothic heartland is from here…
19\84 495.652 8 3 2.667 …to here
7\31 494.118 3 1 3.000 Shōsūshī-Pythagorean ends, Shōsūshī-Superpyth begins
30\133 493.159 13 4 3.250
23\102 492.857 10 3 3.333
16\71 492.308 7 2 3.500
25\111 491.803 11 3 3.667
9\40 490.90 4 1 4.000
20\89 489.796 9 2 4.500
11\49 488.8 5 1 5.000 Shōsūshī-Superpyth ends
13\58 487.500 6 1 6.000
2\9 480.000 1 0 → inf Paucitonic

See also

9L 4s (7/2-equivalent) - septimal Shōsūshī tuning

9L 4s (39/11-equivalent) - neogothic Shōsūshī tuning

9L 4s (18/5-equivalent) - classical Shōsūshī tuning

18L 8s (112/9-equivalent) - septimal Petrushka minor tuning

18L 8s (88/7-equivalent) and 18L 8s (4[math]π[/math]-equivalent) - neogothic Petrushka minor tuning

18L 8s (64/5-equivalent) - classical Petrushka minor tuning

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