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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.
{{Infobox MOS|Tuning=9L 4s<7/2>}}
 
{{MOS intro|Scale Signature=9L 4s⟨7/2⟩}}
 
'''Shōsūshī''' - so named because of the four small steps in its diatonic MOS - is an excellent 8*-limit temperament tempering out 64/63.


== Properties ==
== 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).
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 ==
== 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.
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 = LLsLLLsLLsLLs (Ionian b14), #/b mostly two chromas up/down.
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:
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/け'''
'''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:
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 '''み'''
'''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:
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/けb''' み#/け
'''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
{| class="wikitable"
{| class="wikitable"
|+Cents<ref name=":0">Fractions repeating more than 4 digits written as continued fractions </ref>
|+Cents
!Notation
!Notation
!Supersoft
!Supersoft
Line 32: Line 36:
|-
|-
|か (く#)
|か (く#)
|1\48
|1\48, 45.18
46; 6, 2
|1\35, 61.97
|1\35
|2\57, 76.10
63: 6, 3
| rowspan="2" |1\22, 98.58
|2\57
|3\53, 122.76
77; 2, 2, 1, 1
|2\31, 139.92
| rowspan="2" |1\22
|3\40, 162.66
100
|3\53
124; 7, 4
|2\31
141; 5, 1, 2
|3\40
163.{{Overline|63}}
|-
|-
|そ (すb)
|そ (すb)
|3\48
|3\48, 135.55
138; 3, 4
|2\35, 123.93
|2\35
|3\57, 114.15
126; 3, 6
|2\53, 81.84
|3\57
|1\31, 69.96
116; 7, 1, 3
|1\40, 54.22
|2\53
82; 1, 3, 7
|1\31
70; 1, 1, 2, 3
|1\40
54.{{Overline|54}}
|-
|-
|す
|す
|4\48
|4\48, 180.735
184; 1, 1, 1, 1, 2
|3\35, 185.90
|3\35
|5\57, 190.25
189; 2, 9
|2\22, 197.17
|5\57
|5\53, 204.61
193; 1, 1, 4, 1, 2
|3\31, 209.89
|2\22
|4\40, 218.88
200
|5\53
206; 1, 8, 1, 2
|3\31
211; 1, 3, 4
|4\40
218.{{Overline|18}}
|-
|-
|さ (す#)
|さ (す#)
|5\48
|5\48, 225.92
230; 1, 3, 3
|4\35, 247.87
|4\35
|7\57, 266.35
252; 1, 1, 1, 2, 2
| rowspan="2" |3\22, 295.75
|7\57
|8\53, 327.37
270; 1, 30
|5\31, 349.81
| rowspan="2" |3\22
|7\40, 379.545
300
|8\53
331; 29
|5\31
352; 1, 16
|7\40
381.{{Overline|81}}
|-
|-
|と (つb)
|と (つb)
|7\48
|7\48, 316.29
323; 13
|5\35, 309.83
|5\35
|8\57, 304.40
315; 1, 3, 1, 3
|7\53, 286.45
|8\57
|4\31, 279.85
309; 1, 2, 10
|5\40, 271.10
|7\53
289; 1, 1, 1, 9
|4\31
282; 2, 1, 5
|5\40
272.{{Overline|72}}
|-
|-
|つ
|つ
|8\48
|8\48, 361.47
369; 4, 3
|6\35, 371.80
|6\35
|10\57, 380.495
378; 1, 18
|4\22, 394.33
|10\57
|10\53, 409.21
387; 10, 3
|6\31, 419.77
|4\22
|8\40, 433.765
400
|10\53
413; 1, 3, 1, 5
|6\31
423; 1, 1, 8
|8\40
436.{{Overline|36}}
|-
|-
|た (つ#)
|た (つ#)
|9\48
|9\48, 406.655
415; 2, 2, 2
| rowspan="2" |7\35, 433.765
| rowspan="2" |7\35
|12\57, 456.595
442; 9, 2
|5\22, 492.915
|12\57
|13\53, 531.98
464; 1, 1, 15
|8\31, 559.70
|5\22
|11\40, 596.43
500
|13\53
537; 13, 2
|8\31
564; 1, 2, 2, 2
|11\40
600
|-
|-
|の (ぬb)
|の (ぬb)
|10\48
|10\48, 451.83
461; 1, 1, 6
|11\57, 418.545
|11\57
|4\22, 394.33
425; 1, 4, 6
|9\53, 368.29
|4\22
|5\31, 349.81
400
|6\40, 325.32
|9\53
372; 2, 2, 2, 2
|5\31
352; 1, 16
|6\40
327.{{Overline|27}}
|-
|-
|'''ぬ'''
|'''ぬ'''
|'''11\48'''
|'''11\48,''' '''497.02'''
'''507; 1, 1, 2, 4'''
|'''8\35, 495.73'''
|'''8\35'''
|'''13\57,''' '''494.645'''
'''505; 3, 1, 4'''
|'''5\22, 492.915'''
|'''13\57'''
|'''12\53,''' '''491.055'''
'''503; 4, 2, 3'''
|'''7\31,''' '''489.735'''
|'''5\22'''
|'''9\40,''' '''487.99'''
'''500'''
|'''12\53'''
'''496; 1, 1, 4, 3'''
|'''7\31'''
'''494; 8, 2'''
|'''9\40'''
'''490.{{Overline|90}}'''
|-
|-
|な (ぬ#)
|な (ぬ#)
|12\48
|12\48, 542.21
553; 1, 5, 2
|9\35, 557.70
|9\35
|15\57, 570.74
568; 2, 2, 1, 2
| rowspan="2" |6\22, 591.50
|15\57
|15\53, 613.82
580; 1, 1, 4, 2
|9\31, 629.66
| rowspan="2" |6\22
|12\40, 650.65
600
|15\53
620; 1, 2, 4, 2
|9\31
635; 3, 2, 2
|12\40
654.{{Overline|54}}
|-
|-
|ほ (ふb)
|ほ (ふb)
|14\48
|14\48, 632.57
646; 6, 2
|10\35, 619.665
|10\35
|16\57, 608.79
631; 1, 1, 2, 1, 2
|14\53, 572.90
|16\57
|8\31, 559.70
619; 2, 4, 2
|10\40, 542.21
|14\53
579; 9, 1, 2
|8\31
564; 1, 2, 2, 2
|10\40
545.{{Overline|45}}
|-
|-
|ふ
|ふ
|15\48
|15\48, 677.76
692; 3, 4
|11\35, 681.63
|11\35
|18\57, 684.89
694; 1, 2, 1, 4
|7\22, 690.08
|18\57
|17\53, 695.66
696; 1, 3, 2, 3
|10\31, 699.62
|7\22
|13\40, 704.87
700
|17\53
703; 2, 4, 3
|10\31
705; 1, 7, 2
|13\40
709.{{Overline|09}}
|-
|-
|は (ふ#)
|は (ふ#)
|16\48
|16\48, 722.94
738; 3, 4
|12\35, 743.60
|12\35
|20\57, 760.99
757; 1, 8, 2
| rowspan="2" |8\22, 788.66
|20\57
|20\53, 818.425
774; 5, 6
|12\31, 839.55
| rowspan="2" |8\22
|16\40, 867.53
800
|20\53
827; 1, 1, 2, 2, 2
|12\31
847; 17
|16\40
872.{{Overline|72}}
|-
|-
|も (むb)
|も (むb)
|18\48
|18\48, 813.31
830; 1, 3, 3
|13\35, 805.56
|13\35
|21\57, 799.04
821; 19
|19\53, 777.50
|21\57
|11\31, 769.58
812; 1, 9, 3
|14\40, 759.09
|19\53
786; 4, 1, 5
|11\31
776; 2, 8
|14\40
763.{{Overline|63}}
|-
|-
|む
|む
|19\48
|19\48, 850.49
876; 1, 12
|14\35, 867.53
|14\35
|23\57, 875.14
884; 4, 1, 3
|9\22, 887.25
|23\57
|22\53, 900.27
890; 3, 10
|13\31, 909.51
|9\22
|17\40, 921.75
900
|22\53
910; 2, 1, 9
|13\31
917; 1, 1, 1, 5
|17\40
927.{{Overline|27}}
|-
|-
|よ (む#)
|よ (む#)
|20\48
|20\48, 903.68
923; 13
|15\35, 929.50
|15\35
|25\57, 951.24
947; 2, 1, 2, 2
| rowspan="2" |10\22, 985.83
|25\57
|25\53, 1023.03
967; 1, 2, 1, 7
|15\31, 1049.43
| rowspan="2" |10\22
|20\40, 1084.41
1000
|25\53
1034; 2, 14
|15\31
1058; 1, 4, 1, 2
|20\40
1090.{{Overline|90}}
|-
|-
|け (ゆb)
|け (ゆb)
|22\48
|22\48, 994.045
1015; 2, 2, 2
|16\35, 991.46
|16\35
|26\57, 989.29
1010; 1, 1, 9
|24\53, 982.11
|26\57
|14\31, 979.47
1006; 2, 4, 1, 2
|18\40, 975.97
|24\53
993: 9, 1, 2
|14\31
988; 4, 4
|18\40
981.{{Overline|81}}
|-
|-
|ゆ
|ゆ
|23\48
|23\48, 1039.23
1061; 1, 1, 6
|17\35, 1053.43
|17\35
|28\57, 1065.39
1073; 1, 2, 6
|11\22, 1084.41
|28\57
|27\53, 1104.87
1083; 1, 6, 1, 3
|16\31, 1119.39
|11\22
|21\40, 1138.63
1100
|27\53
1127; 4, 7
|16\31
1129; 2, 2, 3
|21\40
1145.{{Overline|45}}
|-
|-
|や (ゆ#)
|や (ゆ#)
|24\48
|24\48, 1084.41
1107; 1, 1, 2, 4
| rowspan="2" |18\35, 1115.40
| rowspan="2" |18\35
|30\57, 1141.49
1136; 1, 5, 3
|12\22, 1183.00
|30\57
|30\29, 1227.63
1161; 3, 2, 4
|15\31, 1049.43
|12\22
|24\40, 1301.30
1200
|30\29
1241; 2, 1, 1, 1, 3
|15\31
1058; 1, 4, 1, 2
|24\40
1309.{{Overline|09}}
|-
|-
|ま (きb)
|ま (きb)
|25\48
|25\48, 1129.60
1153; 1, 5, 2
|29\57, 1103.44
|29\57
|11\22, 1084.41
1122; 1, 1, 2, 1, 1
|26\53, 1063.95
|11\22
|18\31, 1259.32
1100
|19\40, 1030.19
|26\53
1075; 1, 6, 4
|18\31
1270; 1, 1, 2, 3
|19\40
1036.{{Overline|36}}
|-
|-
|き
|き
|26\48
|26\48, 1174.78
1200
|19\35, 1177.36
|19\35
|31\57, 1179.54
1200
|12\22, 1183.00
|31\57
|29\53, 1186.72
1200
|17\31, 1189.36
|12\22
|22\40, 1192.85
1200
|29\53
1200
|17\31
1200
|22\40
1200
|-
|-
|き#
|き#
|27\48
|27\48, 1219.965
1246; 6, 2
|20\35, 1239.32
|20\35
|33\57, 1255.64
1263: 6, 3
| rowspan="2" |13\22, 1281.58
|33\57
|32\53, 1309.48
1277; 2, 2, 1, 1
|18\31, 1259.32
| rowspan="2" |13\22
|25\40, 1355.52
1300
|32\53
1324; 7, 4
|18\31
1270; 1, 1, 2, 3
|25\40
1363.{{Overline|63}}
|-
|-
|せ (しb)
|せ (しb)
|29\48
|29\48, 1310.33
1338; 3, 4
|21\35, 1301.29
|21\35
|34\57, 1293.69
1326; 3, 6
|31\53, 1268.56
|34\57
|19\31, 1329.28
1316; 7, 1, 3
|23\40, 1247.075
|31\53
1282; 1, 3, 7
|19\31
1341; 5, 1, 2
|23\40
1254.{{Overline|54}}
|-
|-
|し
|し
|30\48
|30\48, 1355.52
1384; 1, 1, 1, 1, 2
|22\35, 1363.26
|22\35
|36\57, 1369.785
1389; 2, 9
|14\22, 1380.16
|36\57
|34\53, 1391.32
1393; 1, 1, 4, 1, 2
|20\31, 1399.24
|14\22
|26\40, 1409.74
1400
|34\53
1406; 1, 8, 1, 2
|20\31
1411; 1, 3, 4
|26\40
1418.{{Overline|18}}
|-
|-
|し#
|し#
|31\48
|31\48, 1400.70
1430; 1, 3, 3
|23\35, 1425.23
|23\35
|38\57, 1445.88
1452; 1, 1, 1, 2, 2
| rowspan="2" |15\22, 1478.745
|38\57
|37\53, 1514.09
1470; 1, 30
|22\31, 1539.17
| rowspan="2" |15\22
|29\40, 1572.40
1500
|37\53
1531; 29
|22\31
1552; 1, 16
|29\40
1581.{{Overline|81}}
|-
|-
|て (ちb)
|て (ちb)
|33\48
|33\48, 1491.07
1523; 13
|24\35, 1487.195
|24\35
|39\57, 1483.93
1515; 1, 3, 1, 3
|36\53, 1473.165
|39\57
|21\31, 1469.205
1509; 1, 2, 10
|27\40, 1463.96
|36\53
1389; 1, 1, 1, 9
|21\31
1482; 2, 1, 5
|27\40
1472.{{Overline|72}}
|-
|-
|ち
|ち
|34\48
|34\48, 1536.25
1569; 4, 3
|25\35, 1549.16
|25\35
|41\57, 1560.03
1578; 1, 18
|16\22, 1577.33
|41\57
|39\53, 1595.93
1587; 10, 3
|23\31, 1609.13
|16\22
|30\40, 1626.62
1600
|39\53
1613; 1, 3, 1, 5
|23\31
1623; 1, 1, 8
|30\40
1636.{{Overline|36}}
|-
|-
|ち#
|ち#
|35\48
|35\48, 1581.44
1615; 2, 2, 2
| rowspan="2" |26\35, 1611.13
| rowspan="2" |26\35
|43\57, 1636.13
1642; 9, 2
|17\22, 1675.91
|43\57
|42\53, 1718.69
1664; 1, 1, 15
|25\31, 1749.05
|17\22
|33\40, 1789.28
1700
|42\53
1737; 13, 2
|25\31
1764; 1, 2, 2, 2
|33\40
1800
|-
|-
|ね (にb)
|ね (にb)
|36\48
|36\48, 1626.62
1661; 1, 1, 6
|42\57, 1598.08
|42\57
|16\22, 1577.33
1625; 1, 4, 6
|38\53, 1555.01
|16\22
|22\31, 1539.17
1600
|28\40, 1518.18
|38\53
1572; 2, 2, 2, 2
|22\31
1552; 1, 16
|28\40
1527.{{Overline|27}}
|-
|-
|'''に'''
|'''に'''
|'''37\48'''
|'''37\48,''' '''1671.80'''
'''1707; 1, 1, 2, 4'''
|'''27\35,''' '''1673.095'''
|'''27\35'''
|'''44\57,''' '''1674.18'''
'''1705; 3, 1, 4'''
|'''17\22, 1675.91'''
|'''44\57'''
|'''41\53,''' '''1677.77'''
'''1703; 4, 2, 3'''
|'''24\31,''' '''1679.09'''
|'''17\22'''
|'''31\40,''' '''1680.84'''
'''1700'''
|'''41\53'''
'''1696; 1, 1, 4, 3'''
|'''24\31'''
'''1694; 8, 2'''
|'''31\40'''
'''1690.{{Overline|90}}'''
|-
|-
|に#
|に#
|38\48
|38\48, 1716.99
1753; 1, 5, 2
|28\35, 1735.06
|28\35
|46\57, 1750.28
1768; 2, 2, 1, 2
| rowspan="2" |18\22, 1774.49
|46\57
|44\53, 1800.535
1780; 1, 1, 4, 2
|26\31, 1819.015
| rowspan="2" |18\22
|34\40, 1843.50
1800
|44\53
1820; 1, 2, 4, 2
|26\31
1835; 3, 2, 2
|34\40
1854.{{Overline|54}}
|-
|-
|へ (ひb)
|へ (ひb)
|40\48
|40\48, 1807.355
1846; 6, 2
|29\35, 1797.03
|29\35
|47\57, 1788.33
1831; 1, 1, 2, 1, 2
|43\53, 1759.61
|47\57
|25\31, 1749.05
1819; 2, 4, 2
|32\40, 1735.06
|43\53
1779; 9, 1, 2
|25\31
1764; 1, 2, 2, 2
|32\40
1745.{{Overline|45}}
|-
|-
|ひ
|ひ
|41\48
|41\48, 1852.54
1892; 3, 4
|30\35, 1858.99
|30\35
|49\57, 1864.43
1894; 1, 2, 1, 4
|19\22, 1873.08
|49\57
|46\53, 1882.37
1896; 1, 3, 2, 3
|27\31, 1888.98
|19\22
|35\40, 1897.72
1900
|46\53
1903; 2, 4, 3
|27\31
1905; 1, 7, 2
|35\40
1909.{{Overline|09}}
|-
|-
|ひ#
|ひ#
|42\48
|42\48, 1897.72
1938; 3, 4
|31\35, 1920.96
|31\35
|51\57, 1940.53
1957; 1, 8, 2
| rowspan="2" |20\22, 1971.66
|51\57
|49\53, 2005.14
1974; 5, 6
|29\31, 2028.90
| rowspan="2" |20\22
|38\40, 2060.385
2000
|49\53
2027; 1, 1, 2, 2, 2
|29\31
2047; 17
|38\40
2072.{{Overline|72}}
|-
|-
|め (みb)
|め (みb)
|44\48
|44\48, 1988.09
2030; 1, 3, 3
|32\35, 1982.93
|32\35
|52\57, 1978.58
2021; 19
|48\53, 1964.22
|52\57
|28\31, 1958.94
2012; 1, 9, 3
|36\40, 1951.94
|48\53
1986; 4, 1, 5
|28\31
1976; 2, 8
|36\40
1963.{{Overline|63}}
|-
|-
|み
|み
|45\48
|45\48, 2033.27
2076; 1, 12
|33\35, 2044.89
|33\35
|54\57, 2054.68
2084; 4, 1, 3
|21\22, 2070.24
|54\57
|51\53, 2086.98
2090; 3, 10
|30\31, 2098.86
|21\22
|39\40, 2114.605
2100
|51\53
2110; 2, 1, 9
|30\31
2117; 1, 1, 1, 5
|39\40
2127.{{Overline|27}}
|-
|-
|み#
|み#
|46\48
|46\48, 2078.46
2123; 13
| rowspan="2" |34\35, 2106.86
| rowspan="2" |34\35
|56\57, 2130.78
2147; 2, 1, 2, 2
|22\22, 2168.83
|56\57
|54\53, 2209.75
2167; 1, 2, 1, 7
|32\31, 2238.79
|22\22
|42\40, 2223.05
2200
|54\53
2234; 2, 14
|32\31
2258; 1, 4, 1, 2
|42\40
2290.{{Overline|90}}
|-
|-
|け (くb)
|こ (くb)
|47\48
|47\48, 2123.64
2169; 4, 3
|55\57, 2092.73
|55\57
|21\22, 2070.24
2129; 31
|50\53, 2046.06
|21\22
|29\31, 2028.90
2100
|37\40, 2006.16
|50\53
2068; 1, 28
|29\31
2047; 17
|37\40
2018.{{Overline|18}}
|-
|-
!く
!く
!48\48
!48\48, 2168.83
2215; 2, 2, 2
!35\35, 2168.83
!35\35
!57\57, 2168.83
2210; 1, 1, 9
!22\22, 2168.83
!57\57
!53\53, 2168.83
2206; 2, 4, 1, 2
!31\31, 2168.83
!22\22
!40\40, 2168.83
2200
!53\53
2193; 9, 1, 2
!31\31
2188; 4, 4
!40\40
2181.{{Overline|81}}
|}
|}
{| class="wikitable"
|+Relative Cents<ref name=":0" />
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|か (く#)
|1\48
''45.8{{Overline|3}}''
|1\35
''62; 1, 6''
|2\57
''77; 5, 5, 2''
| rowspan="2" |1\22
''100''
|3\53
''124; 1, 1, 8, 3''
|2\31
''141; 1, 14, 2''
|3\40
''165''
|-
|そ (すb)
|3\48
''137.5''
|2\35
''125; 1, 2, 2''
|3\57
''115; 1, 3, 1, 3''
|2\53
''83; 53''
|1\31
''70; 1, 30''
|1\40
55
|-
|す
|4\48
''183.{{Overline|3}}''
|3\35
''188; 1, 1, 3''
|5\57
''192; 1, 56''
|2\22
''200''
|5\53
''207; 1, 1, 4, 1, 4''
|3\31
''212; 1, 9, 3''
|4\40
''220''
|-
|さ (す#)
|5\48
''229.1{{Overline|6}}''
|4\35
''251; 2, 3''
|7\57
''270; 5, 1, 2, 3''
| rowspan="2" |3\22
''300''
|8\53
''332; 13, 4''
|5\31
''354; 1, 5, 5''
|7\40
''385''
|-
|と (つb)
|7\48
''320.8{{Overline|3}}''
|5\35
''314; 3, 2''
|8\57
''308; 1, 3, 2, 1, 1, 2''
|7\53
''290; 1, 1, 3, 3, 2''
|4\31
''283; 1, 6, 1, 3''
|5\40
''275''
|-
|つ
|8\48
''366.{{Overline|6}}''
|6\35
''377; 7''
|10\57
''385; 1, 27, 2''
|4\22
''400''
|10\53
''415; 10, 1, 1, 2''
|6\31
''425; 1, 4, 6''
|8\40
''440''
|-
|た (つ#)
|9\48
''412.5''
| rowspan="2" |7\35
''440''
|12\57
''463; 6, 3''
|5\22
''500''
|13\53
''539; 1, 1, 1, 1, 1, 6''
|8\31
''567; 1, 2, 1, 7''
|11\40
''605''
|-
|の (ぬb)
|10\48
''458.{{Overline|3}}''
|11\57
''424; 1, 1, 3, 1, 1, 3''
|4\22
''400''
|9\53
''373; 1, 1, 2, 2, 4''
|5\31
''354; 1, 5, 5''
|6\40
''330''
|-
|'''ぬ'''
|'''11\48'''
'''''504.1{{Overline|6}}'''''
|'''8\35'''
'''''502; 1, 6'''''
|'''13\57'''
'''''501; 1, 3, 14'''''
|'''5\22'''
'''''500'''''
|'''12\53'''
'''''498; 8, 1, 5'''''
|'''7\31'''
'''''496; 1, 3, 2, 3'''''
|'''9\40'''
'''''495'''''
|-
|な (ぬ#)
|12\48
''550''
|9\35
''565; 1, 2, 2''
|15\57
''578; 1, 18''
| rowspan="2" |6\22
''600''
|15\53
''622; 1, 1, 1, 3, 1, 3''
|9\31
''638; 1, 2, 2, 4''
|12\40
''660''
|-
|ほ (ふb)
|14\48
''641.{{Overline|6}}''
|10\35
''628; 1, 1, 3''
|16\57
''617; 1, 1, 5, 5''
|14\53
''581; 7, 1, 1, 3''
|8\31
''567; 1, 2, 1, 7''
|10\40
''550''
|-
|ふ
|15\48
''687.5''
|11\35
''691; 2, 3''
|18\57
''694; 1, 2, 1, 4''
|7\22
''700''
|17\53
''705; 1, 1, 1, 17''
|10\31
''709; 1, 2, 10''
|13\40
''715''
|-
|は (ふ#)
|16\48
''733.{{Overline|3}}''
|12\35
''754; 3, 2''
|20\57
''771; 1, 13, 4''
| rowspan="2" |8\22
''800''
|20\53
''830; 5, 3, 3''
|12\31
''851; 1, 1, 1, 1, 2, 2''
|16\40
''880''
|-
|も (むb)
|18\48
''825''
|13\35
''817; 7''
|21\57
''810; 1, 1, 9''
|19\53
''788; 1, 2, 8, 2''
|11\31
''780; 1, 1, 1, 4, 2''
|14\40
''770''
|-
|む
|19\48
''870.8{{Overline|3}}''
|14\35
''880''
|23\57
''887; 1, 2, 1, 1, 3, 2''
|9\22
''900''
|22\53
''913; 4, 1, 4, 2''
|13\31
''922; 1, 1, 2, 1, 1, 2''
|17\40
''935''
|-
|よ (む#)
|20\48
''916.{{Overline|6}}''
|15\35
''942; 1, 6''
|25\57
''964; 1, 10, 3, 2''
| rowspan="2" |10\22
''1000''
|25\53
''1037; 1, 2, 1, 3, 1, 2''
|15\31
''1064; 1, 1, 15''
|20\40
''1100''
|-
|け (ゆb)
|22\48
''1013.{{Overline|3}}''
|16\35
''1005; 1, 2, 2''
|26\57
''1003; 1, 1, 28''
|24\53
''996: 4, 2, 2, 2''
|14\31
''993; 1, 1, 4, 1, 2''
|18\40
''990''
|-
|ゆ
|23\48
''1059.1{{Overline|6}}''
|17\35
''1068; 1, 1, 3''
|28\57
''1080; 1, 2, 2, 1, 5''
|11\22
''1100''
|27\53
''1130; 1, 3, 13''
|16\31
''1135; 2, 15''
|21\40
''1155''
|-
|や (ゆ#)
|24\48
''1100''
| rowspan="2" |18\35
''1131; 2, 3''
|30\57
''1157; 1, 8, 2''
|12\22
''1200''
|30\29
''1245; 3, 1, 1, 7''
|18\31
''1277; 2, 2, 1, 1, 2''
|24\40
''1320''
|-
|ま (きb)
|25\48
''1145.8{{Overline|3}}''
|29\57
''1119; 3, 2, 1, 5''
|11\22
''1100''
|26\53
''1079; 4, 13''
|15\31
''1064; 1, 1, 15''
|19\40
''1045''
|-
|き
|26\48
''1191.{{Overline|6}}''
|19\35
''1194: 3, 2''
|31\57
''1196; 2, 28''
|12\22
''1200''
|29\53
''1203; 1, 3, 2, 2, 2''
|17\31
''1206; 2, 3, 1, 2''
|22\40
''1210''
|-
|き#
|27\48
''1237.5''
|20\35
''1257: 7''
|33\57
''1273; 1, 2, 6''
| rowspan="2" |13\22
''1300''
|32\53
''1328; 3, 3, 5''
|19\31
''1348; 2, 1, 1, 2, 2''
|25\40
''1375''
|-
|せ (しb)
|29\48
''1329.1{{Overline|6}}''
|21\35
''1320''
|34\57
''1312; 3, 1, 1, 3, 2''
|31\53
''1286; 1, 3, 1, 1, 2''
|18\31
''1277; 2, 2, 1, 1, 2''
|23\40
''1265''
|-
|し
|30\48
''1375''
|22\35
''1382; 1, 6''
|36\57
''1389; 2, 9''
|14\22
''1400''
|34\53
''1411; 3, 8, 2''
|20\31
''1419; 2, 4, 2''
|26\40
''1430''
|-
|し#
|31\48
''1420.8{{Overline|3}}''
|23\35
''1445; 1, 2, 2''
|38\57
''1466.{{Overline|6}}''
| rowspan="2" |15\22
''1500''
|37\53
''1535; 1, 5, 1, 1, 1, 2''
|22\31
''1561; 3, 2, 4''
|29\40
''1595''
|-
|て (ちb)
|33\48
''1512.5''
|24\35
''1508; 1, 1, 3''
|39\57
''1505; 3, 1, 4''
|36\53
''1394; 2, 1, 17''
|21\31
''1490; 3, 10''
|27\40
''1485''
|-
|ち
|34\48
''1558.{{Overline|3}}''
|25\35
''1571; 2, 3''
|41\57
''1582; 2, 5, 5''
|16\22
''1600''
|39\53
''1618; 1, 6, 1, 1, 3''
|23\31
''1632; 3, 1, 7''
|30\40
''1650''
|-
|ち#
|35\48
''1604.1{{Overline|6}}''
| rowspan="2" |26\35
''1633; 3, 2''
|43\57
''1659; 1, 1, 1, 5, 1, 3''
|17\22
''1700''
|42\53
1737; 13, 2
|25\31
''1774; 5, 6''
|33\40
''1815''
|-
|ね (にb)
|36\48
''1650''
|42\57
''1621; 19''
|16\22
''1600''
|38\53
''1578; 2, 1, 3, 1, 3''
|22\31
''1561; 3, 2, 4''
|28\40
''1540''
|-
|'''に'''
|'''37\48'''
'''''1695.8{{Overline|3}}'''''
|'''27\35'''
'''''1696; 7'''''
|'''44\57'''
'''''1699; 4, 14'''''
|'''17\22'''
'''''1700'''''
|'''41\53'''
'''''1701; 1, 7, 1, 5'''''
|'''24\31'''
'''''1703; 4, 2, 3'''''
|'''31\40'''
'''''1705'''''
|-
|に#
|38\48
''1741.{{Overline|6}}''
|28\35
''1760''
|46\57
''1775; 2, 3, 1, 1, 3''
| rowspan="2" |18\22
''1800''
|44\53
''1826; 2, 2,, 2, 4''
|26\31
''1845; 5, 6''
|34\40
''1870''
|-
|へ (ひb)
|40\48
''1833.{{Overline|3}}''
|29\35
''1822; 1, 6''
|47\57
''1814; 28, 2''
|43\53
''1784; 1, 9, 1, 1, 2''
|25\31
''1774; 6, 5''
|32\40
''1760''
|-
|ひ
|41\48
''1879.1{{Overline|6}}''
|30\35
''1885; 1, 2, 2''
|49\57
''1891; 4, 2, 1, 1, 2''
|19\22
''1900''
|46\53
''1909; 2, 3, 3, 2''
|27\31
''1916; 7, 1, 3''
|35\40
''1925''
|-
|ひ#
|42\48
''1925''
|31\35
''1948; 1, 1, 3''
|51\57
''1968; 2, 2, 1, 2''
| rowspan="2" |20\22
''2000''
|49\53
''2033; 1, 25, 2''
|29\31
''2058; 15, 2''
|38\40
''2090''
|-
|め (みb)
|44\48
''2016.{{Overline|6}}''
|32\35
''2011; 2, 3''
|52\57
''2007; 57''
|48\53
''1992; 2, 4, 1, 4''
|28\31
''1987; 10, 3''
|36\40
''1980''
|-
|み
|45\48
''2062.5''
|33\35
''2075; 3, 2''
|54\57
2084; 4, 1, 3
|21\22
''2100''
|51\53
''2116; 1, 52''
|30\31
''2129; 31''
|39\40
''2145''
|-
|み#
|46\48
''2108.{{Overline|3}}''
| rowspan="2" |34\35
''2138; 7''
|56\57
''2161; 2, 2, 11''
|22\22
''2200''
|54\53
''2241; 1, 1, 26''
|32\31
''2270; 1, 30''
|42\40
''2310''
|-
|け (くb)
|47\48
''2154.1{{Overline|6}}''
|55\57
''2122; 1, 4, 5, 2''
|21\22
''2100''
|50\53
''2075; 2, 8, 3''
|29\31
''2058; 15, 2''
|37\40
''2035''
|-
!く
! colspan="6" |''2200''
!
|}
== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
Line 1,234: Line 380:
!Interval category name
!Interval category name
!Generators
!Generators
!Notation of 7/2ish inverse
!Notation of 7/2 inverse
!Interval category name
!Interval category name
|-
|-
Line 1,329: Line 475:
|ね (にb)
|ね (にb)
|diminished eleventh  
|diminished eleventh  
|-
| colspan="6" |The chromatic 22-note MOS (either [[13L 9s (7/2-equivalent)|13L 9s]], [[9L 13s (7/2-equivalent)|9L 13s]] or [[22ed7/2]]) also has these intervals
|-
|-
|13
|13
Line 1,334: Line 482:
|augmented unison  
|augmented unison  
| -13
| -13
|け (くb)
|こ (くb)
|diminished fourteenth  
|diminished fourteenth  
|-
| colspan="6" |The chromatic 22-note MOS (either [[13L 9s (minor fourteenth equivalent)|13L 9s]], [[9L 13s (minor fourteenth equivalent)|9L 13s]] or [[22edXIV]]) also has these intervals
|-
|-
|14
|14
Line 1,395: Line 541:
|doubly diminished twelfth
|doubly diminished twelfth
|}
|}
== Genchain ==
The generator chain for this scale is as follows:
{| class="wikitable"
|+
|
|
|
|
|
|
|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:
{| class="wikitable"
|+
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="12" |Interval type
|-
!name
!pattern
!notation
!2nd
!3rd
!4th
!5th
!6th
!7th
!8th
!9th
!10th
!11th
!12th
!13th
|-
|Lydian Dominant  ♮ 11
|LLLsLLsLLsLLs
|<nowiki>12|0</nowiki>
|M
|M
|A
|P
|M
|M
|P
|M
|M
|P
|P
|M
|-
|Major b14
|LLsLLLsLLsLLs
|<nowiki>11|1</nowiki>
|M
|M
|P
|P
|M
|M
|P
|M
|M
|P
|P
|M
|-
|Mixolydian
|LLsLLsLLLsLLs
|<nowiki>10|2</nowiki>
|M
|M
|P
|P
|M
|m
|P
|M
|M
|P
|P
|M
|-
|Mixolydian b10 (Persian)
|LLsLLsLLsLLLs
|<nowiki>9|3</nowiki>
|M
|M
|P
|P
|M
|m
|P
|M
|m
|P
|P
|M
|-
|Aeolian Dominant ♮6
|LLsLLsLLsLLsL
|<nowiki>8|4</nowiki>
|M
|M
|P
|P
|M
|m
|P
|M
|m
|P
|P
|m
|-
|Dorian b13
|LsLLLsLLsLLsL
|<nowiki>7|5</nowiki>
|M
|m
|P
|P
|M
|m
|P
|M
|m
|P
|P
|m
|-
|Minor
|LsLLsLLLsLLsL
|<nowiki>6|6</nowiki>
|M
|m
|P
|P
|m
|m
|P
|M
|m
|P
|P
|m
|-
|Phrygian ♮2
|LsLLsLLsLLLsL
|<nowiki>5|7</nowiki>
|M
|m
|P
|P
|m
|m
|P
|m
|m
|P
|P
|m
|-
|Locrian ♮2 ♮5
|LsLLsLLsLLsLL
|<nowiki>4|8</nowiki>
|M
|m
|P
|P
|m
|m
|P
|m
|m
|P
|d
|m
|-
|Locrian ♮5
|sLLLsLLsLLsLL
|<nowiki>3|9</nowiki>
|m
|m
|P
|P
|m
|m
|P
|m
|m
|P
|d
|m
|-
|Locrian
|sLLsLLLsLLsLL
|<nowiki>2|10</nowiki>
|m
|m
|P
|d
|m
|m
|P
|m
|m
|P
|d
|m
|-
|Locrian b8
|sLLsLLsLLLsLL
|<nowiki>1|11</nowiki>
|m
|m
|P
|d
|m
|m
|d
|m
|m
|P
|d
|m
|-
|Locrian b8 b11
|sLLsLLsLLsLLL
|<nowiki>0|12</nowiki>
|m
|m
|P
|d
|m
|m
|d
|m
|m
|d
|d
|m
|}
=== Cyclic Permutation Order ===
{| class="wikitable"
|+
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="12" |Interval type
|-
!name
!pattern
!notation
!2nd
!3rd
!4th
!5th
!6th
!7th
!8th
!9th
!10th
!11th
!12th
!13th
|-
|Mixolydian b10 (Persian)
|LLsLLsLLsLLLs
|<nowiki>9|3</nowiki>
|M
|M
|P
|P
|M
|m
|P
|M
|m
|P
|P
|M
|-
|Phrygian ♮2
|LsLLsLLsLLLsL
|<nowiki>5|7</nowiki>
|M
|m
|P
|P
|m
|m
|P
|m
|m
|P
|P
|m
|-
|Locrian b8
|sLLsLLsLLLsLL
|<nowiki>1|11</nowiki>
|m
|m
|P
|d
|m
|m
|d
|m
|m
|P
|d
|m
|-
|Mixolydian
|LLsLLsLLLsLLs
|<nowiki>10|2</nowiki>
|M
|M
|P
|P
|M
|m
|P
|M
|M
|P
|P
|M
|-
|Minor
|LsLLsLLLsLLsL
|<nowiki>6|6</nowiki>
|M
|m
|P
|P
|m
|m
|P
|M
|m
|P
|P
|m
|-
|Locrian
|sLLsLLLsLLsLL
|<nowiki>2|10</nowiki>
|m
|m
|P
|d
|m
|m
|P
|m
|m
|P
|d
|m
|-
|Major b14
|LLsLLLsLLsLLs
|<nowiki>11|1</nowiki>
|M
|M
|P
|P
|M
|M
|P
|M
|M
|P
|P
|M
|-
|Dorian b13
|LsLLLsLLsLLsL
|<nowiki>7|5</nowiki>
|M
|m
|P
|P
|M
|m
|P
|M
|m
|P
|P
|m
|-
|Locrian ♮5
|sLLLsLLsLLsLL
|<nowiki>3|9</nowiki>
|m
|m
|P
|P
|m
|m
|P
|m
|m
|P
|d
|m
|-
|Lydian Dominant ♮11
|LLLsLLsLLsLLs
|<nowiki>12|0</nowiki>
|M
|M
|A
|P
|M
|M
|P
|M
|M
|P
|P
|M
|-
|Aeolian Dominant ♮6
|LLsLLsLLsLLsL
|<nowiki>8|4</nowiki>
|M
|M
|P
|P
|M
|m
|P
|M
|m
|P
|P
|m
|-
|Locrian ♮2 ♮5
|LsLLsLLsLLsLL
|<nowiki>4|8</nowiki>
|M
|m
|P
|P
|m
|m
|P
|m
|m
|P
|d
|m
|-
|Locrian b8 b11
|sLLsLLsLLsLLL
|<nowiki>0|12</nowiki>
|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 <code>root-2(4g-3p)-(3g-2p)-(11g-8p)-2(3g-2p)-(1g)</code> (p = 7/2, g = 8/3) and its minor hexad 12:14:18:21:27:32 or 60:72:90:108:135:160 <code>root-(-5g-p)-(3g-2p)-(-2g)-2(3g-2p)-(1g)</code> (p = 7/2, g = 8/3).
=== Shōsūshī-Superpyth ===
[[Subgroup]]: 7/2.2.3
[[Comma]] list: [[64/63]]
[[POL2]] generator: ~4/3 = 486.8323¢
[[Mapping]]: [{{Val|1 1 2}}, {{Val|0 -2 -5}}]
[[Vals]]: [[22ed7/2]], [[31ed7/2]], [[40ed7/2]], [[49ed7/2]]
== Scale tree ==
The spectrum looks like this:
{{MOS tuning spectrum|Scale Signature=9L 4s <7/2>}}
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/9L_4s_(7/2-equivalent)"