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{{Infobox MOS}}
{{Infobox MOS|Tuning=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 or an excellent 9 (18)-limit temperament tempering out 81/80.
{{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 (ku-su-tsu-nu-fu-mu-yu-ki-shi-chi-ni-hi-mi) = 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/こ'''
'''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 '''こ'''
'''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/こb''' み#/こ
'''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
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|か (く#)
|1\48
46; 6, 2
|1\35
63: 6, 3
|2\57
77; 2, 2, 1, 1
| rowspan="2" |1\22
100
|3\53
124; 7, 4
|2\31
141; 5, 1, 2
|3\40
163.{{Overline|63}}
|-
|そ (すb)
|3\48
138; 2, 6
|2\35
126; 3, 6
|3\57
116; 7, 1, 3
|2\53
82; 1, 3, 7
|1\31
70; 1, 1, 2, 3
|1\40
54.{{Overline|54}}
|-
|す
|4\48
184; 1, 1, 1, 1, 2
|3\35
189; 2, 9
|5\57
193; 1, 1, 4, 1, 2
|2\22
200
|5\53
206; 1, 8, 1, 2
|3\31
211; 1, 3, 4
|4\40
218.{{Overline|18}}
|-
|さ (す#)
|5\48
230; 1, 3, 3
|4\35
252; 1, 1, 1, 2, 2
|7\57
270; 1, 30
| rowspan="2" |3\22
300
|8\53
331; 29
|5\31
352; 1, 16
|7\40
381.{{Overline|81}}
|-
|と (つb)
|7\48
323; 13
|5\35
315; 1, 3, 1, 3
|8\57
309; 1, 2, 10
|7\53
289; 1, 1, 1, 9
|4\31
282; 2, 1, 5
|5\40
272.{{Overline|72}}
|-
|つ
|8\48
369; 4, 3
|6\35
378; 1, 18
|10\57
387; 10, 3
|4\22
400
|10\53
413; 1, 3, 1, 5
|6\31
423; 1, 1, 8
|8\40
436.{{Overline|36}}
|-
|た (つ#)
|9\48
415; 2, 1, 1, 2
| rowspan="2" |7\35
442; 9, 2
|12\57
464; 1, 1, 15
|5\22
500
|13\53
537; 13, 2
|8\31
564; 1, 2, 2, 2
|11\40
600
|-
|の (ぬb)
|10\48
461; 1, 1, 6
|11\57
425; 1, 4, 6
|4\22
400
|9\53
372; 2, 2, 2, 2
|5\31
352; 1, 16
|6\40
327.{{Overline|27}}
|-
|'''ぬ'''
|'''11\48'''
'''507; 1, 2, 4'''
|'''8\35'''
'''505; 3, 1, 4'''
|'''13\57'''
'''503; 4, 2, 3'''
|'''5\22'''
'''500'''
|'''12\53'''
'''496; 1, 1, 4, 3'''
|'''7\31'''
'''494; 8, 2'''
|'''9\40'''
'''490.{{Overline|90}}'''
|-
|な (ぬ#)
|12\48
553; 1, 5, 2
|9\35
568; 2, 2, 1, 2
|15\57
580; 1, 1, 4, 2
| rowspan="2" |6\22
600
|15\53
620; 1, 2, 4, 2
|9\31
635; 3, 2, 2
|12\40
654.{{Overline|54}}
|-
|ほ (ふb)
|14\48
646; 6, 2
|10\35
631; 1, 1, 2, 1, 2
|16\57
619; 2, 4, 2
|14\53
579; 9, 1, 2
|8\31
564; 1, 2, 2, 2
|10\40
545.{{Overline|45}}
|-
|ふ
|15\48
692; 3, 4
|11\35
694; 1, 2, 1, 4
|18\57
696; 1, 3, 2, 3
|7\22
700
|17\53
703; 2, 4, 3
|10\31
705; 1, 7, 2
|13\40
709.{{Overline|09}}
|-
|は (ふ#)
|16\48
738; 2, 6
|12\35
757; 1, 8, 2
|20\57
774; 5, 6
| rowspan="2" |8\22
800
|20\53
827; 1, 1, 2, 2, 2
|12\31
847; 17
|16\40
872.{{Overline|72}}
|-
|も (むb)
|18\48
830; 1, 3, 3
|13\35
821; 19
|21\57
812; 1, 9, 3
|19\53
786; 4, 1, 5
|11\31
776; 2, 8
|14\40
763.{{Overline|63}}
|-
|む
|19\48
876; 1, 12
|14\35
884; 4, 1, 3
|23\57
890; 3, 10
|9\22
900
|22\53
910; 2, 1, 9
|13\31
917; 1, 1, 1, 5
|17\40
927.{{Overline|27}}
|-
|よ (む#)
|20\48
923; 13
|15\35
947; 2, 1, 2, 2
|25\57
967; 1, 2, 1, 7
| rowspan="2" |10\22
1000
|25\53
1034; 2, 14
|15\31
1058; 1, 4, 1, 2
|20\40
1090.{{Overline|90}}
|-
|け (ゆb)
|22\48
1015; 2, 2, 2
|16\35
1010; 1, 1, 9
|26\57
1006; 2, 4, 1, 2
|24\53
993: 9, 1, 2
|14\31
988; 4, 4
|18\40
981.{{Overline|81}}
|-
|ゆ
|23\48
1061; 1, 1, 6
|17\35
1073; 1, 2, 6
|28\57
1083; 1, 6, 1, 3
|11\22
1100
|27\53
1127; 4, 7
|16\31
1129; 2, 2, 3
|21\40
1145.{{Overline|45}}
|-
|や (ゆ#)
|24\48
1107; 1, 2, 4
| rowspan="2" |18\35
1136; 1, 5, 3
|30\57
1161; 3, 2, 4
|12\22
1200
|30\29
1241; 2, 1, 1, 1, 3
|15\31
1058; 1, 4, 1, 2
|24\40
1309.{{Overline|09}}
|-
|ま (きb)
|25\48
1153; 1, 5, 2
|29\57
1122; 1, 1, 2, 1, 1
|11\22
1100
|26\53
1075; 1, 6, 4
|18\31
1270; 1, 1, 2, 3
|19\40
1036.{{Overline|36}}
|-
|き
|26\48
1200
|19\35
1200
|31\57
1200
|12\22
1200
|29\53
1200
|17\31
1200
|22\40
1200
|-
|き#
|27\48
1246; 6, 2
|20\35
1263: 6, 3
|33\57
1277; 2, 2, 1, 1
| rowspan="2" |13\22
1300
|32\53
1324; 7, 4
|18\31
1270; 1, 1, 2, 3
|25\40
1363.{{Overline|63}}
|-
|せ (しb)
|29\48
1338; 2, 6
|21\35
1326; 3, 6
|34\57
1316; 7, 1, 3
|31\53
1282; 1, 3, 7
|19\31
1341; 5, 1, 2
|23\40
1254.{{Overline|54}}
|-
|し
|30\48
1384; 1, 1, 1, 1, 2
|22\35
1389; 2, 9
|36\57
1393; 1, 1, 4, 1, 2
|14\22
1400
|34\53
1406; 1, 8, 1, 2
|20\31
1411; 1, 3, 4
|26\40
1418.{{Overline|18}}
|-
|し#
|31\48
1430; 1, 3, 3
|23\35
1452; 1, 1, 1, 2, 2
|38\57
1470; 1, 30
| rowspan="2" |15\22
1500
|37\53
1531; 29
|22\31
1552; 1, 16
|29\40
1581.{{Overline|81}}
|-
|て (ちb)
|33\48
1523; 13
|24\35
1515; 1, 3, 1, 3
|39\57
1509; 1, 2, 10
|36\53
1389; 1, 1, 1, 9
|21\31
1482; 2, 1, 5
|27\40
1472.{{Overline|72}}
|-
|ち
|34\48
1569; 4, 3
|25\35
1578; 1, 18
|41\57
1587; 10, 3
|16\22
1600
|39\53
1613; 1, 3, 1, 5
|23\31
1623; 1, 1, 8
|30\40
1636.{{Overline|36}}
|-
|ち#
|35\48
1615; 2, 2, 2
| rowspan="2" |26\35
1642; 9, 2
|43\57
1664; 1, 1, 15
|17\22
1700
|42\53
1737; 13, 2
|25\31
1764; 1, 2, 2, 2
|33\40
1800
|-
|ね (にb)
|36\48
1661; 1, 1, 6
|42\57
1625; 1, 4, 6
|16\22
1600
|38\53
1572; 2, 2, 2, 2
|22\31
1552; 1, 16
|28\40
1527.{{Overline|27}}
|-
|'''に'''
|'''37\48'''
'''1707; 1, 2, 4'''
|'''27\35'''
'''1705; 3, 1, 4'''
|'''44\57'''
'''1703; 4, 2, 3'''
|'''17\22'''
'''1700'''
|'''41\53'''
'''1696; 1, 1, 4, 3'''
|'''24\31'''
'''1694; 8, 2'''
|'''31\40'''
'''1690.{{Overline|90}}'''
|-
|に#
|38\48
1753; 1, 5, 2
|28\35
1768; 2, 2, 1, 2
|46\57
1780; 1, 1, 4, 2
| rowspan="2" |18\22
1800
|44\53
1820; 1, 2, 4, 2
|26\31
1835; 3, 2, 2
|34\40
1854.{{Overline|54}}
|-
|へ (ひb)
|40\48
1846; 6, 2
|29\35
1831; 1, 1, 2, 1, 2
|47\57
1819; 2, 4, 2
|43\53
1779; 9, 1, 2
|25\31
1764; 1, 2, 2, 2
|32\40
1745.{{Overline|45}}
|-
|ひ
|41\48
1892; 3, 4
|30\35
1894; 1, 2, 1, 4
|49\57
1896; 1, 3, 2, 3
|19\22
1900
|46\53
1903; 2, 4, 3
|27\31
1905; 1, 7, 2
|35\40
1909.{{Overline|09}}
|-
|ひ#
|42\48
1938; 2, 6
|31\35
1957; 1, 8, 2
|51\57
1974; 5, 6
| rowspan="2" |20\22
2000
|49\53
2027; 1, 1, 2, 2, 2
|29\31
2047; 17
|38\40
2072.{{Overline|72}}
|-
|め (みb)
|44\48
2030; 1, 3, 3
|32\35
2021; 19
|52\57
2012; 1, 9, 3
|48\53
1986; 4, 1, 5
|28\31
1976; 2, 8
|36\40
1963.{{Overline|63}}
|-
|み
|45\48
2076; 1, 12
|33\35
2084; 4, 1, 3
|54\57
2090; 3, 10
|21\22
2100
|51\53
2110; 2, 1, 9
|30\31
2117; 1, 1, 1, 5
|39\40
2127.{{Overline|27}}
|-
|み#
|46\48
2123; 13
| rowspan="2" |34\35
2147; 2, 1, 2, 2
|56\57
2167; 1, 2, 1, 7
|22\22
2200
|54\53
2234; 2, 14
|32\31
2258; 1, 4, 1, 2
|42\40
2290.{{Overline|90}}
|-
|こ (くb)
|47\48
2169; 4, 3
|55\57
2129; 31
|21\22
2100
|50\53
2068; 1, 28
|29\31
2047; 17
|37\40
2018.{{Overline|18}}
|-
!く
!48\48
2215; 2, 2, 2
!35\35
2210; 1, 1, 9
!57\57
2206; 2, 4, 1, 2
!22\22
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
!Notation
!Supersoft
!Supersoft
Line 643: Line 36:
|-
|-
|か (く#)
|か (く#)
|1\48
|1\48, 45.18
''45.8{{Overline|3}}''
|1\35, 61.97
|1\35
|2\57, 76.10
''62; 1, 6''
| rowspan="2" |1\22, 98.58
|2\57
|3\53, 122.76
''77; 5, 5, 2''
|2\31, 139.92
| rowspan="2" |1\22
|3\40, 162.66
''100''
|3\53
''124; 1, 1, 8, 3''
|2\31
''141; 1, 14, 2''
|3\40
''165''
|-
|-
|そ (すb)
|そ (すb)
|3\48
|3\48, 135.55
''137.5''
|2\35, 123.93
|2\35
|3\57, 114.15
''125; 1, 2, 2''
|2\53, 81.84
|3\57
|1\31, 69.96
''115; 1, 3, 1, 3''
|1\40, 54.22
|2\53
''83; 53''
|1\31
''70; 1, 30''
|1\40
55
|-
|-
|す
|す
|4\48
|4\48, 180.735
''183.{{Overline|3}}''
|3\35, 185.90
|3\35
|5\57, 190.25
''188; 1, 1, 3''
|2\22, 197.17
|5\57
|5\53, 204.61
''192; 1, 56''
|3\31, 209.89
|2\22
|4\40, 218.88
''200''
|5\53
''207; 1, 1, 4, 1, 4''
|3\31
''212; 1, 9, 3''
|4\40
''220''
|-
|-
|さ (す#)
|さ (す#)
|5\48
|5\48, 225.92
''229.1{{Overline|6}}''
|4\35, 247.87
|4\35
|7\57, 266.35
''251; 2, 3''
| rowspan="2" |3\22, 295.75
|7\57
|8\53, 327.37
''270; 5, 1, 2, 3''
|5\31, 349.81
| rowspan="2" |3\22
|7\40, 379.545
''300''
|8\53
''332; 13, 4''
|5\31
''354; 1, 5, 5''
|7\40
''385''
|-
|-
|と (つb)
|と (つb)
|7\48
|7\48, 316.29
''320.8{{Overline|3}}''
|5\35, 309.83
|5\35
|8\57, 304.40
''314; 3, 2''
|7\53, 286.45
|8\57
|4\31, 279.85
''308; 1, 3, 2, 1, 1, 2''
|5\40, 271.10
|7\53
''290; 1, 1, 3, 3, 2''
|4\31
''283; 1, 6, 1, 3''
|5\40
''275''
|-
|-
|つ
|つ
|8\48
|8\48, 361.47
''366.{{Overline|6}}''
|6\35, 371.80
|6\35
|10\57, 380.495
''377; 7''
|4\22, 394.33
|10\57
|10\53, 409.21
''385; 1, 27, 2''
|6\31, 419.77
|4\22
|8\40, 433.765
''400''
|10\53
''415; 10, 1, 1, 2''
|6\31
''425; 1, 4, 6''
|8\40
''440''
|-
|-
|た (つ#)
|た (つ#)
|9\48
|9\48, 406.655
''412.5''
| rowspan="2" |7\35, 433.765
| rowspan="2" |7\35
|12\57, 456.595
''440''
|5\22, 492.915
|12\57
|13\53, 531.98
''463; 6, 3''
|8\31, 559.70
|5\22
|11\40, 596.43
''500''
|13\53
''539; 1, 1, 1, 1, 1, 6''
|8\31
''567; 1, 2, 1, 7''
|11\40
''605''
|-
|-
|の (ぬb)
|の (ぬb)
|10\48
|10\48, 451.83
''458.{{Overline|3}}''
|11\57, 418.545
|11\57
|4\22, 394.33
''424; 1, 1, 3, 1, 1, 3''
|9\53, 368.29
|4\22
|5\31, 349.81
''400''
|6\40, 325.32
|9\53
''373; 1, 1, 2, 2, 4''
|5\31
''354; 1, 5, 5''
|6\40
''330''
|-
|-
|'''ぬ'''
|'''ぬ'''
|'''11\48'''
|'''11\48,''' '''497.02'''
'''''504.1{{Overline|6}}'''''
|'''8\35, 495.73'''
|'''8\35'''
|'''13\57,''' '''494.645'''
'''''502; 1, 6'''''
|'''5\22, 492.915'''
|'''13\57'''
|'''12\53,''' '''491.055'''
'''''501; 1, 3, 14'''''
|'''7\31,''' '''489.735'''
|'''5\22'''
|'''9\40,''' '''487.99'''
'''''500'''''
|'''12\53'''
'''''498; 8, 1, 5'''''
|'''7\31'''
'''''496; 1, 3, 2, 3'''''
|'''9\40'''
'''''495'''''
|-
|-
|な (ぬ#)
|な (ぬ#)
|12\48
|12\48, 542.21
''550''
|9\35, 557.70
|9\35
|15\57, 570.74
''565; 1, 2, 2''
| rowspan="2" |6\22, 591.50
|15\57
|15\53, 613.82
''578; 1, 18''
|9\31, 629.66
| rowspan="2" |6\22
|12\40, 650.65
''600''
|15\53
''622; 1, 1, 1, 3, 1, 3''
|9\31
''638; 1, 2, 2, 4''
|12\40
''660''
|-
|-
|ほ (ふb)
|ほ (ふb)
|14\48
|14\48, 632.57
''641.{{Overline|6}}''
|10\35, 619.665
|10\35
|16\57, 608.79
''628; 1, 1, 3''
|14\53, 572.90
|16\57
|8\31, 559.70
''617; 1, 1, 5, 5''
|10\40, 542.21
|14\53
''581; 7, 1, 1, 3''
|8\31
''567; 1, 2, 1, 7''
|10\40
''550''
|-
|-
|ふ
|ふ
|15\48
|15\48, 677.76
''687.5''
|11\35, 681.63
|11\35
|18\57, 684.89
''691; 2, 3''
|7\22, 690.08
|18\57
|17\53, 695.66
''694; 1, 2, 1, 4''
|10\31, 699.62
|7\22
|13\40, 704.87
''700''
|17\53
''705; 1, 1, 1, 17''
|10\31
''709; 1, 2, 10''
|13\40
''715''
|-
|-
|は (ふ#)
|は (ふ#)
|16\48
|16\48, 722.94
''733.{{Overline|3}}''
|12\35, 743.60
|12\35
|20\57, 760.99
''754; 3, 2''
| rowspan="2" |8\22, 788.66
|20\57
|20\53, 818.425
''771; 1, 13, 4''
|12\31, 839.55
| rowspan="2" |8\22
|16\40, 867.53
''800''
|20\53
''830; 5, 3, 3''
|12\31
''851; 1, 1, 1, 1, 2, 2''
|16\40
''880''
|-
|-
|も (むb)
|も (むb)
|18\48
|18\48, 813.31
''825''
|13\35, 805.56
|13\35
|21\57, 799.04
''817; 7''
|19\53, 777.50
|21\57
|11\31, 769.58
''810; 1, 1, 9''
|14\40, 759.09
|19\53
''788; 1, 2, 8, 2''
|11\31
''780; 1, 1, 1, 4, 2''
|14\40
''770''
|-
|-
|む
|む
|19\48
|19\48, 850.49
''870.8{{Overline|3}}''
|14\35, 867.53
|14\35
|23\57, 875.14
''880''
|9\22, 887.25
|23\57
|22\53, 900.27
''887; 1, 2, 1, 1, 3, 2''
|13\31, 909.51
|9\22
|17\40, 921.75
''900''
|22\53
''913; 4, 1, 4, 2''
|13\31
''922; 1, 1, 2, 1, 1, 2''
|17\40
''935''
|-
|-
|よ (む#)
|よ (む#)
|20\48
|20\48, 903.68
''916.{{Overline|6}}''
|15\35, 929.50
|15\35
|25\57, 951.24
''942; 1, 6''
| rowspan="2" |10\22, 985.83
|25\57
|25\53, 1023.03
''964; 1, 10, 3, 2''
|15\31, 1049.43
| rowspan="2" |10\22
|20\40, 1084.41
''1000''
|25\53
''1037; 1, 2, 1, 3, 1, 2''
|15\31
''1064; 1, 1, 15''
|20\40
''1100''
|-
|-
|け (ゆb)
|け (ゆb)
|22\48
|22\48, 994.045
''1013.{{Overline|3}}''
|16\35, 991.46
|16\35
|26\57, 989.29
''1005; 1, 2, 2''
|24\53, 982.11
|26\57
|14\31, 979.47
''1003; 1, 1, 28''
|18\40, 975.97
|24\53
''996: 4, 2, 2, 2''
|14\31
''993; 1, 1, 4, 1, 2''
|18\40
''990''
|-
|-
|ゆ
|ゆ
|23\48
|23\48, 1039.23
''1059.1{{Overline|6}}''
|17\35, 1053.43
|17\35
|28\57, 1065.39
''1068; 1, 1, 3''
|11\22, 1084.41
|28\57
|27\53, 1104.87
''1080; 1, 2, 2, 1, 5''
|16\31, 1119.39
|11\22
|21\40, 1138.63
''1100''
|27\53
''1130; 1, 3, 13''
|16\31
''1135; 2, 15''
|21\40
''1155''
|-
|-
|や (ゆ#)
|や (ゆ#)
|24\48
|24\48, 1084.41
''1100''
| rowspan="2" |18\35, 1115.40
| rowspan="2" |18\35
|30\57, 1141.49
''1131; 2, 3''
|12\22, 1183.00
|30\57
|30\29, 1227.63
''1157; 1, 8, 2''
|15\31, 1049.43
|12\22
|24\40, 1301.30
''1200''
|30\29
''1245; 3, 1, 1, 7''
|18\31
''1277; 2, 2, 1, 1, 2''
|24\40
''1320''
|-
|-
|ま (きb)
|ま (きb)
|25\48
|25\48, 1129.60
''1145.8{{Overline|3}}''
|29\57, 1103.44
|29\57
|11\22, 1084.41
''1119; 3, 2, 1, 5''
|26\53, 1063.95
|11\22
|18\31, 1259.32
''1100''
|19\40, 1030.19
|26\53
''1079; 4, 13''
|15\31
''1064; 1, 1, 15''
|19\40
''1045''
|-
|-
|き
|き
|26\48
|26\48, 1174.78
''1191.{{Overline|6}}''
|19\35, 1177.36
|19\35
|31\57, 1179.54
''1194: 3, 2''
|12\22, 1183.00
|31\57
|29\53, 1186.72
''1196; 2, 28''
|17\31, 1189.36
|12\22
|22\40, 1192.85
''1200''
|29\53
''1203; 1, 3, 2, 2, 2''
|17\31
''1206; 2, 3, 1, 2''
|22\40
''1210''
|-
|-
|き#
|き#
|27\48
|27\48, 1219.965
''1237.5''
|20\35, 1239.32
|20\35
|33\57, 1255.64
''1257: 7''
| rowspan="2" |13\22, 1281.58
|33\57
|32\53, 1309.48
''1273; 1, 2, 6''
|18\31, 1259.32
| rowspan="2" |13\22
|25\40, 1355.52
''1300''
|32\53
''1328; 3, 3, 5''
|19\31
''1348; 2, 1, 1, 2, 2''
|25\40
''1375''
|-
|-
|せ (しb)
|せ (しb)
|29\48
|29\48, 1310.33
''1329.1{{Overline|6}}''
|21\35, 1301.29
|21\35
|34\57, 1293.69
''1320''
|31\53, 1268.56
|34\57
|19\31, 1329.28
''1312; 3, 1, 1, 3, 2''
|23\40, 1247.075
|31\53
''1286; 1, 3, 1, 1, 2''
|18\31
''1277; 2, 2, 1, 1, 2''
|23\40
''1265''
|-
|-
|し
|し
|30\48
|30\48, 1355.52
''1375''
|22\35, 1363.26
|22\35
|36\57, 1369.785
''1382; 1, 6''
|14\22, 1380.16
|36\57
|34\53, 1391.32
''1389; 2, 9''
|20\31, 1399.24
|14\22
|26\40, 1409.74
''1400''
|34\53
''1411; 3, 8, 2''
|20\31
''1419; 2, 4, 2''
|26\40
''1430''
|-
|-
|し#
|し#
|31\48
|31\48, 1400.70
''1420.8{{Overline|3}}''
|23\35, 1425.23
|23\35
|38\57, 1445.88
''1445; 1, 2, 2''
| rowspan="2" |15\22, 1478.745
|38\57
|37\53, 1514.09
''1466.{{Overline|6}}''
|22\31, 1539.17
| rowspan="2" |15\22
|29\40, 1572.40
''1500''
|37\53
''1535; 1, 5, 1, 1, 1, 2''
|22\31
''1561; 3, 2, 4''
|29\40
''1595''
|-
|-
|て (ちb)
|て (ちb)
|33\48
|33\48, 1491.07
''1512.5''
|24\35, 1487.195
|24\35
|39\57, 1483.93
''1508; 1, 1, 3''
|36\53, 1473.165
|39\57
|21\31, 1469.205
''1505; 3, 1, 4''
|27\40, 1463.96
|36\53
''1394; 2, 1, 17''
|21\31
''1490; 3, 10''
|27\40
''1485''
|-
|-
|ち
|ち
|34\48
|34\48, 1536.25
''1558.{{Overline|3}}''
|25\35, 1549.16
|25\35
|41\57, 1560.03
''1571; 2, 3''
|16\22, 1577.33
|41\57
|39\53, 1595.93
''1582; 2, 5, 5''
|23\31, 1609.13
|16\22
|30\40, 1626.62
''1600''
|39\53
''1618; 1, 6, 1, 1, 3''
|23\31
''1632; 3, 1, 7''
|30\40
''1650''
|-
|-
|ち#
|ち#
|35\48
|35\48, 1581.44
''1604.1{{Overline|6}}''
| rowspan="2" |26\35, 1611.13
| rowspan="2" |26\35
|43\57, 1636.13
''1633; 3, 2''
|17\22, 1675.91
|43\57
|42\53, 1718.69
''1659; 1, 1, 1, 5, 1, 3''
|25\31, 1749.05
|17\22
|33\40, 1789.28
''1700''
|42\53
1737; 13, 2
|25\31
''1774; 5, 6''
|33\40
''1815''
|-
|-
|ね (にb)
|ね (にb)
|36\48
|36\48, 1626.62
''1650''
|42\57, 1598.08
|42\57
|16\22, 1577.33
''1621; 19''
|38\53, 1555.01
|16\22
|22\31, 1539.17
''1600''
|28\40, 1518.18
|38\53
''1578; 2, 1, 3, 1, 3''
|22\31
''1561; 3, 2, 4''
|28\40
''1540''
|-
|-
|'''に'''
|'''に'''
|'''37\48'''
|'''37\48,''' '''1671.80'''
'''''1695.8{{Overline|3}}'''''
|'''27\35,''' '''1673.095'''
|'''27\35'''
|'''44\57,''' '''1674.18'''
'''''1696; 7'''''
|'''17\22, 1675.91'''
|'''44\57'''
|'''41\53,''' '''1677.77'''
'''''1699; 4, 14'''''
|'''24\31,''' '''1679.09'''
|'''17\22'''
|'''31\40,''' '''1680.84'''
'''''1700'''''
|'''41\53'''
'''''1701; 1, 7, 1, 5'''''
|'''24\31'''
'''''1703; 4, 2, 3'''''
|'''31\40'''
'''''1705'''''
|-
|-
|に#
|に#
|38\48
|38\48, 1716.99
''1741.{{Overline|6}}''
|28\35, 1735.06
|28\35
|46\57, 1750.28
''1760''
| rowspan="2" |18\22, 1774.49
|46\57
|44\53, 1800.535
''1775; 2, 3, 1, 1, 3''
|26\31, 1819.015
| rowspan="2" |18\22
|34\40, 1843.50
''1800''
|44\53
''1826; 2, 2,, 2, 4''
|26\31
''1845; 5, 6''
|34\40
''1870''
|-
|-
|へ (ひb)
|へ (ひb)
|40\48
|40\48, 1807.355
''1833.{{Overline|3}}''
|29\35, 1797.03
|29\35
|47\57, 1788.33
''1822; 1, 6''
|43\53, 1759.61
|47\57
|25\31, 1749.05
''1814; 28, 2''
|32\40, 1735.06
|43\53
''1784; 1, 9, 1, 1, 2''
|25\31
''1774; 6, 5''
|32\40
''1760''
|-
|-
|ひ
|ひ
|41\48
|41\48, 1852.54
''1879.1{{Overline|6}}''
|30\35, 1858.99
|30\35
|49\57, 1864.43
''1885; 1, 2, 2''
|19\22, 1873.08
|49\57
|46\53, 1882.37
''1891; 4, 2, 1, 1, 2''
|27\31, 1888.98
|19\22
|35\40, 1897.72
''1900''
|46\53
''1909; 2, 3, 3, 2''
|27\31
''1916; 7, 1, 3''
|35\40
''1925''
|-
|-
|ひ#
|ひ#
|42\48
|42\48, 1897.72
''1925''
|31\35, 1920.96
|31\35
|51\57, 1940.53
''1948; 1, 1, 3''
| rowspan="2" |20\22, 1971.66
|51\57
|49\53, 2005.14
''1968; 2, 2, 1, 2''
|29\31, 2028.90
| rowspan="2" |20\22
|38\40, 2060.385
''2000''
|49\53
''2033; 1, 25, 2''
|29\31
''2058; 15, 2''
|38\40
''2090''
|-
|-
|め (みb)
|め (みb)
|44\48
|44\48, 1988.09
''2016.{{Overline|6}}''
|32\35, 1982.93
|32\35
|52\57, 1978.58
''2011; 2, 3''
|48\53, 1964.22
|52\57
|28\31, 1958.94
''2007; 57''
|36\40, 1951.94
|48\53
''1992; 2, 4, 1, 4''
|28\31
''1987; 10, 3''
|36\40
''1980''
|-
|-
|み
|み
|45\48
|45\48, 2033.27
''2062.5''
|33\35, 2044.89
|33\35
|54\57, 2054.68
''2075; 3, 2''
|21\22, 2070.24
|54\57
|51\53, 2086.98
2084; 4, 1, 3
|30\31, 2098.86
|21\22
|39\40, 2114.605
''2100''
|51\53
''2116; 1, 52''
|30\31
''2129; 31''
|39\40
''2145''
|-
|-
|み#
|み#
|46\48
|46\48, 2078.46
''2108.{{Overline|3}}''
| rowspan="2" |34\35, 2106.86
| rowspan="2" |34\35
|56\57, 2130.78
''2138; 7''
|22\22, 2168.83
|56\57
|54\53, 2209.75
''2161; 2, 2, 11''
|32\31, 2238.79
|22\22
|42\40, 2223.05
''2200''
|54\53
''2241; 1, 1, 26''
|32\31
''2270; 1, 30''
|42\40
''2310''
|-
|-
|こ (くb)
|こ (くb)
|47\48
|47\48, 2123.64
''2154.1{{Overline|6}}''
|55\57, 2092.73
|55\57
|21\22, 2070.24
''2122; 1, 4, 5, 2''
|50\53, 2046.06
|21\22
|29\31, 2028.90
''2100''
|37\40, 2006.16
|50\53
''2075; 2, 8, 3''
|29\31
''2058; 15, 2''
|37\40
''2035''
|-
|-
!く
!く
! colspan="7" |''2200''
!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 ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
Line 1,235: 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 1,331: Line 476:
|diminished eleventh  
|diminished eleventh  
|-
|-
| 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
| 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 1,611: Line 756:
|M
|M
|-
|-
|Aeolian Dominant ♮6
|Aeolian Dominant ♮6
|LLsLLsLLsLLsL
|LLsLLsLLsLLsL
|<nowiki>8|4</nowiki>
|<nowiki>8|4</nowiki>
Line 1,659: Line 804:
|m
|m
|-
|-
|Phrygian ♮2
|Phrygian ♮2
|LsLLsLLsLLLsL
|LsLLsLLsLLLsL
|<nowiki>5|7</nowiki>
|<nowiki>5|7</nowiki>
Line 1,675: Line 820:
|m
|m
|-
|-
|Locrian ♮2 ♮5
|Locrian ♮2 ♮5
|LsLLsLLsLLsLL
|LsLLsLLsLLsLL
|<nowiki>4|8</nowiki>
|<nowiki>4|8</nowiki>
Line 1,688: Line 833:
|m
|m
|P
|P
|P
 
|d
|m
|m
|-
|-
|Locrian ♮5
|Locrian ♮5
|sLLLsLLsLLsLL
|sLLLsLLsLLsLL
|<nowiki>3|9</nowiki>
|<nowiki>3|9</nowiki>
Line 1,704: Line 850:
|m
|m
|P
|P
|P
|d
|m
|m
|-
|-
|Locrian  
|Locrian
|sLLsLLLsLLsLL
|sLLsLLLsLLsLL
|<nowiki>2|10</nowiki>
|<nowiki>2|10</nowiki>
Line 1,720: Line 866:
|m
|m
|P
|P
|P
|d
|m
|m
|-
|-
Line 1,756: Line 902:
|}
|}


== Temperaments ==
=== Cyclic Permutation Order ===
The '''Shōsūshī''' temperament spells its major hexad 24:30:36:45:54:64 or 84:108:126:162:189:224 <code>root-2(4g-3p)-(3g-2p)-(11g-8p)-2(3g-2p)-(1g)</code> (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 <code>root-(-5g-p)-(3g-2p)-(-2g)-2(3g-2p)-(1g)</code> (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]]: [{{Val|1 1 2}}, {{Val|0 -2 -5}}]
 
[[Vals]]: [[22edXIV]], [[35edXIV]], [[48edXIV]], [[61edXIV]]
 
=== 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]]: [[22edXIV]], [[31edXIV]], [[40edXIV]], [[49edXIV]]
 
== Scale tree ==
The spectrum looks like this:
{| class="wikitable"
{| class="wikitable"
|+
|+
! colspan="3" |Generator
!Mode
!<u>Normalized<ref name=":0" /></u>
!Scale
!''ed22\12 (→ed11\6)<ref name=":0" />''
![[Modal UDP Notation|UDP]]
!L
! colspan="12" |Interval type
!s
!L/s
!Comments
|-
|-
|3\13
!name
|
!pattern
|
!notation
|<u>514; 3, 2</u>
!2nd
|''507; 1, 3, 3''
!3rd
|1
!4th
|1
!5th
|1.000
!6th
|Equalised
!7th
!8th
!9th
!10th
!11th
!12th
!13th
|-
|-
|
|Mixolydian b10 (Persian)
|17\74
|LLsLLsLLsLLLs
|
|<nowiki>9|3</nowiki>
|<u>510</u>
|M
|''505; 2, 2, 7''
|M
|6
|P
|5
|P
|1.200
|M
|
|m
|P
|M
|m
|P
|P
|M
|-
|-
|
|Phrygian ♮2
|14\61
|LsLLsLLsLLLsL
|
|<nowiki>5|7</nowiki>
|<u>509.{{Overline|09}}</u>
|M
|''504; 1, 11, 5''
|m
|5
|P
|4
|P
|1.250
|m
|
|m
|P
|m
|m
|P
|P
|m
|-
|-
|
|Locrian b8
|
|sLLsLLsLLLsLL
|25\109
|<nowiki>1|11</nowiki>
|<u>508; 2, 9, 3</u>
|m
|''504; 1, 1, 2, 2, 1, 2, 2''
|m
|9
|P
|7
|d
|1.286
|m
|
|m
|d
|m
|m
|P
|d
|m
|-
|-
|
|Mixolydian
|11\48
|LLsLLsLLLsLLs
|
|<nowiki>10|2</nowiki>
|<u>507; 1, 3, 3</u>
|M
|''504.1{{Overline|6}}''
|M
|4
|P
|3
|P
|1.333
|M
|
|m
|P
|M
|M
|P
|P
|M
|-
|-
|
|Minor
|
|LsLLsLLLsLLsL
|19\83
|<nowiki>6|6</nowiki>
|<u>506.{{Overline|6}}</u>
|M
|''503; 1, 1, 1, 1, 2, 6''
|m
|7
|P
|5
|P
|1.400
|m
|
|m
|P
|M
|m
|P
|P
|m
|-
|-
|
|Locrian
|8\35
|sLLsLLLsLLsLL
|
|<nowiki>2|10</nowiki>
|<u>505; 3, 1, 4</u>
|m
|''502; 1, 6''
|m
|3
|P
|2
|d
|1.500
|m
|Shōsūshī-Meantone starts here
|m
|P
|m
|m
|P
|d
|m
|-
|-
|
|Major b14
|
|LLsLLLsLLsLLs
|21\92
|<nowiki>11|1</nowiki>
|<u>504</u>
|M
|''502; 5, 1, 3''
|M
|8
|P
|5
|P
|1.600
|M
|
|M
|P
|M
|M
|P
|P
|M
|-
|-
|
|Dorian b13
|
|LsLLLsLLsLLsL
|13\57
|<nowiki>7|5</nowiki>
|<u>503; 4, 2, 3</u>
|M
|''501; 1, 3, 14''
|m
|5
|P
|3
|P
|1.667
|M
|
|m
|P
|M
|m
|P
|P
|m
|-
|-
|
|Locrian ♮5
|
|sLLLsLLsLLsLL
|18\79
|<nowiki>3|9</nowiki>
|<u>502; 3, 14</u>
|m
|''501; 3, 1, 3, 5''
|m
|7
|P
|4
|P
|1.750
|m
|
|m
|P
|m
|m
|P
|d
|m
|-
|-
|
|Lydian Dominant ♮11
|
|LLLsLLsLLsLLs
|23\101
|<nowiki>12|0</nowiki>
|<u>501.{{Overline|81}}</u>
|M
|''500.{{Overline|0099}}''
|M
|9
|A
|5
|P
|1.800
|M
|
|M
|P
|M
|M
|P
|P
|M
|-
|-
|
|Aeolian Dominant ♮6
|5\22
|LLsLLsLLsLLsL
|
|<nowiki>8|4</nowiki>
|<u>500</u>
|M
|''500''
|M
|2
|P
|1
|P
|2.000
|M
|Shōsūshī-Meantone ends, Shōsūshī-Pythagorean begins
|m
|P
|M
|m
|P
|P
|m
|-
|-
|
|Locrian ♮2 ♮5
|
|LsLLsLLsLLsLL
|27\119
|<nowiki>4|8</nowiki>
|<u>498; 2, 6</u>
|M
|''499; 6, 3, 1, 4''
|m
|11
|P
|5
|P
|2.200
|m
|
|m
|P
|m
|m
|P
|d
|m
|-
|-
|
|Locrian b8 b11
|
|sLLsLLsLLsLLL
|22\97
|<nowiki>0|12</nowiki>
|<u>498; 8, 1, 5</u>
|m
|''498; 1, 31, 3''
|m
|9
|P
|4
|d
|2.250
|m
|
|m
|-
|d
|
|m
|
|m
|17\75
|d
|<u>497; 1, 1, 3, 1, 1, 2</u>
|d
|''498.{{Overline|6}}''
|m
|7
|3
|2.333
|
|-
|
|
|12\53
|<u>496; 1, 1, 4, 3</u>
|''498; 8, 1, 5''
|5
|2
|2.500
|Shōsūshī-Neogothic heartland is from here…
|-
|
|
|19\84
|<u>495; 1, 1, 1, 7</u>
|''497; 1, 1, 1, 1, 1, 2''
|8
|3
|2.667
|…to here
|-
|
|7\31
|
|<u>494; 8, 2</u>
|''496; 1, 3, 2, 3''
|3
|1
|3.000
|Shōsūshī-Pythagorean ends, Shōsūshī-Superpyth begins
|-
|
|
|30\133
|<u>493; 6, 1, 1, 1, 3</u>
|''496; 4, 6, 2, 2''
|13
|4
|3.250
|
|-
|
|
|23\102
|<u>492.8{{Overline|3}}</u>
|''496; 12, 1, 3''
|10
|3
|3.333
|
|-
|
|
|16\71
|<u>492; 3, 4</u>
|''495; 1, 3, 2, 3, 2''
|7
|2
|3.500
|
|-
|
|
|25\111
|<u>491; 1, 4, 12</u>
|''495; 2, 55''
|11
|3
|3.667
|
|-
|
|9\40
|
|<u>490.{{Overline|90}}</u>
|''495''
|4
|1
|4.000
|
|-
|
|
|20\89
|<u>489; 1, 3, 1, 9</u>
|''494; 2, 1, 1, 1, 1, 1, 1, 2''
|9
|2
|4.500
|
|-
|
|11\49
|
|<u>488.{{Overline|8}}</u>
|''493; 1, 7, 6''
|5
|1
|5.000
|Shōsūshī-Superpyth ends
|-
|
|13\58
|
|<u>487.5</u>
|''493; 9, 1, 2''
|6
|1
|6.000
|
|-
|2\9
|
|
|<u>480</u>
|''488.{{Overline|8}}''
|1
|0
|→ inf
|Paucitonic
|}
|}
== 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)"