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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 fan” Japanese value.
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
{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|か (く#)
|1\48, 45.18
|1\35, 61.97
|2\57, 76.10
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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
| rowspan="2" |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 ==
{| class="wikitable"
|+
!Generators
!Fourteenth notation
!Interval category name
!Generators
!Notation of 7/2 inverse
!Interval category name
|-
| colspan="6" |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
|-
| 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
|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:
{| 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)"