User:Moremajorthanmajor/9L 4s (7/2-equivalent)
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 = 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/け
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 み
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/けb み#/け
| 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 |
1\22
100 |
3\53
124; 7, 4 |
2\31
141; 5, 1, 2 |
3\40
163.63 |
| そ (すb) | 3\48
138; 3, 4 |
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
54.54 | |
| 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
218.18 |
| さ (す#) | 5\48
230; 1, 3, 3 |
4\35
252; 1, 1, 1, 2, 2 |
7\57
270; 1, 30 |
3\22
300 |
8\53
331; 29 |
5\31
352; 1, 16 |
7\40
381.81 |
| 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
272.72 | |
| 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
436.36 |
| た (つ#) | 9\48
415; 2, 2, 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
327.27 | |
| ぬ | 11\48
507; 1, 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
490.90 |
| な (ぬ#) | 12\48
553; 1, 5, 2 |
9\35
568; 2, 2, 1, 2 |
15\57
580; 1, 1, 4, 2 |
6\22
600 |
15\53
620; 1, 2, 4, 2 |
9\31
635; 3, 2, 2 |
12\40
654.54 |
| 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
545.45 | |
| 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
709.09 |
| は (ふ#) | 16\48
738; 3, 4 |
12\35
757; 1, 8, 2 |
20\57
774; 5, 6 |
8\22
800 |
20\53
827; 1, 1, 2, 2, 2 |
12\31
847; 17 |
16\40
872.72 |
| 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
763.63 | |
| 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
927.27 |
| よ (む#) | 20\48
923; 13 |
15\35
947; 2, 1, 2, 2 |
25\57
967; 1, 2, 1, 7 |
10\22
1000 |
25\53
1034; 2, 14 |
15\31
1058; 1, 4, 1, 2 |
20\40
1090.90 |
| 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
981.81 | |
| 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
1145.45 |
| や (ゆ#) | 24\48
1107; 1, 1, 2, 4 |
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
1309.09 |
| 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
1036.36 | |
| 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 |
13\22
1300 |
32\53
1324; 7, 4 |
18\31
1270; 1, 1, 2, 3 |
25\40
1363.63 |
| せ (しb) | 29\48
1338; 3, 4 |
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
1254.54 | |
| 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
1418.18 |
| し# | 31\48
1430; 1, 3, 3 |
23\35
1452; 1, 1, 1, 2, 2 |
38\57
1470; 1, 30 |
15\22
1500 |
37\53
1531; 29 |
22\31
1552; 1, 16 |
29\40
1581.81 |
| 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
1472.72 | |
| 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
1636.36 |
| ち# | 35\48
1615; 2, 2, 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
1527.27 | |
| に | 37\48
1707; 1, 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
1690.90 |
| に# | 38\48
1753; 1, 5, 2 |
28\35
1768; 2, 2, 1, 2 |
46\57
1780; 1, 1, 4, 2 |
18\22
1800 |
44\53
1820; 1, 2, 4, 2 |
26\31
1835; 3, 2, 2 |
34\40
1854.54 |
| 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
1745.45 | |
| 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
1909.09 |
| ひ# | 42\48
1938; 3, 4 |
31\35
1957; 1, 8, 2 |
51\57
1974; 5, 6 |
20\22
2000 |
49\53
2027; 1, 1, 2, 2, 2 |
29\31
2047; 17 |
38\40
2072.72 |
| 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
1963.63 | |
| 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
2127.27 |
| み# | 46\48
2123; 13 |
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
2290.90 |
| け (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
2018.18 | |
| 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
2181.81 |
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard |
|---|---|---|---|---|---|---|---|
| か (く#) | 1\48
45.83 |
1\35
62; 1, 6 |
2\57
77; 5, 5, 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
183.3 |
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
229.16 |
4\35
251; 2, 3 |
7\57
270; 5, 1, 2, 3 |
3\22
300 |
8\53
332; 13, 4 |
5\31
354; 1, 5, 5 |
7\40
385 |
| と (つb) | 7\48
320.83 |
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
366.6 |
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 |
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
458.3 |
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
504.16 |
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 |
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
641.6 |
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
733.3 |
12\35
754; 3, 2 |
20\57
771; 1, 13, 4 |
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
870.83 |
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
916.6 |
15\35
942; 1, 6 |
25\57
964; 1, 10, 3, 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
1013.3 |
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
1059.16 |
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 |
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
1145.83 |
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
1191.6 |
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 |
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
1329.16 |
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
1420.83 |
23\35
1445; 1, 2, 2 |
38\57
1466.6 |
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
1558.3 |
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
1604.16 |
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
1695.83 |
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
1741.6 |
28\35
1760 |
46\57
1775; 2, 3, 1, 1, 3 |
18\22
1800 |
44\53
1826; 2, 2,, 2, 4 |
26\31
1845; 5, 6 |
34\40
1870 |
| へ (ひb) | 40\48
1833.3 |
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
1879.16 |
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 |
20\22
2000 |
49\53
2033; 1, 25, 2 |
29\31
2058; 15, 2 |
38\40
2090 |
| め (みb) | 44\48
2016.6 |
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
2108.3 |
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
2154.16 |
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 | 2200 | ||||||
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 |
| 13 | か (く#) | augmented unison | -13 | け (くb) | diminished fourteenth |
| The chromatic 22-note MOS (either 13L 9s, 9L 13s or 22edXIV) also has these intervals | |||||
| 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 |