30edo

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The 30 equal division divides the octave into 30 equal steps of precisely 40 cents each. Its patent val is a doubled version of the patent val for 15edo through the 11-limit, so 30 can be viewed as a contorted version of 15. In the 13-limit it supplies the optimal patent val for quindecic temperament.

However, 5\30 is 200 cents, which is a good (and familiar) approximation for 9/8, and hence 30 can be viewed inconsistently, as having a 9' at 95\30 as well as a 9 at 96\30. Instead of the 18\30 fifth of 720 cents, 30 also makes available a 17\30 fifth of 680 cents. This is an ideal tuning for pelogic (5-limit mavila), which tempers out 135/128. When 30 is used for pelogic, 5\30 can again be used inconsistently as a 9/8.

Below is a plot of the Z function around 30:

plot30.png

Intervals

Step Cents pions 7mus ups and downs notation
0 ¢ 0 P1 unison, minor 2nd D, Eb
1 40 42.4 51.2 (33.316) ^1, ^m2 up unison, upminor 2nd D^, Eb^
2 80 84.8 102.4 (66.616) ^^1, v~2 double-up unison, downmid 2nd D^^, Eb^^
3 120 127.2 153.6 (99.A16) ~2 mid 2nd Ev3
4 160 169.6 204.8 (CC.D16) ^~2 upmid 2nd Evv
5 200 212 256 (10016) vM2 downmajor 2nd Ev
6 240 254.4 307.2 (133.316) M2, m3 major 2nd, minor 3rd E, F
7 280 296.8 358.4 (166.616) ^m3 upminor 3rd F^
8 320 339.2 409.6 (199.A16) v~3 downmid 3rd F^^
9 360 381.6 460.8 (1CC.D16) ~3 mid 3rd F^3
10 400 424 512 (20016) ^~3 upmid 3rd F#vv
11 440 466.4 563.2 (233.316) vM3, v4 downmajor 3rd, down 4th F#v, Gv
12 480 508.8 614.4 (266.616) P4 major 3rd, perfect 4th F#, G
13 520 551.2 665.6 (299.A16) ^4, ^d5 up 4th, updim 5th G^, Ab^
14 560 593.6 716.8 (2CC.D16) ^^4, ^^d5 double-up 4th, double-up dim 5th G^^, Ab^^
15 600 636 768 (30016) ^34, v35 triple-up 4th, triple-down 5th G^3, Av3
16 640 677.4 819.2 (333.316) vvA4, vv5 double-down aug 4th, double-down 5th G#vv, Avv
17 680 720.8 870.4 (366.616) vA4, v5 downaug 4th, down 5th G#v, Av
18 720 763.2 921.6 (399.A16) P5 perfect 5th, minor 6th A, Bb
19 760 805.6 972.8 (3CC.D16) ^5, ^m6 up 5th, upminor 6th A^, Bb^
20 800 848 1024 (40016) v~6 downmid 6th Bb^^
21 840 890.4 1095.2 (433.316) ~6 mid 6th Bv3
22 880 932.8 1126.4 (466.616) ^~6 upmid 6th Bvv
23 920 975.2 1177.6 (499.A16) vM6 downmajor 6th Bv
24 960 1017.6 1228.8 (4CC.D16) M6. m7 major 6th, minor 7th B, C
25 1000 1260 1280 (50016) ^m7 upminor 7th C^
26 1040 1102.4 1331.2 (533.316) v~7 downmid 7th C^^
27 1080 1144.8 1382.4 (566.616) ~7 mid 7th C^3
28 1120 1187.2 1433.6 (599.A16) ^~7, vv8 upmid 7th, double-down 8ve C#vv, Dvv
29 1160 1227.6 1484.8 (5CC.D16) vM7, v8 downmajor 7th, down 8ve C#v, Dv
30 1200 1272 1536 (60016) P8 major 7th, 8ve C#, D

Commas

30 EDO tempers out the following commas. (Note: This assumes the val < 30 48 70 84 104 111 | .)

Comma Monzo Value (Cents) Name 1 Name 2 Name 3
256/243 | 8 -5 > 90.22 Limma Pythagorean Minor 2nd
250/243 | 1 -5 3 > 49.17 Maximal Diesis Porcupine Comma
128/125 | 7 0 -3 > 41.06 Diesis Augmented Comma
15625/15552 | -6 -5 6 > 8.11 Kleisma Semicomma Majeur
1029/1000 | -3 1 -3 3 > 49.49 Keega
49/48 | -4 -1 0 2 > 35.70 Slendro Diesis
64/63 | 6 -2 0 -1 > 27.26 Septimal Comma Archytas' Comma Leipziger Komma
64827/64000 | -9 3 -3 4 > 22.23 Squalentine
875/864 | -5 -3 3 1 > 21.90 Keema
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
4000/3969 | 5 -4 3 -2 > 13.47 Octagar
1029/1024 | -10 1 0 3 > 8.43 Gamelisma
6144/6125 | 11 1 -3 -2 > 5.36 Porwell
250047/250000 | -4 6 -6 3 > 0.33 Landscape Comma
100/99 | 2 -2 2 0 -1 > 17.40 Ptolemisma
121/120 | -3 -1 -1 0 2 > 14.37 Biyatisma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
65536/65219 | 16 0 0 -2 -3 > 8.39 Orgonisma
385/384 | -7 -1 1 1 1 > 4.50 Keenanisma
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Wizardharry
3025/3024 | -4 -3 2 -1 2 > 0.57 Lehmerisma

Music

Fifteen Short Pieces by Todd Harrop