30edo

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Theory

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 ups and downs notation
0 P1 unison, minor 2nd D, Eb
1 40 ^1, ^m2 up unison, upminor 2nd ^D, ^Eb
2 80 ^^1, v~2 double-up unison, downmid 2nd ^^D, ^^Eb
3 120 ~2 mid 2nd v3E
4 160 ^~2 upmid 2nd vvE
5 200 vM2 downmajor 2nd vE
6 240 M2, m3 major 2nd, minor 3rd E, F
7 280 ^m3 upminor 3rd ^F
8 320 v~3 downmid 3rd ^^F
9 360 ~3 mid 3rd ^3F, v3F#
10 400 ^~3 upmid 3rd vvF#
11 440 vM3, v4 downmajor 3rd, down 4th vF#, vG
12 480 P4 major 3rd, perfect 4th F#, G
13 520 ^4, ^d5 up 4th, updim 5th ^G, ^Ab
14 560 v~4, v~d5 downmid 4th, downmid 5th ^^G, ^^Ab
15 600 ~4, ~5 mid 4th, mid 5th ^3G, v3A
16 640 ^~A4, ^~5 upmid 4th, upmid 5th vvG#, vvA
17 680 vA4, v5 downaug 4th, down 5th vG#, vA
18 720 P5 perfect 5th, minor 6th A, Bb
19 760 ^5, ^m6 up 5th, upminor 6th ^A, ^Bb
20 800 v~6 downmid 6th ^^Bb
21 840 ~6 mid 6th v3B
22 880 ^~6 upmid 6th vvB
23 920 vM6 downmajor 6th vB
24 960 M6. m7 major 6th, minor 7th B, C
25 1000 ^m7 upminor 7th ^C
26 1040 v~7 downmid 7th ^^C
27 1080 ~7 mid 7th ^3C
28 1120 ^~7, vv8 upmid 7th, double-down 8ve vvC#, vvD
29 1160 vM7, v8 downmajor 7th, down 8ve vC#, vD
30 1200 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 | .)

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

Music

Fifteen Short Pieces by Todd Harrop