30edo
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:
Intervals
| Step | Cents | ups and downs notation | ||
|---|---|---|---|---|
| 0 | 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 | Ev3 |
| 4 | 160 | ^~2 | upmid 2nd | Evv |
| 5 | 200 | vM2 | downmajor 2nd | Ev |
| 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 | F^3 |
| 10 | 400 | ^~3 | upmid 3rd | F#vv |
| 11 | 440 | vM3, v4 | downmajor 3rd, down 4th | F#v, Gv |
| 12 | 480 | P4 | major 3rd, perfect 4th | F#, G |
| 13 | 520 | ^4, ^d5 | up 4th, updim 5th | G^, Ab^ |
| 14 | 560 | ^^4, ^^d5 | double-up 4th, double-up dim 5th | G^^, Ab^^ |
| 15 | 600 | ^34, v35 | triple-up 4th, triple-down 5th | G^3, Av3 |
| 16 | 640 | vvA4, vv5 | double-down aug 4th, double-down 5th | G#vv, Avv |
| 17 | 680 | vA4, v5 | downaug 4th, down 5th | G#v, Av |
| 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 | Bv3 |
| 22 | 880 | ^~6 | upmid 6th | Bvv |
| 23 | 920 | vM6 | downmajor 6th | Bv |
| 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 | C^3 |
| 28 | 1120 | ^~7, vv8 | upmid 7th, double-down 8ve | C#vv, Dvv |
| 29 | 1160 | vM7, v8 | downmajor 7th, down 8ve | C#v, Dv |
| 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 | .)
| 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