30edo
30edo, the 30 equal division, divides the octave into 30 equal steps of precisely 40 cents each.
Theory
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 | 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 |
Rank Two Temperaments
As 30edo is highly composite, only 7, 11 and 13 steps create mode of symmetry scales that cover every interval using one period per octave. 7/30 produces Lovecraft, in which 2 generators is a moderately sharp 11/8, 3 a near perfect 13/8 and 5 the familiar mildly flat 9/8 from 12edo, creating the possibility of ignoring the 3rd & 5th entirely to use those harmonics as the primary building blocks of harmony in a similar way to orgone. 11 produces a flat sensi scale. 13 is an excellent higher order Mavila tuning that functions the closest to the familiar diatonic scale you can get in this edo.
- MOS scales
- Lovecraft[5] - 77772
- Lovecraft[9] - 525252522
- Lovecraft[13] - 3223223223222
- Lovecraft[17] - 22221222122212221
- Sensi[5] - 83838
- Sensi[8] - 53353353
- Sensi[11] - 33323332332
- Sensi[19] - 2121212212121221212
- Mavila[5] - 94944
- Mavila[7] - 5445444
- Mavila[9] - 444414441
- Mavila[16] - 3131313113131311
- Mavila[23] - 21121121121112112112111
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