30edo
| ← 29edo | 30edo | 31edo → |
30 equal divisions of the octave (30edo) is the tuning system derived by dividing the octave into 30 equal steps of 40 ¢ each.
Theory
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18.0 | +13.7 | -8.8 | -3.9 | +8.7 | -0.5 | -8.3 | +15.0 | -17.5 | +9.2 | +11.7 |
| Relative (%) | +45.1 | +34.2 | -22.1 | -9.8 | +21.7 | -1.3 | -20.7 | +37.6 | -43.8 | +23.0 | +29.3 | |
| Steps (reduced) |
48 (18) |
70 (10) |
84 (24) |
95 (5) |
104 (14) |
111 (21) |
117 (27) |
123 (3) |
127 (7) |
132 (12) |
136 (16) | |
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 30edo can be viewed inconsistently, as having a 9/1 at 95\30 as well as 96\30.
Instead of the 18\30 fifth of 720 cents, 30edo 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 30edo is used for pelogic, 5\30 can again be used inconsistently as a 9/8.
Subsets and supersets
30edo has subset edos 1, 2, 3, 5, 6, 10, 15 and it is a largely composite edo.
30edo is the 3rd primorial edo, being the product of first three primes and thus the smallest number with three distinct prime factors. As a corollary, 30edo is the smallest EDO that supports perfectly balanced scales that are minimal and not equally spaced. See the article on perfect balance.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 17\30) |
Ups and downs notation (Dual sharp fifth 18\30) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 40 | D♯, E♭♭♭ | ^D, ^E♭, ^F♭ | |
| 2 | 80 | 21/20, 22/21, 23/22, 24/23, 25/24 | D𝄪, E♭♭ | ^^D, ^^E♭, ^^F♭ |
| 3 | 120 | 14/13 | D♯𝄪, E♭ | ^3D, v3E, v3F |
| 4 | 160 | 11/10, 12/11, 23/21 | E | vvD♯, vvE, vvF |
| 5 | 200 | E♯, F♭♭♭♭ | vD♯, vE, vF | |
| 6 | 240 | 8/7, 23/20 | E𝄪, F♭♭♭ | E, F |
| 7 | 280 | 13/11, 20/17 | E♯𝄪, F♭♭ | ^E, ^F, ^G♭ |
| 8 | 320 | 6/5 | E𝄪𝄪, F♭ | ^^E, ^^F, ^^G♭ |
| 9 | 360 | 16/13, 21/17, 26/21 | F | ^3E, ^3F, v3G |
| 10 | 400 | F♯, G♭♭♭ | vvE♯, vvF♯, vvG | |
| 11 | 440 | 22/17 | F𝄪, G♭♭ | vE♯, vF♯, vG |
| 12 | 480 | 21/16 | F♯𝄪, G♭ | G |
| 13 | 520 | 19/14, 23/17 | G | ^G, ^A♭ |
| 14 | 560 | 11/8, 25/18 | G♯, A♭♭♭ | ^^G, ^^A♭ |
| 15 | 600 | 17/12, 24/17 | G𝄪, A♭♭ | ^3G, v3A |
| 16 | 640 | 16/11 | G♯𝄪, A♭ | vvG♯, vvA |
| 17 | 680 | A | vG♯, vA | |
| 18 | 720 | A♯, B♭♭♭ | A | |
| 19 | 760 | 17/11 | A𝄪, B♭♭ | ^A, ^B♭, ^C♭ |
| 20 | 800 | A♯𝄪, B♭ | ^^A, ^^B♭, ^^C♭ | |
| 21 | 840 | 13/8, 21/13 | B | ^3A, v3B, v3C |
| 22 | 880 | 5/3 | B♯, C♭♭♭♭ | vvA♯, vvB, vvC |
| 23 | 920 | 17/10, 22/13 | B𝄪, C♭♭♭ | vA♯, vB, vC |
| 24 | 960 | 7/4 | B♯𝄪, C♭♭ | B, C |
| 25 | 1000 | B𝄪𝄪, C♭ | ^B, ^C, ^D♭ | |
| 26 | 1040 | 11/6, 20/11 | C | ^^B, ^^C, ^^D♭ |
| 27 | 1080 | 13/7 | C♯, D♭♭♭ | ^3B, ^3C, v3D |
| 28 | 1120 | 21/11, 23/12 | C𝄪, D♭♭ | vvB♯, vvC♯, vvD |
| 29 | 1160 | C♯𝄪, D♭ | vB♯, vC♯, vD | |
| 30 | 1200 | 2/1 | D | D |
Rank-2 temperaments
As 30edo is largely composite, only 7, 11 and 13 steps create MOS 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].)
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 3 | 256/243 | [8 -5⟩ | 90.22 | Sawa | Limma, Pythagorean minor sedond |
| 5 | 250/243 | [1 -5 3⟩ | 49.17 | Triyo | Maximal diesis, Porcupine comma |
| 5 | 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Diesis, augmented comma |
| 5 | 15625/15552 | [-6 -5 6⟩ | 8.11 | Tribiyo | Kleisma, semicomma majeur |
| 7 | 1029/1000 | [-3 1 -3 3⟩ | 49.49 | Trizogu | Keega |
| 7 | 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Slendro diesis |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal comma, Archytas' comma, Leipziger Komma |
| 7 | 64827/64000 | [-9 3 -3 4⟩ | 22.23 | Laquadzo-atrigu | Squalentine |
| 7 | 875/864 | [-5 -3 3 1⟩ | 21.90 | Zotriyo | Keema |
| 7 | 126/125 | [1 2 -3 1⟩ | 13.79 | Zotrigu | Septimal semicomma, Starling comma |
| 7 | 4000/3969 | [5 -4 3 -2⟩ | 13.47 | Rurutriyo | Octagar |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.43 | Latrizo | Gamelisma |
| 7 | 6144/6125 | [11 1 -3 -2⟩ | 5.36 | Saruru-atrigu | Porwell |
| 7 | (12 digits) | [-4 6 -6 3⟩ | 0.33 | Trizogugu | Landscape comma |
| 11 | 100/99 | [2 -2 2 0 -1⟩ | 17.40 | Luyoyo | Ptolemisma |
| 11 | 121/120 | [-3 -1 -1 0 2⟩ | 14.37 | Lologu | Biyatisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma |
| 11 | 65536/65219 | [16 0 0 -2 -3⟩ | 8.39 | Satrilu-aruru | Orgonisma |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Luzozogu | Werckisma |
| 11 | 4000/3993 | [5 -1 3 0 -3⟩ | 3.03 | Triluyo | Wizardharry |
| 11 | 3025/3024 | [-4 -3 2 -1 2⟩ | 0.57 | Loloruyoyo | Lehmerisma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Music
- Edolian - Shift (2020)