30edo
← 29edo | 30edo | 31edo → |
30 equal divisions of the octave (abbreviated 30edo or 30ed2), also called 30-tone equal temperament (30tet) or 30 equal temperament (30et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 30 equal parts of exactly 40 ¢ each. Each step represents a frequency ratio of 2^{1/30}, or the 30th root of 2.
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
Step | Cents | Ups and downs notation | Armodue Notation | ||
---|---|---|---|---|---|
0 | 0 | P1 | unison, minor 2nd | D, Eb | 1 |
1 | 40 | ^1, ^m2 | up unison, upminor 2nd | ^D, ^Eb | 2b |
2 | 80 | ^^1, v~2 | dup unison, downmid 2nd | ^^D, ^^Eb | 9# |
3 | 120 | ~2 | mid 2nd | v3E | 1# |
4 | 160 | ^~2 | upmid 2nd | vvE | 2 |
5 | 200 | vM2 | downmajor 2nd | vE | 3b |
6 | 240 | M2, m3 | major 2nd, minor 3rd | E, F | 1x, 4bb |
7 | 280 | ^m3 | upminor 3rd | ^F | 2# |
8 | 320 | v~3 | downmid 3rd | ^^F | 3 |
9 | 360 | ~3 | mid 3rd | ^3F, v3F# | 4b |
10 | 400 | ^~3 | upmid 3rd | vvF# | 5b |
11 | 440 | vM3, v4 | downmajor 3rd, down 4th | vF#, vG | 3# |
12 | 480 | M3, P4 | major 3rd, perfect 4th | F#, G | 4 |
13 | 520 | ^4 | up 4th | ^G | 5 |
14 | 560 | v~4, v~d5 | downmid 4th, downmid 5th | ^^G, ^^Ab | 6b |
15 | 600 | ~4, ~5 | mid 4th, mid 5th | ^3G, v3A | 4# |
16 | 640 | ^~4, ^~5 | upmid 4th, upmid 5th | vvG#, vvA | 5# |
17 | 680 | v5 | down 5th | vA | 6 |
18 | 720 | P5, m6 | perfect 5th, minor 6th | A, Bb | 7b |
19 | 760 | ^5, ^m6 | up 5th, upminor 6th | ^A, ^Bb | 5x, 8bb |
20 | 800 | v~6 | downmid 6th | ^^Bb | 6# |
21 | 840 | ~6 | mid 6th | v3B | 7 |
22 | 880 | ^~6 | upmid 6th | vvB | 8b |
23 | 920 | vM6 | downmajor 6th | vB | 6x, 9bb |
24 | 960 | M6. m7 | major 6th, minor 7th | B, C | 7# |
25 | 1000 | ^m7 | upminor 7th | ^C | 8 |
26 | 1040 | v~7 | downmid 7th | ^^C | 9b |
27 | 1080 | ~7 | mid 7th | ^3C | 1b |
28 | 1120 | ^~7, vv8 | upmid 7th, dud 8ve | vvC#, vvD | 8# |
29 | 1160 | vM7, v8 | downmajor 7th, down 8ve | vC#, vD | 9 |
30 | 1200 | P8 | major 7th, 8ve | C#, D | 1 |
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
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
Subsets of Mavila[16]
- Arcade (approximated from 32afdo): 9 3 5 8 5
- Blackened skies (approximated from Compton in [72edo]]): 8 5 2 3 2 8 2
- Carousel (this is the original/default tuning): 9 4 4 9 4
- Dewdrops (this is the original/default tuning): 4 4 4 5 4 4 5
- Geode (approximated from 6afdo): 7 6 4 9 4
- Lost spirit (approximated from Meantone in 31edo): 7 5 2 3 5 3 5
- Lost phantom (this is the original/default tuning): 8 5 2 2 6 2 5
- Mechanical (approximated from 16afdo): 7 2 8 8 5
- Mushroom (approximated from 30afdo): 7 5 5 3 10
- Nightdrive (this is the original/default tuning): 8 5 4 9 4
- Pelagic (this is the original/default tuning): 8 4 2 4 7 5
- Bathypelagic (this is the original/default tuning): 8 4 2 3 8 5
- Underpass (approximated from 10afdo): 8 9 5 3 5
- Volcanic (approximated from 16afdo): 3 6 8 8 5
Subsets of 15edo
- Augmented[6] MOS: 8 2 8 2 8 2
- Equipentatonic (exact from 5edo): 6 6 6 6 6
- Rockpool (approximated from 47zpi): 2 8 2 6 6 6
Other notable scales
- Approximation of Pelog lima: 3 4 10 3 10
Music
- Edolian - Shift (2020)