# 30edo

 ← 29edo 30edo 31edo →
Prime factorization 2 × 3 × 5
Step size 40¢
Fifth 18\30 (720¢) (→3\5)
Semitones (A1:m2) 6:0 (240¢ : 0¢)
Dual sharp fifth 18\30 (720¢) (→3\5)
Dual flat fifth 17\30 (680¢)
Dual major 2nd 5\30 (200¢) (→1\6)
Consistency limit 5
Distinct consistency limit 5

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 21/30, or the 30th root of 2.

## Theory

Approximation of odd harmonics in 30edo
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.

A plot of the Z function around 30.

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
1. 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.

## Scales

### 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

Bryan Deister
Todd Harrop
Micronaive
NullPointerException Music