30edo

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 40.000 ¢ each. Each step represents a frequency ratio of 2^1/30, 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
Error	relative (%)	+45	+34	-22	-10	+22	-1	-21	+38	-44	+23	+29
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
0	0	P1	unison, minor 2nd	D, Eb
1	40	^1, ^m2	up unison, upminor 2nd	^D, ^Eb
2	80	^^1, v~2	dup 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, dud 8ve	vvC#, vvD
29	1160	vM7, v8	downmajor 7th, down 8ve	vC#, vD
30	1200	P8	major 7th, 8ve	C#, 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