30edo

30 equal divisions of the octave (30edo) is the tuning system derived by dividing the octave into 30 equal steps of 40 ¢ each.

← 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

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.

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.

Being the smallest positive integer with three distinct prime factors, 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

JI approximation

Zeta function

Below is a plot of the Z function around 30:

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	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

Music

Todd Harrop

• Fifteen Short Pieces

Micronaive

• No.27.62