49edo

← 48edo

49edo

50edo →

Prime factorization	72
Step size	24.4898 ¢
Fifth	29\49 (710.204 ¢)
Semitones (A1:m2)	7:2 (171.4 ¢ : 48.98 ¢)
Dual sharp fifth	29\49 (710.204 ¢)
Dual flat fifth	28\49 (685.714 ¢) (→ 4\7)
Dual major 2nd	8\49 (195.918 ¢)
Consistency limit	7
Distinct consistency limit	7

49 equal divisions of the octave (abbreviated 49edo or 49ed2), also called 49-tone equal temperament (49tet) or 49 equal temperament (49et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 49 equal parts of about 24.5 ¢ each. Each step represents a frequency ratio of 2^1/49, or the 49th root of 2.

Theory

49edo is very much on the sharp side of things, with sharp tunings of harmonics 3, 5, 7, and 11. It is the optimal patent val for superpyth temperament in the 7- and 11-limit, archytas (7-limit), and ares (11-limit) planar temperaments, being almost exactly equal to 3⁄10-comma superpyth. It tempers out 64/63, 245/243, and 3125/3087 in the 7-limit, and 100/99, 540/539, and 1375/1372 in the 11-limit. In the 13-limit, its patent val ⟨49 78 114 138 170 181], has a rather flat (by relative error) harmonic 13, which leads to inconsistent mappings; but using the 49f val ⟨49 78 114 138 170 182] improves 13-limit consistency, and in this val it tempers out 364/363 and 847/845.

Harmonics

Approximation of odd harmonics in 49edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+8.2	+5.5	+10.8	-8.0	+11.9	-7.9	-10.7	-7.0	-3.6	-5.5	+8.5
	Relative (%)	+33.7	+22.6	+44.0	-32.6	+48.8	-32.2	-43.8	-28.6	-14.8	-22.4	+34.5
Steps (reduced)		78 (29)	114 (16)	138 (40)	155 (8)	170 (23)	181 (34)	191 (44)	200 (4)	208 (12)	215 (19)	222 (26)

Subsets and supersets

Since 49 factors into primes as 7², 49edo contains 7edo as its only nontrivial subset. 49edo is the first square edo with a non-enfactored diatonic fifth. Doubling it produces 98edo, a respectable (if overly complex) meantone tuning.

Intervals

#	Cents	Approximate ratios*	Ups and downs notation
0	0.000	1/1	D
1	24.490	50/49	^D, vE♭
2	48.980	28/27, 36/35, 49/48, 81/80	^^D, E♭
3	73.469	22/21, 25/24, 33/32	^3D, ^E♭
4	97.959	16/15, 21/20	v3D♯, ^^E♭
5	122.449	15/14	vvD♯, ^3E♭
6	146.939	12/11	vD♯, v3E
7	171.429	10/9, 11/10	D♯, vvE
8	195.918	28/25	^D♯, vE
9	220.408	8/7, 9/8, 25/22	E
10	244.898	125/108, 144/125	^E, vF
11	269.388	7/6	F
12	293.878	25/21, 33/28	^F, vG♭
13	318.367	6/5	^^F, G♭
14	342.857	11/9	^3F, ^G♭
15	367.347	27/22	v3F♯, ^^G♭
16	391.837	5/4	vvF♯, ^3G♭
17	416.327	14/11	vF♯, v3G
18	440.816	9/7	F♯, vvG
19	465.306	125/96, 162/125	^F♯, vG
20	489.796	4/3, 21/16	G
21	514.286	75/56	^G, vA♭
22	538.776	15/11, 27/20	^^G, A♭
23	563.265	11/8	^3G, ^A♭
24	587.755	7/5	v3G♯, ^^A♭
25	612.245	10/7	vvG♯, ^3A♭
26	636.735	16/11	vG♯, v3A
27	661.244	22/15, 40/27	G♯, vvA
28	685.714	112/75	^G♯, vA
29	710.204	3/2, 32/21	A
30	734.694	125/81, 192/125	^A, vB♭
31	759.184	14/9	^^A, B♭
32	783.673	11/7	^3A, ^B♭
33	808.163	8/5	v3A♯, ^^B♭
34	832.653	44/27	vvA♯, ^3B♭
35	857.143	18/11	vA♯, v3B
36	881.633	5/3	A♯, vvB
37	906.122	42/25, 56/33	^A♯, vB
38	930.612	12/7	B
39	955.102	125/72, 216/125	^B, vC
40	979.592	7/4, 16/9, 44/25	C
41	1004.082	25/14	^C, vD♭
42	1028.571	9/5, 20/11	^^C, D♭
43	1053.061	11/6	^3C, ^D♭
44	1077.551	28/15	v3C♯, ^^D♭
45	1102.041	15/8, 40/21	vvC♯, ^3D♭
46	1126.531	21/11, 48/25, 64/33	vC♯, v3D
47	1151.020	27/14, 35/18, 96/49, 160/81	C♯, vvD
48	1175.510	49/25	^C♯, vD
49	1200.000	2/1	D

* Based on 49edo's 11-limit patent val ⟨49 78 114 138 170] mapping

Notation

Ups and downs notation

49edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Sharp symbol
Flat symbol

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol
Flat symbol

Sagittal notation

Evo flavor

Revo flavor

Approximation to JI

Selected 19-limit intervals approximated in 49edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 49edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 49edo (direct approximation, even if inconsistent)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/9, 18/13	0.117	0.5
11/7, 14/11	1.181	4.8
15/11, 22/15	1.825	7.5
7/6, 12/7	2.517	10.3
5/3, 6/5	2.726	11.1
15/13, 26/15	2.843	11.6
15/14, 28/15	3.006	12.3
11/6, 12/11	3.698	15.1
11/9, 18/11	4.551	18.6
13/11, 22/13	4.668	19.1
7/5, 10/7	5.243	21.4
5/4, 8/5	5.523	22.6
9/7, 14/9	5.732	23.4
13/7, 14/13	5.849	23.9
11/10, 20/11	6.424	26.2
13/8, 16/13	7.875	32.2
9/8, 16/9	7.992	32.6
3/2, 4/3	8.249	33.7
13/12, 24/13	8.366	34.2
15/8, 16/15	10.718	43.8
7/4, 8/7	10.766	44.0
9/5, 10/9	10.975	44.8
13/10, 20/13	11.092	45.3
11/8, 16/11	11.947	48.8

15-odd-limit intervals in 49edo (patent val mapping)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/7, 14/11	1.181	4.8
15/11, 22/15	1.825	7.5
7/6, 12/7	2.517	10.3
5/3, 6/5	2.726	11.1
15/14, 28/15	3.006	12.3
11/6, 12/11	3.698	15.1
11/9, 18/11	4.551	18.6
7/5, 10/7	5.243	21.4
5/4, 8/5	5.523	22.6
9/7, 14/9	5.732	23.4
11/10, 20/11	6.424	26.2
13/8, 16/13	7.875	32.2
3/2, 4/3	8.249	33.7
7/4, 8/7	10.766	44.0
9/5, 10/9	10.975	44.8
11/8, 16/11	11.947	48.8
13/10, 20/13	13.398	54.7
15/8, 16/15	13.772	56.2
13/12, 24/13	16.124	65.8
9/8, 16/9	16.498	67.4
13/7, 14/13	18.641	76.1
13/11, 22/13	19.822	80.9
15/13, 26/15	21.647	88.4
13/9, 18/13	24.373	99.5

15-odd-limit intervals by 49f val mapping

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
13/9, 18/13	0.117	0.5
11/7, 14/11	1.181	4.8
15/11, 22/15	1.825	7.5
7/6, 12/7	2.517	10.3
5/3, 6/5	2.726	11.1
15/13, 26/15	2.843	11.6
15/14, 28/15	3.006	12.3
11/6, 12/11	3.698	15.1
11/9, 18/11	4.551	18.6
13/11, 22/13	4.668	19.1
7/5, 10/7	5.243	21.4
5/4, 8/5	5.523	22.6
9/7, 14/9	5.732	23.4
13/7, 14/13	5.849	23.9
11/10, 20/11	6.424	26.2
3/2, 4/3	8.249	33.7
13/12, 24/13	8.366	34.2
7/4, 8/7	10.766	44.0
9/5, 10/9	10.975	44.8
13/10, 20/13	11.092	45.3
11/8, 16/11	11.947	48.8
15/8, 16/15	13.772	56.2
9/8, 16/9	16.498	67.4
13/8, 16/13	16.615	67.8

Zeta peaks

The strongest local zeta peak around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for Triple BP.

Approximation to irrational intervals

Acoustic ϕ and ϕ^ϕ−1

49edo has a very close approximation of both acoustic phi and ϕ^ϕ-1, a kind of logarithmic phi that divides acoustic phi logarithmically by phi (instead of dividing 2/1).

The phith root of phi (ϕ^ϕ-1) has interesting applications as Metallic MOS, and in particular the fractal-like possibilities of self-similar subdivision of musical scales within acoustic phi.

Direct approximation

Interval	Error (abs, ¢)	#\49
ϕ / ϕϕ−1 = ϕ(2 − ϕ)	0.155	13
ϕ	−0.437	34
ϕϕ−1	−0.592	21

Not until 592 do we find a better edo in terms of relative error on these two intervals (but whose edo-steps upon which these intervals are mapped are not based on the Fibonacci sequence, unlike 49edo).

Music

• Sevish - Star Nursery uses a scale based on acoustic phi and ϕ^ϕ−1. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[78 -49⟩	[⟨49 78]]	−2.60	2.60	10.62
2.3.5	15625/15552, 20480/19683	[⟨49 78 114]]	−2.53	2.12	8.69
2.3.5.7	64/63, 245/243, 3125/3087	[⟨49 78 114 138]]	−2.85	1.92	7.87
2.3.5.7.11	64/63, 100/99, 245/243, 1331/1323	[⟨49 78 114 138 170]]	−2.97	1.74	7.11

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	1\49	24.5	99/98	Sengagen
1	4\49	98.0	16/15	Passion
1	6\49	146.9	12/11	Bohpier
1	8\49	195.9	28/25	Didacus
1	11\49	269.4	7/6	Infraorwell
1	12\49	293.9	25/21	Kleiboh
1	13\49	318.4	6/5	Catalan
1	16\49	391.8	5/4	Magus
1	17\49	416.3	14/11	Sqrtphi
1	18\49	440.8	9/7	Clyde
1	19\49	465.3	55/36	Semisept
1	20\49	489.8	4/3	Superpyth
7	20\49 (1\49)	489.8 (24.5)	4/3 (250/243)	Sevond (49)
			4/3 (25/24)	Seville (49c)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Octave stretch or compression

49edo's primes 3, 5, 7 and 11 are all tuned sharp, so 49edo can benefit from octave shrinking. Some compressed-octave tunings of 49edo include (least to most compression): 176ed12, 114ed5, 233zpi, 127ed6, 138ed7 and 78edt.

Nonoctave temperament

The TE-optimized triple Bohlen–Pierce scale is obtained by taking every second degree of 49edo with the octave compressed by 3.861 cents to 1196.139 cents. It realizes the Tenney–Euclidean regular temperament on the 3.5.7.11.13 subgroup mapped as [⟨78 114 138 170 182]]. Under this compression, the primes map to the 49fgh val in the 23-limit.

Scales

MOS scales

Main article: List of MOS scales in 49edo

• Bohpier[8]: 6 6 6 6 7 6 6 6
• Catalan[7]: 3 10 3 10 3 10 10 (vaugely diminished-like)
• Catalan[11]: 3 7 3 3 7 3 3 7 3 3 7
• Catalan[19]: 3 3 1 3 3 3 3 1 3 3 3 1 3 3 3 3 1 3 3
• Clyde[5]: 5 13 5 13 13 (mysterious, adventurous)
• Didacus[6]: 8 8 8 8 8 9 (like the whole tone scale)
• Didacus[13]: 1 7 1 7 1 7 1 7 1 7 1 7 1
• Infraorwell[5]: 11 11 5 11 11
• Infraorwell[22]: 1 4 1 4 1 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
• Kleiboh[5]: 12 12 12 12 1
• Kleiboh[13]: 1 10 1 1 10 1 1 1 10 1 1 10 1
• Magus[7]: 1 15 1 15 1 15 1 (vaguely augmented-like)
• Passion[12]: 4 4 4 4 4 4 5 4 4 4 4 4 (like 12edo)
• Passion[23]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
• Sevond[21]/Seville[21]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
• Superpyth[5]: 11 9 9 11 9 (in between minor pentatonic and equipentatonic)
• Superpyth[7]: 9 2 9 9 9 2 9 (Dorian mode; rotate for other modes)
• Superpyth[12]: 2 7 2 7 2 2 7 2 7 2 7 2 (same melodic shape as 12edo but much more xenharmonic harmonies)
• Superpyth[27]: 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2

Other scales

• The 6ed7/3+7edo scale (non-octave-repeating)

Instruments

Lumatone

• Lumatone mapping for 49edo

Skip fretting

Skip fretting system 49 3 7 is a skip fretting system for 49edo. All examples are for 5-string bass.

Harmonics

1/1: string 2 open

2/1: not easily accessible

3/2: string 4 fret 5 and string 1 fret 12

5/4: string 3 fret 3

7/4: string 3 fret 11

11/8: string 3 fret 5

Music

Modern renderings

The Cure

• Boys Don't Cry (1979) – Lumatone cover by YoVariable (2025)

21st century

Bryan Deister

• Deltarune – Man (cover) (2023)
• microtonal improv in 49edo (2024)
• weathergirl - FLAVOR FOLEY (microtonal cover in 49edo) (2025)
• I'm Your Captain Now (The Ancients) - The Recovery System (microtonal cover in 49edo) (2026)
• 49edo improv (2026)
• 49edo riff (2026)
• 49edo groove (2026)
• 49edo prelude (2026)

Mercury Amalgam

• Wrong Generation (2022 demo version)

Cam Taylor

• 49-equal: 7-equal meets superpyth (2023)