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

Template:EDO intro

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 and the e-based analog of Lucy tuning. It tempers out 64/63, 245/243, and 3125/3087 in the 7-limit, and 100/99 and 1375/1372 in the 11-limit.

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
Error	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)

Miscellany

49edo is the first square equal division with a "real" 3 of step coprime to its cardinality.

Intervals

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

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

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

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).

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 / ϕ^{ϕ^-1} = ϕ^(2-ϕ)	0.155	13
ϕ	-0.437	34
ϕ^{ϕ^-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 Sstretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Sstretch (¢)	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

Rank-2 temperaments by generators
Periods per 8ve	Generator	Associated Ratio	Temperaments
1	1\49	99/98	Sengagen
1	4\49	16/15	Passion
1	6\49	12/11	Bohpier
1	8\49	28/25	Didacus
1	11\49	7/6	Infraorwell
1	12\49	25/21	Kleiboh
1	13\49	6/5	Catalan
1	16\49	5/4	Magus
1	17\49	14/11	Sqrtphi
1	18\49	9/7	Clyde
1	19\49	55/36	Semisept
1	20\49	4/3	Superpyth
7	20\49	4/3	Sevond (49) Seville (49c)