50edo

Prime factorization

2 × 5²

Step size

24¢

Fifth

29\50 (696¢)

Semitones (A1:m2)

3:5 (72¢ : 120¢)

Consistency limit

9

Distinct consistency limit

7

50 equal divisions of the octave (50edo), or 50-tone equal temperament (50tet), 50 equal temperament (50et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 50 equal parts of exactly 24 ¢ each.

Theory

In the 5-limit, 50edo tempers out 81/80, making it a meantone system, and in that capacity has historically has drawn some notice. In "Harmonics or the Philosophy of Musical Sounds" (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the least squares tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While 31edo extends meantone with a 7/4 which is nearly pure, 50 has a flat 7/4 but both 11/8 and 13/8 are nearly pure. It is the highest edo which maps 9/8 and 10/9 to the same interval in a consistent manner, with two stacked fifths falling almost precisely in the middle of the two.

50edo tempers out 126/125, 225/224 and 3136/3125 in the 7-limit, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the 11-limit and 105/104, 144/143 and 196/195 in the 13-limit, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the coblack (15&50) temperament, and provides the optimal patent val for 11 and 13-limit bimeantone. It is also the unique equal temperament tempering out both 81/80 and the vishnuzma, [23 6 -14⟩;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.

Odd harmonics

Approximation of odd harmonics in 50edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-6.0	-2.3	-8.8	-11.9	+0.7	-0.5	-8.3	-9.0	-9.5	+9.2	-4.3
Error	relative (%)	-25	-10	-37	-50	+3	-2	-34	-37	-40	+38	-18
Steps (reduced)		79 (29)	116 (16)	140 (40)	158 (8)	173 (23)	185 (35)	195 (45)	204 (4)	212 (12)	220 (20)	226 (26)

Relations

The 50edo system is related to 7edo, 12edo, 19edo, 31edo as the next approximation to the "Golden Tone System" (Das Goldene Tonsystem) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "A Theory of Evolving Tonality").

Intervals

#	Cents	Ratios*	Ups and Downs Notation
0	0	1/1	Perfect 1sn	P1	D
1	24	45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168	Up 1sn	^1	^D
2	48	27/26, 33/32, 36/35, 50/49, 55/54, 64/63	Dim 2nd, Downaug 1sn	d2, vA1	Ebb, vD#
3	72	21/20, 25/24, 26/25, 28/27	Aug 1sn, Updim 2nd	A1, ^d2	D#, ^Ebb
4	96	22/21	Downminor 2nd	vm2	vEb
5	120	16/15, 15/14, 14/13	Minor 2nd	m2	Eb
6	144	13/12, 12/11	Upminor 2nd	^m2	^Eb
7	168	11/10	Downmajor 2nd	vM2	vE
8	192	9/8, 10/9	Major 2nd	M2	E
9	216	25/22	Upmajor 2nd	^M2	^E
10	240	8/7, 15/13	Downaug 2nd, Dim 3rd	vA2, d3	vE#, Fb
11	264	7/6	Updim 3rd, Aug 2nd	^d3, A2	^Fb, E#
12	288	13/11	Downminor 3rd	vm3	vF
13	312	6/5	Minor 3rd	m3	F
14	336	27/22, 39/32, 40/33, 49/40	Upminor 3rd	^m3	^F
15	360	16/13, 11/9	Downmajor 3rd	vM3	vF#
16	384	5/4	Major 3rd	M3	F#
17	408	14/11	Upmajor 3rd	^M3	^F#
18	432	9/7	Downaug 3rd, Dim 4th	vA3, d4	vFx, Gb
19	456	13/10	Updim 4th, Aug 3rd	A3, ^d4	^Gb, Fx
20	480	33/25, 55/42, 64/49	Down 4th	v4	vG
21	504	4/3	Perfect 4th	P4	G
22	528	15/11	Up 4th	^4	^G
23	552	11/8, 18/13	Downaug 4th	vA4	vG#
24	576	7/5	Aug 4th	A4	G#
25	600	63/44, 88/63, 78/55, 55/39	Upaug 4th, Downdim 5th	^A4, vd5	^G#, vAb
26	624	10/7	Dim 5th	d5	Ab
27	648	16/11, 13/9	Updim 5th	^d5	^Ab
28	672	22/15	Down 5th	v5	vA
29	696	3/2	Perfect 5th	P5	A
30	720	50/33, 84/55, 49/32	Up 5th	^5	^A
31	744	20/13	Downaug 5th, Dim 6th	vA5, d6	vA#, Bbb
32	768	14/9	Updim 6th, Aug 5th	^d6, A5	^Bbb, A#
33	792	11/7	Downminor 6th	vm6	vBb
34	816	8/5	Minor 6th	m6	Bb
35	840	13/8, 18/11	Upminor 6th	^m6	^Bb
36	864	44/27, 64/39, 33/20, 80/49	Downmajor 6th	vM6	vB
37	888	5/3	Major 6th	M6	B
38	912	22/13	Upmajor 6th	^M6	^B
39	936	12/7	Downaug 6th, Dim 7th	vA6, d7	vB#, Cb
40	960	7/4	Updim 7th, Aug 6th	^d7, A6	^Cb, B#
41	984	44/25	Downminor 7th	vm7	vC
42	1008	16/9, 9/5	Minor 7th	m7	C
43	1032	20/11	Upminor 7th	^m7	^C
44	1056	24/13, 11/6	Downmajor 7th	vM7	vC#
45	1080	15/8, 28/15, 13/7	Major 7th	M7	C#
46	1104	21/11	Upmajor 7th	^M7	^C#
47	1128	40/21, 48/25, 25/13, 27/14	Downaug 7th, Dim 8ve	vA7, d8	vCx, Db
48	1152	52/27, 64/33, 35/18, 49/25, 108/55, 63/32	Updim 8ve, Aug 7th	^d8, A7	^Db, Cx
49	1176	88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169	Down 8ve	v8	vD
50	1200	2/1	Perfect 8ve	P8	D

* using the patent val

JI approximation

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 50edo (ordered by absolute error). Prime harmonics are in bold; inconsistent intervals are in italic.

Direct approximation (even if inconsistent)
Interval, complement	Error (abs, ¢)
16/13, 13/8	0.528
15/14, 28/15	0.557
11/8, 16/11	0.682
13/11, 22/13	1.210
13/10, 20/13	1.786
5/4, 8/5	2.314
7/6, 12/7	2.871
11/10, 20/11	2.996
9/7, 14/9	3.084
6/5, 5/3	3.641
13/12, 24/13	5.427
4/3, 3/2	5.955
7/5, 10/7	6.512
12/11, 11/6	6.637
15/13, 26/15	7.741
16/15, 15/8	8.269
14/13, 13/7	8.298
8/7, 7/4	8.826
15/11, 22/15	8.951
14/11, 11/7	9.508
10/9, 9/5	9.596
18/13, 13/9	11.382
11/9, 18/11	11.408
9/8, 16/9	11.910

Regular temperament properties

Temperament measures

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-79 50⟩	[⟨50 79]]	+1.88	1.88	7.83
2.3.5	81/80, [-27 -2 13⟩	[⟨50 79 116]]	+1.58	1.59	6.62
2.3.5.7	81/80, 126/125, 84035/82944	[⟨50 79 116 140]]	+1.98	1.54	6.39
2.3.5.7.11	81/80, 126/125, 245/242, 385/384	[⟨50 79 116 140 173]]	+1.54	1.63	6.76
2.3.5.7.11.13	81/80, 105/104, 126/125, 144/143, 245/242	[⟨50 79 116 140 173 185]]	+1.31	1.57	6.54

Commas

50edo tempers out the following commas. (Note: This assumes the val ⟨50 79 116 140 173 185 204 212 226], comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

Prime Limit	Ratio^[1]	Monzo	Cents	Name
5	81/80	[-4 4 -1⟩	21.51	Syntonic comma
5	(20 digits)	[-27 -2 13⟩	18.17	Ditonma
5	(20 digits)	[23 6 -14⟩	3.34	Vishnuzma
7	59049/57344	[-13 10 0 -1⟩	50.72	Harrison's comma
7	126/125	[1 2 -3 1⟩	13.79	Starling comma
7	225/224	[-5 2 2 -1⟩	7.71	Marvel comma
7	3136/3125	[6 0 -5 2⟩	6.08	Hemimean
7	(24 digits)	[11 -10 -10 10⟩	5.57	Linus
7	(12 digits)	[-11 2 7 -3⟩	1.63	Meter
7	(12 digits)	[-6 -8 2 5⟩	1.12	Wizma
11	245/242	[-1 0 1 2 -2⟩	21.33	Cassacot
11	385/384	[-7 -1 1 1 1⟩	4.50	Keenanisma
11	540/539	[2 3 1 -2 -1⟩	3.21	Swetisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Wizardharry
11	9801/9800	[-3 4 -2 -2 2⟩	0.18	Kalisma
13	105/104	[-3 1 1 1 0 -1⟩	16.57	Animist comma
13	144/143	[4 2 0 0 -1 -1⟩	12.06	Grossma
13	196/195	[2 -1 -1 2 0 -1⟩	8.86	Mynucuma
13	1188/1183	[2 3 0 -1 1 -2⟩	7.30	Kestrel Comma
13	364/363	[2 -1 0 1 -2 1⟩	4.76	Gentle comma
13	2200/2197	[3 0 2 0 1 -3⟩	2.36	Petrma
17	170/169	[1 0 1 0 0 -2 1⟩	10.21
17	221/220	[-2 0 -1 0 -1 1 1⟩	7.85
17	289/288	[-5 -2 0 0 0 0 2⟩	6.00	Septendecimal semitones comma
17	375/374	[-1 1 3 0 -1 0 -1⟩	4.62
19	153/152	[-3 2 0 0 0 0 1 -1⟩	11.35	Ganassisma
19	171/170	[-1 2 -1 0 0 0 -1 1⟩	10.15
19	210/209	[1 1 1 1 -1 0 0 1⟩	8.26
19	324/323	[2 4 0 0 0 0 -1 -1⟩	5.35	Nusu comma
19	361/360	[-3 -2 -1 0 0 0 0 2⟩	4.80	Go comma
19	495/494	[-1 2 1 0 1 -1 0 -1⟩	3.50
23	1288/1287	[3 -2 0 1 -1 -1 0 0 1⟩	1.34	Triaphonisma

↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per Octave	Generator	Cents	Associated Ratio	Temperament
1	1\50	24.00	686/675	Sengagen
1	9\50	216.00	17/15	Tremka
1	11\50	264.00	7/6	Septimin
1	13\50	312.00	6/5	Oolong
1	17\50	408.00	15625/12288	Ditonic / coditone
1	19\50	456.00	125/96	Qak
1	21\50	504.00	4/3	Meantone / meanpop
1	23\50	552.00	11/8	Emka
2	2\50	48.00	36/35	Pombe
2	3\50	72.00	25/24	Vishnu / vishnean
2	4\50	96.00	35/33	Bimeantone
2	6\50	144.00	12/11	Bisemidim
2	9\50	216.00	17/15	Wizard / lizard / gizzard
2	12\50	288.00	13/11	Vines
5	21\50 (1\50)	504.00 (24.00)	4/3	Cloudtone
5	3\50	72.00	21/20, 25/24	Coblack
10	21\50 (1\50)	504.00 (24.00)	4/3	Decic
10	2\50 (3\50)	48.00 (72.00)	36/35 (25/24)	Decavish