50edo divides the octave into 50 equal parts of precisely 24 cents each. In the 5-limit, it 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.

50 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 15&50 temperament (Coblack), 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.

Relations[edit]

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[edit]

Degrees of 50edo	Cents value	Ratios*	Generator for*
0	0	1/1
1	24	45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168	Sengagen
2	48	33/32, 36/35, 50/49, 55/54, 64/63
3	72	21/20, 25/24, 26/25, 27/26, 28/27	Vishnu (2/oct), Coblack (5/oct)
4	96	22/21	Injera (50d val, 2/oct)
5	120	16/15, 15/14, 14/13
6	144	13/12, 12/11
7	168	11/10
8	192	9/8, 10/9
9	216	25/22	Tremka, Machine (50b val)
10	240	8/7, 15/13
11	264	7/6	Septimin (13-limit)
12	288	13/11
13	312	6/5
14	336	27/22, 39/32, 40/33, 49/40
15	360	16/13, 11/9
16	384	5/4	Wizard (2/oct)
17	408	14/11	Ditonic
18	432	9/7	Hedgehog (50cc val, 2/oct)
19	456	13/10	Bisemidim (2/oct)
20	480	33/25, 55/42, 64/49
21	504	4/3	Meantone/Meanpop
22	528	15/11
23	552	11/8, 18/13	Barton, Emka
24	576	7/5
25	600	63/44, 88/63, 78/55, 55/39
26	624	10/7
27	648	16/11, 13/9
28	672	22/15
29	696	3/2
30	720	50/33, 84/55, 49/32
31	744	20/13
32	768	14/9
33	792	11/7
34	816	8/5
35	840	13/8, 18/11
36	864	44/27, 64/39, 33/20, 80/49
37	888	5/3
38	912	22/13
39	936	12/7
40	960	7/4
41	984	44/25
42	1008	16/9, 9/5
43	1032	20/11
44	1056	24/13, 11/6
45	1080	15/8, 28/15, 13/7
46	1104	21/11
47	1128	40/21, 48/25, 25/13, 52/27, 27/14
48	1152	64/33, 35/18, 49/25, 108/55, 63/32
49	1176

using the 13-limit patent val except as noted

Selected just intervals by error[edit]

The following table shows how some prominent just intervals are represented in 50edo (ordered by absolute error).

Best direct mapping, even if inconsistent[edit]

Interval, complement	Error (abs., in cents)
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

Patent val mapping[edit]

Interval, complement	Error (abs., in cents)
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
9/8, 16/9	11.910
11/9, 18/11	12.592

Commas[edit]

50 EDO 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.

Monzo	Cents	Ratio	Name 1	Name 2
\| -4 4 -1 >	21.51	81/80	Syntonic comma	Didymus comma
\| -27 -2 13 >	18.17		Ditonma
\| 23 6 -14 >	3.34		Vishnu comma
\| 1 2 -3 1 >	13.79	126/125	Starling comma	Small septimal comma
\| -5 2 2 -1 >	7.71	225/224	Septimal kleisma	Marvel comma
\| 6 0 -5 2 >	6.08	3136/3125	Hemimean	Middle second comma
\| -6 -8 2 5 >	1.12		Wizma
\|-11 2 7 -3 >	1.63		Meter
\| 11 -10 -10 10 >	5.57		Linus
\|-13 10 0 -1 >	50.72	59049/57344	Harrison's comma
\| 2 3 1 -2 -1 >	3.21	540/539	Swets' comma	Swetisma
\| -3 4 -2 -2 2 >	0.18	9801/9800	Kalisma	Gauss' comma
\| 5 -1 3 0 -3 >	3.03	4000/3993	Wizardharry	Undecimal schisma
\| -7 -1 1 1 1 >	4.50	385/384	Keenanisma	Undecimal kleisma
\| -1 0 1 2 -2 >	21.33	245/242	Cassacot
\| 2 -1 0 1 -2 1 >	4.76	364/363	Gentle comma
\| 2 -1 -1 2 0 -1 >	8.86	196/195	Mynucuma
\| 2 3 0 -1 1 -2 >	7.30	1188/1183	Kestrel Comma
\| 3 0 2 0 1 -3 >	2.36	2200/2197	Petrma	Parizek comma
\| -3 1 1 1 0 -1 >	16.57	105/104	Animist comma	Small tridecimal comma
\| 4 2 0 0 -1 -1 >	12.06	144/143	Grossma
\| 3 -2 0 1 -1 -1 0 0 1 >	1.34	1288/1287	Triaphonisma