editing disabled

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

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

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

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

Commas

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


Music

Twinkle canon – 50 edo by Claudi Meneghin
Fantasia Catalana by Claudi Meneghin
Fugue on the Dragnet theme by Claudi Meneghin
the late little xmas album by Cam Taylor
Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor
Long improvisation 2 in 50EDO by Cam Taylor
Chord sequence for Difference tones in 50EDO by Cam Taylor
Enharmonic Modulations in 50EDO by Cam Taylor
Harmonic Clusters on 50EDO Harpsichord by Cam Taylor
Fragment in Fifty by Cam Taylor

Additional reading

Robert Smith's book online
More information about Robert Smith's temperament

50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor
iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor