50edo

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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).

Best direct mapping, even if inconsistent

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

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

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