50edo

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50edo divides the octave into 50 equal parts of precisely 24 cents 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.

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* 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 Oolong
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 88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169
50 1200 2/1

* Using the 13-limit patent val, except as noted.

Just approximation

Selected just intervals

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13
Error absolute (¢) 0.0 -6.0 -2.3 -8.8 +0.7 -0.5
relative (%) 0.0 -24.8 -9.6 -36.8 +2.8 -2.2

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 mapping (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
Patent val mapping
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
9/8, 16/9 11.910
11/9, 18/11 12.592

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 50et.

3-limit 5-limit 7-limit 11-limit 13-limit
Octave stretch (¢) +1.88 +1.58 +1.98 +1.54 +1.31
Error absolute (¢) 1.88 1.59 1.54 1.63 1.57
relative (%) 7.83 6.62 6.39 6.76 6.54

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.

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

Music

Additional reading