# 50edo

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 ← 49edo 50edo 51edo →
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 (abbreviated 50edo or 50ed2), also called 50-tone equal temperament (50tet) or 50 equal temperament (50et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 50 equal parts of exactly 24 ¢ each. Each step represents a frequency ratio of 21/50, or the 50th root of 2.

## Theory

As an equal temperament, 50et tempers out 81/80 in the 5-limit, making it a meantone system, and in that capacity has historically 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 also the highest edo where the mapping of 9/8 and 10/9 to the same interval is consistent, with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It also maps all 15-odd-limit intervals consistently, with the sole exceptions of 11/9 and 18/11.

It 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 temperament (15 & 50), 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 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.

### 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
Relative (%) -24.8 -9.6 -36.8 -49.6 +2.8 -2.2 -34.5 -37.3 -39.6 +38.4 -17.8
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

Selected 29-limit intervals approximated in 50edo

### 15-odd-limit mappings

The following tables show how 15-odd-limit intervals are represented in 50edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 50edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.528 2.2
15/14, 28/15 0.557 2.3
11/8, 16/11 0.682 2.8
13/11, 22/13 1.210 5.0
13/10, 20/13 1.786 7.4
5/4, 8/5 2.314 9.6
7/6, 12/7 2.871 12.0
11/10, 20/11 2.996 12.5
9/7, 14/9 3.084 12.9
5/3, 6/5 3.641 15.2
13/12, 24/13 5.427 22.6
3/2, 4/3 5.955 24.8
7/5, 10/7 6.512 27.1
11/6, 12/11 6.637 27.7
15/13, 26/15 7.741 32.3
15/8, 16/15 8.269 34.5
13/7, 14/13 8.298 34.6
7/4, 8/7 8.826 36.8
15/11, 22/15 8.951 37.3
11/7, 14/11 9.508 39.6
9/5, 10/9 9.596 40.0
13/9, 18/13 11.382 47.4
11/9, 18/11 11.408 47.5
9/8, 16/9 11.910 49.6
15-odd-limit intervals in 50edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.528 2.2
15/14, 28/15 0.557 2.3
11/8, 16/11 0.682 2.8
13/11, 22/13 1.210 5.0
13/10, 20/13 1.786 7.4
5/4, 8/5 2.314 9.6
7/6, 12/7 2.871 12.0
11/10, 20/11 2.996 12.5
9/7, 14/9 3.084 12.9
5/3, 6/5 3.641 15.2
13/12, 24/13 5.427 22.6
3/2, 4/3 5.955 24.8
7/5, 10/7 6.512 27.1
11/6, 12/11 6.637 27.7
15/13, 26/15 7.741 32.3
15/8, 16/15 8.269 34.5
13/7, 14/13 8.298 34.6
7/4, 8/7 8.826 36.8
15/11, 22/15 8.951 37.3
11/7, 14/11 9.508 39.6
9/5, 10/9 9.596 40.0
13/9, 18/13 11.382 47.4
9/8, 16/9 11.910 49.6
11/9, 18/11 12.592 52.5

## Regular temperament properties

### Temperament measures

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
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
3 (20 digits) [-79 50 297.75 50-comma
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 16807/16384 [-14 0 0 5 44.13 Cloudy comma
7 3645/3584 [-9 6 1 -1 29.22 Schismean 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 31213/31104 [-7 -5 0 4 0 1 6.06 Praveensma
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 Major naiadma
17 221/220 [-2 0 -1 0 -1 1 1 7.85 Minor naiadma
17 289/288 [-5 -2 0 0 0 0 2 6.00 Semitonisma
17 375/374 [-1 1 3 0 -1 0 -1 4.62 Ursulisma
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 Malcolmisma
19 210/209 [1 1 1 1 -1 0 0 1 8.26 Spleen comma
19 324/323 [2 4 0 0 0 0 -1 -1 5.35 Photisma
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 Eulalisma
23 507/506 2.3.11.13.23 [-1 1 -1 2 -1 3.42 Laodicisma
23 529/528 2.3.11.23 [-4 -1 -1 2 3.28 Preziosisma
23 576/575 2.3.5.23 [6 2 -2 -1 3.01 Worcester comma
23 1288/1287 [3 -2 0 1 -1 -1 0 0 1 1.34 Triaphonisma
1. Ratios longer than 10 digits are presented by placeholders with informative hints

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperament
1 1\50 24.0 686/675 Sengagen
1 9\50 216.0 17/15 Tremka
1 11\50 264.0 7/6 Septimin
1 13\50 312.0 6/5 Oolong
1 17\50 408.0 325/256 Coditone
1 19\50 456.0 125/96 Qak
1 21\50 504.0 4/3 Meantone / meanpop
1 23\50 552.0 11/8 Emka
2 2\50 48.0 36/35 Pombe
2 3\50 72.0 25/24 Vishnu / vishnean
2 6\50 144.0 12/11 Bisemidim
2 9\50 216.0 17/15 Wizard / lizard / gizzard
2 12\50 288.0 13/11 Vines
2 21\50
(4\50)
504.0
(96.0)
4/3
(35/33)
Bimeantone
5 21\50
(1\50)
504.0
(24.0)
4/3
(49/48)
Cloudtone
5 23
(3\50)
552.0
(72.0)
11/8
(21/20)
Coblack
10 7\50
(3\50)
168.0
(72.0)
54/49
(25/24)
Decavish
10 21\50
(1\50)
504.0
(24.0)
4/3
(78/77)
Decic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Lumatone

## Music

### Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns
Gabriel Fauré
• Pavane, op. 50 (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020)
Akira Kamiya

Francium
Claudi Meneghin
Cam Taylor