50edo

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 2^1/50, 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
Error	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^{[note 1]}	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

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
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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^{[note 2]}	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

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

Instruments

Lumatone

See Lumatone mapping for 50edo

Music

Modern renderings

Johann Sebastian Bach

"Contrapunctus 4" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)
"Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)

Nicolaus Bruhns

Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)

Gabriel Fauré

Pavane, op. 50 (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020)

Akira Kamiya

funfunfun ta yo (2007) – rendered by MortisTheneRd (2024)

21st century

Francium

On My Way To Somewhere (2023)

Claudi Meneghin

Twinkle canon – 50 edo ^{[dead link]}
Blue Fugue for Organ (2018)
La Petite Poule Grise - Fugue (2019)
Happy Birthday Canon, 6-in-1 Canon in 50edo (2019)
Fantasia Catalana (2020)
Preludi Nocturn i Fuga sobre la Lluna la Pruna (2020)
Fugue on the Dragnet theme (2020)
Canon at the Semitone on The Mother's Malison Theme, for Organ (2022)
Fugue on an Original Theme, for Baroque Ensemble (2023) (for Organ)
Catalan Fugue (La Santa Espina) (2023)
Canon in C= for Baroque Wind Ensemble (2023)
Fantasia Catalana, for Baroque Ensemble (2023)

Cam Taylor

the late little xmas album (2014)
Harpsichord meantone improvisation 1 in 50EDO (2014)
Long improvisation 2 in 50EDO (2014)
Chord sequence for Difference tones in 50EDO (2014)
Enharmonic Modulations in 50EDO (2014)
Harmonic Clusters on 50EDO Harpsichord (2014)
Fragment in Fifty (2014)

Additional reading

Notes

↑ Based on treating 13-limit as a temperament; other approaches are also possible.
↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

[1] Based on treating 13-limit as a temperament; other approaches are also possible.

[2] Ratios longer than 10 digits are presented by placeholders with informative hints.

[note 1]

[note 2]