50edo: Difference between revisions

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m →Selected just intervals: 50 EDO is essentially 23-limit meantone because all the primes tend flat and thus a lot of error cancellation opportunities arise, furthermore, the prime 3 being almost perfectly 25% off justifies extra precision
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m Added EDO info box and notation
Line 1: Line 1:
{{Infobox ET
| Prime factorization = 2 x 5<sup>2</sup>
| Step size = 24.000
| Fifth = 29\50 = 696¢
| Major 2nd = 8\50 = 192¢
| Minor 2nd = 5\50 = 120¢
| Augmented 1sn = 3\50 = 72¢
}}
'''50edo''' divides the [[octave]] into 50 equal parts of precisely 24 [[cent|cents]] each.  
'''50edo''' divides the [[octave]] into 50 equal parts of precisely 24 [[cent|cents]] each.  


Line 17: Line 26:
! Cents
! Cents
! Ratios*
! Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! Generator for*
! Generator for*
|-
|-
Line 22: Line 32:
| 0
| 0
| 1/1
| 1/1
| Perfect 1sn
| P1
| D
|  
|  
|-
|-
Line 27: Line 40:
| 24
| 24
| 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168
| 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168
| Up 1sn, Dim 2nd
| ^1, d2
| ^D, Ebb
| [[Hemimean_clan#Sengagen|Sengagen]]
| [[Hemimean_clan#Sengagen|Sengagen]]
|-
|-
Line 32: Line 48:
| 48
| 48
| 33/32, 36/35, 50/49, 55/54, 64/63
| 33/32, 36/35, 50/49, 55/54, 64/63
|  
| Downaug 1sn, Updim 2nd
| vA1, ^d2
| vD#, ^Ebb
|-
|-
| 3
| 3
| 72
| 72
| 21/20, 25/24, 26/25, 27/26, 28/27
| 21/20, 25/24, 26/25, 27/26, 28/27
| Aug 1sn
| A1
| D#
| [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [[Starling temperaments #Coblack temperament|Coblack]] (5/oct)
| [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [[Starling temperaments #Coblack temperament|Coblack]] (5/oct)
|-
|-
Line 42: Line 63:
| 96
| 96
| 22/21
| 22/21
| Downminor 2nd
| vm2
| vEb
| [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
| [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
|-
|-
Line 47: Line 71:
| 120
| 120
| 16/15, 15/14, 14/13
| 16/15, 15/14, 14/13
| Minor 2nd
| m2
| Eb
|  
|  
|-
|-
Line 52: Line 79:
| 144
| 144
| 13/12, 12/11
| 13/12, 12/11
| Upminor 2nd
| ^m2
| ^Eb
|  
|  
|-
|-
Line 57: Line 87:
| 168
| 168
| 11/10
| 11/10
| Downmajor 2nd
| vM2
| vE
|  
|  
|-
|-
Line 62: Line 95:
| 192
| 192
| 9/8, 10/9
| 9/8, 10/9
| Major 2nd
| M2
| E
|  
|  
|-
|-
Line 67: Line 103:
| 216
| 216
| 25/22
| 25/22
| Upmajor 2nd
| ^M2
| ^E
| [http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&limit=2.3.5.11.13 Tremka], [[Subgroup_temperaments#x2.9.7.11-Machine|Machine]] (50b val)
| [http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&limit=2.3.5.11.13 Tremka], [[Subgroup_temperaments#x2.9.7.11-Machine|Machine]] (50b val)
|-
|-
Line 72: Line 111:
| 240
| 240
| 8/7, 15/13
| 8/7, 15/13
| Downaug 2nd, Dim 3rd
| vA2, d3
| vE#, Fb
|  
|  
|-
|-
Line 77: Line 119:
| 264
| 264
| 7/6
| 7/6
| Updim 3rd, Aug 2nd
| ^d3, A2
| ^Fb, E#
| [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
| [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
|-
|-
Line 82: Line 127:
| 288
| 288
| 13/11
| 13/11
| Downminor 3rd
| vm3
| vF
|  
|  
|-
|-
Line 87: Line 135:
| 312
| 312
| 6/5
| 6/5
| Minor 3rd
| m3
| F
| [[Oolong]]
| [[Oolong]]
|-
|-
Line 92: Line 143:
| 336
| 336
| 27/22, 39/32, 40/33, 49/40
| 27/22, 39/32, 40/33, 49/40
| Upminor 3rd
| ^m3
| ^F
|  
|  
|-
|-
Line 97: Line 151:
| 360
| 360
| 16/13, 11/9
| 16/13, 11/9
| Downmajor 3rd
| vM3
| vF#
|  
|  
|-
|-
Line 102: Line 159:
| 384
| 384
| 5/4
| 5/4
| Major 3rd
| M3
| F#
| [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
| [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
|-
|-
Line 107: Line 167:
| 408
| 408
| 14/11
| 14/11
| Upmajor 3rd
| ^M3
| ^F#
| [[Ditonmic_family|Ditonic]]
| [[Ditonmic_family|Ditonic]]
|-
|-
Line 112: Line 175:
| 432
| 432
| 9/7
| 9/7
| Downaug 3rd, Dim 4th
| vA3, d4
| vFx, Gb
| [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
| [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
|-
|-
Line 117: Line 183:
| 456
| 456
| 13/10
| 13/10
| Updim 4th, Aug 3rd
| A3, ^d4
| ^Gb, Fx
| [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
| [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
|-
|-
Line 122: Line 191:
| 480
| 480
| 33/25, 55/42, 64/49
| 33/25, 55/42, 64/49
| Down 4th
| v4
| vG
|  
|  
|-
|-
Line 127: Line 199:
| 504
| 504
| 4/3
| 4/3
| Perfect 4th
| P4
| G
| [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
| [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
|-
|-
Line 132: Line 207:
| 528
| 528
| 15/11
| 15/11
| Up 4th
| ^4
| ^G
|  
|  
|-
|-
Line 137: Line 215:
| 552
| 552
| 11/8, 18/13
| 11/8, 18/13
| Downaug 4th
| vA4
| vG#
| [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
| [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
|
|-
|-
| 24
| 24
| 576
| 576
| 7/5
| 7/5
| Aug 4th
| A4
| G#
|  
|  
|-
|-
Line 147: Line 232:
| 600
| 600
| 63/44, 88/63, 78/55, 55/39
| 63/44, 88/63, 78/55, 55/39
| Upaug 4th, Downdim 5th
| ^A4, vd5
| ^G#, vAb
|  
|  
|-
|-
Line 152: Line 240:
| 624
| 624
| 10/7
| 10/7
| Dim 5th
| d5
| Ab
|  
|  
|-
|-
Line 157: Line 248:
| 648
| 648
| 16/11, 13/9
| 16/11, 13/9
| Updim 5th
| ^d5
| ^Ab
|  
|  
|-
|-
Line 162: Line 256:
| 672
| 672
| 22/15
| 22/15
| Down 5th
| v5
| vA
|  
|  
|-
|-
Line 167: Line 264:
| 696
| 696
| 3/2
| 3/2
| Perfect 5th
| P5
| A
|  
|  
|-
|-
Line 172: Line 272:
| 720
| 720
| 50/33, 84/55, 49/32
| 50/33, 84/55, 49/32
| Up 5th
| ^5
| ^A
|  
|  
|-
|-
Line 177: Line 280:
| 744
| 744
| 20/13
| 20/13
|  
| Downaug 5th, Dim 6th
| vA5, d6
| vA#, Bbb
|-
|-
| 32
| 32
| 768
| 768
| 14/9
| 14/9
| Updim 6th, Aug 5th
| ^d6, A5
| ^Bbb, A#
|  
|  
|-
|-
Line 187: Line 295:
| 792
| 792
| 11/7
| 11/7
| Downminor 6th
| vm6
| vBb
|  
|  
|-
|-
Line 192: Line 303:
| 816
| 816
| 8/5
| 8/5
| Minor 6th
| m6
| Bb
|  
|  
|-
|-
Line 197: Line 311:
| 840
| 840
| 13/8, 18/11
| 13/8, 18/11
| Upminor 6th
| ^m6
| ^Bb
|  
|  
|-
|-
Line 202: Line 319:
| 864
| 864
| 44/27, 64/39, 33/20, 80/49
| 44/27, 64/39, 33/20, 80/49
| Downmajor 6th
| vM6
| vB
|  
|  
|-
|-
Line 207: Line 327:
| 888
| 888
| 5/3
| 5/3
| Major 6th
| M6
| B
|  
|  
|-
|-
Line 212: Line 335:
| 912
| 912
| 22/13
| 22/13
| Upmajor 6th
| ^M6
| ^B
|  
|  
|-
|-
Line 217: Line 343:
| 936
| 936
| 12/7
| 12/7
| Downaug 6th, Dim 7th
| vA6, d7
| vB#, Cb
|  
|  
|-
|-
Line 222: Line 351:
| 960
| 960
| 7/4
| 7/4
| Updim 7th, Aug 6th
| ^d7, A6
| ^Cb, B#
|  
|  
|-
|-
Line 227: Line 359:
| 984
| 984
| 44/25
| 44/25
| Downminor 7th
| vm7
| vC
|  
|  
|-
|-
Line 232: Line 367:
| 1008
| 1008
| 16/9, 9/5
| 16/9, 9/5
| Minor 7th
| m7
| C
|  
|  
|-
|-
Line 237: Line 375:
| 1032
| 1032
| 20/11
| 20/11
| Upminor 7th
| ^m7
| ^C
|  
|  
|-
|-
Line 242: Line 383:
| 1056
| 1056
| 24/13, 11/6
| 24/13, 11/6
| Downmajor 7th
| vM7
| vC#
|  
|  
|-
|-
Line 247: Line 391:
| 1080
| 1080
| 15/8, 28/15, 13/7
| 15/8, 28/15, 13/7
| Major 7th
| M7
| C#
|  
|  
|-
|-
Line 252: Line 399:
| 1104
| 1104
| 21/11
| 21/11
| Upmajor 7th
| ^M7
| ^C#
|  
|  
|-
|-
Line 257: Line 407:
| 1128
| 1128
| 40/21, 48/25, 25/13, 52/27, 27/14
| 40/21, 48/25, 25/13, 52/27, 27/14
| Downaug 7th, Dim 8ve
| vA7, d8
| vCx, Db
|  
|  
|-
|-
Line 262: Line 415:
| 1152
| 1152
| 64/33, 35/18, 49/25, 108/55, 63/32
| 64/33, 35/18, 49/25, 108/55, 63/32
| Updim 8ve, Aug 7th
| ^d8, A7
| ^Db, Cx
|  
|  
|-
|-
Line 267: Line 423:
| 1176
| 1176
| 88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169
| 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
|  
|  
|-
|-
Line 272: Line 431:
| 1200
| 1200
| 2/1
| 2/1
| Perfect 8ve
| P8
| D
|  
|  
|}
|}

Revision as of 16:15, 7 June 2021

← 49edo 50edo 51edo →
Prime factorization 2 x 52
Step size 24 ¢ 
Fifth 29\50 (696 ¢)
Semitones (A1:m2) 3:5 (72 ¢ : 120 ¢)
Consistency limit 9
Distinct consistency limit 7

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* Ups and Downs Notation Generator for*
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, Dim 2nd ^1, d2 ^D, Ebb Sengagen
2 48 33/32, 36/35, 50/49, 55/54, 64/63 Downaug 1sn, Updim 2nd vA1, ^d2 vD#, ^Ebb
3 72 21/20, 25/24, 26/25, 27/26, 28/27 Aug 1sn A1 D# Vishnu (2/oct), Coblack (5/oct)
4 96 22/21 Downminor 2nd vm2 vEb Injera (50d val, 2/oct)
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 Tremka, Machine (50b val)
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# Septimin (13-limit)
12 288 13/11 Downminor 3rd vm3 vF
13 312 6/5 Minor 3rd m3 F Oolong
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# Wizard (2/oct)
17 408 14/11 Upmajor 3rd ^M3 ^F# Ditonic
18 432 9/7 Downaug 3rd, Dim 4th vA3, d4 vFx, Gb Hedgehog (50cc val, 2/oct)
19 456 13/10 Updim 4th, Aug 3rd A3, ^d4 ^Gb, Fx Bisemidim (2/oct)
20 480 33/25, 55/42, 64/49 Down 4th v4 vG
21 504 4/3 Perfect 4th P4 G Meantone/Meanpop
22 528 15/11 Up 4th ^4 ^G
23 552 11/8, 18/13 Downaug 4th vA4 vG# Barton, Emka
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, 52/27, 27/14 Downaug 7th, Dim 8ve vA7, d8 vCx, Db
48 1152 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 13-limit patent val, except as noted.

Just approximation

Selected just intervals

Script error: No such module "primes_in_edo".

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[1] Monzo Cents Name(s)
5 81/80 [-4 4 -1⟩ 21.51 Syntonic comma, Didymus comma
5 (20 digits) [-27 -2 13⟩ 18.17 Ditonma
5 (20 digits) [23 6 -14⟩ 3.34 Vishnuzma, Vishnu comma
7 59049/57344 [-13 10 0 -1⟩ 50.72 Harrison's comma
7 126/125 [1 2 -3 1⟩ 13.79 Starling comma, Small septimal comma
7 225/224 [-5 2 2 -1⟩ 7.71 Septimal kleisma, Marvel comma
7 3136/3125 [6 0 -5 2⟩ 6.08 Hemimean, Middle second comma
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, undecimal kleisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Swets' comma, Swetisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Wizardharry, undecimal schisma
11 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
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
13 364/363 [2 -1 0 1 -2 1⟩ 4.76 Gentle comma
13 2200/2197 [3 0 2 0 1 -3⟩ 2.36 Petrma, Parizek comma
17 170/169 [1 0 1 0 0 -2 1⟩ 10.21
17 221/220 [-2 0 -1 0 -1 1 1⟩ 7.85
17 289/288 [-5 -2 0 0 0 0 2⟩ 6.00 minor seconds comma
17 375/374 [-1 1 3 0 -1 0 -1⟩ 4.62
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
19 210/209 [1 1 1 1 -1 0 0 1⟩ 8.26
19 324/323 [2 4 0 0 0 0 -1 -1⟩ 5.35
19 361/360 [-3 -2 -1 0 0 0 0 2⟩ 4.80
19 495/494 [-1 2 1 0 1 -1 0 -1⟩ 3.50
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

Music

Additional reading

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