Xenharmonic Wiki

50edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
URL (talk | contribs)
114 edits
m Bolded "50edo".
Line 1: Line 1:
__FORCETOC__
__FORCETOC__
'''50edo''' divides the [[Octave|octave]] into 50 equal parts of precisely 24 [[cent|cent]]s each. In the [[5-limit|5-limit]], it tempers out 81/80, making it a [[Meantone|meantone]] system, and in that capacity has historically has drawn some notice. In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "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 [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While [[31edo|31edo]] extends meantone with a [[7/4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8|11/8]] and [[13/8|13/8]] are nearly pure.
'''50edo''' divides the [[Octave|octave]] into 50 equal parts of precisely 24 [[cent|cent]]s each. In the [[5-limit|5-limit]], it tempers out 81/80, making it a [[Meantone|meantone]] system, and in that capacity has historically has drawn some notice. In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "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 [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While [[31edo|31edo]] extends meantone with a [[7/4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8|11/8]] and [[13/8|13/8]] are nearly pure.


Line 13: Line 14:
! | Degrees of 50edo
! | Degrees of 50edo
! | Cents value
! | Cents value
!pions
!7mus
! | Ratios*
! | Ratios*
! | Generator for*
! | Generator for*
|-
|-
| | 0
| | 0
| | 0
| colspan="3"| 0
| | 1/1
| | 1/1
| |  
| |  
Line 23: Line 26:
| | 1
| | 1
| | 24
| | 24
|25.44
|30.72 (1E.B8<sub>16</sub>)
| | 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
| | [[Hemimean_clan#Sengagen|Sengagen]]
| | [[Hemimean_clan#Sengagen|Sengagen]]
Line 28: Line 33:
| | 2
| | 2
| | 48
| | 48
|50.88
|61.44 (3D.71<sub>16</sub>)
| | 33/32, 36/35, 50/49, 55/54, 64/63
| | 33/32, 36/35, 50/49, 55/54, 64/63
| |  
| |  
Line 33: Line 40:
| | 3
| | 3
| | 72
| | 72
|76.32
|92.16 (5C.29<sub>16</sub>)
| | 21/20, 25/24, 26/25, 27/26, 28/27
| | 21/20, 25/24, 26/25, 27/26, 28/27
| | [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11 Coblack] (5/oct)
| | [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11 Coblack] (5/oct)
Line 38: Line 47:
| | 4
| | 4
| | 96
| | 96
|101.76
|122.88 (7A.E1<sub>16</sub>)
| | 22/21
| | 22/21
| | [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
| | [[Meantone_family#Injera|Injera]] (50d val, 2/oct)
Line 43: Line 54:
| | 5
| | 5
| | 120
| | 120
|127.2
|153.6 (99.9A<sub>16</sub>)
| | 16/15, 15/14, 14/13
| | 16/15, 15/14, 14/13
| |  
| |  
Line 48: Line 61:
| | 6
| | 6
| | 144
| | 144
|152.64
|184.32 (B8.52<sub>16</sub>)
| | 13/12, 12/11
| | 13/12, 12/11
| |  
| |  
Line 53: Line 68:
| | 7
| | 7
| | 168
| | 168
|178.08
|215.04 (D7.0A<sub>16</sub>)
| | 11/10
| | 11/10
| |  
| |  
Line 58: Line 75:
| | 8
| | 8
| | 192
| | 192
|203.52
|245.76 (F5.C3<sub>16</sub>)
| | 9/8, 10/9
| | 9/8, 10/9
| |  
| |  
Line 63: Line 82:
| | 9
| | 9
| | 216
| | 216
|228.96
|276.48 (114.7B<sub>16</sub>)
| | 25/22
| | 25/22
| | [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 68: Line 89:
| | 10
| | 10
| | 240
| | 240
|254.4
|307.2 (133.33<sub>16</sub>)
| | 8/7, 15/13
| | 8/7, 15/13
| |  
| |  
Line 73: Line 96:
| | 11
| | 11
| | 264
| | 264
|279.88
|337.92 (151.EB8<sub>16</sub>)
| | 7/6
| | 7/6
| | [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
| | [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]
Line 78: Line 103:
| | 12
| | 12
| | 288
| | 288
|305.28
|368.64 (170.A4<sub>16</sub>)
| | 13/11
| | 13/11
| |  
| |  
Line 83: Line 110:
| | 13
| | 13
| | 312
| | 312
|330.72
|399.36 (18F.5C<sub>16</sub>)
| | 6/5
| | 6/5
| |  
| |  
Line 88: Line 117:
| | 14
| | 14
| | 336
| | 336
|356.16
|430.08 (1AE.148<sub>16</sub>)
| | 27/22, 39/32, 40/33, 49/40
| | 27/22, 39/32, 40/33, 49/40
| |  
| |  
Line 93: Line 124:
| | 15
| | 15
| | 360
| | 360
|381.6
|460.8 (1CC.CD<sub>16</sub>)
| | 16/13, 11/9
| | 16/13, 11/9
| |  
| |  
Line 98: Line 131:
| | 16
| | 16
| | 384
| | 384
|407.04
|491.52 (1EB.85<sub>16</sub>)
| | 5/4
| | 5/4
| | [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
| | [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)
Line 103: Line 138:
| | 17
| | 17
| | 408
| | 408
|432.48
|522.24 (20A.4<sub>16</sub>)
| | 14/11
| | 14/11
| | [[Ditonmic_family|Ditonic]]
| | [[Ditonmic_family|Ditonic]]
Line 108: Line 145:
| | 18
| | 18
| | 432
| | 432
|457.92
|552.96 (228.F6<sub>16</sub>)
| | 9/7
| | 9/7
| | [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
| | [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)
Line 113: Line 152:
| | 19
| | 19
| | 456
| | 456
|483.36
|583.68 (247.AE<sub>16</sub>)
| | 13/10
| | 13/10
| | [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
| | [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)
Line 118: Line 159:
| | 20
| | 20
| | 480
| | 480
|508.8
|614.4 (266.66<sub>16</sub>)
| | 33/25, 55/42, 64/49
| | 33/25, 55/42, 64/49
| |  
| |  
Line 123: Line 166:
| | 21
| | 21
| | 504
| | 504
|534.24
|645.12 (285.1F<sub>16</sub>)
| | 4/3
| | 4/3
| | [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
| | [[Meantone|Meantone]]/[[Meanpop|Meanpop]]
Line 128: Line 173:
| | 22
| | 22
| | 528
| | 528
|559.68
|675.84 (2A3.D7<sub>16</sub>)
| | 15/11
| | 15/11
| |  
| |  
Line 133: Line 180:
| | 23
| | 23
| | 552
| | 552
|585.12
|706.56 (2C2.8F<sub>16</sub>)
| | 11/8, 18/13
| | 11/8, 18/13
| | [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
| | [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|Emka]]
Line 138: Line 187:
| | 24
| | 24
| | 576
| | 576
|610.56
|737.28 (2E1.48<sub>16</sub>)
| | 7/5
| | 7/5
| |  
| |  
Line 143: Line 194:
| | 25
| | 25
| | 600
| | 600
|636
|768 (300<sub>16</sub>)
| | 63/44, 88/63, 78/55, 55/39
| | 63/44, 88/63, 78/55, 55/39
| |  
| |  
Line 148: Line 201:
| | 26
| | 26
| | 624
| | 624
|661.44
|798.72 (31E.B8<sub>16</sub>)
| | 10/7
| | 10/7
| |  
| |  
Line 153: Line 208:
| | 27
| | 27
| | 648
| | 648
|686.88
|829.44 (33D.71<sub>16</sub>)
| | 16/11, 13/9
| | 16/11, 13/9
| |  
| |  
Line 158: Line 215:
| | 28
| | 28
| | 672
| | 672
|712.32
|860.16 (35C.29<sub>16</sub>)
| | 22/15
| | 22/15
| |  
| |  
Line 163: Line 222:
| | 29
| | 29
| | 696
| | 696
|737.76
|900.88 (37A.E1<sub>16</sub>)
| | 3/2
| | 3/2
| |  
| |  
Line 168: Line 229:
| | 30
| | 30
| | 720
| | 720
|763.2
|921.6 (399.9A<sub>16</sub>)
| | 50/33, 84/55, 49/32
| | 50/33, 84/55, 49/32
| |  
| |  
Line 173: Line 236:
| | 31
| | 31
| | 744
| | 744
|788.64
|952.32 (3B8.52<sub>16</sub>)
| | 20/13
| | 20/13
| |  
| |  
Line 178: Line 243:
| | 32
| | 32
| | 768
| | 768
|814.08
|983.04 (3D7.0A<sub>16</sub>)
| | 14/9
| | 14/9
| |  
| |  
Line 183: Line 250:
| | 33
| | 33
| | 792
| | 792
|239.52
|1013.76 (3F5.C3<sub>16</sub>)
| | 11/7
| | 11/7
| |  
| |  
Line 188: Line 257:
| | 34
| | 34
| | 816
| | 816
|864.96
|1044.48 (414.7B<sub>16</sub>)
| | 8/5
| | 8/5
| |  
| |  
Line 193: Line 264:
| | 35
| | 35
| | 840
| | 840
|890.4
|1095.2 (433.33<sub>16</sub>)
| | 13/8, 18/11
| | 13/8, 18/11
| |  
| |  
Line 198: Line 271:
| | 36
| | 36
| | 864
| | 864
|915.84
|1105.92 (451.EB8<sub>16</sub>)
| | 44/27, 64/39, 33/20, 80/49
| | 44/27, 64/39, 33/20, 80/49
| |  
| |  
Line 203: Line 278:
| | 37
| | 37
| | 888
| | 888
|941.28
|1136.64 (470.A4<sub>16</sub>
| | 5/3
| | 5/3
| |  
| |  
Line 208: Line 285:
| | 38
| | 38
| | 912
| | 912
|966.72
|1167.36 (48F.5C<sub>16</sub>)
| | 22/13
| | 22/13
| |  
| |  
Line 213: Line 292:
| | 39
| | 39
| | 936
| | 936
|992.16
|1198.08 (4AE.148<sub>16</sub>)
| | 12/7
| | 12/7
| |  
| |  
Line 218: Line 299:
| | 40
| | 40
| | 960
| | 960
|1017.6
|1228.8 (4CC.CD<sub>16</sub>)
| | 7/4
| | 7/4
| |  
| |  
Line 223: Line 306:
| | 41
| | 41
| | 984
| | 984
|1043.04
|1261.52 (4EB.85<sub>16</sub>)
| | 44/25
| | 44/25
| |  
| |  
Line 228: Line 313:
| | 42
| | 42
| | 1008
| | 1008
|1068.48
|1290.24 (50A.4<sub>16</sub>)
| | 16/9, 9/5
| | 16/9, 9/5
| |  
| |  
Line 233: Line 320:
| | 43
| | 43
| | 1032
| | 1032
|1093.92
|1320.96 (528.F6<sub>16</sub>)
| | 20/11
| | 20/11
| |  
| |  
Line 238: Line 327:
| | 44
| | 44
| | 1056
| | 1056
|1119.36
|1351.68 (547.AE<sub>16</sub>)
| | 24/13, 11/6
| | 24/13, 11/6
| |  
| |  
Line 243: Line 334:
| | 45
| | 45
| | 1080
| | 1080
|1144.8
|1382.4 (566.66<sub>16</sub>)
| | 15/8, 28/15, 13/7
| | 15/8, 28/15, 13/7
| |  
| |  
Line 248: Line 341:
| | 46
| | 46
| | 1104
| | 1104
|1170.24
|1413.12 (585.1F<sub>16</sub>)
| | 21/11
| | 21/11
| |  
| |  
Line 253: Line 348:
| | 47
| | 47
| | 1128
| | 1128
|1195.68
|1443.84 (5A3.D7<sub>16</sub>)
| | 40/21, 48/25, 25/13, 52/27, 27/14
| | 40/21, 48/25, 25/13, 52/27, 27/14
| |  
| |  
Line 258: Line 355:
| | 48
| | 48
| | 1152
| | 1152
|1221.12
|1474.56 (5C2.8F<sub>16</sub>)
| | 64/33, 35/18, 49/25, 108/55, 63/32
| | 64/33, 35/18, 49/25, 108/55, 63/32
| |  
| |  
Line 263: Line 362:
| | 49
| | 49
| | 1176
| | 1176
|1246.56
|1505.28 (5E1.48<sub>16</sub>)
| | 88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169
| |  
| |  
| |  
|-
|50
|1200
|1272
|1536 (600<sub>16</sub>)
|2/1
|
|}
|}
*Using the 13-limit patent val, except as noted.
*Using the 13-limit patent val, except as noted.
Line 442: Line 550:
! | Name 2
! | Name 2
|-
|-
| | | -4 4 -1 &gt;
| |<nowiki> | -4 4 -1 </nowiki>&gt;
| style="text-align:right;" | 21.51
| style="text-align:right;" | 21.51
| style="text-align:center;" | 81/80
| style="text-align:center;" | 81/80
Line 448: Line 556:
| | Didymus comma
| | Didymus comma
|-
|-
| | | -27 -2 13 &gt;
| |<nowiki> | -27 -2 13 </nowiki>&gt;
| style="text-align:right;" | 18.17
| style="text-align:right;" | 18.17
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 454: Line 562:
| |  
| |  
|-
|-
| | | 23 6 -14 &gt;
| |<nowiki> | 23 6 -14 </nowiki>&gt;
| style="text-align:right;" | 3.34
| style="text-align:right;" | 3.34
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 460: Line 568:
| |  
| |  
|-
|-
| | | 1 2 -3 1 &gt;
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| style="text-align:right;" | 13.79
| style="text-align:right;" | 13.79
| style="text-align:center;" | 126/125
| style="text-align:center;" | 126/125
Line 466: Line 574:
| | Small septimal comma
| | Small septimal comma
|-
|-
| | | -5 2 2 -1 &gt;
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| style="text-align:right;" | 7.71
| style="text-align:right;" | 7.71
| style="text-align:center;" | 225/224
| style="text-align:center;" | 225/224
Line 472: Line 580:
| | Marvel comma
| | Marvel comma
|-
|-
| | | 6 0 -5 2 &gt;
| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| style="text-align:right;" | 6.08
| style="text-align:right;" | 6.08
| style="text-align:center;" | 3136/3125
| style="text-align:center;" | 3136/3125
Line 478: Line 586:
| | Middle second comma
| | Middle second comma
|-
|-
| | | -6 -8 2 5 &gt;
| |<nowiki> | -6 -8 2 5 </nowiki>&gt;
| style="text-align:right;" | 1.12
| style="text-align:right;" | 1.12
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 484: Line 592:
| |  
| |  
|-
|-
| | |-11 2 7 -3 &gt;
| |<nowiki> |-11 2 7 -3 </nowiki>&gt;
| style="text-align:right;" | 1.63
| style="text-align:right;" | 1.63
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 490: Line 598:
| |  
| |  
|-
|-
| | | 11 -10 -10 10 &gt;
| |<nowiki> | 11 -10 -10 10 </nowiki>&gt;
| style="text-align:right;" | 5.57
| style="text-align:right;" | 5.57
| style="text-align:center;" |  
| style="text-align:center;" |  
Line 496: Line 604:
| |  
| |  
|-
|-
| | |-13 10 0 -1 &gt;
| |<nowiki> |-13 10 0 -1 </nowiki>&gt;
| style="text-align:right;" | 50.72
| style="text-align:right;" | 50.72
| style="text-align:center;" | 59049/57344
| style="text-align:center;" | 59049/57344
Line 502: Line 610:
| |  
| |  
|-
|-
| | | 2 3 1 -2 -1 &gt;
| |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| style="text-align:right;" | 3.21
| style="text-align:right;" | 3.21
| style="text-align:center;" | 540/539
| style="text-align:center;" | 540/539
Line 508: Line 616:
| | Swetisma
| | Swetisma
|-
|-
| | | -3 4 -2 -2 2 &gt;
| |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| style="text-align:right;" | 0.18
| style="text-align:right;" | 0.18
| style="text-align:center;" | 9801/9800
| style="text-align:center;" | 9801/9800
Line 514: Line 622:
| | Gauss' comma
| | Gauss' comma
|-
|-
| | | 5 -1 3 0 -3 &gt;
| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
| style="text-align:right;" | 3.03
| style="text-align:right;" | 3.03
| style="text-align:center;" | 4000/3993
| style="text-align:center;" | 4000/3993
Line 520: Line 628:
| | Undecimal schisma
| | Undecimal schisma
|-
|-
| | | -7 -1 1 1 1 &gt;
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:right;" | 4.50
| style="text-align:right;" | 4.50
| style="text-align:center;" | 385/384
| style="text-align:center;" | 385/384
Line 526: Line 634:
| | Undecimal kleisma
| | Undecimal kleisma
|-
|-
| | | -1 0 1 2 -2 &gt;
| |<nowiki> | -1 0 1 2 -2 </nowiki>&gt;
| style="text-align:right;" | 21.33
| style="text-align:right;" | 21.33
| style="text-align:center;" | 245/242
| style="text-align:center;" | 245/242
Line 532: Line 640:
| |  
| |  
|-
|-
| | | 2 -1 0 1 -2 1 &gt;
| |<nowiki> | 2 -1 0 1 -2 1 </nowiki>&gt;
| style="text-align:right;" | 4.76
| style="text-align:right;" | 4.76
| style="text-align:center;" | 364/363
| style="text-align:center;" | 364/363
Line 538: Line 646:
| |  
| |  
|-
|-
| | | 2 -1 -1 2 0 -1 &gt;
| |<nowiki> | 2 -1 -1 2 0 -1 </nowiki>&gt;
| style="text-align:right;" | 8.86
| style="text-align:right;" | 8.86
| style="text-align:center;" | 196/195
| style="text-align:center;" | 196/195
Line 544: Line 652:
| |  
| |  
|-
|-
| | | 2 3 0 -1 1 -2 &gt;
| |<nowiki> | 2 3 0 -1 1 -2 </nowiki>&gt;
| style="text-align:right;" | 7.30
| style="text-align:right;" | 7.30
| style="text-align:center;" | 1188/1183
| style="text-align:center;" | 1188/1183
Line 550: Line 658:
| |  
| |  
|-
|-
| | | 3 0 2 0 1 -3 &gt;
| |<nowiki> | 3 0 2 0 1 -3 </nowiki>&gt;
| style="text-align:right;" | 2.36
| style="text-align:right;" | 2.36
| style="text-align:center;" | 2200/2197
| style="text-align:center;" | 2200/2197
Line 556: Line 664:
| | Parizek comma
| | Parizek comma
|-
|-
| | | -3 1 1 1 0 -1 &gt;
| |<nowiki> | -3 1 1 1 0 -1 </nowiki>&gt;
| style="text-align:right;" | 16.57
| style="text-align:right;" | 16.57
| style="text-align:center;" | 105/104
| style="text-align:center;" | 105/104
Line 563: Line 671:
| |  
| |  
|-
|-
| | | 4 2 0 0 -1 -1 &gt;
| |<nowiki> | 4 2 0 0 -1 -1 </nowiki>&gt;
| style="text-align:right;" | 12.06
| style="text-align:right;" | 12.06
| style="text-align:center;" | 144/143
| style="text-align:center;" | 144/143
Line 569: Line 677:
| |  
| |  
|-
|-
| | | 3 -2 0 1 -1 -1 0 0 1 &gt;
| |<nowiki> | 3 -2 0 1 -1 -1 0 0 1 </nowiki>&gt;
| style="text-align:right;" | 1.34
| style="text-align:right;" | 1.34
| style="text-align:center;" | 1288/1287
| style="text-align:center;" | 1288/1287
Line 604: Line 712:
[https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]
[https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]


[http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]      [[Category:50edo]]
[http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]       
[[Category:50edo]]
[[Category:edo]]
[[Category:edo]]
[[Category:golden]]
[[Category:golden]]
Retrieved from "https://en.xen.wiki/w/50edo"