50edo: Difference between revisions

''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.

50 tempers out 126/125, 225/224 and 3136/3125 in the [[7-limit|7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit|11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit|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 15&50 temperament ([http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11 Coblack]), and provides the optimal patent val for 11 and 13 limit [[Meantone_family#Septimal meantone-Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma|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|7edo]], [[12edo|12edo]], [[19edo|19edo]], [[31edo|31edo]] as the next approximation to the "Golden Tone System" ([[Das_Goldene_Tonsystem|Das Goldene Tonsystem]]) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "[http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").

=Intervals=

 

|-

! | Degrees of 50edo

! | Cents value

! | 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

| | [[Hemimean_clan#Sengagen|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

| | [[Vishnuzmic_family#Vishnu|Vishnu]] (2/oct), [http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11 Coblack] (5/oct)

|-

| | 4

| | 96

| | 22/21

| | [[Meantone_family#Injera|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

| | [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)

|-

| | 10

| | 240

| | 8/7, 15/13

| |  

|-

| | 11

| | 264

| | 7/6

| | [[Marvel_temperaments#Septimin-13-limit|Septimin (13-limit)]]

|-

| | 12

| | 288

| | 13/11

| |  

|-

| | 13

| | 312

| | 6/5

| |  

|-

| | 14

| | 336

| | 27/22, 39/32, 40/33, 49/40

| |  

|-

| | 15

| | 360

| | 16/13, 11/9

| |  

|-

| | 16

| | 384

| | 5/4

| | [[Marvel_temperaments#Wizard-11-limit|Wizard]] (2/oct)

|-

| | 17

| | 408

| | 14/11

| | [[Ditonmic_family|Ditonic]]

|-

| | 18

| | 432

| | 9/7

| | [[Porcupine_family#Hedgehog|Hedgehog]] (50cc val, 2/oct)

|-

| | 19

| | 456

| | 13/10

| | [[Starling_temperaments#Bisemidim|Bisemidim]] (2/oct)

|-

| | 20

| | 480

| | 33/25, 55/42, 64/49

| |  

|-

| | 21

| | 504

| | 4/3

| | [[Meantone|Meantone]]/[[Meanpop|Meanpop]]

|-

| | 22

| | 528

| | 15/11

| |  

|-

| | 23

| | 552

| | 11/8, 18/13

| | [[Chromatic_pairs#Barton|Barton]], [[Hemimean_clan#Emka|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

| |  

| |  

|}

==Selected just intervals by error==

 

=Commas=

 

=Music=

[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 Twinkle canon – 50 edo] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]

 

[http://soonlabel.com/xenharmonic/archives/1118 Fantasia Catalana by Claudi Meneghin]

 

[http://soonlabel.com/xenharmonic/archives/1929 Fugue on the Dragnet theme by Claudi Meneghin]

 

[https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album the late little xmas album by Cam Taylor]

[https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor]

[https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo Long improvisation 2 in 50EDO by Cam Taylor]

[https://soundcloud.com/camtaylor-1/chord-sequence-for-difference Chord sequence for Difference tones in 50EDO by Cam Taylor]

[https://soundcloud.com/camtaylor-1/enharmonic-modulations-in Enharmonic Modulations in 50EDO by Cam Taylor]

[https://soundcloud.com/camtaylor-1/fragment-in-fifty Fragment in Fifty] by Cam Taylor

[http://www.archive.org/details/harmonicsorphilo00smit Robert Smith's book online]

[https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]