50edo: Difference between revisions

In the [[5-limit|5-limit]], 50edo 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. 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|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.

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]").

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

50 EDO tempers out the following commas. (Note: This assumes the val {{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.

[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/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq Harmonic Clusters on 50EDO Harpsichord 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]

 

[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html More information about Robert Smith's temperament]

 

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

 

[http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]