50edo: Difference between revisions

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'''50edo''' divides the [[octave]] into 50 equal parts of precisely 24 [[cent|cents]] each.

~~'''50edo''' divides the [[Octave|octave]] into 50 equal parts of precisely 24 [[cent|cents]] 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. 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.

== Theory ==

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.

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+'''50edo''' divides the [[octave]] into 50 equal parts of precisely 24 [[cent|cents]] each.
-'''50edo''' divides the [[Octave|octave]] into 50 equal parts of precisely 24 [[cent|cents]] 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 &ndash; 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.
+== Theory ==
+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 &ndash; 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.
 edo 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&amp;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&gt;, 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.