---------------------- multipart/alternative attachment Don, Ed, Thanks for the information and direction to websites that explain these things. The one below is from the Explore-Music site. Thought others might be interested as well. Dave Stahl "The syntonic comma, also known as the comma of _Didymus_ (http://www.explore-biography.com/musicians/D/Didymus_the_Musician.html) or _Ptolemaic_ (http://www.explore-biography.com/astronomers/P/Ptolemy.html) comma, is a small _interval_ (http://www.explore-music.com/music_theory/I/Interval_(music).html) between two _music_ (http://www.explore-music.com/music/M/Music.html) al notes, equal to the frequency ratio 81:80, or around 21.51 _cents_ (http://www.explore-music.com/music_theory/C/Cent_(music).html) . It is the difference between four _justly_ (http://www.explore-music.com/music/J/Just_intonation.html) tuned _perfect fifths_ (http://www.explore-music.com/music_theory/P/Perfect_fifth.html) , and two _octaves_ (http://www.explore-music.com/music_theory/O/Octave.html) plus a justly tuned _major third_ (http://www.explore-music.com/music_theory/M/Major_third.html) . A just perfect fifth has its notes in the _frequency_ (http://www.explore-physics.com/physics/F/Frequency.html) ratio _3:2_ (http://www.explore-math.com/mathematics/S/Sesquialterum.html) , which is equal to 702.96 cents, and four of them are equal to 2807.82 cents. A just major third has its notes in the frequency ratio _5:4_ (http://www.explore-math.com/mathematics/S/Sesquiquartum.html) , which is equal to 386.31 cents, and one of them plus two octaves is equal to 2786.31 cents. The difference between these is 21.51, a syntonic comma. This difference is significant because on a _piano_ (http://www.explore-music.com/musical_instruments/P/Piano.html) keyboard, four fifths is equal to two octaves plus a major third. Starting from a C, both combinations of intervals will end up at E. The fact that using justly tuned intervals yields two slightly different notes is one of the reasons compromises have to be made when deciding which system of _musical tuning_ (http://www.explore-music.com/musical_instruments/M/Musical_tuning.html) to use for an instrument. _Pythagorean tuning_ (http://www.explore-music.com/music_theory/P/Pythagorean_tuning.html) tunes the fifths as exact 3:2s, but uses the relatively complex ratio of 81:64 for major thirds. Quarter comma meantone, on the other hand, uses exact 5:4s for major thirds, but flattens each of the fifths by a quarter of a syntonic comma. Other systems use different compromises." ---------------------- multipart/alternative attachment An HTML attachment was scrubbed... URL: https://www.moypiano.com/ptg/pianotech.php/attachments/73/b5/bc/2c/attachment.htm ---------------------- multipart/alternative attachment--
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