Just tuning and "sweetness"

[Piannaman@aol.com]

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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."

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