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<P>Don, Ed,
<P>Thanks for the information and direction to websites that explain these
things. The one below is from the Explore-Music site. Thought ot=
hers
might be interested as well.
<P>Dave Stahl
<P>
<P>"The <B>syntonic comma</B>, also known as the <B>comma of <A class=a-lk=
title=Didymus
href="http://www.explore-biography.com/musicians/D/Didymus_the_Musician.ht=
ml">Didymus</A></B>
or <B><A class=a-lk title=Ptolemaic
href="http://www.explore-biography.com/astronomers/P/Ptolemy.html">Ptolema=
ic</A>
comma</B>, is a small <A class=a-lk title=interval
href="http://www.explore-music.com/music_theory/I/Interval_%28music%29.htm=
l">interval</A>
between two <A class=a-lk title=music
href="http://www.explore-music.com/music/M/Music.html">music</A>al notes, =
equal
to the frequency ratio 81:80, or around 21.51 <A class=a-lk title=cents
href="http://www.explore-music.com/music_theory/C/Cent_%28music%29.html">c=
ents</A>.
<P>It is the difference between four <A class=a-lk title=justly
href="http://www.explore-music.com/music/J/Just_intonation.html">justly</A=
>
tuned <A class=a-lk title="perfect fifths"
href="http://www.explore-music.com/music_theory/P/Perfect_fifth.html">perf=
ect
fifths</A>, and two <A class=a-lk title=octaves
href="http://www.explore-music.com/music_theory/O/Octave.html">octaves</A>=
plus
a justly tuned <A class=a-lk title="major third"
href="http://www.explore-music.com/music_theory/M/Major_third.html">major
third</A>. A just perfect fifth has its notes in the <A class=a-lk
title=frequency
href="http://www.explore-physics.com/physics/F/Frequency.html">frequency</=
A>
ratio <A class=a-lk title=3:2
href="http://www.explore-math.com/mathematics/S/Sesquialterum.html">3:2</A=
>,
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 <A class=a-lk title=
=5:4
href="http://www.explore-math.com/mathematics/S/Sesquiquartum.html">5:4</A=
>,
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.
<P>This difference is significant because on a <A class=a-lk title=piano=
href="http://www.explore-music.com/musical_instruments/P/Piano.html">piano=
</A>
keyboard, four fifths is equal to two octaves plus a major third. Starting f=
rom
a C, both combinations of intervals will end up at E. The fact that using ju=
stly
tuned intervals yields two slightly different notes is one of the reasons
compromises have to be made when deciding which system of <A class=a-lk
title="musical tuning"
href="http://www.explore-music.com/musical_instruments/M/Musical_tuning.ht=
ml">musical
tuning</A> to use for an instrument. <A class=a-lk title="Pythagorean tu=
ning"
href="http://www.explore-music.com/music_theory/P/Pythagorean_tuning.html"=
>Pythagorean
tuning</A> tunes the fifths as exact 3:2s, but uses the relatively complex r=
atio
of 81:64 for major thirds. Quarter comma meantone, on the other hand, uses e=
xact
5:4s for major thirds, but flattens each of the fifths by a quarter of a
syntonic comma. Other systems use different
compromises."</P></DIV></FONT></BODY></HTML>