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       #Post#: 114--------------------------------------------------
       MUSIC MATH HARMONY
       By: eba95 Date: July 30, 2010, 6:45 am
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       Music Math Harmony
       It is a remarkable(!) coincidence that
       27/12 is very close to 3/2.
       Why?
       Harmony occurs in music when two
       pitches vibrate at frequencies in
       small integer ratios.
       For instance, the notes of middle C
       and high C sound good together
       (concordant) because the latter has
       TWICE the frequency of the former.
       Middle C and the G above it sound
       good together because the
       frequencies of G and C are in a 3:2
       ratio.
       Well, almost!
       In the 16th century the popular
       method for tuning a piano was to a
       just-toned scale. What this means is
       that harmonies with the
       fundamental note (tonic) of the scale
       were pure; i.e., the frequency ratios
       were pure integer ratios. But
       because of this, shifting the melody
       to other keys would make the music
       sound different (and bad) because
       the harmonies in other keys were
       impure!
       So, the equal-tempered scale (in
       common use today), popularized by
       Bach, sets out to "even out" the
       badness by making the frequency
       ratios the same between all 12 notes
       of the chromatic scale (the white and
       the black keys on a piano). Thus,
       harmonies shifted to other keys
       would sound exactly the same,
       although a really good ear might be
       able to tell that the harmonies in the
       equal-tempered scale are not quite
       pure.
       So to divide the ratio 2:1 from high C
       to middle C into 12 equal parts, we
       need to make the ratios between
       successive note frequencies 21/12:1.
       The startling fact that 27/12 is very
       close to 3/2 ensures that the interval
       between C and G, which are 7 notes
       apart in the chromatic scale, sounds
       "almost" pure! Most people cannot
       tell the difference!
       What a harmonious coincidence!
       The Math Behind the Fact:
       It is possible that our octave might
       be divided into something other
       than 12 equal parts if the above
       coincidence were not true!
       It is worth noting that on a stringed
       instrument, a player has complete
       control over the frequency of notes.
       So she can produce pure harmonies.
       Very good string players will actually
       play A-sharp and B-flat differently.
       Sometimes they don't even realize
       they are doing it--- they just do what
       their ear tells them. Playing pure
       harmonies on stringed instruments
       also means that the same note will
       sound different depending on the
       key the music is played in; for
       instance, notes based on 'C' will
       produce slightly different
       frequencies than those based on 'A'.
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