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The answer is that I don't think it is ! The perfect 5th is ( in theory)
!/ 2 the distane from the root. (E.g. : C up to G is a 7 semitone
distance, while G to C ( one octave above the first C ), is only a 5
semi-tone distance. This means that the distance from root note to
"perfect" fifth, may be ( very close to ) !/ 2 the distance to the
octave.
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However, the spacing of notes between root to fifth, and fifth to octave
is not equal ! The perfect fifth should be be 6 semi-tones sharp of the
root, The distance from that fifth, to the octave, should also be 6
semi-tones, not 5. This means that all of the intervening notes are not
spaced evenly, as equal temperament suggests. If the calculations are
correct to determine 1/2 of the distance from root to fifth, then
fifth to octave, as well as fractions of the octave based on a
denominator of 12 ( root frequency to octave ), the actual frequency of
the individual notes becomes vastly different from equal temper.This
would result in a perfect fifth bieng based on what we now call a
diminished fifth. So, a perfect fifth from A to E, is now, on a
keyboard, A to D#.( six semi-tones). D# to the octave ( A ) is also six
semi-tones.( D# to A ). This would spread out the remaining notes as per
the following calculations : ( using analog X frquency converter )
Octave =A 440 hz , root = A 220 hz
A#= 1/12 from octave =(220 x 0.08333…=18.333…)+220=238.333….
Converts to : A# ( equal temp.); + 39 cents.
B=2/12 ( or 1/6) from octave,=(220 x 0.1666…=36.666…)+220=256.666…
Converts to : C - 33 cents ( or B + 67)
C=3/12(or 1/ 4 )from octave=(220 x 0.25=55)+220=270
Converts to C#. -45 cents ( or C +55 )
C#=4/12(or 1/3) from octave=(220 x 0.333….=73.333….)+220=293.333….
Converts to D - 2 cents ( or C# +98)
D=5/12 from octave =(220 x 0.41666….=91.666….)+220=311.666…..
Converts to D# + 3 cents.
D#=6/12 (or 1/ 2 ) from octave = (220 x 0.5 =110 )+220=330
Converts to E + 2 cents.
E=7/12 from octave = (220 x 0.58333….= 128.333…) + 220 = 348.333…
Converts to F -4 cents
F=8/12 (or 2/3 ) from octave = (220 x 0.666….= 146.666…) + 220 =
366.666…
Converts to F#. -16 cents.
F#=9/12 (or 3/ 4 ) from octave = (220 x 0.75 = 165 ) + 220 = 385.
Converts to G -31 cents.
G= 10/12 ( or 5/6) from octave = (220 x 0.8333… = 183.333…) + 220 =
403.333…
Converts to G + 49 cents.
G# = 11/12 from octave = (220 x 0.91666…= 201.666…) + 220 = 421.666…
Converts to G# + 26 cents.
This scale results in even spacing from root to fifth, fifth to octave,
as well as an even distribution of notes in between each, regardless of
the rules of equal temperament. An A magor chord now needs to be played
as an A magor DIMINISHED. ( eg: A + C# + D# ).to be harmonically "pure."
Any questions or comments, e-mail me at jeffcarter@hotmail.com.
