Entry

why is the perfect 5th 7 tones from the root, and 5 tones from the octave?

Nov 10th, 2008 04:13
Asrar Khan, jeff carter,

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.