It seems that the maths was wrong! For all the mathematicians, here is a
mathematically correct version.




"
Putting aside the wonderful world of just temperment for a moment, think
of the piano keyboard as the embodiment of equal temperment, which it is.
The distance from one note to it's octave, a ratio of 1 to 2, is divided
into twelve equal steps. But the way that is done is to multiply each note
by a factor that is the value 2 to the 1/12 power, or 1.059463094. For my
table below I'm going to call this factor "f".

Lets start with a hypothetical note of 100 Hz, because it's a pretty
number, helping us to recognize the fifth and octave of 150 Hz, and 200 Hz,
respectively. (Ed, forgive me if I use the wrong name for a solfeg note)

DO = 100
DI = 100 * f = 105.946
RE = 105.946 * f = 112.246 NOTE: Equal temp. 2nd is flat (just = 112.5)
RI = 112.246 * f = 118.921
MI = 118.921 * f = 125.992 NOTE: Equal tmp. 3rd is sharp (just = 125.0)
FA = 125.992 * f = 133.484 NOTE: Equal temp. 4th is sharp (just = 133.34)
FI = 133.484 * f = 141.421
SO = 141.421 * f = 149.830 NOTE: Equal temp. 5th is flat (just = 150.0)
SI = 149.830 * f = 158.740
LA = 158.740 * f = 168.179 NOTE: Equal temp. 6th is sharp (just = 166.67)
LI = 168.179 * f = 178.180 NOTE: Equal temp. minor 7th is waaay sharp!
(just = 175.0)
TI = 178.180 * f = 188.775
DO = 188.775 * f = 200

Anyway, because these notes are a geometric progression, each figured by
mutliplying a factor by it's previous neighbor, rather than a linear
progression (just taking twelfths of the octave), the frequency that is 1.5
the root is actually 7 of the twelve steps up.

I'm not sure I explained that in a way that will make you happy, but I
tried.

"

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