Frequencies of Notes

[James Ellis]

To: PTG-CAUT
From: James Ellis <claviers@nxs.net>
Subject: Frequencies of Notes, etc.
Bcc: Jim Ellis

Elwood Doss asked for a table of the frequencies of "un-tempered" notes.
Don Mannino gave a good illustration of the problem here.  For any readers
on this list who might still be confused, let me illustrate it another way:

Let's pretend that inharmonicity does not exist, and let's calculate the
frequencies of pure contiguous Major thirds from A3 to A4.  A4 is 440, so
A3 is 220.  The ratio of a Major third is 4:5.  We calculate a series of
4:5 contiguous intervals, and we end up with:  A3=220, C#=275, F=373.75,
and finally, A4=429.6875.  A4 came out very flat, so we have to stretch the
contiguous Major thirds if we want to end up with an octave.

We can do the same thing with contiguous minor thirds, ratio of 5:6, and we
end up with A3=220, C=264, D#=316.8, F#=380.16, and A4=456.192.  A4 is now
sharp.  If we want to end up with an octave, we have to compress the
contiguous minor thirds.  When we go through these exercises, we find that
we have to "temper" EVERY interval except the octave.  That's why we call
it a "temperament".  Although Equal Temperament what we use most now, it's
only one of many tempered scales.

Now, enter "inharmonicity", and the overtones are no longer pure harmonics,
so we have to temper everything, including the octave.

I suspect that Elwood's question had to do with what the frequencies would
be if inharmonicity did not exist, and we could tune a pure "Equal
Temperament".  All we have to do here is multiply every chromatic step by
the 12th root of 2 if we are going up scate, or divide by the 12th root of
2 if we are going down scale.  The 12th root of 2 is 1.059463094, carried
to 10 significant figures (nine decimal places).  That's as far as I took
it, but that's plenty.  Start with A4 = 440 Hz and go up scale, one
chromatic step at a time, and down scale, one chromatic step at a time, and
you will end up with almost perfect equally tempered chromatic steps and
almost pure octaves.

I do have a table of frequencies, rounded off to four decimal places, that
I made up about 10 years ago when I was at the height of my research
project with longitudinal modes.  You end up with a lot of numbers.  But
you can calculate your own.  Just round off 1.059463094 to however many
places your calculator has, multiply each chromatic step by that number if
you are going up scale, and divide if you are going dowm, and you will come
out very close to a theoretical equal temperament, with NO correction for
inharmonicity.  I think that's what you were asking for.

Jim Ellis, RPT, Oak Ridge, TN