Frequencies of Notes

[Richard E. West]

Hey, Jim

Good writing!  You have given the  most succinct and accurate description of
our tuning dilemmas and how we deal with them in pianos.

Richard West, University of Nebraska

James Ellis wrote:

> 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
>
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