Pure 5ths ET.

[Kent Swafford]

On 4/11/01 4:40 PM, "Richard Moody" <remoody@midstatesd.net> wrote:

>   Using the cents approach the "2 cents fifth" is not accurate
> enough.  The actual figure is 1.95500....x   which gets rounded off
> to 1.96, which again gets rounded off to 2 cents.    Using 1.955 times
> 12 pure 5ths gets 23.46 cents.  Since this covers 7 octaves, each
> octave would have to be 3.514 cents wide. (23.46/7)  Now,  in one
> definition of ET, each semitone must be the same so that requires a
> value of the twelfth root NOT of two, but of the expanded octave!.
> The pure octave has a ratio of 2/1, the expanded octave will have to
> be 2.x/1.     Can we figure out the ratio of an octave expanded by
> 3.514 cents?  IF 2 were an actual frequency then using
> F*2^(3.5143/1200)  equals 2.003875474.   Plugging that in to my beat
> spread sheet, I get zero beats.

This is a bit more complex than needed. Use the _fifth_ as the basis for
calculating the frequencies of pure 5ths ET, not the octave.

In the mathematical model, two notes an octave apart will have frequencies
in the ratio of 2 to 1, and two notes a fifth apart will have frequencies in
the ratio of 1.5 to 1.

So - just as the 12-tone-to-the-octave ET scale consists of notes whose
frequencies increase by the ratio of one to the 12th root of 2, the
7-tone-to-the-pure-5th ET has notes whose frequencies increase by the ratio
of 1 to the 7th root of 1.5.

Kent Swafford