Pure 5ths ET.

[Richard Moody]

----- Original Message -----
From: Kent Swafford <kswafford@earthlink.net>
To: pianotech list <pianotech@ptg.org>
Sent: Wednesday, April 11, 2001 5:30 PM
Subject: Re: Pure 5ths ET.

Richard Moody wrote.....
| > 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

|
| 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

Quite right and good point.   I would have never thought of that.   My
spread sheet was set up for the 2:1 or 1200 cent octave so I numbly
went from there.  (And I did want to know the ratio of the stretched
octave to get pure 5ths)(but you can get that easier by using your
method)

Using cents you don't need the log calculations (since cents are logs
to begin with.)
So by adding the cents value of 1/7 of the comma of Pythagorus on to
the 1200 cent octave  and divide by 12.   you get 1203.514/12 = 100.28
cents.
 Or on your suggestion simply divide 701.995 by 7.  You get the same
value with one less step, and easier to comprhend I think.  The
701.995 is the precise cent value of a pure 5th and 7 notes
(semitines) makes up a 5th.  Each one then has a value of 100.28.
Twelve of these notes makes a pure 5th ET in a stretched octave.

With this you can off set the ETD by adding .28 cents succesively to
each note starting at A4 ......A#(4) + .28  , B + .56 and so on.
While of course subtracting the same way going down from A4.    Makes
me wish I had one. (as if...)  But Jim Coleman has posted his aural
procedures (in the archives), and my spreadsheet gives the beats.

BTW if anyone wants the spreadsheet that calculates the frequencies
and beats (including the octaves) of the Coleman Pure Fifth
Temperament, or simply the readout drop me an email.  ---ric

| 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.