Towards Pure(er) 5ths in ET

[Richard Moody]

Tom Cole wrote
| I've always thought that the fourth beating faster than the fifth
was
| indicative of an octave stretched beyond pure


If you check Braid White p.68 you will see that the 4th on the top
beats twice as fast the 5th on the bottom.   It is when the 4th is on
the bottom that they both beat the same.
I never realized this until it caught my eye. Or maybe I heard it and
Braid White comfirmed it.
     It is not intuitive because in a pure octave, a pure 5 gives a
pure 4 and vice versa.  And when you contract the 5th the 4 must widen
and you would think by the same rate.   However this is true only when
the 4th is at the bottom (root).   C4--F4 beats the same as F4-C5.
BUT C4--G4   is one half the rate of G4--C5    The coincident partials
are higher, but I don' know if I would call them an octave higher.
All I know is that the arithmetic works out.
    In the Coleman Pure 5th all this all goes out the window because
all the octaves are stretched, more than an ET 4th.   If you have a
beatless 5th (bottom) and the 4th (top)is beating you have a stretched
octave but that is what you are supposed to have in the Pure 5th ET.
`But in regular ET in a pure octave the 5th and 4th have a beat rate.
The partials of one configuration are 4--3 to 3--2...here they beat
the same.  In the other, 3--2 to 4--3.  the 4th is beating twice the
rate of the 5th.    I suppose if you were fussy about 4:2 octaves you
would be better off tuning the 4-3-2 configuration. ---ric

----- Original Message -----
From: Tom Cole <tcole@cruzio.com>
To: <pianotech@ptg.org>
Sent: Wednesday, April 11, 2001 6:33 PM
Subject: Re: Towards Pure(er) 5ths in ET


| I've always thought that the fourth beating faster than the fifth
was
| indicative of an octave stretched beyond pure, that, as Jim just
said
| about his pure fifths temperament, you are expanding not only the
octave
| but all the intervals within. Thus the (narrowed) fifths slow down
and
| the (expanded) fourths speed up.
|
| Another point that might explain the fourth being faster is that the
| coincident partials of G4 - C5 are an octave higher than C4 - G4.
|
| But as you say, it is a lofty subject.
|
| Tom Cole
|
| Richard Moody wrote:
| >
| > |On the contrary, if you tune an
| > | octave, then expand the fourth within that octave, you will
| > | automatically contract the fifth.
| > |
| > | Does this cloudify the subject?
| > |
| > | Tom Cole
| >
| > Well it is a lofty subject esp when you wonder why the G4--C5
fourth
| > will beat twice as fast as the C4--G4 fifth.  And this is not the
same
| > thing as saying the 4th on top is twice as sharp as the 5th on the
| > bottom.  ---ric