Pure 5ths ET.

[Richard Moody]

Jim, List

    It is always interesting to explore the possiblilities and
challanges of a pure 5ths ET.
Of course in the same breath it might also be stated Wide Octaves. So
we might call it
WOP5ET, or P5WOET to make it even more "significant".
|
| The confusion arises because some fail to understand that a pure
5ths
| temperament requires a larger stretching of the octave. When one
does
| this, the 5ths do indeed become pure and as a consequence the 4ths
get
| wider.

Yes it is the consequence of the wider octave that the 4ths get wider.
However there is sort of a conundrum here, or rather a multitude of
effects that makes it hard to figure out what is actually going on.
The first question is how much to widen the octave.  From the aural
point one might suggest tuning an ET 5th on the root of an octave then
lowering that root until a pure 5th (P5) is obtained.  However that
would leave the same 4th and that cannot be right becase we would
expect a different  4th in an expanded octave than a pure octave.
    I am not sure if the past theorists had a name for the amount a
5th needs to be contracted to get ET  (schisma is close I tink) but
ever since Ellis it has been called 2 cents so if the bottom note of
the 5th and octave are the same and that note is lowered 2 cents to
get a pure 5th then we have  expanded the octave by  2 cents.  If you
tune a pure 5th down from the top note do you do now have a 4th to the
bottom note of the right width? But it is not as simple as because
when I figure it out on a spread sheet I get an octave expanded by
3.37 cents.  This I got by putting numbers until all the 5th had zero
beats.     But what is the theoritical way to figure it out, and can
it be done on accoustic tuning principles?
    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.
    Well it was faster by fiddling with the 2 point something octave
than figuring it all out.  But so much for the efficiency and accuracy
of ordinary spread sheets.  So using 3.5 cents for widening the octave
is close enough.  It is a little larger than the intuitive figure of 2
cents since that is how narrow the 5th is.
    Getting back to the aural considerations the spread sheet shows
C4-C5 to beat at 1.01 bps.    So if you start with that at one per
second you are on your way. But you would have to set C5 at 523.504
instead of 523.251. if you want A to be 440.

|
| If one tuned a pure 5th down from A4 to D4 and then a pure 5th down
| from D4 to G3, the G3 would be almost 4 cents flatter than it would
have
| been if TEMPERED 5ths were tuned. since G3 is very close to the
| note A3,  the A3-A4 octave is also stretched about 3 cents in a pure
5ths
| tuning. In this kind of tuning all the intervals progress in beat
rate
| evenly just as in regular ET except the wide intervals are a tiny
bit
| faster and the narrow intervals are a tiny bit slower. To say it
another
| way, in the Pure 5ths type of tuning which I wrote about 3 years
ago,
| all intervals are equally widened including the octave.
|
| Jim Coleman, Sr.