Hi Fred... just for clarity (please excuse me copying your entire
post...) I find everything you state below to be agreeable with what I
have been saying. No conflicts in thought per sé. I would point out tho
that in your first paragraph... tho it is true one can <<emulate>> the
results of a P 12th one in fact doesnt do this in a usual temperament
and octave priority tuning. Just do your usual temperament... A3-A4 is
it ??.. in anycase when you have it done extend the thing so that you
can compare your first 12th. Measure the resultant 3:1 relationship
between the two. You'll more then likely find its off a bit.. depending
on your style as much as a half a cent I can imagine... it also depends
a bit on the piano. Transfering 440 from A4-to A3 as is usually
required, then tuning as usual the D3:A3 5th most often puts the D3:A4
12th a bit narrow. Extending this kind of priority over a whole tuning
results in a different tuning... naturally enough. Yes one CAN emulate a
P-12th tuning using octave priorities... but number one this is not in
fact done, and number two why would you do so when you can much more
easily just tune a P-12th directly ?
Nice discussion !
Cheers
RicB
Hi Ric,
Comments below.
On Oct 13, 2008, at 4:49 AM, Richard Brekne wrote:
> Fred writes
> I frankly don't think dividing a 12th evenly into 19 is
> significantly different from dividing an octave evenly into 12
>
>
> Actually it is... tho the distinction in the temperament area is a
> bit subtle. If you require that D3 and A4 have a pure 3:1
> relationship... then require that A4 and A3 have a pure 6:3
> relationship then your 3:2 D3 and A3 is not necessarily going to
> work out quite like you want it... and you may be forced to fudge.
My statement assumes that the same stretch is selected for both the
12th and the octave which are to be divided. How you arrive at that
"sameness of stretch" may be a bit convoluted, but it can certainly be
done. We are talking about beginning with a bottom pitch and a top
pitch and generating the half steps between them, nothing more and
nothing less. The differences may be significant to a mathematician,
but they are in the range of hundredth of a hertz or smaller, hence
utterly insignificant in any practical way. Your statement about "then
require A4 and A3 have a pure 6:3 relationship" is directly opposed to
what I was trying to convey. The 6:3 relationship is almost certainly
wider than the stretch called for in 3:1.
> Course that kind of thing develops around the temperament octave as
> you go.
> But more then this, and evenly divided 12ths temperament extended
> into the treble so that all notes above this area are tuned to pure
> 3:1 to their respective 12ths below then the treble curve around the
> F5 - F6 area is significantly different then in an Octave type based
> curve. Jim Coleman commented on this when my own P 12ths for Tune
> Lab was released back around the turn of the century. Indeed, it was
> a characteristic he liked quite abit.
I think that a great deal of this difference you describe can be
attributed to the partial level at which the tuning is being done.
Tuning first partials to the 3rd partial of notes a 12th lower will
yield a different curve than tuning using a smooth curve at the 4th
partial, for instance, particularly if large inharmonic jumps occur.
But this is a separate issue. It is certainly possible to emulate
closely the same stretch as a 3:1 using other parameters. A 4:1 plus
0.5 cents or thereabouts will do a reasonable job.
Focusing on a single interval is a way of achieving a consistent
sound. Octave, 12th, double octave, 19th, triple octave, all can be
used. And all can be used in more than one way (eg, 3:1 plus 0.3 cents
as opposed to "pure"). It is also possible to achieve a consistent
sound using some sort of progression (eg, narrower in the middle
becoming wider in the extremes). Lots of possibilities out there.
Regards,
Fred Sturm
University of New Mexico
fssturm at unm.edu
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