[pianotech] Do fourths beat faster?

[Jeff Deutschle]

I think Ron has hit the nail on the head, as usual.

Apparently I do not tune with the same stretch as those that have
equal beating (or close enough to appear that way) 4ths and 5ths. It
probably is because of the limit I impose on single octaves. I've been
"pushing the envelope" and I notice that the 4ths and 5ths still beat
faster toward the treble, but not as much.

Brian said "Now before you start telling me about inharmonicity, …"
but neither he nor anyone else has addressed how inharmonicity could
make any difference. I expect that the articles that Ed mentioned do.
I do not have access to them, but I think I may now understand the
basic concept.

To determine the effect of inharmonicity on the frequency of the
partials, the iH constant is multiplied by the square of the partial
number. For a fifth the partial numbers are 3 and 2. The squares are 9
and 4. So iH affects the lower partial of a fifth over twice as much
as the upper partial. But for a fourth, the partial numbers are 4 and
3. The squares are 16 and 9. So iH affects the lower partial of a
fourth less than twice as much as upper partial. Also a fifth is a
larger part of an octave than a fourth. So any stretch done by the
tuner will affect a fifth more than a fourth.

I would have to do additional math to understand this thoroughly, but
I can now see how 4ths and 5ths might be tuned so that they beat at
the same rate across the keyboard (or close enough to appear that way)
if the octave stretch is large enough.


On Fri, Feb 6, 2009 at 11:17 PM, Ron Nossaman <rnossaman at cox.net> wrote:
>
> I've gotta wonder what planet I'm on this week when something like this
> comes up. In a decent aural tuning, where the 3rds, 4ths, 5ths, 6ths,
> octaves, 10ths, 12ths, double octaves, 17ths, and even triple octaves all
> fall into some reasonably sane relationship, the 4ths and 5ths beat at
> pretty much the same speed, respectively, throughout the scale. The
> determination of this is a result of listening rather than math and theory.
> Remember listening? Hands? This would be in the coin of the realm, or what
> nominally passes for ET. Do any of you actually produce tunings that have
> 4ths and 5ths audibly increasing in beat rate as you go up scale? If so,
> how?
>
> Eternally mystified,
> Ron N
>
>



-- 
Regards,
Jeff Deutschle

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