perfect fifth questions

[Jim Coleman, Sr.]

Hi Kent again:

You are correct about the theory of equal temperament. However, all of
that theory was developed when we couldn't even spell inharmonicity. It
was not yet invented, but it was present in even greater amounts since
the harpsichord days.

If one tuned the center octave as a pure 2-1 type octave, there would
still be inharmonicity and the octave would be stretched. When one
progresses to the 3-2 type octave, that is, where the 4th partial of the
lower note equals the 2nd partial of the upper note, the octave is 
stretched more (this involves the 4th-5th test or the 3rd 10th test 
being equal). When one utilizes the 6-3 type octave where the 6th 
partial of te lower note equals the 3rd partial of the upper note, the
octave is stretched even more (this involves the the lower minor 3rd
being equal to its upper complementary Major 6ths - complementary means
completing the octave). So from these examples, you can see that we
are never tuning a simple octave where the upper note is exactly twice
the frequency of the lower note, hence the 12th root of 2 is not used
in practice in piano tuning except in a few rare cases where the slope
of the inharmonicity curve just happens to make it so (these are usually
pretty poor pianos to try to tune).

I like to look at inharmonicity as our friend. When properly balanced,
it can do wonderful things for the tone and tuning.

As far as the stability of wide tunings is concerned, I sense a greater
stability in them after the 2nd time. The first time is kinda like a 
minor pitch raise. We have an excellent aural tuner in our area, Brent
Fischer, whose tunings exhibit great stability. In his words he tunes
the octaves on the very edge of wideness and his 5ths are not 
necessarily pure, but much more quiet. I just heard two of his pianos
today in concert in a two piano concerto competition. The tuning was
impeccable. Every time I have heard his tunings, they have been like
that. Many of the Steinway tuners over the years have tuned with wide
stretch. The recordings sound great. Please note, I'm not accusing 
them of tuning "Pure 5ths tuning", but approaching that. It's the 
only way to try to keep the high treble from sounding flat IMHO.

Jim Coleman, Sr.

On Sun, 5 Oct 1997, Kent Swafford wrote:

> Jim Coleman, Sr. wrote:
> 
> >Hi Kent:
> >
> >Since my name has been used, may I humbly disagree with a few of your
> >comments?
> >
> >Equal temperament on pianos is not the 12th root of 2 due to the effect
> >of inharmonicity which in and of itself stretches the octaves some. 
> >This is why the 5ths in normal equal temp. tunings are less than 2 cents
> >and the 4ths are more than 2 cents. You can easily check this with your
> >RCT or the SAT (I have done both).
> 
> Thank you for your reply to my (perhaps overly-brash) post.  I am eager 
> to try out your suggestions.
> 
> The mathematical _model_ for 12-tone to the octave equal temperament, as 
> represented in our VTD's by "0.0" cents deviations for each note of the 
> musical scale, is indeed a tuning in which the frequencies of the 
> fundamentals of ascending half-steps increase by a factor equal to the 
> twelfth root of two.  This model for equal temperament is completed by 
> assuming zero inharmonicity in the placement of partials and calculating 
> the beat rates of the coincident partials of the tuning intervals.  
> Pianos apparently have been scaled so that their beat rates may resemble 
> that of the zero inharmonicity mathematical model of equal temperament;  
> at least that is my understanding of what has been taught.
> 
> My point was, I think, that if we completely throw out the theoretical 
> beat rates as the ideal, we may run into unexpected problems.  An example 
> of these problems would be that wide tunings might be less able to 
> survive the effects of normal drift, and wouldn't "last" as long as our 
> more "normal" tunings.  (I may sound like a conservative fuddy-duddy, but 
> some of our customers, I think, wouldn't be very sympathetic to what wide 
> tunings can sound like after a few weeks.)
> 
> Kent Swafford
>