ETD Question

[Kent Swafford]

on 6/11/00 12:25 AM, Marc Damashek at mdamashek@erols.com wrote:

  
> If I were to plot up the number of cents each partial was
> sharp, the points would not lie on a smoothly rising curve, but would
> instead deviate randomly from that curve, with some of the deviations
> being pretty sizable. The Strobotuner showed me that the successively
> faster advance rates of the upper partials on some strings wasn't
> uniform: some partials were obviously slower or faster than they
> ought to have been, and didn't interpolate the frequencies of their
> neighbors.
> 
> Is this higher-order effect well known?

Yes. The actual term is paraharmonicity, to quote Dean Reyburn:

"This is part of the effect I call "paraharmonicity" which is inharmonicity
which is not explained by the ususal formulae (it can be negative or
positive).

> I can't recall ever seeing it discussed.

Paraharmonicity has been discussed here on pianotech. Lower partials
(particularly noticeable in the tenor) can be very inconsistent. RCT's
Pianalyzer function clearly demonstrates this. The higher partials are
actually more consistent than lower ones. The point in the partial ladder at
which partials smooth out has been a subject of spirited discussion.  :)
 
> So the hierarchy of tuning subtlety goes something like this:
> 
> perfect strings (harmonics at integer multiples of the fundamental)
> 
> --> strings with inharmonicity (harmonics at successively
> higher frequencies than integer multiples -- stretched scales)
> 
> --> strings with inevitable random imperfections (random
> deviations from smoothly stretched scales)

Congratulations on your fine analysis.

> Every piano tuning is a compromise in which we try to
> accommodate those random deviations as best we can, and that's what
> really separates the sheep from the goats.
> 
> Now, how do existing ETD's decide what the 'best' frequency
> is for a string? Obviously not by simply targeting the equal-tempered
> fundamentals, because of inharmonicity (first-order correction). But
> do any of them explicitly optimize their targets based on accurate
> measurements of the partials on all strings (at least in octaves 2-5,
> say), which can exhibit random deviations from a smoothly stretched
> scale (second-order correction)? Probably not -- note that this is
> different from measuring just the A's up the keyboard, for instance,
> in order to find an appropriate stretching curve (which is a Good
> Thing as far as it goes).

Exactly. RCT measures multiple partials on multiple A's. The more partials
and the more strings measured the better. However, any system which makes
assumptions about the inharmonicity of strings not measured will be wrong to
one extent or another. This is why visual tunings done by such calculations
must be aurally verified/tweaked.

> While it might seem awfully complicated to obtain a
> self-consistent tuning solution that minimizes the bad effects of
> random irregularities, it's not impossible, especially when you're
> toting around a fast laptop computer and can already accurately
> measure the partials. (The solution involves some straightforward
> linear algebra.) I strongly suspect that it is precisely because
> ETD's ignore random irregularities in real strings that the best
> aural tunings still outstrip the best ETD tunings, especially on the
> worst pianos. If this situation were to change -- and I claim that it
> can -- we would hear the improvement immediately.

A "self-consistent tuning solution that minimizes the bad effects of random
irregularities" has been developed by Steve Fairchild. It measures all the
partials quite a way up the partial ladder on _all_ strings of the piano. He
calls it the "Aural Tuning Emulator". I believe it exists as a spread sheet
in a computer program.

Fascinatingly, the first version of Reyburn CyberTuner which I saw almost 4
1/2 years ago now had an interface programmed into it for a planned
inclusion of the Fairchild Aural Tuning Emulator. The programming eventually
took a different turn and the ATE was never completed.  I hope I don't get
into trouble for mentioning this.   :)

The programming direction that RCT _has_ taken in the last 4 years has been
to make the most of the "A" data that _is_ taken from each piano. The
improvements achieved in the last 4 years by taking this tack have been
substantial, and in practice, have provided some of the improvements that we
thought would require the Aural Tuning Emulator. Also, another projected
feature of RCT at one time was the ability to calculate a tuning at more
than one partial per note and to display two partials at once while actually
tuning. The improvements in the single partial per note calculations have
been so substantial in the past 4 years, that I no longer covet that 2nd
partial capability.

I can't argue that the ATE and/or a 2nd partial display would not bring
improvements, but, gee, RCT tunings are mighty good these days.

Kent Swafford