source file: m1408.txt
Date: Thu, 7 May 1998 18:13 +0100 (BST)

Subject: Reply to Paul Erlich

From: gbreed@cix.compulink.co.uk (Graham Breed)

> I maintain that any systematic inharmonicity in brass instruments, which
> is either nonexistent or extremely small, has little or nothing to do
> with the failure of the resonant modes of the instrument to form an
> exact harmonic series.

Now you're getting suitably equivocal.  Before you were saying that no
inharmonicity could possibly occur, which places too much faith in the
simplified models.  Ideally, we should develop more complicated models to
give a quantitative estimate of the (upper limit of the) inharmonicity.

Inharmonicity will cause problems with JI whether or not it's systematic.

Any noise in the input will cause peaks at the resonant modes.  The result
will be inharmonicity.  The level of that inharmonicity can be estimated
from the noise level of the input.  I don't know of a good model of the
human embouchure (that's the word, isn't it?) although there may be one
that is too complicated to escape from the academic literature.

If the overtones are not centered at the resonant peaks, the amplitude of
each overtone on output will depend on it's pitch.  That means, where
vibrato is present, the corrected waveform will not be perfectly periodic.
 This will probably show up as a subharmonic of the heterodyned overtone.
 Whatever, it isn't important for tuning as the effect will be swamped by
the vibrato itself.

I've had a look at some books on this today.  The pitch from the
embouchure is determined by the mode of resonance, but overtones can still
arise.  Also, the nature of the mouthpiece is such that noise may not
propagate to the conical bit.  Another thing is that it is not only brass
instruments that have conical tubes.  Examples of recorded resonances were
given for a range of wind instruments, and they're all significantly off
from the integer ratios.  This means that the flute, where the initial
sound is not periodic, should show the most inharmonicity.  Fortunately,
this is one case where experiment agrees with theory, at least on the
basis of the one flute sample analysed and explained here recently.  I
couldn't find any discussion in books of the inharmonicity of the sound of
any instruments.

The resonances of flutes were also given relative to 12-equal.  They were
up to 20 cents out!  It's only the flautist's skill that adjusts them to
the desired scale.