Perfect Tuning

> Rather than try to devise a scheme which tries to reduce the average
> error of tuning a tiny bit, why not have absolutely perfect tuning?
> Why not actually play the notes at frequencies that are the ones that
> you want and have the harmonic relations to each other that are desired?

Assuming that, by "absolutely perfect tuning" you mean Just Intonation, then there are two reasons:
Historical: To be able to do that in a large variety of keys
requires you to play an extremely large number of pitches
in an octave's span. That's an exceptionally difficult
proposition for pianos, harpsichords, harps, etc. Even
on electronic instruments it's not trivial, and electronic
instruments still don't have the naturalness and expres-
siveness of "real" instruments.
More modern: The cummulative effect of each tempera-
ment's inaccuracies gives music a "mood" that is inte-
resting in its own right. Some would argue that Just
Intonation, in its phase-locked perfection sounds clean
to the point of being almost boring. I don't agree as
such, but can readily see why they'd make that claim.

[Gary Morrison TD186.1]

> Assuming that, by "absolutely perfect tuning" you mean Just
> Intonation, then there are two reasons:
>Historical: To be able to do that in a large variety of keys
> requires you to play an extremely large number of pitches
> in an octave's span. That's an exceptionally difficult
> proposition for pianos, harpsichords, harps, etc. Even
> on electronic instruments it's not trivial,

Definitely I am talking about electronic instruments. It is possible to
make a piano do this by fitting tension or string length adjusters but
it would clearly be a very expensive exercise (although pehaps justifies
for concert pianos).

>... and electronic
> instruments still don't have the naturalness and expres-
> siveness of "real" instruments.

Gary, you are right again. I believe that there are multiple reasons
for this and they need to be addressed in order to get a result that is
the perfect instrument. The most important of these is that many
electronic instruments use waveforms played at various speeds to produce
notes. Sometimes they have one sample per octave to recognise that the
waveform changes with frequency. However playing a waveform even 50%
faster than recorded at still has an entirely different set of
resonances. This must be properly done to get a correct result. I have
worked out how to do this and a couple of simple parameters allow
accurate reproduction of many instruments.

Another factor is that real instrument waveforms continue to develop
with time and don't settle down to a static situation as many electronic
instruments force on us. Again there are ways to address this.

>More modern: The cummulative effect of each tempera-
> ment's inaccuracies gives music a "mood" that is inte-
> resting in its own right. Some would argue that Just
> Intonation, in its phase-locked perfection sounds clean
> to the point of being almost boring. I don't agree as
> such, but can readily see why they'd make that claim.

Of course if we played a very pure instrument (say a sine wave
generator) in perfect harmony, then it would be quite boring.
That is why I stated from the start that there were additional
considerations regarding modulation and variation that must also be
allowed for. This partly overlaps the electronic factor. It is
actually more interesting to have a note that has various levels of
variation and modulation including tremello (Yeah I know, not too much
like modern opera singers) and other slower variations. It is important
that such variations are not random however but desirable that they be
co-ordinated with the harmony and rhythm of the music.

If we have a piece of music in 4/4 time with 120 bpm and in C (let us
say for simplicity that C=256 Hz) then the rhythm at 120 bpm is also at
2 Hz which is exactly in tune with the key. Any periodic variations in
the instruments should be in tune also, such as at 4 Hz or 8 Hz or maybe
6 Hz for a little bit of spice.

If notes are to be deliberately out of tune, as you suggest, then the
amount should be chosen also deliberately and not dictated randomly as
in ET. For example, having say C=256 Hz and G=386 Hz (instead of 384)
gives a 2Hz beat which is in tune with and probably with the rhythm.
These dynamical variations occur in real music and prevent it being
sterile. They should apply to many things, volume, tempo, pitch,
ratios, etc etc and they should occur at many levels of scale from fast
to medium to slow and even very slow. The Gods and Magic speak to us in
the frequencies which are below human hearing.

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