Re: [Phys-L] music scales

[John Denker <jsd@av8n.com>]

On 2/17/25 13:16, Anthony Lapinski via Phys-l wrote:
> So this equal temperament (ET) scale - used for all pianos. All instruments
> today, too? So why are all "physics" tuning
> fork sets different? I know they were developed in the 1700s. Why aren't
> they ET to match all instruments?

A piano is *not* tuned to the mathematical equal intervals.
The octaves (and other intervals) are *stretched* to account
for the stiffness of the strings, which is small but nonzero.

The idea is that you want the first harmonic of C4 (middle C)
to line up with the fundamental of C5 (the next one up), which
is slightly higher than twice C4.

If you don't stretch the octaves, the piano sounds badly out
of tune. You get a nasty "beat frequency" when playing C4 and
C5 together. The nastiness is most pronounced in the higher
octaves, where the stiffness has a bigger effect.

   https://en.wikipedia.org/wiki/Stretched_tuning

Along the same lines: Every brass or woodwind instrument ever
made is mismatched to the mathematical equal intervals. Also
mismatched to the piano. This is why you have B♭ trumpets, C
trumpets, et cetera. Choose one that comes close to matching
the key of whatever piece you are playing.

Singers and violinists accompanied by a piano fudge their
intervals to match the piano. Brass and woodwind players can
also fudge their intervals, within limits.

============

Please do not imagine that equal temperament is the "best"
tuning. If you are playing chords in the key of C, a piano
optimized for that key sounds very much nicer than equal
temperament. For starters, the major thirds are quite
conspicuously nicer. It's painful to re-tune (aka de-tune)
such a piano back to equal temperament, once you know how
much better it "could" sound.

The downside of single-key optimized tuning is that chords
in other keys sound extra-awful.

Equal temperament is "least bad" in the minimax sense: It
spreads the badness equally across all key signatures. Das
Wohltemperierte Klavier intentionally modulates from one
key to another, precisely to illustrate this point.

On a synthesizer, you can switch from one optimal temperament
to another, so you can have the best of both worlds. This
assumes the notes are truly synthesized, not sampled from a
real instrument.

===========

Max Mathews and John Pierce studied the psychophysical basis
for the perception of harmony. They built some instruments
with very weird partials and very weird temperament ... which
sounded fine because the partials lined up.
  Harmony and Nonharmonic Partials
  http://articles.ircam.fr/textes/Mathews80b/