Why pianos are out of tune

I recommended How
Equal Temperament Ruined Harmony (and Why You Should Care)
by <a href="Ross
Duffin
last week
without having read it, so this week I decided to read it.

The description of “how” equal temperament took over is a bit
vague, but the “when” is extremely detailed without being dry
and scholarly, largely thanks to the entertaining biographies of
the major players.

Where I would have really liked more detail is in where to go
to listen to non-equal temperaments. He does recommend 6
degrees of tonality and Beethoven
in the temperaments by Enid Katahn as CD’s for hearing a
piano tuned in non-equal temperaments. But his
arguments about “why you should care” seem to be the standard
“early music” ones: Beethoven did it on this kind of piano and
so should we. I’d be surprised if they convinced any of the
people who believe that Beethoven really wanted his sonatas
played on a modern Steinway, and was just stuck with those silly
fortepianos that were always breaking strings.

I actually think you can make a case that it isn’t equal
temperament that makes modern pianos out of tune, but rather the
other way around — there’s no possible way to make a modern piano
in tune, so that’s why equal temperament, which is “easier” in
ways that Duffin explains in detail, became accepted.

Piano tuning

I think Ross Duffin doesn’t really realize how out of tune any
modern piano is, even when just tuned by a good tuner to exactly
the frequencies that are theoretically accepted as the best
ones.

Octaves

There’s one issue that he does explain in detail, and that is
that the octaves are in fact wider than the doubled frequency
Pythagoras and Helmholz and all tuners before the metal framed
piano believed in.

Many people’s eyes glaze over when I try to explain this, even
though I think it’s one of the most elegantly complicated
explanations in the history of musical acoustics. So if your
eyes glaze over on complicated explanations, feel free to skip
to the next section.

The short answer for why a note on a piano is more than twice
the frequency of the note an octave below it is that with a
string as stiff as a piano string the
overtones are sharper than the harmonics.

That is, with a light string like a harpsichord or guitar has,
when the string vibrates in two sections to produce the first
overtone, the lengh is in fact almost exactly half the length of the
string, making the frequency twice the frequency of the string’s
fundamental tone.

On a piano, however, the string is so stiff that when it
vibrates in two sections, the actual vibrating length is
noticeably less that half the length of the string. And the
difference is even more pronounced with the higher
overtones.

So if you tuned a piano so that the fundamental of a string
was precisely twice the fundamental of the string an octave below
it, you would have horrible beats between the first overtone of
the lower string and the fundamental of the higher string, and
even more horrible beats between other pairs of overtones.

So one of the things piano tuners do is figure out how much
they have to “stretch” each octave to minimize these beats
formed by the out-of-tune harmonics.

Unisons

If you’ve looked at piano pieces, you can see that pianists
play octaves all the time — there are whole genres of piano
music where the left hand is doing nothing but play a walking
bass line in octaves. So if you have to tune octaves out of
tune, there’s no way anyone is going to ever hear a piano as in
tune no matter what theoretical temperament the tuner uses.

But it gets worse than that — not only are the octaves all
sharp — all the unisons are deliberately tuned out of tune.

Only the bottom notes of the piano are played by one string —
the others are have two or three strings (usually) hit by the
hammer. (The soft pedal works by shifting the hammers over so
that only one string is played instead of all two or three.)

Most piano tuners believe that the piano sound is richer if the
two or three strings that play one note are tuned a little bit
differently from each other, to produce something like one beat
per second.

And of course, if you think about the description above of why
the octaves have to be out of tune, you can see that even one
string played all by itself is producing overtones that are “out
of tune” by any theoretical tuning system based on simple
ratios.

Alternate history

So I think the history of the acceptance of equal temperament
as the dominant tuning system may be something like this:

During the late Renaissance and Baroque eras, people played
music that became more chromatic and more based on harmonies and
played in a wider variety of keys. So tuning systems wer invented with
more compromises in order to play
the wider variety of notes and intervals. This is much better described in Ross Duffin’s 150 page
book than I can do here.

During the nineteenth century, pianos became larger and louder,
and therefore needed to use stiffer strings, so tuning them to any
system based on single frequencies and their ratios became
impossible.

Pianos also became the dominant instrument, so that most
singers and other instrumentalists were most likely to perform
with a piano as accompaniment rather than with an organ or a cello.

It became increasingly difficult to tell the difference between
the non-equal temperaments favored by the nineteenth century piano
tuners (even when they said they were tuning equal temperaments)
and an equal temperament. And the equal temperament is easier to
train people to tune. So starting in 1917, all piano tuning
manuals advocated equal temperament, and most instrumental
instruction included at least methods for dealing with playing
with an equal tempered instrument, even if they believed some
other kind of tuning was preferable for solo playing.

However, piano tuners (and pianists) do in fact believe that
piano tuning is an art, not a science, so when they’ve finished
tempering all their fifths and stretching all their octaves and
detuning all their unisons the way the manual or their tuning
course told them to, then they play the piano and fix
anything that doesn’t sound right to them. I haven’t looked up
the literature, but I’m pretty sure that this often results in a
tuning where a very large fraction of the strings are vibrating a
a frequency very different from what a computer program will tell
you is an equal tempered scale.

Summary

None of which is to imply that I didn’t enjoy Ross Duffin’s book a lot, or that you shouldn’t read it if you’re interested in its subject matter.

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