Some theory aspects of piano tuning.

1. What is tuning? How is it done?
“No tuning is ever finished - just left behind....” Ron Nossaman, piano builder.
How is a piano tuned? What is piano tuning? Here’s my own definition:
Part arithmetic and part flower arranging.
In this article we’ll look first at the physical/mechanical aspects of tuning a piano, and then consider some
theory about the musical scale and what it means to be “in tune”.
MECHANICAL ASPECTS
A piano’s sound is produced when little wooden hammers covered in highly compressed wool felt strike
steel wire “strings” and set them in motion. The strings are secured at each end, and they pass over a
raised bridge that is attached to the soundboard. The soundboard is a large slightly domed board made
(usually) of spruce. When the string moves, its motion is transferred through the bridge, and sets the
soundboard in motion. This arrangement might be called an acoustic transformer. Loudness is gained, as
the soundboard sets a much larger volume of air in motion than the string could on its own. Taking energy
from the string to do this, means that the string vibrates for a shorter time than it would without the
connection to the bridge and soundboard.
At the bottom of an upright piano or the far end of a grand the string is secured round a “hitch pin”
integral to the cast iron plate. The other end of the string, at the top of an upright, or near the pianist in a
grand piano, is wrapped (coiled) around a metal tuning pin like this one:
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2. You can see the little hole in the tuning pin, where the end of the string gets inserted.
More than two hundred strings are needed for the eighty-eight notes of the piano and the end of each
string is coiled around its own tuning pin. The tuning pins fit very tightly into a laminated wood plank which
in UK piano terminology is called the Wrest Plank and in USA parlance, the Pinblock.
In the photo above, the top grey and gold part represents the cast iron plate. The steel tuning pins are a
very tight fit in the wooden Pinblock. Tuning consists of adjusting the tension of each string by using a
special wrench called a Tuning Lever to very slightly move the tuning pins. The movements are extremely
subtle and much of the skill in tuning lies in ability to manipulate the tuning lever appropriately.
Shown here are are some different tuning levers. In the first photo the top one is of a traditional design,
and the lower one can take different heads so as to accommodate some older styles of tuning pins. In
general tuning pins have square ends, but some older ones are occasionally found that are oblong. In
recent years some excellent engineering has gone into a re-think of the traditional tuning lever, notably by
Steve Fujan, and the lower picture on the next page shows my Fujan carbon fibre tuning lever, which is
lightweight and extremely rigid, and makes the job easier.
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4. The Fujan lever can take different tips for different tuning pin sizes. Tuning pin tips, or heads, have a “star”
profile so as to fit square section pins in various positions.
The technique of manipulating the pin and string is subtle. The strings pass over and under various
components, including the bridge, and these create pressure points. The strings have to be moved with
some vigour so that the tension in the segments of the string evens out. In this photo you can see that
when the strings come from the coils around the pins, they quickly pass under a pressure bar, and then
over a V-bar. These help to position the strings. You can readily imagine that there is friction at these
points, and the string has to be coaxed to move such that the tension is evened out along the full length.
The tuner has to know and use the right techniques to “set the pin” and “set the string”. In other words, to
ensure that things stay put when he has finished with one note and moves to the next. Poor technique
with these aspects can result in a tuning that is not very stable. It is difficult to describe just how the
tuning lever is manipulated, and we won’t attempt it here. It’s not especially necessary for understanding
what’s to follow. Could an amateur musician tune a piano using a mechanic’s socket set, with a small
enough socket? The short answer is no. It is just about possible to correct a single string which for some
reason has gone badly “out”. But to achieve an entire piano tuning is such a manner is an impossibility.
We are going to move on now from the physical, mechanical aspects, to the other part:
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5. THEORY OF THE SCALE AND TUNING
Books have been written about how our Western musical scale is derived, and we can’t go too deeply into
that here. Pythagoras in ancient Greece is noted for carrying out experiments and making discoveries
about musical sounds. We like a scale of musical notes in which we go from one pitch, or frequency, to
exactly double that pitch, or frequency, in a series of twelve steps. All of western music is based on such
scales, or arrangements of notes.
An item which moves four hundred and forty times per second, shoves the air around it that many times
per second, and the air shoves our eardrums that many times per second, and the movement of the
eardrum is transmitted by three tiny bones to the inner ear and in a mechanism of great beauty and
complexity, converted into electrical nerve impulses for interpretation by the brain. The sensation we
experience, is what we call the note A above Middle C. If we go down twelve steps from there, we get
another A, which moves two hundred and twenty times per second. Going up instead of down, we get an A
at eight hundred and eighty movements per second. These vibrations, or “cycles per second” are named
after the physicist Hertz, with the abbreviation Hz. Thus we say that the note A above Middle C has a
frequency of 440 Hz.
The first note on the piano is an A, and its frequency is 27.5Hz. If we go up the piano in seven “doublings”
or seven octaves, calling the first note A0, we get:
A0 27.5 Hz
A1 55 Hz
A2 110 Hz
A3 220 Hz
A4 440 Hz
A5 880 Hz
A6 1760 Hz
A7 3520 Hz
This is a geometric progression, but let’s not get too far into maths!
What about the other notes in the scale, between the A notes? Pythagroras by experiment found pleasing
mathematical ratios that worked well to give pleasant sounding notes. He found that if you multiply the
frequency of a note – say A3, 220Hz, by 1.5, you get the musical interval we call a Fifth; the note E above
middle C. Using Pythagoras’ multiplier of 1.5, the frequency of that E would be 330Hz.
If we start at the first note on the piano keyboard, A0, and go up in a series of “Fifths”, we find that on the
12th jump, we are at the same place as on the seventh Octave jump, back to an A. The notes would be
(follow it on the piano keyboard if it’s easier):
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6. Starting point A0,
First jump E1
Second jump B1
Third jump F#2
Fourth jump C#3
Fifth jump Ab3
Sixth jump Eb4
Seventh jump Bb4
Eighth jump F5
Ninth jump C6
Tenth jump G6
Eleventh jmp D7
Twelfth jump A7
However, if we use Pythagoras’ multiplier of 1.5 to get the frequency of each fifth from the fifth below,
something odd happens! Here are the frequencies starting at A0 and multiplying by 1.5 each time:
Starting point A0, 27.50
First jump E1 41.25
Second jump B1 61.88
Third jump F#2 92.82
Fourth jump C#3 139.23
Fifth jump Ab3 208.85
Sixth jump Eb4 313.28
Seventh jump Bb4 469.92
Eighth jump F5 704.88
Ninth jump C6 1057.32
Tenth jump G6 1585.98
Eleventh jump D7 2378.97
Twelfth jump A7 3568.46
Oh dear! When we get to the A seven octaves up from the starting point, the frequency, calculated using
12 jumps of 1.5 (the “right” ratio for an interval of a Fifth), that A is 48.46Hz different!
A perhaps simpler example, would be to consider the musical interval of a Major Third. Three Major Thirds
make up an octave. Starting at A 220Hz, the first Major Third up is C# then F then A. Pythagoras
determined that to go up by a Major third, you should multiply the starting note by 1.25. So, we would
have A 220Hz, C# 275, F 343.75, A 429.69.
Erk! The next A up is supposed to be an Octave, double the frequency, 440Hz. Not 429.69 Hz!
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7. These differences are sometimes called the Pythagorean Comma. This problem happens with all the
different ratios Pythagoras worked out for the series of notes in a scale. Pythagoras wasn’t wrong. It’s a
problem in physics. If you tune each of the thirds above to the frequencies given, you will get three
beautiful thirds when played individually. But they will sound awful played together, and won’t add up to
an actave!
In order to get a scale on the piano to “fit” properly, it is necessary to “temper” the intervals – to alter
them to fit, as it were. Taking the example of the three Thirds, you can see that there is a “shortfall” of
10.31 Hz; they fail to add up to an octave by that amount. The Thirds therefore need to be “widened” a bit,
so that they add up to an octave.
The generally accepted system of “tempering” the scale, or adjusting the spaces between the notes away
from Pythagoras’ theoretical intervals, such that they will all “fit” into a scale, is called Equal Temperament.
(Again, books have been written on the theory of temperament, and we can’t go too deeply into it here).
The idea in Equal Temperament is to go from one note to the same note an Octave above, is a series of 12
steps of equal proportion. To do that, you multiply each time, by the 12th root of 2. (The number which,
multiplied by itself 12 times, gives 2). That number (to seven decimal places) is 1.0594631.
If we multiply A 220 by 1.0594631, we get the frequency of the next note in the scale, A# (B flat). Doing this
twelve times brings us to A440 (almost – for all practical purposes anyway!)
This is all complicated by the fact that musical strings in a piano, like any other vibrating object, do not
vibrate in a simple manner. A string vibrates in sections as well as along its full length, and each of these
“sections” contributes to the sound. The string for A220 also moves as two halves, three thirds, four
quarters, etc. Thus it produces sounds, “harmonics”, at 440Hz, 660Hz, 880Hz etc.
Every string in the piano behaves in this way. Each string produces a “harmonic series”, a whole set of
frequencies. The frequency of a particular harmonic of one note, can be close to the frequency of a
harmonics of another note and this gives rise to “beats”.
When two sources of sound have a frequency that is near, but not identical, an effect called “beating” is
produced. You might have observed this with two propeller engines in a plane; the drone has a kind of
“pulse” in it, when they are not at identical revs. Or two motor lawnmowers, as local authority workmen
cut the grass in summer!
For a very clear demonstration of this effect of “beating” or “beats”, go to
http://www.indiana.edu/~emusic/acoustics/phase.htm
At the bottom of the page, are buttons to switch on sounds for 440Hz, 441Hz and 442Hz. Try 440 with 441,
then 440 with 442. You will clearly hear the “beats”, a pulse in the sound, once per second, and twice per
second, respectively.
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8. This phenomenon of “beats”, is the basis for tuning a piano. Most notes have three strings per note (two
in the lower tenor section and one in the low bass notes). Each of the three strings should be at absolutely
identical frequency, therefore “beatless”.
Where does the “flower arranging” come in, that we referred to at the start? No piano conforms to a
perfect mathematical model. There are compromises with scale design, so that strings are not impossibly
long. And within an individual string, as the string vibrates in fractions, the fractions have a higher relative
stiffness, which can make the harmonic have a higher frequency that the theoretical (this is called
inharmonicity).
In addition, there is a tendency, which varies from person to person, for the ear to prefer a gradual
increasing of the frequencies of the notes beyond their theoretical positions, in the top register of the
piano. And in the very low notes, the fundamental, “whole string” frequency is not very dominant; our ear
extrapolates it down from the higher harmonics.
All of this (it’s a lot more complex than this brief explanation!) adds up to the fact that the tuner has to
arrange the tuning, to place the notes, as it were, in a way that is best for the particular instrument (and
client). It is never a matter of conforming to some mathematical formula. That is why, while the tuner has
to understand the arithmetic of the theoretical scale, he or she must also be a “flower arranger”.
(With thanks to Chuck Behm for the use of his Pinblock photo).
© David Boyce 2012
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