| Tuning In
| Lesson Idea by:
David Ward, Rutland Senior Secondary School,
Central Okanagan School District
Some people think piano tuners have something
called "perfect pitch," says Paul Brown, a registered piano repair
technician. But it's just a catch phrase someone must have dreamed up
If someone had perfect pitch they could correctly
tune the first note tuned on a piano, A 440, without using a tuning
fork or an electronic device. This is simply impossible. (The 440
relates to the number of cycles per second that the piano string
completes while vibrating.)
In a band or orchestra setting, many instruments
may need to "tune in" with the A 440 note before practicing scales and
certainly prior to playing together. Whether played on a piano or an
oboe, it is vital that the A 440 note be perfectly tuned to
ensure the concert is made up of an enjoyable, harmonious series of
Since proper tuning is so vital for good music,
one can't rely on the myth of perfect pitch. Therefore, piano repair
technicians set the first note properly with a tuning fork or an
electronic device. The rest of the piano is tuned by comparing the
notes to that first one.
The goal is to equally space all the other notes
so that the sounds made relate to one another in a harmonious way. In
other words, the sounds must relate with harmonicity. Without this
harmonicity, the sounds made would be harsh to the ears.
Make a visit to the music room and have someone
play the A 440 note on a piano. Listen to it on its own and as part of
Observe that for the more standard scales in the
middle of a piano, playing most keys activates a hammer that strikes
three wires to form a single note! Is this a surprise?
What is of concern to a tuner here? How about a
If possible, have someone play some scales on a
guitar, a clarinet, a trombone, or on other instruments. What do you
How do you think math relates to a range of notes
played on a piano or any other instrument?
Ask if it is possible to see and hear a tuning
fork. If you can, observe the way an electronic device may be used for
tuning a piano or guitar.
The first job of the tuner is to set the A 440,
described in terms of the cycles or "vibrations per second." Then the
tuner must equally space all the other notes in relation to A 440 and
to each other.
One of our early mathematical "stars," Pythagoras
(sixth century BC), discovered some of the relationships between the
length of strings in musical instruments and harmonious intervals. As
methods of measuring the frequency of vibrations were developed,
Galileo (1564-1642) and Mersenne (1558-1648) established some important
Today, several quite complex "laws of strings"
govern the many special factors that affect control of musical notes in
a piano. However, all notes are essentially separated by one special
The frequency of any higher note is calculated
by multiplying the frequency of the previous note by the . Dividing
the frequency of a note by
establishes the number of vibrations per second of a lower note.
To accomplish the task above on most calculators,
you would press 2, followed by the 2nd layer function button, then the
"x sq. root sign y" button, followed by 12. Try it. The result should
Next, multiply 440 by the expression .
Did you get 466.1637615? This is the frequency of
the note A#.
Answer these questions:
- Are the multiple decimal places necessary?
- Is the answer a rational or irrational
(Remember: All rational numbers are made up of the set of natural
numbers, whole numbers and integers. They can be written as decimals
that are either terminating or repeating. Irrational numbers are those
numbers with decimal values where there is not a clear pattern -- that
is, neither terminating nor repeating decimals. The value "pi" is an
example of an irrational number.)
- Is it wise to consider the recording of a
"rational explanation" to describe an irrational number? Would
"rounding off" to a certain decimal place provide an accurate answer?
For the purpose of the exercise on the Student
Activity Sheet, simply record the first two decimal values reached in
your calculations. You do not need to round off or change the answers
on the calculator before each successive operation of multiplication or
division. (See instructions that follow.)
Apply the simple calculation described above in
order to determine the number of string vibrations per second for the
notes of a piano, as set out in the Student Activity Sheet. You may
start at the C (261.63) position and move upward to complete the chart.
To check your work, start at the A 440 value and
divide by each time
you arrive at an answer to get values in a descending order. Do you
notice any differences? Discuss your observations with other class
members and the teacher.
Check your work against the solution below.
Piano tuner Paul Brown notes: "If an electronic
device were used for all tuning, the spacing of notes would be very
rigid in terms of separation. Fortunately, the human brain is such a
marvelous creation that it can be used to help space notes apart
equally, at the same time taking into account slight imperfections in
the piano wire!" After special training and lots of practice, it is
possible for piano repair technicians to listen for certain "intervals"
or "beat rates" that distinguish notes from one another. For example, a
tuner has to be able to count from one to 10 beats per second on
occasion. With plenty of experience, piano tuners truly gain a sense of
"feeling" when notes are in proper tune.
While many of the standard octaves can be tuned
"aurally," an electronic device may be used for the higher octaves
because of the high-level frequencies involved. The length, thickness
and tension of good quality steel wire are changed to produce notes of
a higher pitch.
The "hertz" or cycles per second can be as high as
1,568.0 for G in the A octave directly above A 440; 2,793.8 for F in
the A-7th octave, and 4,186.0 for the highest C note of an 88 key piano
(the last key on the right).
Discuss: How do you think changes in temperature
could create problems for musicians?
Perhaps the next time you hear music, you will
consider for a moment the clever and special mathematical way that
instruments work together in order to produce the series of sounds that
we so often take for granted! And, any time that someone tells you they
have "perfect pitch" and can sing any note without a comparison first,
you'll know that "perfect pitch" is a myth!
Tuning In Student Activity Sheet
Course: _________________________ Block: _______________
| Curriculum Organizer:
| Curriculum Sub-organizer(s):
Understand differences between rational and irrational numbers
- Describe the difference between exact and approximate values for
- Use a calculator to evaluate work involving a radical
- Relate math to other subject disciplines
Determine whether or not a number is rational
- Basic proficiency in use of a calculator for work involving rational
Solution for Student Activity Sheet
The frequencies for the notes are:
C# = 277.18
D = 293.66
D# = 311.13
E = 329.63
F = 349.23
F# = 370.00
G = 392.00
G# = 415.31
A = 440.00
B = 493.89
C = 523.25
These 12 notes, along with C (261.63),
comprise the chromatic scale of C.
Partnership by the Center for Applied Academics, Bridges
Transitions Inc., a Xap Corporation company and The
B.C. Ministry of Education, Skills and Training. Copyright
© 2002 Center for Applied Academics