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 years ago.

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 sounds.

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 a scale.

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 listener?

If possible, have someone play some scales on a guitar, a clarinet, a trombone, or on other instruments. What do you hear?

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 relationship rules.

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 formula:

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 be 1.059463094.

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 number?
  (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? Explain.

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.

Additional Activity

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 Name: _______________________________________________ Course: _________________________ Block: _______________ Teacher: ______________________________________________

Curriculum Organizer: - Number operations	Curriculum Sub-organizer(s): - Understand differences between rational and irrational numbers - Describe the difference between exact and approximate values for irrational numbers - Use a calculator to evaluate work involving a radical - Relate math to other subject disciplines
Prerequisites: - Determine whether or not a number is rational - Basic proficiency in use of a calculator for work involving rational numbers	Resources: - Pen (or pencil) - Calculator

 

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 A#= 466.17 B = 493.89 C = 523.25 These 12 notes, along with C (261.63), comprise the chromatic scale of C.

Published in 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