By: Darrin Clark, Pam Duncan, and Damon Kelley

Dr. Carl Lee, MA 502

University of Kentucky

 PREFACE

Choosing a math topic for a research project leaves the door wide open to so many interesting subjects. As future math teachers we realize that our "audience" will not always be filled with bright-eyed interested left-brained students. In fact even some of the higher advanced math students will see a need to merely "get through" math in high school and simply choose only university study related courses. Also, unfortunately, at age 19 may never ever pick up another math book-expressing relief that they lived through it and luckily it is over.

So how exactly do we strike an interest in students and young adults? How do we answer the never-ending question-"when will I ever use this?" or "I am going to be a farmer; why do I care about this subject?" Sure it is easy to point out check book balancing and banking related topics, maybe tax time and government issues, or possibly the famous line of staying a step ahead of your offspring when they are bringing home algebraic word problems. But what about a look into a more liberal arts related area of life? As our research project we have chosen a scientific approach to music.

Many school systems are choosing to make cuts in their liberal arts curriculum. This forces students into a three R (reading, ‘riting, ‘rithmetic) course study. A history of mathematics in music and a look into various musical mathematical methods could turn a future mathematical "dropout" into an interested student. Plus, it could not hurt to "culture" those who are actually interested in math to take a look into a liberal arts education.

Combining subjects and teaching across the curriculum benefits all and hopefully could be a "break from the norm". Read onward to get a better look into ……… The Physics and Math of Music.

 THE HISTORY OF MATHEMATICS IN MUSIC

"….Adding up the total of a love that’s true, multiply life by the power of two…" -Power of Two by the Indigo Girls

  The song "Power of Two" by the Indigo Girls is a song about love and friendship centered on a pun in mathematical terminology. However, the Indigo Girls are not alone. Musicians ranging from the modern Violent Femmes with the song "Add it Up" to country singers Doug Supernaw with "Count the Costs" have belted out tunes with a somewhat mathematical flavor. Could any child of the 1960’s and beyond get through a Saturday cartoon session without singing along with such School House Rocks favorites as "Three, oh it’s the magic number, oh yes it is"? It is very clear that musicians and listeners have a respect for the power of mathematics, but what about mathematicians?

Backing up a couple of years to the sixth century B.C. in Greece, Pythagoras and the Pythagoreans began an in depth look into the abstract study of numbers. At the age of 50, Pythagoras founded a school whose pupils concentrated on four subjects of study: arithmetica (arithmetic in the sense of number theory as opposed to calculating), harmonia (music), geometria (geometry), and astrologia (astronomy). (The History of Mathematics, An Introduction. David M. Burton, p. 90-93).

Pythagoras spent much of his life devoted to the mathematics of music. One of the great Pythagorean accomplishments in music was the construction of the diatonic scale. To construct a diatonic scale begins with an arbitrary tone of frequency represented by 1, and ascends in steps of perfect 5ths:

1 3/2 (3/2)^2 (3/2)^3 (3/2)^4 (3/2)^5

OR

1 3/2 9/4 27/8 81/16 243/32

If these were based on C4 the notes would be C4 G4 D5 A5 E6 B6

  Step 2: Reduce the ratios obtained in step 1 into the range of a single octave by descending from these notes in whole octave steps: Keep in mind that doubling the frequency raises an octave and halving the frequency will drop an octave.

1 3/2 (9/4)*(1/2) (27/8)*(1/2) (81/16)*(1/4) (243/32)*(1/4) 2

OR

1 3/2 9/8 27/16 81/64 243/128 2

OR approximately

1 1.5 1.125 1.6875 1.265665 1.8984375 2

  Step 3: Arrange the notes obtained in step 2 in ascending order:

1 9/8 81/64 3/2 27/16 243/128 2

Step 4: The above sequence of notes is missing one important note the one corresponding to a perfect 4^th from the beginning point. This can be obtained by descending from the beginning point by a perfect fifth (2/3) and then moving up one octave (2/3) x 2 =4/3 = 1.3333…. Placing it in the ascending sequence as arranged in step 3, we have the following sequence:

1 9/8 81/64 4/3 3/2 27/16 243/128 2

  This is known as the Pythagorean diatonic scale. Examining the ratios between each step,

1 9/8 81/64 4/3 3/2 27/16 243/128 2

\/ \/ \/ \/ \/ \/ \/

9/8 9/8 256/243 9/8 9/8 9/8 256/243

  We can see that the scale consists of the familiar major scale (where the whole step is 9/8 and the ½ step is 256/243).

 ELEMENTARY MUSIC

Inversions, fractions, roots…. is this the beginning of an algebra class? It could be quite possibly that this is a look into the beginning of basic music theory.

A beginning music learners’ first step to the Royal Philharmonic would be a lesson in fractions. Let’s begin with notes.

Whole notes- receive 4 beats or counts

Dotted Half notes- receive 3 beats or counts

Half notes- receive 2 beats or counts

Quarter notes- receive 1 beat or count

Eighth notes- receive ½ beat or count

 We are now ready to begin our introduction to scales. There have been numerous ways devised to divide the

octave (frequency ratio of 2/1 sometimes written explicitly as 2:1) into smaller intervals. All the scales thus produced

have their own characteristics and are thus more or less suited for the playing of a particular type of music (or

particular types of music).

  Recent Western music has used a scale with 12 steps- semitone intervals- called the chromatic scale (from

the Greek khroma, meaning colour). Other cultures use scales with a different number steps, for example

4,5,7,11,12,13.

A scale is a discrete set of pitch relationships (or intervals), most often arranged in such a way as to yield a

maximum possible number of consonant combinations (or minimum possible number of dissonances) when two or

more of these intervals are sounded together.

The most consonant interval after the octave is the fifth (3/2) and the next most consonant is the fourth (4/3).

The difference between these two intervals

(3/2)/(4/3) = 3/2 * 3/4 = 9/8

is defined as the interval of a whole-step. This means that in a whole-step interval the frequency of the higher tone is

increased by a factor of 9/8 of the lower, or the lower is decreased by a factor of 8/9 of the upper.

 THE PHYSICS OF MUSIC

Mathematical principles and formulas are the roots of the scientific area of physics. Upon taking a closer look into

the actual physics of music gives a better understanding of what is actually heard. Sound is what makes up music.

There are many different sounds in music and many different instruments that make them, but what makes music?

The source of all sounds is vibration. These vibrations are making little zones of compressed air. This air pushes

against the air around itself which pushes that air and so forth. The combination of this compressed air and refraction

of matter produces longitudinal waves or sound waves. A vibrating source that produces rhythmic variations in air

pressure is the source of sound waves.

Ears pick up the many different characteristics of sound, like amplitude, frequency, and wavelength. Amplitude is

the loudness of the sound. Frequency is the pitch of the sound, and wavelength is just as it sounds, the length of the

sound wave, or the distance between successive compressions.

When something resonates it increases the amplitude of a vibration by repeatedly applying a small external force at

the same natural frequency of the object. If a vibration of an object causes sound waves that excite another object

into vibrating, the second object is said to resonate.

  Now for the grand finale of this look into the math and physics of music, let's look at the physics behind the ever-

popular string instrument of the guitar. Lately no song is ever fully complete without the harmonic sounds of the guitar.

The guitar crosses all areas of music, ranging from the twangy bluegrass to the up beat pop sound.

Vibrating a string to produce sound operates the guitar like all other string instruments. The pitch or frequency of

the sound created by the string depends on both the lengths of the string and its tension. A guitarist has the ability to

change both. They can change the length by moving their finger up the fret board. This movement is what changes the

pitch allowing music to be played. The tension of the string, unlike the length, is rarely changed. This is only done to

tune the guitar.

When a guitar string is plucked, transverse waves are created within the string. These transverse waves are sent

down the guitar string to the bridge and they are then reflected back. The reflected waves then interfere with the

waves traveling in the opposite direction, producing standing waves.

TRANSVERSE WAVE

  These transverse waves vibrate the surrounding air to produce longitudinal sound waves. However, the string

alone cannot vibrate enough air to produce a loud note. The little air that vibrated travels through the sound hole and

into the sound box. Once inside, the longitudinal waves resonate. This causes the entire sound box to vibrate,

increasing the amplitude of the note.