Download Paige Studlack Malone Middle School Ms

Paige Studlack
Malone Middle School
Ms. Matthews
Mathematics and Music
Your Algebra teacher may have said to your class, "Math is everywhere." The way I see it, there is two
reactions to this sentence; you roll your eyes and continue doodling in your notebook or your eyes may
have truly opened and started to see (and hear) it, too. Mathematics is everywhere. You use it and are
exposed to it all the time and you don't even realize that as you are listening to your favorite c.d. or writing
your eight measure long duet for band homework that your brain is doing the math almost unconsciously.
Music is a series of sound waves, pure and simple. Sound is measured in frequencies (cycles per
second) labeled in Hertz. When you hear different, overlapping notes that sound good together, it is
because their frequencies are consonant and the sound waves meet and cross at regular intervals. This
produces a pleasing sound (harmony). For example, a piano's middle C's second wave and the 3rd wave of
the G directly above always match in a pattern and thusly, sound good together. These notes are illustrated
below. Middle C has about 262 cycles per second and G's frequency is 392 Hz so the notes have a ratio of
approximately 3/2. A note combination that makes even someone nearly tone-deaf cringe is the same
Middle C and the F# of that same octave. The sound waves seem to never coincide, although they probably
do by chance long after the human ear can hear it and there is no apparent pattern. The first person to see
that ratios could be applied to nearly everything (including music) was Pythagoras of Southern Italy in the
fifth century. He developed the Pythagorean Theorem and noticed the ratios of the Greek scale in a harp.
The A note below the Middle C was a full string and the Middle C was 4/5 of the full string. The D string
was 3/4, E equaled 2/3, and the F string was 3/5 compared to the original A string. The octave higher A
was one half of the lower A and the fractions continued in this arrangement. Anyone that knows a simple
scale knows that there is also a G and B in the octave. Some unaccredited philosopher decided to change
the Greek scale into today's norm by taking E squared and multiplying the product by two to get the B
above octave A. The same person also found the note known as G by taking B's "string length" of 8/9's
inverse (9/8) and .5 to get the product of 9/16, known as G. A regular scale today can be transcribed using
the irregular ratio of 2 to the 1/12th power. The unit of measurement for sound is frequency, which is the
basis of music.
Other aspects of sound affect the way we hear music. Acoustics, for example, is the way sound waves
bounces off surfaces and influences the speed of which we hear the notes. The angles of the sound waves as
they hit a surface before reaching you also have an effect on the way music sounds to you. There is even
trigonometry in the air molecules after they come out of an instrument. The formula is p (t) =A sin (Bt +C).
The t is time, p is pitch, A is volume, and B is frequency. Since the sound waves move in a back-and-forth
fashion, this formula is the reason we hear notes and not just noise. Frequency is not the only sound
property that influences music.
Not only are there patterns in the mechanical sense of music, but when a composer writes a piece of
music, he or she uses some mathematical systems. For example, the golden mean tends to apply to a
musical work. The song will have an emotional or volume-wise peak approximately .89 of the way
through. Although not as popular in today's music, look at almost any Baroque era composition and you
will find a peak. Pieces from the Baroque and Renaissance eras were very mathematical and were
considered math, not an art. Another musical property used was probability. In a two century year old dice
game, anyone could roll a dice, which picked, out of eleven, the first seven bars of a Viennese minuet.
Then they would roll for the eighth measure, picking one of two choices. This was repeated for the second
half of the minuet also. The idea was that you could make a random song and have it be pleasing and
harmonically correct using sheer probability. Mathematics is needed in composing music also.
Music has beats and rhythms. Both are based on math. A tempo is the number of beats per minute and it
is the speed at which music is played. There is a certain number of beats per measure and which kind of
note gets the beat. For example, in common time (4/4), there is four beats between the bar lines and the
quarter note gets the beat. If you were playing a march with a tempo of 120 and used common time, you
would play two quarter notes' worth of notes per second. Music uses fractions in the writing of the notes.
Say we're in 4/4. A whole note is four beats, a half note is two beats, a quarter note is 1 beat, an eighth note
is ½ a beat, and so on until you get to about 32nd notes (the farthest I've seen). Math is present in tempo,
beats, and note structure.
Whole Notes
4 beats in a common time measure
Half
Notes
2 beats in a common time measure
Quarter
Notes
1 beat in a common time measure
Eighth Notes
½ beat in a common time measure
Sixteenth Notes
¼ beat in a common time measure
32nd
Notes
1/8 beat in a common time measure
Harmonics are chords and also have a basis on math. A full harmonic note has its intentional note the
loudest and then its major third at half the volume at the same time. It also plays the major fifth at a quarter
of the volume of the first note and the pattern continues upward until human ears cannot hear it anymore.
For example, if I played a middle C and held it, I would also be playing E at half the volume and G at a
quarter of the volume and so on and so forth. The harmonics of a chord are matched by frequencies and
ratios, so harmonics can sound heavenly or off. Mathematics greatly influences harmonics as well as the
other aspects of music.
I think that the most important math property in music as far as playing a good melody or harmony is
ratios. If you play a chord with off ratios, it will sound terrible. The D to middle C ratio is about 9/8 and E:
C ratio is 5/4. The E and C will sound better together because the waves cross at the middle more often.
This is true for all twelve notes of a chromatic octave; the more the waves meet, the more pleasing a chord
to our ears.
C to D
C to E
C to F
C to G
C to A
C to B
Approximate ratio: 9/8
5/4
4/3
3/2
5/3
17/9
While music is incredibly an artistic thing, as with everything, there has to be math involved. The math
we use in music without thinking about it includes: ratios, fractions, trigonometry, angles, speed, and math
having to do with sound waves. The difference between a great sounding chord and your cat walking on
your piano keys has to do with math.
Bibliography
Heimiller, Joseph. "Where Math Meets Music." Musicmasterworks.com. 2002. 3
April 2005.
www.musicmasterworks.com/WhereMathMeetsMusic.html
Henderson, Nancy. "Music and Math a la Mozart." Study Works! Online. 3 April
2005.
www.studyworksonline.com
"Math and Music: A Primer." 3 April 2005.
www.members.cox.net
Rusin, Dave. "Mathematics and Music." 1/9/2004. 3 April 2005