Mathematics is such a fundamental subject that it tends to play a part in just about every discipline, scientific and otherwise. As a certified math and music teacher, I had a diverse course load throughout my college days. The fun part about having two foci in college is that the hard lines that divide the subjects in your head begin to blur, and the connections that then develop are really special and unique. One of the significant connections between math and music that I had the chance to study in depth was the origin of our musical scale.
Most of us are familiar with the tune “Doe, A Deer,” which Julie Andrews made famous in The Sound of Music. This is our starting point of reference for what I mean when I say “musical scale” – Do, Re, Mi, Fa, Sol, La, Ti, Do. These eight pitches have been with us as a species well before Julie Andrews even stepped foot into Hollywood (though I did have a professor who liked to joke that Mrs. Andrews was, in fact, the creator of Solfege). There is no consensus about when this scale appeared in our history, but it certainly has origins at least as far back as the Pythagorean society in Ancient Greece (Circa 500 BC). After having studied Ancient Chinese music in depth for a research paper, my opinion is that this scale has been with us well before recorded history, but that’s a whole ‘nother blog post.
The Math
The method for achieving this eight-note scale really is quite simple. The math involved would fit perfectly into a 6th grader’s curriculum because the process requires only fractions and ratios. The problem, removed from its context, would look like this:
If I were a very studious 6th grader who read all of my directions carefully, my work might look like this:
The Octave
Now lets dissect this and give this problem some meaning. One very special, very fundamental musical idea is that of the “octave.” Sound is all about vibrations, and the speed at which something is vibrating is called its “frequency.” If two musical sounds are an octave apart, one is vibrating exactly twice as fast as the other. In other words, if you double a sound’s frequency, you will get a sound an octave higher; cut it in half, and you get a sound an octave lower. Musically speaking, octaves sound like the same note, just higher or lower. All around the world, men and women sing together in octaves, often never thinking twice that they are singing different notes. If you’ve ever taken a close look at a piano, the pattern of the notes repeats every octave, and we even call notes that are an octave apart by the same name.
This idea of the octave gives some meaning to our problem above. If we ever get a fraction higher than 2/1, we’ve gone higher than an octave away from our starting fraction (1/1), so we will cut the fraction in half, lowering it by one octave.
The Fifth
The distance from one musical note to another is called an interval. The octave is one example of an interval. The other interval we need to know to understand the math above is called the “fifth.” Notes that are a fifth apart have the ratio of 2:3. In other words, if two strings sound a fifth apart, one string is vibrating two times in the same time it takes the other string to vibrate three times. The sound of the fifth is very pure and resonant, and this interval is important in most musical cultures around the world. This interval is so important to our musical scale that it serves as the foundation to the math in our problem above. Each time we multiply our note by 3/2, we will achieve a note a fifth above.
Where’s the Music?
Finally, the answer to the problem needs interpretation. The seven fractions we achieve at the end of the problem represent the seven notes in our scale, in order from lowest sounding to highest sounding. Each note’s fraction can be thought of as the ratio of the lowest note in the scale to the given note. If we wanted to tune a piano’s white notes (F-G-A-B-C-D-E), we would simply let our starting pitch be F, which is what we call a note that vibrates at 174.61 times every second (and at every octave above and below!), and multiply that frequency by the fractions we boxed as our final answers above. That would look like this:
If we had a fancy program that could tune piano strings just by telling the program the exact number of times we wanted the strings to vibrate every second (if you know of one, please tell me!), the above numbers would give you a familiar sounding scale. Notice that I found the last “F” by multiplying our first “F” by 2 – these notes are an octave apart.
Now that we have these numbers, we could tune all the white notes on a piano by simply multiplying or dividing by 2 to find notes an octave above or below any of the notes we already found.
Deep Thoughts
The mathematical method you saw above has laid the foundation for thousands of years of music all across the world. These simple fractions allow countless composers and performers the opportunity to make beautiful works of art, which have mystified and captivated audiences for millennia. These numbers are echoed in nearly every song you’ve ever heard – every note in a Haydn symphony, every riff in a Justin Beiber hit single, and every syllable in the plainchant songs of ancient monks.
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