What are Twelve Notes Doing in the Octave?

In a memorable movie scene, a young Indiana Jones bursts into the house with news. His father holds up his hand, and Indiana obediently stops and begins counting slowly to ten. In Greek. In past times, a knowledge of Greek and Latin was an attribute of an educated person.

Many teachers today encourage students to learn about the ancient Greek and Latin roots of English words to help them with vocabulary. So, an educated student might guess that there are eight notes in a musical octave, right? Right, and wrong.

An octave is the difference in pitch between two notes where one has twice the frequency of the other. Frequency is the number describing the rate at which the physical waveform is traveling (or speed of vibration), whereas pitch describes how high or low the note sounds. The higher the frequency, the higher the pitch, and so for practical purposes, people use the two terms interchangeably. The note A above middle C has a frequency of 440 cycles/second or Hertz (Hz). This is the note used by orchestras to tune their instruments before performance. An octave higher, high A has a pitch of 2 * 440 Hz or 880 Hz.

If you look at a piano keyboard, there are eight white notes in the scale C, D, E, F, G, A, B, C. This is an octave, although without the second C there are only really seven white notes, hence the idea of the seven-note scale. But wait. There are also five black notes, which means this is a twelve-note scale, which is also known as the chromatic (coming from the Greek word khrōma meaning color) scale. Confused yet?

The number of notes in an octave depends on how the frequencies between f and 2f are quantized into discrete notes. On the piano keyboard, each successive note is said to be a semitone (or half step) apart. The C major scale is made up of only white notes played in a pattern of semitones (S) and tones (T) which looks like TTSTTTS. The modern piano has 88 keys, 36 black and 52 white, organized in 88/12 octaves to help the pianist find her way around the keyboard.

So what is the frequency progression up a scale? Let’s think. We already know that twice the starting note’s frequency f(0) is 12 semitones higher, or 2 * f(0) = f(12). We could say that 12 semitones represent the “doubling time” for the frequency f(0). We already know that the frequency progression between semitones is not linear, but exponential. Letting “x^n” mean “x raised to the n power”, we can write the progression between semitones as:

f(1) = f(0) * x
f(2) = f(1) * x = f(0) * (x^2)
f(3) = f(2) * x = f(1) * (x^2) = f(0) * (x^3)
…

f(12) = f(11)*x = … = f(0) * (x^12)

Combining this with our doubling time condition, we can conclude that:

2 * f(0) = f(0) * (x^12)  x^12 = 2  x = 2^(1/12) ~ 1.059463.

So the associated frequencies (in Hz) of the chromatic scale from middle C to high C look like:

C = 261.63
C#/Db = 277.18
D = 293.66
D#/Eb = 311.13
E = 329.63
F = 349.23
F3/Gb = 369/.99
G = 392.00
G#/Ab = 415.30
A = 440.00
A#/Bb = 466.16
B = 493.88
C = 523.35.

When a piano key is played, it strikes a piano string tuned to a specific frequency. The resulting vibration of the string is periodic. We learned a couple weeks ago that a periodic function of frequency f can be represented, using Fourier analysis, by sinusoidal waves with frequencies at integer multiples of f. These frequencies are called harmonics. Notes which share a lot of harmonics are said to sound sweet, or consonant, when played together.

For example, consider the notes C and G. The ratio of their frequencies f(G)/f(C) = 3/2. This ratio can be rewritten as 2*f(G) = 3*f(C), demonstrating that any two notes this number of semitones apart share many harmonics. Such “perfect fifths”, as they are called in music theory, are often played together because their sound is harmonious to the ear.

Music has the magical ability to touch human emotion, something we rarely discuss in the rational world of STEM. An understanding of the mathematics (and physics) behind music helps us understand why simultaneously played notes can sound consonant or dissonant. Yes, science can enhance our technical understanding of music, but it does nothing to diminish its wonder.