Music Theory Primer

This page is designed to give basic information about music theory that gets covered in lessons, so that students can consult and review this information at their leisure. I am gradually building this page. Be patient with me!

Reading Music

Cellos have four strings, and they are written in musical notation as follows:

The C-string is the lowest string, followed by the G, D and A. The C-string sounds exactly two octaves lower than middle C. Note that the open strings all fall on the lines of the bass clef staff, not the spaces. Some students find it helpful to envision these lines as the strings themselves: the top line is the A-string, the middle line is the D-string, and the bottom line is the G-string. The lowest string, C, is two ledger lines below the staff.

There are no good shortcuts. Learning to read music simply requires lots of reading of music. It goes slowly at first, but very soon it becomes second nature. Eventually, cellists also have to learn to read tenor and treble clefs (and even the further variant of treble clef down one octave for certain orchestra parts, such as Dvorak).

Harmonic Series

It's useful to know a little bit about the physics of sound (because it turns out to be the basis for all musical relationships, the content of music theory). When we hear a specific note, say the open A-string, what we are actually hearing is a fundamental waveform traveling through the air vibrating at a speed (or frequency) of 220 waves per second, plus a whole lot of softer-sounding but faster waves, called partials. This is a little complex, so let me unpack it some.

What makes musical sounds different from other random sounds is that they vibrate at a constant frequency. As a result, our ears can discern a specific pitch. Random noises don't vibrate this way, so our ears only hear an incoherent jumble of many pitches. When an orchestra tunes at the beginning of each rehearsal and concert, it uses a musical pitch that vibrates 440 times per second (which is measured in units called Hertz, notated as 440 Hz).

As it turns out, we not only hear that frequency, but many others at the same time--only the higher notes, called partials, are so soft (have much less amplitude) that usually we can't consciously hear them. But those partials are there, and they make all the difference in the tone quality of the sound we're hearing.

When we collect all the partials, sometimes called harmonics, that sound when we hear a fundamental note, we end up with something called the harmonic series. Every note has its own harmonic series, but they all include the same sequence of intervals (I'll have much more to say about intervals later). In other words, the harmonic series is a universal physical feature of musical pitch. Here is what one looks like:

So, in this example, the note we play on the cello is our lowest open string, C (the "fundamental"). All the other notes you see on this chart also sound, though much more softly. The "second harmonic" (or "first partial," depending on what terminology you prefer) is the C one octave higher, followed by the G above that, followed by the C above that (middle C), followed by the next E, then G, then B-flat, then C, and then step by step, in smaller and smaller increments as it goes higher. All of these notes are vibrating in the air whenever you play a low C on the cello.

This simple but universal fact of acoustics accounts for the way we humans organize musical space. While different cultures have responded differently to the harmonic series (see here for more about that), I will focus on the specific way Western ears have heard and organized musical sound.

The Law of Octave Equivalence

The first, most basic, principle to emerge out of the harmonic series is the division of musical space into octaves. As we ascend through the universe of all possible pitches, we begin to notice that higher notes sound very similar to certain lower ones. Specifically, whenever we double or halve a musical frequency, we produce a sound that seems the same. That's why the note the orchestra tunes to (the note A, at 440 Hz) sounds the same as the top string of the cello (the note A, at 220 Hz). In fact, they are different pitches, one significantly higher than the other. But, still, they sound virtually identical, so we give them a common name, A.

Here are all the pitches named C, all equivalent, throughout the whole of musical space:

The best explanation for why our ears hear it this way lies with the harmonic series. Every time we hear a pitch, the closest and often loudest partial we hear simultaneously is always exactly one octave higher. This tight synchronicity leads our ear into sensing that the two notes are, in some inexplicable way, the same (or, more accurately, equivalent).

For this reason, early in our musical history we recognized that the universe of different (as opposed to equivalent) notes was relatively small, occupying the space within an octave. Higher and lower octaves offer no entirely new notes, just the same notes and relationships repeated at higher and lower strata of musical space. This is why we only use seven letters of the alphabet (A-G) to describe all the possible notes.

The Basic Scale

If we take all the notes of the harmonic series and collapse them into a single octave, we get the following series, ascending from the fundamental note: C, D, E, G, B-flat. This almost gives us a full seven-note scale. The F and the A fill in the gaps to give us a scale form, called Mixolydian mode (C, D, E, F, G, A, B-flat, C), common in the earliest Western music:

The F, A (and B-natural) missing from the harmonic series based on C can be derived from taking the harmonic series of G, the note closest to C in its own harmonic series. The harmonic series based on G yeilds: G, A, B, D, F. Indeed, by combining the notes of these two harmonic series, all the notes of the basic scale forms (and all the white notes of the piano, plus B-flat) are accounted for:

The music theory primer continues...