Understanding the Harmonic Series

Most approaches to music theory tend to be quite dry, at least in my experience of studying it, so it is refreshing to encounter another way of getting the hang of how music “works”. Ron Gorow’s book, Hearing and Writing Music, seems to offer a path into music for the aspiring composer that involves really getting inside the fundamental principles.

The first thing that has started to fascinate me is the harmonic series, which can easily be found on a string instrument. I happen to have an old guitar hanging about, so I had a go with that, as I’d already learned how to play harmonics. It’s a pleasant sound, but a closer look at what’s going on reveals something interesting about the structure of music.

By placing your finger lightly on the guitar string half way along, you force the string to vibrate in the second harmonic, which sounds an octave higher than the fundamental. The tone is closely related to the fundamental, and it is percieved as being the same in some way, although “higher”. Moving the finger two thirds of the way up the string produces the third harmonic, sounding a perfect fifth above the second.

The perfect fifth is generally considered the most “stable” interval, and across numerous cultures it is generally regarded as having a good sound. The Chinese tuned instruments using perfect fifths and derived their pentatonic scale in that way. The fifth seems to gravitate towards the root or fundamental tone, relating to the concept of “resolution” in theory. What sounds “final” to the ear has a strong relation to the properties of harmonics.

The fourth harmonic completes the second octave with a perfect fourth. The fourth isn’t as stable as the fifth, and feels like it wants to go “up” to the root note. It is said to have an ambiguous quality, or indeterminate state, but I feel that the ascending fourth has a slightly more satisfying resolution.

In the equal tempered system, that enables compositions to use any of the keys, the fifth and the fourth are the only natural intervals, maintaining the tone of their pure harmonics. The other intervals are progressively detuned to produce accurate fourths and fifths up and down the scale.

The natural thirds emerge from the 5th and 6th harmonics, a major third and a minor third. These fit into the space of a perfect fifth, and in fact form a major triad. At this point we have the main foundations of diatonic music, and the familiar 12-semitone scale.

When you think about music in such fundamental terms, it really aids understanding, and it is easier to get the hang of older musical conventions, such as the modes used in plainchant of the middle ages. Modes seem to make much more sense when you consider this predated the equal-temperament system, and tuning was only done to one key. The modes allowed a variation in mood without the diatonic system of major and minor keys, because if you tried to do that with an instrument tuned to natural intervals the result would sound wrong. The rules of modes are designed so the composer of melodies can achieve the desired feel of a piece.

I may say “designed” but my thinking is more towards the idea of discovery. Before the idea of equal temperament, musicians were familiar with the process of tuning with perfect fifths (platonic tuning) and using the ear as a guide rather than mathematics. What we now call the major scale could be formed from that, but there would usually just be the one scale used at any one time, tuned to whatever sounded best. Modes were probably discovered as musicians studied the sounds of the intervals produced in the scale, noticing that a different “colour” emerged if the scale was played from a certain tone.

Modes can be emulated easily by making use of only the white notes on a keyboard. The basic “major” mode on C is essentially a major scale, but starting on E instead results in a minor third emerging, giving the sound of a minor key, although it is not quite the same. Because the mood can only be maintained by staying with the note order, composing with modes leads to a kind of static quality in music. Before the advent of equal temperament, most music would tend to be like this, and mostly melodic as harmony is fairly “limited”, or minimal to give that kind of music its due.

This is pretty much what you might learn in music theory books, but what Gorow suggests is to spend time familiarising yourself with the sound of each interval so you can identify it instantly and “feel” what a fifth or fourth is before considering symbols on a stave. Music theory tends to start with a discussion of notes rather than what they are supposed to represent. The phenomenon of a fifth is the same no matter where it appears in the audible frequency spectrum, and has no fundamental relationship with standardised tuning (i.e the A above middle C being set at 440Hz).

Ultimately, perception of intervals is the key to perceiving melody and harmony, and intimate knowledge of each interval is essential for the composer. Talented individuals may get there faster without necessarily being concious of the fact, but systematic study is just as valid.