Intervals and the Math of Music

Music is so many things to society. It is an activity, hobby, profession, art and cultural exchange of meaning. It is also a mathematical construct which can be studied scientifically and defined very precisely.

Acoustic music is rarely as precise as mathematics. Jazz drummers, for example, are taught from their earliest performances to "hang back" a little on the offbeats, and the other members of the band are taught to follow along. Symphony orchestras almost never perform to a metronome, even though most virtuosos and soloists are capable of very precise measurments of tempo, acceleration and deceleration of time.

But the recent advent of digital music makes the math of sound much more important, and understanding the western diatonic form far more relevant to the theory of music as opposed to the practical considerations of the art form.

Tonality, which is the study of certain sounds at given frequencies, is abstracted from the art of music, since tonality could be applied non-musically to any sound. Certain musicians with perfect pitch, for example, can tell you with unerring accuracy the frequency and piano note of the creaking sound made by a door hinge as the door opens or closes. Those with relative pitch are a little more common than people with perfect pitch. They are very good at comparing sounds, telling you whether two sounds are "in tune" or how far apart two sounds are on a diatonic scale.

Those measurements are called "intervals" and are extraordinarily important, because the language of intervals are how music and tonality are translated from sound to language.

All musical tonality takes place within a linear series of 12 progressively higher frequencies known as a chromatic scale. On a piano keyboard, these tones can be measured by the black keys. (Warning: Technical explanation approaching, but in four easy steps!) At one end of the scale are two black keys separated by one white key, and at the other end are three black keys each seperated by one white key. Here's how to construct a chromatic scale:

1. Start with the white key to the left of the set of two black keys

2. Find the white key just to the right of the rightmost black key in the set of three black keys

3. Add the next white key to the right of that one

4. Put all of those notes together

Congratulations, you just built a chromatic scale, starting on the note of 'C' up one octave to the next note of 'C.' This scale includes all of the written notes, regardless of key, within one octave.

Now, a diatonic scale drops five of these notes, and only includes the tones native to a certain key, but in a certain progression. This progression is the same for every major scale, regardless of key. The progression is (proceeding up the scale) whole step, whole step, half step, whole step, whole step, whole step, half step. This progression is known as a "major scale" because it includes all of the notes in a major key. If you start on 'C,' you'll notice that this progression includes none of the black keys: and it the only such major key.

If you start on any other note, and use the same progression, you will be constructing a major scale, just in a different key. For example, if you start on the right-most black key in the group of two black keys, you'll be constructing a major scale in the key of E-flat, and you'll find that you'll have to use three black keys to make your scale: E-flat, A-flat and B-flat.

The "steps" noted in the major scale progression have a more precise name when discussing their "intervals." Starting from 'C,' on a piano, the "whole step" up to the next note, the note of 'D' is known as an interval of a "major second" because that note is the "second" note in the major scale. The next note up, the note of 'E,' is called an interval of a major third.

Well, that's easy! So it's just second, third and fourth! You might think so, until you realize there is no such thing as a major fourth, or fifth. For reasons dating back many centuries which could easily become another article, there is a special relationship between the first note of a scale, known in the language of chords, or combinations of sounds, as a "tonic," and the fourth and fifth major intervals on that scale, known as "sub-dominant" and "dominant" respectively. Because of this, fourth intervals and fifth intervals are known as "perfect" intervals instead of major intervals.

Also, two notes with exactly the same frequencies, called a "unison" interval, are known as a "perfect unison," while two notes which are precisely an octave apart are known as a "perfect octave." An octave is the next note up or down from any given note on a piano with the same name. For example, another 'C' key is located 12 tones to the right of middle 'C' on a piano. That 'C' is said to be one "octave" from middle 'C,' and playing both notes together is an interval of a "perfect octave."

In part two, we'll see how intervals can be used to construct chords and chord progressions, which are the building blocks of nearly all genres of music.