The foundations of harmony are not based in “tritone substitutions”, or even “tritone resolution”, nor in “proper voice leading”, nor “roman numerals” or even simply in “triads”.

The foundations of harmony, or harmonic theory, lie in the natural overtones which are produced in a musical tone. As a guitarist, you can find these natural overtones by playing harmonics on an open string. The exact position of each harmonic can vary depending on your guitars setup, but try to find the following 12 harmonics on the lowest (E) string:

1: open string
2: above 12th fret
3: above 7th fret
4: above 5th
5: a bit behind 4th fret
6: a bit in front of 3rd fret
7: a bit behind 3rd fret
8: halfway between 2nd and 3rd fret, a bit close to the 2nd fret
9: a bit in front of 2nd fret
10: just a bit behind 2nd fret
11: just a bit behind 10th harmonic
12: just a bit behind 11th harmonic
etc…

The series of overtones can be extended infinitely, but you probably found that they became more difficult to produce the higher you went. We will concern ourselves primarily with the harmonic tones through the 8th harmonic.

In our exercise, we ordered and numbered each harmonic played. The first harmonic was not really a harmonic at all, but simply the open string. The second harmonic was produced at the 12th fret and is “one octave” higher than the first. The important thing here is the relationship between the two different tones. In music, this relationship can be referred to as an “interval”, in mathematics a “ratio”. Western musicians refer to this first interval as an octave, and mathematically the ratio between the two tones is 2-to-1, or 1-to-2, or 2/1, or 1/2, all essentially being the same, being “inversions” of each other.

Each harmonic is related to every other harmonic and can be measured this way. 2/3, the relationship between the second and third harmonic, is called a “perfect fifth” in western theory.

Here is a list of the same harmonics, with the corresponding interval in western theory:

1: fundamental
2: 1st “octave” of the 1st harmonic
3: “perfect fifth”
4: 2nd “octave” of the 1st harmonic
5: “major third”
6: 2nd “octave” of the 3rd harmonic
7: “minor seventh”
8: 3rd “octave” of the 1st harmonic
9: “perfect fifth” of the 3rd harmonic
10: 1st “octave” of the 5th harmonic
11: “tritone”
12: 3rd “octave” of the 3rd harmonic

We established that the relationship of 2/1 is an octave doubling. If we apply this relationship/ratio to any harmonic, we will end up one octave higher. This is an example of what can be called a “derived interval” or the stacking of two or more intervals to produce a “new note” that is not based on a new prime number, or new identity.

Two important things must be understood going forward. When we add two intervals together, say ‘perfect 5th’ and ‘perfect 5th’, we must multiply their ratios, 3/2 * 3/2. Also, when we talk about ratios, we usually want the ratio to fit within one octave – I would rather write 2/1 instead of 4/1, 8/1, 16/1, or 32/1. See Kyle Gann’s great article Just Intonation Explained for further elaboration on working with musical ratios. The best part about Gann’s page is all of the audio demonstrations he has put together. If nothing else, listen to those and let your ear do the work for you.

Let’s look at another example of a compound interval to find a “new note”, the ‘major 9th’. When we multiply the ratio of 3/2 to any harmonic, we will end up a ‘perfect fifth’ higher (in fact, a perfect fifth the next octave higher). Multiply 3/2 by 3/2, and you will have 9/4. Multiply 9/4 by 1/2 to raise it an octave higher and you have 9/8, the 9th harmonic, which is also a ‘perfect fifth’ of the 3rd harmonic, or a ‘perfect fifth’ of a ‘perfect fifth’. Over 1000 years of western music was based entirely on the stacking of perfect fifths – more will be discussed about Pythagorean Theory in another post.

What about intervals based on prime numbers? Go back to the image – see the numbers with the colored circles? Each of these numbers is a prime number. Prime numbers represent new “identities” of tones, sort of like different colors or different moods… 1 is the fundamental. 2 is the octave, both a different note but also the same identity, as the fundamental. 3 is the ‘perfect fifth’, strong and powerful, being closely related to the fundamental. 5 is the ‘major third’, strident, colorful, active, and warm. 7 is the ‘flat minor seventh’ bluesy, soulful, more complex. 11 is a kind of ‘tritone’, and to my ear has a flat and nasal quality that is tough to describe. When we reach prime 13 or above, personally my ear has a tough time understanding and gets lost.

These harmonics do NOT correspond precisely with the notes produced by the frets on your guitar, nor by the notes played by the piano. A explanation and critique of ‘equal temperament’ will follow.