Harmonic Theory

If a loose definition of musical harmony is "The study of notes played together", to gain a deeper understanding we can ask:

1. What is a note?
2. How do we know which notes to play together at what time?

A note is a disturbance of air molecules created by the vibration of some physical matter between initial time and final time . This disturbance may be described as a wave that consists of chunks of single-period waves, all with period .

Each of the chunks from to may be represented as a fourier series (shown here in amplitude-phase form)

where:

• and is called the fundamental angular frequency.
• and is called the fundamental frequency. (Most texts call the fundamental frequency)
• is the DC offset of chunk
• is the amplitude of the harmonic of chunk
• is the phase angle of the harmonic of chunk

Since all of it's chunks have period , the note may be said to have period and fundamental frequency .

The key point from section 1 is to realize a note is a wave that consists of chunks of waves, where each chunk has fundamental frequency and can be represented by an infinite sum of integer multiple sinusoidal frequencies, each of some arbitrary amplitude and phase, known as harmonics.

We can now begin to think about other notes in relation to each other.

Notation: A Note with fundamental frequency is denoted by

Consider 2 notes and where and are any positive integers. shares every of it's harmonics with every harmonic of .