Mixing Two Tones (Take 3)

Instead of using ω₀ as the reference frequency for the superposition one can use ϖ = ½(ω₀ + ω) and derive a new expression for the sum. The computing both expressions over a number of beats one finds that there difference is zero.


So the epicycles were the result of using ω₀ as the reference frequency which resulted in a steady increase of phase φ = ½Δωt.

In general one can simplify the above result a little more.


There is a steady tone at the reference frequency ω₀ and one at the average of the two frequencies ϖ which is the one that fluctuates in amplitude. When a and b are equal the steady tone is not present and one just hears the warbling tone. The second term shifts in phase and amplitude relative to the first. That's the "epicycle".

Try listening to the shorter intervals such as the tone or semitone in the Wikipedia particular number article. There's a bit of a quaver present. One could also try simultaneously playing two neighboring keys on a piano or notes on a guitar.