Finding Harmonic Relationships Between Frequencies in Experimental Data

[DeathbyGreen]

I'm trying to relate some different frequencies together in an experiment. Say I have 3 different frequencies, $\omega_1,\omega_2, \omega_3$. Omega 3 is the large envelope, and the other two must fit inside of it, and so they are integer multiples of each other. Is there some way to express $\omega_1, \omega_2$ as equal approximately, or at least in terms of the third without having a mess of different constants? I've looked through some harmonic theory without much luck. All I've been able to think is

$\omega_1 = n_1\omega_3\\ \omega_2 = n_2\omega_3\\ \omega_1 = n_3\omega_2\\$

And just mix and match from there. What I would like though is for $\omega_1\approx.\omega_2$, maybe using a decomposition or something.

 

[Mark44]

I'm trying to relate some different frequencies together in an experiment. Say I have 3 different frequencies, $\omega_1,\omega_2, \omega_3$. Omega 3 is the large envelope, and the other two must fit inside of it, and so they are integer multiples of each other. Is there some way to express $\omega_1, \omega_2$ as equal approximately, or at least in terms of the third without having a mess of different constants? I've looked through some harmonic theory without much luck. All I've been able to think is

$\omega_1 = n_1\omega_3\\ \omega_2 = n_2\omega_3\\ \omega_1 = n_3\omega_2\\$

And just mix and match from there. What I would like though is for $\omega_1\approx.\omega_2$, maybe using a decomposition or something.

Based on what you wrote, it depends on the frequencies.
As an example, consider the fundamental frequency of a guitar string -- call this $F = \omega$. One harmonic can be sounded by lightly pressing the middle of the string, at the 12th fret. This tone is an octave above the fundamental tone, with a frequency of $F_1 = 2\omega$. Another harmonc can be sounded by lightly pressing at the 7th fret, a third of the length of the string -- $F_2 = 3\omega$.

Although $F_1$ and $F_2$ are integer multiples of the fundamental tone F, the other two tones in my example are not integer multiples of each other.

There's another harmonic that can be sounded -- the one by lightly pressing the fifth fret, a quarter of the string length. This frequency, $F_3 = 2F_1 = 4\omega$, so here's an example where the frequency of one of the harmonics is an integer multiple of the frequency of another.

 

[Baluncore]

Given an envelope frequency ω3, you know that ω1 and ω2 will be integer multiples of ω3. Assuming ω1 and ω2 must be different frequencies, simply select two close integers, n1 and n2, then make ω1 = n1⋅ω3 and ω2 = n2⋅ω3

If you make sure that the integers n1 and n2 are mutually prime, then the relative phase of ω1 and ω2 will not repeat within the period of the envelope ω3. The obvious way to select n1 and n2 to be close but not equal, would be to pick a prime for n1, then make n2 = n1 ± 1.

The bigger n1 is, the closer ω2 can be to ω1 and the more cycles of ω1 and of ω2 there will be within the period of the ω3 envelope.