# Resonance and Pitch classes

Suppose we denote the frequency of the note $\textbf{C}_{0}$ by $f$ then we know that $f,2f,3f,...$ are all resonating frequencies. However the frequencies $2f,4f,...,2^{m}f$ belong to the same pitch class $\textbf{C}$ having the "same" sound(higher pitch) whereas the frequency $3f$ is the note $\textbf{G}_{1}$ and belongs to the pitch class $\textbf{G}$. What is the reason(physical) for this ?

• essentially you are asking is: Why is $\textbf{G}_{1}$ a harmonic to $\textbf{C}_{0}$? (or very close to in in the equal-tempered scale.) is that it? the answer is that $2^{19/12}$ is very close to 3. that's the reason a perfect fifth sound so harmonic. and the reason a major third sounds pretty good is that $2^{29/12}$ is pretty close to 5. – robert bristow-johnson Jan 6 '16 at 17:05

I think you gave the reason in your question. A periodic function can be written as a Fourier series, i.e. as a weighted sum of phase shifted sinusoids, where the frequencies of the individual sinusoids are integer multiples of the fundamental frequency. And the odd multiples obviously give you other pitches than the pitch corresponding to the fundamental frequency. This is what causes the same note to sound different when played on different instruments. It's called timbre.

The harmonics of the note C consist of all possible notes (and also in-between notes). The strongest harmonics with pitches different from C are G (the perfect fifth at $3f$), and E (the major third at $5f$).

EDIT: Your comment clarified your question for me. The phenomenon you're addressing is called octave equivalence. The most common argument for explaining octave equivalence is that the perfect octave is the most basic interval after the perfect unison, due to the simple relation between the harmonics of two notes an octave apart. But the assumption of octave equivalence is by no means universally accepted. As an example, every musician knows from experience that certain chord voicings which sound good in some range sound different (usually not as good) in a lower range. For some interesting thoughts on octave equivalence have a look at this page.

• What I am trying to understand is the reason for 2f to be so different from 3f. Why is is that 2f or 4f sound "exactly" the same (just higher pich) but not 3f. – midi Jan 6 '16 at 10:57
• @midi: I've updated my answer. – Matt L. Jan 6 '16 at 13:13

Pitch is a psychoacoustic phenomena that is different from pure spectral frequency. Under certain conditions, a human ear-brain will combine (fuse) multiples spectral frequencies into what sounds like a single pitch. This likely helps simplify recognizing an single source producing an overtone rich audio periodicity (e.g. a single very large critter in the wild is very different from a bunch of little ones.) Thus pure 1f plus 3f (etc.) sinusoids of the right volumes and timing will sound like one pitch to human listeners, while a 3f sinusoid alone will sound like a different pitch.

e.g. pitch and pitch class are not purely physical, but perceptual.

• Thus the existence of auditory illusions, just like optical illusions. Lots of research papers can be found on the topic. – hotpaw2 Jan 6 '16 at 13:26

Neat question! If you took any note's frequency, twice its value is the same note in a higher octave:

A4 --> 440 Hz

A5 --> A4*2 == 880 Hz

A6 --> A5*2 == 1760 Hz... and so on

This is simply how nature works. Any fundamental waveform and a multiple which is an exponent of 2 is perceived as the same note, which it is.

Now, if you take any note which is just an integer multiple, it is one of the harmonics of the note.

A major scale has the notes: A, C# and E

3*A4 = 1320 Hz $\approx$ E6 = 1318.51 Hz

5*A4 = 2200 Hz $\approx$ C#7 = 2217.46 Hz

which is a crude example to differentiate harmonics from the same note in a different octave.