# Fractional harmonics in musical Instruments

I recently did some Fourier transforms on different audio files containing saxophone or trumpet (John Coltrane/Clifford Brown).

I found that with the saxophone, the frequency spectrum occasionally contained half-integer and even quarter-integer harmonics. With the trumpet, I saw half-integer harmonics, which I was able to reproduce with my own trumpet. Also, harmonics were found in both instruments below the fundamental frequency.

Is there any explanation for these phenomena in terms of treating the instruments as pipes with standing waves, or does the existence of fractional harmonics mean that a different method must be used?

• That's cool and should give rise to a better understanding of the sound synthesis for these instruments. I think one would have to look at a trumpet as, at least, multiple coupled pipes rather than one pipe. Did you see this for the lowest frequency that the instrument can produce? – CuriousOne Jan 7 '16 at 3:54
• I did not do it for the lowest frequency, but coupled pipes is kind of what I had in mind. If the fundamental frequency generate even and odd harmonics, a separate wave that is generated at half the frequency that contains only odd harmonics will produce half-integer harmonics. However, the amplitude profile is not distributed like the sawtooth and square waves. – soultrane Jan 7 '16 at 3:59
• It would be interesting to know at what frequencies this is strongest and then one should try to match that to the effective pipe lengths in the instrument... that would be my first guess before thinking about non-linearities. An interesting experiment could be to excite the instrument with a loudspeaker at the horn end and see what its frequency response is. That would also allow you to test the hypothesis if something non-linear is happening e.g. in the valves. Non-linearities should be strongly amplitude dependent. – CuriousOne Jan 7 '16 at 4:05
• Do you mean blowing air in the horn and recording the response at the mouthpiece? I'm not sure if that would work... – soultrane Jan 7 '16 at 4:13
• Could you please post a plot of the data? – DanielSank Jan 7 '16 at 6:37

Edit after providing the plot: Please note, that the simple model prediction of harmonics frequency position does not say anything about its strength in actual sound. It is typical feature of brass instruments in middle and lower registers that the fundamental frequency is not the strongest.

Original answer: I am sorry guys, but that's well-known and studied phenomena (that's for the comments). In fact, the question is too broad to be answered simply without examinating the concrete recording, but let me here give some notes on why it could be like that.

• Resonant frequencies of the waveguides don't have to be strictly harmonic. The key equation for simple understanding is the Webster equation for acoustic pressure dealing with the 1D waveguide along the $x$ with cross-section area $S(x)$:

$$\frac{\partial^2 p}{\partial x^2} + \frac{\partial (\ln S(x))}{\partial x}\frac{\partial p}{\partial x}+ k^2x=0$$

Apparently, for a cylindrival duct the middle term is zero and we would have simple LHO equation. And the mouthpiece cavity give the significant compliant element to the system. Other distortion of the harmonic behavior are termoviscous wall losses.

• In the frequency domain you can see the reflections from instrument discontinuities. Typically the frequency corresponding to the first open tonehole $\lambda$ and the frequency of the whole instrument $\lambda$ as well.

• There are points on the instrument with so constricted cross section that the mean flow of the breath is not negligibly small. In short: bye, the linear acoustics. We must the solve the complete Navier-Stokes eq. or at least use Burger's equation. Generally, in driven non-linear systems there could be resonances on subharmonics.

• The driving of the system is nonlinear as well.

• There is much of uncertainty when it comes to feedback loop of oscillator-resonator in these cases.

For the future reference I strongly recommend you The Physics of Musical Instruments by Fletcher and Rossing.