# Does a harmonic become a fundamental of its own harmonic series?

Simple question, hopefully there's a simple answer. I'm about half a piano tuner, not a physicist.

A musical tone has a fundamental frequency, say $220\,\text{Hz}$. Its second harmonic is $440\,\text{Hz}$, its third harmonic is $660\,\text{Hz}$, etc.

My question is:

Does a harmonic have its own harmonic series with itself as the fundamental? For example, does a $220\,\text{Hz}$ vibration, a 2nd harmonic which exists only because someone banged on a piano string that sounded a $110\,\text{Hz}$ fundamental, have its own second harmonic that is $440\,\text{Hz}$, a third harmonic, etc.

If not why not? It seems to me that if harmonics are real which I know they are because I learned to hear them, they must have their own harmonic series too.

If so, do these other harmonics have a name? I couldn't find them on google or wikipedia. I would think that if they exist, they must have a relatively low amplitude.

If I denote by $f$ the fundamental frequency, then the $n$-th harmonics has a frequency $n\times f$. So the $m$-th harmonics of that $n$-th harmonics would have a frequency $m\times n\times f$: this is the $(m\times n)$-th harmonics of the fundamental, which does exist. So that nomenclature you devised is consistent. We don't use it in Physics afaik.

A periodic waveform has a fundamental period $T$ (the length in time of the repeating pattern). By Fourier's theorem, such a periodic waveform can be decomposed into a fundamental component, with (fundamental) frequency $f_1 \equiv \frac{1}{T}$, plus components with frequencies that are integer multiples of $f_1$ which are called harmonics.

Note that it's not the fundamental component that 'has' harmonics (it doesn't, it's a pure tone), it's the waveform itself that has (contains) harmonics.

So I don't really grok the notion of a "harmonic [having] its own harmonic series with itself as a fundamental". It is the (periodic) waveform itself that has a harmonic series, not the components (which are pure tones) of the waveform.

• I would disagree slightly with your 2nd paragraph. A harmonic series is a mathematical construction. The waveform itself contains a fundamental (the actual longest wavelength standing wave, and lowest resonant frequency) and overtones. For many vibrating systems, the overtones may correspond to frequencies of a harmonic series built on the fundamental. In other cases such as tympani and flat bars, the overtones don't match the harmonics. – Bill N Sep 1 '17 at 14:37
• I'm curious has to how actual musical overtones which are not integer multiples of the musical fundamental would be reflected in a Fourier spectrum which is strictly integer multiples of the musical fundamental. I believe that operational FFT analyzers base their spectra on 1 or 2 Hz false fundamental bin widths. – Bill N Sep 1 '17 at 14:48
• @BillN, you may disagree if you wish but note that I began my answer with "A periodic waveform" (emphasis added) and Fourier's theorem is, well, a theorem. But, the waveform produced by, e.g., a tympani is not periodic. – Alfred Centauri Sep 1 '17 at 15:10
• There are no "periodic waveforms" (in the strict mathematical sense of the term) that exist in the real world, because they would have to start at an infinitely distant time in the past and continue for ever. And a piano tuner should know something about "inharmonicity," which means that even for a piano, the harmonic frequencies are not exactly in the ratio 220, 440, 660, 880, etc. The higher harmonics are progressively more sharp. Piano notes also decay just like a timp or a flat bar, except they decay slower. There are a lot of half-truths stated and taught about Fourier analysis! – alephzero Sep 1 '17 at 22:58
• @alephzero, it's true that there are no physical periodic waveforms (as I state in quite a bit more detail in the 2nd part of my answer here) but there are physical waveforms that are, in some sense, good approximations of periodic waveforms and those that aren't. But I don't really see that this fact is relevant to the essential point that I make in my answer above which is this: waveforms have Fourier components, Fourier components don't have Fourier components. Do you disagree? – Alfred Centauri Sep 1 '17 at 23:31

If you play the A below middle C, you get 220Hz and it has harmonics at 440, 660, 880, etc. 440 should match the A above middle C. 660 will be the just tempered E above that which is nearly but not quite the well tempered E. 880 will the next A etc.

If you play the A above middle C then you get 440Hz and it will have harmonics at 880, 1320, etc. These will be a subset of the first set of harmonics, the nth harmonic of this series will be the 2nth of the previous set. The odd harmonics of the lower note, e.g. 660 and 1100, will not appear in this set.

Similarly, if you play the the E near the top of the treble clef then you will get nearly 660Hz (not exact with well temperament) and its harmonics will be nearly every third of the original A at 220.

I am not aware of any special name for this relationship. I guess that no one has ever felt the need.