# Can we quantify the pitch of a sound that is a mixture of many frequencies?

How is the pitch of a sound defined quantitatively when it is a mixture of many frequencies? For example, the sound emitted by a plucked guitar string, or say, the pitch of somebody's (normal) voice. I know that female voices are generally of higher pitch. But can we quantify the notion of pitch? My guess is that the pitch of a complex sound depends on the relative amplitudes of the different frequency components.

• Have you read the article Perception of musical pitch varies across cultures? Jun 28 at 22:15
• Pitch perception is a complicated biosocial phenomenon. For instance, by selectively removing overtones from a harmonic series, you can create the perception of at least two pitches. This artist is exceptionally good at it, but it’s a pretty great party trick.
– rob
Jun 28 at 22:56
• @rob overtone singing doesn't work by removing overtones, it works by creating a resonance that amplifies certain overtones. Basically, it's not just the perception of multiple pitches, it is multiple pitches: two oscillators, though one of them happens to be excited by the other. Jun 29 at 14:46
• @leftaroundabout “Removing” vs “amplifying” is a distinction without a difference. The transition from “ooooo” to “eeeee” is managed by amplifying the higher-frequency part of the overtone series. The overtone singer amplifies more selectively than a person producing a more standard vowel. There is only one oscillator, the vocal fold, in a human overtone singer; a second oscillator would have its own overtone series, which are missing from the spectrographs in the link. For a two-oscillator system try activating your vocal folds while whistling; the effect is very different.
– rob
Jun 29 at 22:26
• @rob I mean, this is to some degree a matter of terminology-lawyering. In fact every audio filter (except digital FIRs) consists of a series of (more or less) damped oscillators that are (more or less) linearly driven by the input signal. As you make the damping weaker (overtone singing vs normal speech), it'll behave more and more like a standalone oscillator that keeps on ringing after the input signal has stopped, and if there are nonlinearities (as with the rapid air stream through the lips when you whistle) it can even self-excitate at a stable amplitude. It's still all an oscillator. Jun 30 at 7:45

Pitch can be described as a subjective perception of an auditory stimulus which cannot be objectively, unambiguously quantified. It is strongly related to the objective physical property of frequency such that higher frequencies typically correspond to higher perceived pitch, but even notes with identical frequency can be perceived as having different pitches, depending on how loud they are, what other frequencies are played at the same time or in close proximity, and other factors. A pure sine wave is usually fairly easily mapped to its perceived pitch, but more complex sounds may not be.

A mixture of many frequencies may have a dominant frequency which is perceived as the overall pitch of the sound, or the many frequencies may mix in a way that are not perceived as any particular pitch at all. The sound of a snare drum, for example, is not usually perceived as having any particular pitch, but almost everyone would agree that a snare drum has a higher pitch than a bass drum. The bass drum resonates more strongly with lower frequencies than the snare drum does, but neither sound is very well described as having a particular pitch.

You may be interested in auditory illusions like the Shepard–Risset glissando, which is perceived as having ever-ascending pitch, despite the fact that the frequencies remain within a fixed window. We can objectively quantify the underlying frequencies, but it is not always simple to map a waveform to the subjective experience of pitch. As pointed out by @march, it's even possible to perceive a pitch from a complex tone when the corresponding frequency is damped or even completely absent from the waveform. The pitch produced by a timpano, for example, is implied by the harmonics it produces, with the fundamental frequency resonating much more weakly than higher frequencies - the perceived pitch of the drum is not simply the frequency with the highest amplitude. Practical applications of this effect can be seen in the design of some sound systems that have small speakers - by using a particular combination of higher frequenies, it allows the listener to experience low pitches that the speaker is not even physically capable of producing!

• There's also the auditory illusion of the "missing fundamental." This one's interesting because provided that the frequencies making up a sound are all members of a harmonic series, the pitch is often associated with the fundamental frequency of that harmonic series, even if that particular frequency is missing from the Fourier spectrum of the sound. Jun 28 at 20:46
• What @march said. This "missing fundamental" effect occurs with bells & chimes, see en.wikipedia.org/wiki/Strike_tone Jun 28 at 21:51
• The subjective perception of complex tones is, um, complex. ;) There has been a lot of research on this, particularly in regard to speech sounds. See en.wikipedia.org/wiki/Formant Jun 28 at 21:56
• More on Shepard tone (Wikipedia). Jun 29 at 7:50

Use the lowest of your frequencies, also known as the fundamental, to represent the pitch.

First off, it seems worth noting that many sounds that consist of many frequencies don't have a pitch at all. For example, consider the sound of a piano chord or the hiss of a radio with bad reception. In cases like these, you do have many frequencies, but you do not have a pitch. There is too much going on with the sound to represent it with a single number.

Now let's move on to sounds where you can identify a pitch, as in the examples you gave. In a plucked guitar string or a human voice, the sound would consist of a fundamental and its harmonics. The frequencies of the harmonics would be integer multiples of the fundamental's. And the amplitudes of the harmonics, relative to the fundamental's, collectively make up the timbre. They are why the same middle-C sounds different when played by a guitar than when sung by a human.

For sounds like these, the one number describing their pitch is the frequency of the fundamental. That's the number you're looking for.

A Postscript On "The Missing Fundamental"

In a comment, Michael Seifert pointed out the curious case of the missing fundamental. It is possible to synthesize a sound that contains only the overtones but omits the fundamental. (The Wikipedia page on the phenomenon also mentions some naturally-occurring sounds where the fundamental is greatly attenuated.)

When that happens, an auditory illusion in the human brain can make a human hear a fundamental even though it is not in the spectrum. Here is a YouTube video showcasing the phenomenon.

When the fundamental is missing, the perceived pitch is often, but not always, the highest common divisor of the overtones. Informally, this is what the fundamental would have been if it wasn't missing.

• Not all pitched sounds consist of a fundamental and its harmonics; sometimes the fundamental is missing. Jul 1 at 17:54
• I did not know that, Michael Seifert. Thank you, I'll look into it! Jul 3 at 6:01

The technical, quantitative definition of pitch only applies to "single sounds" such as music notes. Here, the pitch is the fundamental frequency.

However, when applied to "multiple sounds" collectively such as a voice or all the notes an instrument is capable of, the meaning is only qualitative. In this context, the more correct term for the quantifying "pitch" would the "range" which is self-explanatory whereas "pitch" refers to a vague qualitative sense of what is dominant in the range.

If you want quantitative metrics more specific than the range then you'll probably need to make one up and just make the metric known.

For example, you could narrow down the aformentioned range down a bit to where more than 50% of the time-averaged power is contained within some standard deviation during "normal sound production" (i.e. no falsetto).

If you're after a single number then perhaps the frequency component that results in the largest time-average power during normal sound production. You may also need to scale it against the sensitivity of the human ear.

• If we want to be really pedantic, only an eternal pure sine wave has an exact frequency. ;) Somewhat ironically, it's (usually) easier to identify the pitch of a sound that contains at least a few strong components of the harmonic series, rather than a more pure sine-like tone. Eg, it can be easier to tune a guitar to a note from a harmonica or pitch pipe than from a tuning fork. Jun 29 at 21:44
• @PM2Ring I'm all for being pedantic. Is it easier to identify with harmonics than a pure sine tone? I've never gotten a chance to try to tune an instrument to a pure tone. I should get a tuning fork Jun 29 at 21:48
• A tuning fork is fine in a quiet environment, but it's not so great if there are competing sound sources (even if the tuning fork is loud enough). It's a bit like trying to use a laser as a flashlight. :) It's easier to hear when the guitar string is in tune when its harmonic series matches up with the harmonic series of your tuning device. The electronic version of this is that it's easier to tune to a square wave than a sine wave, since the square wave has a nice big fat harmonic series. Jun 29 at 22:13

In physics, high pitch is translated to high frequencies and low pitch is lower frequencies in sound frequency spectrum. for example in the spectrum chart below:

the signal has most of its power gathered around 300KHz. the way to achieve such spectrum for a signal, be it voice signal or not, is to use Fourier transform on the given signal. for your question, human hearing spectrum is from 20Hz-20KHz. although speaking spectrum is much lower, being for females from 165-255Hz and for males 85-155Hz for seeing music notes pitches frequencies, see: https://en.wikipedia.org/wiki/Scientific_pitch_notation this will hopefully give a hint for what you are looking for.

• If the frequency vs intensity plot has several peaks (say, one around 100kHz, one around 200 kHz, one around 300 kHz, 400 kHz etc), what would be the pitch? At least, intuitively? Jun 28 at 19:47
• @Solidification 100 kHz. Any string, tube (e.g. vocal cords), etc. make some base frequency and then a bunch of harmonics above that which are some multiple of the base frequency. The base frequency is the pitch you hear, and the harmonics help give the instrument its distinct quality. Jun 28 at 19:57
• @SeñorO Why does the base frequency determine the pitch of the sound? Is it because it carries the most power? In that case, the power spectrum will have a very tall peak concentrated around base frequency compared to the peaks at other harmonics. Is that right? Jun 28 at 20:03
• @Solidification no, it's because we're used to the exact harmonics. For example, a guitar string with fundamnetal 100 Hz, it'll produce 100, 200, 300, 400, 500, etc. In fact, if your brain hears 200, 300, 400, 500, etc. without the base 100Hz, your brain will still perceive it as a 100Hz note and just assume the fundamental is weak, because your brain knows those harmonics can only be produced by a 100Hz fundamental in physical objects. Jun 28 at 21:31
• Actually i see that @march mentioned this concept above, called "the missing fundamental" Jun 28 at 21:45

Forget about Fourier series for a while. Pitch isn't really about frequencies of sinusoidal components. It's just the frequency of how often the signal's oscillation repeats per time unit. We have a signal $$u(t)$$, and we're interested in its periodicity, i.e. the constant $$\tau$$ such that $$u(t+\tau) = u(t)\quad \forall t.$$ The frequency, or pitch, of $$u$$ is then simply the reciprocal: $$\nu = \tfrac1\tau$$.

There are at least three problems with that:

• In the real world, no signal is truely periodic. There will always be at least small pertubations: literally noise but also effects like amplitude decay over time or similar. So what we should rather look for is $$u(t+\tau) = u(t)+\varepsilon\ \forall t$$, for some suitably small $$\varepsilon$$.
• $$\tau$$ is not unique. In particular, a signal that repeats after time $$\tau$$ also repeats after time $$2\times\tau$$ etc. The $$\varepsilon$$ allowance makes it even worse, since a continuous signal will always change only by a small amount in sufficiently short time.
• In reality we also can't have infinitely long signals. Right at the start of a guitar note it isn't periodic at all, rather you have a transient.

But still: for signals like voice or flute or whatever, we actually do have periodicity over a substantial time (on the order of a second) with hundreds of complete oscillations that are to a good approximation the same. So pitch as repeat-frequency is a sensible notion. In practice, to determine $$\tau$$ one uses the autocorrelation of the signal.

Again, none of this relies on Fourier decomposition, although typical implementations of autocorrelation do use a fast Fourier transform under the hood because that's computionally more efficient than directly carrying out the integration in time-space.

For most e.g. musical instrument signals, the periodicity-frequency happens to equal the lowest strong Fourier partial, aka the fundamental, which in many cases is also the one with the strongest amplitude. But this is by no means universal: in fact it is possible to completely remove the fundamental while changing neither the autocorrelation-periodicity nor the human-perceived pitch. It only changes the timbre of the sound.

The description of pitch consisting of multiple frequencies that are harmonics of the fundamental frequency is called timbre. It is the timbre of a note that gives the difference in tone between a guitar playing an A at 440 Hz and a trumpet playing an A at 440 Hz. Since harmonics are integer multiples of the fundamental frequency at lower amplitudes, when added to the fundamental, these harmonics change the shape of the waveform without altering the frequency of the fundamental allowing pitch to be retained. It is possible to quantify harmonic content (number and relative intensity).

In addition to harmonic content, timbre also includes the attack-decay envelope of the note, and vibrato/tremolo. So a full quantification of timbre is difficult.