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How can people be sure that the A4 note of a Piano is 440Hz? I'm asking in a more practical/physical way. As far as I know, any notes on the keyboard are made of different frequencies, so A4 isn't made by 440Hz alone (if it does make up the sound).

This is how you can have different instruments and say them all play A4 and yet they sound different, they're made of different frequencies with different combination of amplitudes (which makes up the timbre).

Is there some sort of device or way that receives a sound wave and returns a single frequency?

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  • Are you asking about how a piano tuner might tune a piano, for example (ie: what tools or technology do they use to ensure perfect tuning), or are you asking about humans with perfect pitch (ie: how does the ear and brain identify pitches?). It's not clear from your question which you mean. I think either way this probably is a better fit on another stack, either related to technology or human biology, depending on what you're after. – J... Dec 2 '20 at 17:25
  • Yeah in short a practical way would be to represent the signal by a fourier series. This gives you the amplitudes of component frequencies. The highest amplitudes could be indicative of the note moat times. Now obviously you can get into harmony, where you have multiple notes which take front stage. So. There are things called the fundamental. That is the note, then there can be secondary notes as well, which are harmonic. That mean multiple of the period of the fundamental. There could be overtones etc. Lots of stuff. – marshal craft Dec 2 '20 at 18:16
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You are right that each note contains both a main frequency that defines its pitch and additional frequencies that create the color or timbre of the note. The main pitch is called the fundamental, and it is the strongest frequency that sounds. So no matter what instrument is making the sound, the fundamental frequency will correspond to the note name (A = 440Hz, for instance). An electric tuner can hear a complex instrument, and it will identify the fundamental.

The additional frequencies are known as harmonics. These are multiples of the fundamental frequency, and the amplitude of these various frequencies creates the sound of unique sound of each instrument. Real life sound waves are incredibly complex and also contain frequencies that are not multiples of the fundamental, and there are other factors that determine an instruments character besides the frequency content - most notably attack, but I digress.

There is a tool called a spectrum analyzer that allows you to see the amplitudes of these various frequencies. They were originally pieces physical equipment, but today they are readily available in digital form. They are usually included in DAW packages and even as visualizer plugins for playback software.

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    The fundamental is not necessarily the strongest frequency. Indeed, for some instruments it is effectively absent. – Bob says reinstate Monica Dec 2 '20 at 18:31
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    @BobsaysreinstateMonica This is true, but it does not really affect the correctness of my answer. Notice that I said "strongest sounding." While it may or not be the the frequency with the highest amplitude, it is the pitch that our ear recognizes. – Peter Dec 2 '20 at 18:44
  • No, that's not what you wrote in your answer. And fwiw an electronic tuner will not always identify the same fundamental as the human ear, it will only generally do so in the case of harmonic sounds. Many tuned percussion instruments for example have non-harmonic overtone series, electronic tuners can be very bad at determining the correct pitch as they assume a harmonic overtone series. – Bob says reinstate Monica Dec 2 '20 at 19:42
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    The question specifically references pitch (A = 440Hz), so I'm addressing pitched instruments. I believe it is counter-productive to include too much additional info in a basic. Feel free to edit my answer if you want to add something specific. – Peter Dec 2 '20 at 19:50
  • Basic spectrum analysers are available as free smartphone apps (e.g. FFT spectrum analyser on Android). For example if I whistle normally I get a very clean spectrum: the fundamental (with side lobes I'll ignore) and a first harmonic at twice the frequency (plus a bit of broadband low frequency from the rushing air). If I whistle the same note through my teeth I get more harmonics. If I hum I get more, including one at half the dominant frequency, and if I sing I get the same as humming but much stronger at higher frequencies. They're not exactly quantitative but the graphs help – Chris H Dec 3 '20 at 17:56
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The best tool for analysing what pitch will be perceived from a particular complex waveform is... the human ear and brain. Because it IS about what is perceived, not always about what's actually there!

Mix a set of sine waves corresponding to the upper harmonics of a note, the fundamental will be perceived, even though that frequency doesn't actually exist!

Apparently the sound of timpani lacks the fundamental. This link has far more than you want to know on the subject.

https://wtt.pauken.org/

Bells are peculiar too. Their overtones are non-harmonic. This link is very tecchie, but note that the fundamental isn't always the strongest frequency, or the perceived one. In fact, perception of a bell's pitch, without the clues of a harmonic set of overtones, can be ambiguous.

http://www.hibberts.co.uk/phd/phdart3.htm

But yes, more generally, the fundamental is there, and is heard as the pitch. It's not always the strongest component though. Which is why electronic tuners generally get the name of the note right but are easy to fool about what octave it's in.

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    Glad you mentioned bells. I read Aaron's answer and questioned the last statement. Most instruments - not all. – Tim Dec 2 '20 at 12:34
  • I too am glad to see something outside the harmonic sequence pointed out. – ggcg Dec 2 '20 at 18:29
  • Also, people seem to think it's all about the instrument but the attack matters too. The physics of the instrument may determine what is possible but the attack determines what from those possibilities is actually produced. – ggcg Dec 2 '20 at 18:30
  • Your right, but the OP asked specifically about frequency, not pitch. – Bob says reinstate Monica Dec 2 '20 at 18:33
  • How would a person recognise a frequency other than by its pitch? – Laurence Payne Dec 3 '20 at 0:27
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A long time ago, an instrument called the tuning fork was invented. It was 1711, by a trumpeter and lutenist John Shore, according to Wikipedia. Its tines are a specific length, so that when hit against something hard, it would vibrate (for an A=440Hz) making a pretty pure A note. Held against a wooden surface, it was easily audible.

By using this, and playing the suspect note, it is simple to hear any differences. If the suspect note is within a semitone, beats - fluctuations in volume - will be heard between the two notes.

Something I do with students on guitar is fold a tiny piece of paper, half a stamp, say, and rest it on the A string. Hit the tuning fork and place it on the guitar body.When the string is exactly right, the paper will fall off. Sympathetic vibration causes the string to move. Although here, the string is actually two octaves lower - we can tell just by listening - but it still works. For a more acurate one, capo top string on 5th fret.

Otherwise - electronic tuners tell exactly what note, in what octave it is too.

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A device called a tuner can detect pitch. Although any pitch contains several frequencies, they vary in amplitude (volume). The strongest of these is called the "fundamental", and that is what the tuner detects.

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    This is simply false, the strongest is NOT always the fundamental and it's pretty easy to excite wave forms where the fundamental is not the strongest. I'd recommend editing this or deleting it. – ggcg Dec 2 '20 at 18:28
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    The fundamental isn't necessarily the strongest. – Bob says reinstate Monica Dec 2 '20 at 18:28
  • Comments are not for extended discussion; this conversation has been moved to chat. – Doktor Mayhem♦ Dec 4 '20 at 9:45
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You seem to be asking several different question.

"How can people be sure that the A4 note of a Piano is 440Hz?"

Based on the title I thought you were referring to person identifying the Hz of a tone by hearing it. In general, from what I've been taught, most people CANNOT identify absolute frequencies. We are very good at identifying relative frequencies, i.e. intervals. But it takes a special gift to Id the frequency by hearing. I read an article about 20 years ago that claimed people who speak languages that require tone for definition (specifically Chinese) are highly likely to have perfect pitch in adulthood even if they do not show musical talent or interest.

"I'm asking in a more practical/physical way. As far as I know, any notes on the keyboard are made of different frequencies, so A4 isn't made by 440Hz alone (if it does make up the sound)."

This statement hints at two things. The first being, is there a device that is capable of measuring frequency (or many frequencies). The second being that an instrument produces many frequencies to produce a tone. To the first point, yes there are devices that are capable of registering a frequency and they are very easy to make. Even before electronic tuners people could dissect the multiple harmonics in a complex tone using resonators. The idea is based on simple physics. For a complex system that produces waves we usually have many (possibly and infinite number) of harmonics all mixed up in the wave field. If you create a mechanical system that has one and only one resonant frequency, say a pendulum, or mass on a spring, and figure out a way to excite that with the acoustic field then it will only move with a large amplitude when its resonant frequency is present in the field. This type of experimental set up was used by Helmholtz to analyze acoustics. The method probably goes back thousands of years.

Now we have sophisticated electronic devices that can capture the full signal and dissect it using a mathematical signal processing algorithm called a Fast Fourier Transform.

"This is how you can have different instruments and say them all play A4 and yet they sound different, they're made of different frequencies with different combination of amplitudes (which makes up the timbre)."

This comment speaks to the fact that in complex systems a single excitation will generally produce hundreds or thousands of frequencies related to the vibrational mechanics of the system. And yes, the tone or timbre of the instrument, i.e. that characteristic sound, comes from two things (1) the mixture of harmonics that is allowed by the instrument and (2) the attack of the musician.

"Is there some sort of device or way that receives a sound wave and returns a single frequency?"

This may NOT be the right thing to do! It depends, the resonator I described will respond noticeably at one and only one frequency (Newton's second law says it will move in the presence of any force, but resonance means that even a small input at the right frequency produces a very large motion so there is really no chance of confusing the response). Since it is hard to produce a pure tone without harmonics this type of device is really the only thing that will output a single pure tone in response to an acoustic vibration. Modern electronic tuners probably respond to either the highest amplitude frequency in the complex waveform (which may NOT be the fundamental) or if there is some sophisticated circuitry in there it may be doing some signal processing to guess at the fundamental. Vibrating strings and resonant air columns in a tube come pretty close to obeying the "harmonic" relationship fn = n * f1. And we make instruments to follow this relation. If a {microphone + data acquisition card + computer} system picked up a wave and found 440Hz and 660Hz you would NOT have a harmonic relation, you would have a perfect 5th and that may mean that you truly have 2 fundamental tones. The only way to know for sure is some other data, like you are a piano tuner (the person not the device) and you know for a fact that you only hit one key while the others were dampened. Then perhaps these are 2 harmonics from the sequence (220Hz, 440Hz, 660Hz) and the real note being played was 220Hz. This can actually happen! There is some possibility for ambiguity in what is really being produced. The system could have been excited in such a way that the fundamental was very weak and the first two harmonics were strong, and the noise floor was high enough that the fundamental could not be determined in the spectrum. There is really no way to know. If you truly had two notes being played, 440 and 660 then you may have the harmonics of each intertwined. This produces a signal fingerprint and we can separate the sequences out, inferring that 440 and 660 were really each separately present. There are a couple things you should know about humans: (1) our ears are non-linear and produce aural harmonics when excited by a single pure tone. Thus is is not possible for a human to have ever experienced a single frequency, and (2) the brain has the ability to discover the harmonic sequence and determine what the fundamental is even if it's missing. This is called fundamental tracking. Because of this you can trick a human into hearing something that is not there, not present in the acoustic field and NOT created by any source. This makes human's unreliable in an uncontrolled environment.

Lastly, some vibrating systems do NOT follow the harmonic sequence, they have harmonics but they follow some other sequence. I have personally done research with the acoustics of such objects and can tell you that when you excite multiple harmonics you hear them as different tones rather than a complex wave. This is probably due to the fact that we have evolved to respond to the harmonic sequence and anything off that our brains handle differently.

When you FFT a complex wave form and try to infer the fundamental using some type of logic there are ways to get the wrong answer.

To answer your question there are in fact devices that will take a complex wave as the input and spit out a single result but they may either be too limited (like the mechanical resonator) or error prone.

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Depends on who you mean by "people".

Professional piano tuners often use a strobe tuner to measure whether the A4 key on a piano produces a sound (within 1 cent or less) of 440 Hz in pitch. A stobe tuner doesn't display the frequency of the sound. Instead it shows how close the periodicity of the sound waveform matches a repeat rate of 440 per second, by the drift against the strobe. Lots of waveforms can repeat at 440 time per seconds, even though they may not look at all like a sine wave with a frequency of 440 Hz.

The highest peak in a sound frequency plot is very often not the pitch of the piano key (or string bass or guitar string or male voice, etc.) So that information may or may not be useful.

Regular humans (not using mechanical or electronic tuners, etc.) detect pitch by completely different (subconscious/ear-brain) psycho-acoustic mechanisms, for which there are many hypothesis. One of the better hypothesis, IMO, for sounds with a rich timbre (not pure sinusoids) is that the ear-brain combination looks at the spacing between overtones, and uses that information to guess the pitch, as rich sounds have an overtone series that corresponds well to the mechanical mechanism producing pitched sounds.

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