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People, instruments, objects, etc. all are capable of producing "sounds" that sound different (not sure the proper term for this) but can produce the same note.

Given that sound can be represented with an oscillating wave, and the wider the peaks and valleys of the wave determine the pitch, how is it possible to have a given note which has a different sound.

For example the same note could be produced in different ways:

  • Someone could hum it
  • It could be produced by any number of instruments
  • It could be produced when two pieces of metal clang together

Each of these would yield a "different sound" despite being the same note.

How does the "sound" and the pitch coexist in one wave?

Side note that I think this is a related concept: A person can say words (sing lyrics) at a given pitch. How is that possible since each minute portion of those words in themselves have varying tones. Is singing at pitch essentially just an "average" of the minute and rapid tones produced by the person?

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    I feel like the word timbre should also be explained somewhere.
    – trlkly
    Commented Oct 31, 2022 at 1:34
  • @trlkly - it is, on page 9 of tags, but not particularly clearly. Would be great if those tags were in alphabetical order...
    – Tim
    Commented Oct 31, 2022 at 10:08
  • It is explained e.g., in Wikipedia. By the way, human singing doesn't need to have a clearly defined pitch, and I don't mean just noises, for example there also exist mind-blowing things like this.
    – dtldarek
    Commented Nov 2, 2022 at 12:48

7 Answers 7

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Simple answer - overtones or harmonics. Each sound producing machine (instrument) has the propensity to produce not only the fundamental (first harmonic), but others too. The mix of whichever these are will determine the 'quality' of that note.

That note will have a base, fundamental, which gives it its name - say C4 - and that mix of overtones, and the volume of each, will help make the 'timbre'.

Other factors may well come into play also. ADSR - attack, decay, sustain and release. They won't affect the pitch of the note, but will affect how we hear it.

Then there's vibrato that occurs naturally in some instruments, particularly voice, which will vary the volume, thus our perception of that note. Some singers will also use a sort of tremolo for the same ends. Or even a tonal variation, which again will have a similar effect.

Put all these factors together, and it's easy to see how, say, a violin, with many overtones, making a rich sound, will appear (?) very different from a flute, with a much purer sine wave sound.

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    The stability of the pitch has a considerable effect. Even without deliberate vibrato, some instruments, e.g. the piano, have a very stable pitch while others do not. I remember long a cassette tape of piano music. It no longer sounded at all like a piano; it sounded like a musical saw. The overtones and most other aspects will have been retained, just the pitch varied
    – badjohn
    Commented Oct 31, 2022 at 15:21
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    @user7868 having sung ee-ay-ah-oh-oo into an app on my phone that lets you visualize the the frequencies (X-axis is time, Y-axis is frequency, and color denotes intensity), I can definitely say you can see a difference in which harmonics are present. Not sure if it's enough to make a full answer, though, but it's an interesting alternate way of visualizing the result compared to what other answers here show (the resulting waveform or its components) Commented Nov 1, 2022 at 5:03
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    You've left out a very important factor, the attack portion of the sound. A surprising amount of timbral identity is contained in that. Chop the front off a selection of instrumental notes and see what I mean. Differences between the sustain portion are often much less marked.
    – Laurence
    Commented Nov 1, 2022 at 19:11
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    @Tim The 'signature' attack of a musical note is a lot more complex than just ADSR applied to a constant waveform!
    – Laurence
    Commented Nov 1, 2022 at 19:38
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    @Tim I'd say para 3 is a woefully short treatment of attack. Depending on the instrument, the attack can be a semi-broadband transient mixture of frequencies, even outside of the harmonics. Saying it doesn't affect pitch is very misleading IMO.
    – mbrig
    Commented Nov 2, 2022 at 2:34
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There's more to an instrument's waveform than the over-riding 'perceived pitch' frequency. There's lots of other frequencies mixed in there.

https://brilliant.org/wiki/sound/#musical-sound-and-noise

And apart from the sustain portion of a musical note (which is what is usually shown when discussing this topic) there's also the attack - a much more chaotic portion of the waveform that has a LOT to do with defining the instrument. So much so that in the days when sample memory was at a premium, a very successful generation of synths (Roland D-50, Yamaha SY series) used 'Hybrid synthesis' where a short sampled attack was grafted on to a synthesised sustain. It worked surprisingly well.

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The essential terms you want to use are timbre, overtones, and fundamental.

The "sound" difference between voice, flute, guitar, etc. is called timbre.

Different timbres are the results of overtones a concept which comes from acoustical science. It's like a blend of many pitches that combine together to create a timbre.

The pitch that stands out from the spectrum of pitches in a "sound", which we might give a name like C4 or A 440, etc. is called the fundamental.

So, a "sound" might have a fundamental, with a collection of overtones, which results in particular timbre.

If you are already using terms like "oscillating wave", you should be able to look up timbre, overtones, and fundamental, and follow a general encyclopedic overview easily.

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  • +1 for giving OP some new (to them, presumably) and useful terms in the space where they're fumbling around, empowering them to go and continue their learning!
    – nitsua60
    Commented Nov 1, 2022 at 17:09
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Because of harmonics.

When we hear a note on an instrument, we're actually hearing a mix of waves at different frequencies playing together. One frequency, the lowest and usually loudest is called the fundamental frequency. This is what we refer to when we say a note has a pitch, and is why a piano playing middle C will sound similar to a person singing middle C.

The rest of the frequencies are called harmonics and will vibrate faster than the fundamental - often in multiples like 2x, 3x 10x... It is the blend of these harmonics that makes each instrument sound different.

Harmonics summing to make a note with 50 Hz fundamental frequency

We design instruments to give distinctive pleasing blends of harmonics. This is why a guitar sounds different to a harp even though they both use a plucked string.

Try it yourself!

Sing a note with an 'ooooh' sound. Keeping that same note, change your mouth to make an "eeeee" sound. You will hear the sound become 'brighter', even though you are singing the same note. That is because because your new mouth shape has changed the harmonics that form as the air bounces around your mouth. You've just made some new harmonics!

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Some details to add to Tim's answer:

Many musical instruments, and also human voice, produce also non-harmonic components. For example, the knock of the piano hammer on the string, air whistling in the flute; in human voice: plosive sounds (p, g...), fricatives (z, v...). They don't always affect the perceived pitch, but they contribute to the timbre of the sound.

The initial phase of the sound called attack may alter the perception of the following sound a lot. The frequency spectrum of the attack often includes a continuous component.

Some instruments, drums in particular, produce a continuous frequency spectrum, and even if the human ear can interpret the sound as having a particular pitch, the timbre is quite different from the pure sine wave.

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  • Glad that you mentioned timbre, but I think it would be better to explain what the word means.
    – trlkly
    Commented Oct 31, 2022 at 1:35
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sound can be represented with an oscillating wave...

It's true for a pure tone (e.g. an electronic beep), but musical sounds are always made of multiple waves, theoretically an infinite number of sine waves.

Musical instruments actually produce complex sounds which are a distribution of pure tones with different frequencies and amplitudes. This distribution model, referred to as timbre, is particular to each instrument, as it's actually the result of a distortion created by the instrument. When playing a A, the 440Hz pure tone is actually played along with various additional waves, all multiple of 440Hz, called harmonics. The exact distribution varies from one instrument to the other, hence the sound heard is different. I'll explain in detail.

Notes would be pure tones only if instruments were able to create perfect sine waves, it's not case. It's very fortunate, as this is a pure sine wave (but your computer will still create a bit of distortion).

It could be produced when two pieces of metal clang together...

I'll comment this only briefly here as it is a different topic. For percussion instruments, except for a few ones, of which timpani and xylophone, the sound has no definite pitch. The sound is not even a periodic wave, it doesn't repeat at regular interval. Such sound which is a random mix is known as acoustic noise.


Pure tone vs complex tone

A sound can be anything, and in some cases it can be a pure tone, a sound made of a single perfect sine wave:

enter image description here

Pure tone: Time domain view

A diagram like above, where the horizontal axis is time, and the vertical axis shows how amplitude varies with time, is appropriate to see the shape of the signal. There is another useful view, where the main axis is the range of observed frequencies. This is called a spectral view (a view of the audio spectrum). On such a diagram, a pure tone appears as an isolated peak at its own frequency, and the rest of the spectrum is flat:

enter image description here

Pure tone: Frequency domain view

As long as the sound is steady, these two views are equivalent, they show exactly the same information.

One might think: The spectrum view shows only the frequency and amplitude of the signal, not its shape, maybe it's a square wave, who knows? Actually if it were a square wave there would be many additional peaks in the spectrum. The spectrum really tells us all the smallest details of the shape as we'll see further down.

Hence, the short answer to your question is three-fold:

  • Sounds from vibrating devices tend to be sine waves, two instruments should therefore create the same sine wave sounds. It's not true, it means the shapes of the waves have been altered, and there is a cause to it.

  • The difference is visible on the spectrum, by looking at peaks. I'll show further down informative spectra of the same note for different instruments. Alternatively the difference can be seen by looking at the shape of the waves, but for complex sounds this is not practical, nor very informative. Nevertheless, as soon as the spectrum / the shape is different, the sound is different.

  • This thing that changes waves is known under the general wording distortion from an engineering point of view, and timbre from a musical point of view for which distortion is actually desired. Each particular instrument distorts sounds according to its timbre, adding peaks into the spectrum.

So let's dig into these two aspects: Wave spectrum (next section) and timbre (one more section further).

Harmonics: Mathematical proof

The most common distortion, which is at work in musical instruments, is related to the amplification of the perfect sine. The wave can be clipped because its amplitude grows beyond the capabilities of the amplifier, resulting in the top and bottom of the wave clipped at a certain level:

enter image description here

Clipped sine wave: Time domain view

This sound is still periodic, and the frequency is still the same, but it's no more a sine wave.

It can be proved mathematically any complex sound which repeats its own shape at frequency f, regardless of how chaotic this shape is, is just the sum of perfect sine waves of frequency f, 2f, 3f, 4f, etc, each with its unique amplitude.

Indeed our clipped sine wave takes up more space in the audio spectrum than a perfect sine wave:

enter image description here

F is the fundamental frequency (first peak above), the one the signal repeats itself over time, and 2f, 3F, 4f, etc, are the harmonic frequencies.

The shape equation and the set of sine waves are mathematically the same thing, the two sides of the same coin, we can look at a sound either way, none is more correct and they are physically indistinguishable. As soon as a sine wave is distorted, its spectral view is changed and the harmonics distribution adjusted to match the new shape.

This conversion from the shape view of the wave to its equivalent spectral view is mathematically done using a Fourier transform (FT), the computational method to do it is the fast Fourier transform FFT.

There are lots of web pages where sine harmonics materialized by fixed-length arms rotating at constant frequencies just combine to reproduce any hand-drawn shape.

Now we know harmonics are the result of sine distortion, let's see the origin of the distortion.

Harmonics physical origin: Instrument timbre

String vibration and other oscillatory phenomena are naturally sinusoidal. It cannot be different. The reason is when a string or another vibrating element is moved, it develops a restoring force which is in theory proportional to the displacement, the resulting motion is a sinusoidal oscillation (principle of the harmonic oscillator). It's like the trajectory of a planet orbiting its sun: It has to be elliptical else either the planet falls on its sun or it escapes its sun.

However if this sun is not a perfect sphere, with a perfectly symmetrical density, then the gravity field is not equally distributed anymore and the perfect elliptical motion is distorted. This is the same with a sine wave, its shape is influenced by the material while transmitted, and most of time, the perfect sine is distorted.

By itself, a piano string may generate a not so distorted signal close to a sine wave. Note this sound is weak, and is not the one heard by the audience. The string vibration is transmitted to the wooden soundboard which role is to amplify the sound and make it audible.

Wood is not a perfect material for sound transmission, technically we will say it's not a linear system. Because of this non-linearity, wood will transmit the wave differently, e.g. instead of transmitting the dashed blue sine which is a perfect amplification of the input, the amplifier creates this orange wave where the gain depends on the amplitude:

enter image description here

The non-linearity of the amplifying element distorts the sine wave

The way the soundboard affects the string sound, or the saxophone brass body affects the reed sound is unique, due to the unique physical characteristics of the instrument. This is called the instrument timbre. Timbre in music domain is the same than the impulse response in audio engineering. A piano timbre is comparable to how a room creates an echo, a temporal distortion. Instruments with different timbres (different impulse responses) transmit differently the original sound, exactly like different rooms show different echoes, transmitting the sound differently.

Different piano makes have different timbres due to the choice of materials, the design of the parts, and many other details. Timbres can even have differences among the same design, because all trees are not equal.

While a computer beep has a simple timbre (close to a single frequency spectrum), musical instruments have elaborated timbres with various harmonic amplitudes. See these spectral views for the same note (A of octave 4) from a tuning fork, from pianos and from a saxophone:

enter image description here

A4: Frequency domain view. Stolen from this question

Note also the initial wave created by a guitar string is already different from the original wave created by a piano string, because even simple strings are non-linear systems and therefore distortion occurs and harmonics appear.

For a string to be a linear system, it should exert a restoring force strictly proportional to its displacement when moved by the performer. Unfortunately for the science, and fortunately for music, such perfect elastic material doesn't exist, and the restoring force becomes larger than it should as the amplitude increases.

Harmonics perception by our ear

Our ear is not very reactive to a sound which is not a sine wave. We do not sense the sound as a whole because of its chaotic shape. It's strange when thinking at it, but our ear actually detects individual harmonic frequencies present in the sound. Our brain immediately makes a distinction between a C played on a piano and the same C played on a violin, due to the different harmonics distributions (the different timbres).

However while our brain cares to process harmonics, it keeps the result for itself, it's actually difficult to voluntarily hear individual harmonics (see comment by Ian Goldby). It seems strange we cannot easily isolate the different harmonic frequencies from the whole sound, because on the other hand, we are perfectly able to distinguish two notes which are not consonant, e.g. D and A. Harmonics consonance hide them to us. Still our brain is nice enough to share with us identified dissonances, frequencies occurring simultaneously but which are not multiple of some other frequency.

One more strange thing about what we hear: If the fundamental frequency is artificially removed from the spectrum, our brain, which has been trained to perceive the relationship between a fundamental and its harmonics, will just recreate the missing fundamental and process the sound as if it were complete.

Chords are based on their harmonic spectrum

Harmonics are fundamental (if I may say) in music. When notes match, we just say they are in harmony, or they are consonant. This happens when harmonics overlap.

Notes C E G are played simultaneously are consonant. Their frequency is in the ratio 4:5:6 (they are all multiple of some frequency f). The triad sound contains:

  • C note 4f and its harmonics 8f, 12f, ...,
  • E note 5f and its harmonics 10f, 15f, ...,
  • G note 6f and its harmonics 12f, 18f, .....

enter image description here

CEG triad: Frequency domain view

We see most of the harmonics overlap in some way. This is why they are perceived as a whole and are harmonious.

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    "String vibration and other oscillatory phenomena are naturally sinusoidal. It cannot be different." is not true at all. It is true that periodic waveforms can be broken into sum of sines and cosines, but it is not a good approach to understanding how musical instruments work.
    – ojs
    Commented Nov 2, 2022 at 8:33
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    The majority of this essay is correct, but typically acoustic instruments (and subtractive synthetizers) work the other way: the sound source has more harmonics than the final sound, and the rest of the instrument filters and amplifies them.
    – ojs
    Commented Nov 2, 2022 at 8:40
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    @IanGoldby - one reason (that and the reverb) that makes me sound so good in the bath.
    – Tim
    Commented Nov 2, 2022 at 12:08
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    Excellent answer and probably the most helpful in terms of amount of information learned. Definitely expands on the others and was eye-opening for me. As I've read these answers, I realized there is another concept here that I've inadvertently eluded to when referencing human voice: the idea of an entire composition being represented as a sign wave. An entire composition of many instruments and sounds can be represented in a SINGLE waveform. Maybe this was answered and I haven't understood it, but is this just the sum of each wave? Commented Nov 2, 2022 at 12:31
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    Just to amplify (ahem) the first two comments - harmonics arise naturally in certain oscillators (such as a vibrating string or air in a pipe) without requiring distortion. This is because there is more than one possible mode of vibration. A string can vibrate as a whole, or in two halves, or thirds, etc. (It's true that distortion is another way to create harmonics from a single frequency signal, but that's more in the realm of electronic instruments.)
    – Ian Goldby
    Commented Nov 3, 2022 at 12:35
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This is like asking, “why is it that several things that are green can still look different to me?”

Sounds can be nearly anything. Far from all of them have a clearly defined pitch or near-constant pitch through a long enough period for us to notice. Explosions, rockets or sea waves are examples of not particulary “pitch”-y sounds.

In general there is no reason to believe that the pitch we experience when listening to “musical” sounds is a good general framework to describe any kind of sound, or even that it captures the essence of “musical” sounds.

As a psychoacoustic parameter of interest, pitch has the additional benefit that it in many cases can easily be measured and reasoned about with numbers (they even did so 1000s of years ago). But there are many other psychoacoustic parameters, that are just as important for our ears and brain, whether they can be easily measured or reasoned about with numbers, or not.

Leaves and green cars are different. Even though they are both green.
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    This answer is rather unsatisfactory as an explanation, especially when compared to the detail in the answers that already existed when you posted it. ("Sounds sound different because they are different" doesn't particularly help clarify why/how they are different, nor does it appreciably add to the existing explanations.)
    – R.M.
    Commented Oct 31, 2022 at 17:44

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