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It is my understanding from so far limited research that all whole number integer ratios sound simultaneously with the fundamental, but has this been proven beyond all doubt? Is it possible that the partials sound in succession even if it so rapid that to the human ear it sounds as a chord rather than an arpeggio? Are there any ways to measure this?

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    I’m voting to close this question because this is better asked on a physics.SE.
    – Aaron
    Apr 23, 2021 at 14:56
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    I have answered this in from what I thought was a similar perspective to user77599, but I now see that Michael Curtis has answered from (I think?) a slightly different perspective, so perhaps the question could do with clarifying. @Aaron despite those concerns, I can't see anything here that isn't solidly relevant to music? Apr 23, 2021 at 15:31
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    I believe musical acoustics should be on topic here, if they aren’t already. Perhaps a meta post is in order. Apr 23, 2021 at 18:52
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    @Aaron, I agree with topo Reinstate Monica and think harmonics are usually very on topic here. I think the question about "proof" seeks to ask if this has been verified experimentally. User77573, it's not required for all harmonics to be present in all contexts. In fact, it depends on the shape of the guitar string when plucked (or piano string when hit). The volume/amplitude of the harmonics depends on that initial shape. Check out the answers to this question.
    – jdjazz
    Apr 24, 2021 at 5:50
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    There is no such "proof" , and even if there were, it would fail in the presence of physical instruments which have limited capability (e.g., vibrational range of a string). Apr 26, 2021 at 17:41

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It is my understanding from so far limited research that all whole number integer ratios sound simultaneously with the fundamental

It depends on exactly what situation/instrument you're talking about. With many acoustic instruments, you often see an initial 'chaotic' part of the sound that quickly turns into a pitched sound as the energy driving the sound becomes subject to the resonances of the system. It's entirely possible that one partial might reach some threshold of audibility before another, so I don't think your statement is necessarily true.

Is it possible that the partials sound in succession even if it so rapid that to the human ear it sounds as a chord rather than an arpeggio?

Do you mean such that each partial stops before the next starts, or just that they start in succession? Either is possible, but both are probably easier to achieve with electronic than acoustic instruments.

Are there any ways to measure this?

The fourier transform is the most usual way to see where in the frequency spectrum the energy in an audio signal is.

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Sound consists of individual sine waves, meaning that any waveform that occurs in the physical world can be decomposed into a number of single-frequency waves.

A sound can consist of only a single frequency. For example, a sine wave produced by a synthesizer's sine oscillator is like that. In that case there are no overtones, there are no partials. Only a fundamental.

If there are overtones in addition to a fundamental in an instrument's sound, then the overtones do sound simultaneously together with the fundamental.

This can be seen in a spectrum analyzer. And it can be heard by the human ear https://en.wikipedia.org/wiki/Hearing#Inner_ear The ear contains small hairs which react to the incoming sound, and different areas or cells are sensitive to different frequencies.

Whatever kind of sound waveform you produce, say, by playing an instrument, using a synthesizer, or by drawing a waveform (that you could store in a WAV file) in an audio application and playing it back with your computer, the sound output will "contain" one or more partials, sine waves. This can be "proven" in many ways. You could for example filter the sound with very narrow band filters and notice that more than one of the filters lets something through it. Or you could have resonators that resonate at different frequencies, and you would see that a sound with overtones excites several of them, not just one. Or you could use a spectrum analyzer, if you believe what it's showing to you.

Let's take a square wave as an example https://en.wikipedia.org/wiki/Square_wave

square wave

(Picture from Wikipedia https://en.wikipedia.org/wiki/Square_wave#/media/File:Fourier_series_for_square_wave.gif)

A perfect square wave has infinitely sharp edges, the corners of the squares. Such a sound cannot "exist" in the real world - any observer with finite resolution would perceive it as consisting of a number of sine waves, without being able to tell if the sound was really a square, or if the non-squareness is just beyond their capabilities to perceive. In the above picture, a square wave (blue curve) is synthesized by adding more and sine partials with different frequencies, and as more and more partials are added, the sum of the partials gets closer and closer to a theoretical square. Or as Wikipedia puts it, "An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth."

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    Most sound actually consists of a pretty complex waveform. The fact that Fourier Analysis can describe it as the sum of many sine waves doesn't mean that's what it IS.
    – Laurence
    Apr 23, 2021 at 21:53
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    @LaurencePayne Your comment has veered into the philosophical. The waveform is the waveform is the waveform. It can be parametrized by decomposition into sines, or into Zernickes, or lots of other complete function sets. Apr 26, 2021 at 17:47
  • I tried to avoid the word "is" in the first paragraph. Consists of ... and then a sentence about what I mean with that consisting. The message did not get through. Maybe I should have added: in the sense that the individual waves, when combined, would produce an identical waveform. If someone wants to find a definition saying what something "is", without opening that up in terms of behaviors or actions or however it could be explicated from other perspectives, then it's going to be a useless dead-end. I wonder how Laurence would have translated his remark into a language without the word "is". Apr 27, 2021 at 5:53
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I don't think it's always presented that way. The Wikipedia page has three different graphical representations.

Of those representations the one in staff notation presents it like a series I think to simply allow space for labeling. I know that I don't read it and think it's literally presenting pitches in time, like it was a melody.

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