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I'm trying to understand what chords are in terms of frequency.

So I get that each note represents a frequency range. For instance, an "A" note in the third octave (A2) is ~110hz.

An A major chord (in third octave) is constructed with an A, C#, E, with corresponding frequencies 220.5, 138.5, 164.5. Since we are playing this in a single stroke, one might assume that this blends into a single frequency that is equal to the average of the three, which is 174.5hz. But an F3 note is also equal to 174.5hz, and it doesn't sound just like an F, so that must be incorrect. Chords can also span multiple octaves, which definitely throws out the method of averaging note frequencies

What are chords in terms of frequencies? Why does a (major, for instance) chord use a third and a fifth? Most books/articles I've read regarding music theory just take this as a given and don't properly explain why a chord has these frequencies meshed together.

From the comments below, the brain does analyze the notes individually. What is it about the sound that makes this possible?

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    "I assume that this blends into a single frequency that is equal to the average of the three, which is 174.5hz." I don't know why you would assume that, but it's dead wrong. Musical instruments, acoustical physics, our ears, and our brains do not work like that at all. There's no averaging. It's the opposite - all the different frequencies present are separated and analyzed by our brains, more or less. Some frequencies are more important than others when our brains decode them. – Todd Wilcox Mar 13 '16 at 16:35
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    Right. Obviously my assumption did not hold; that's why there is a question here at all. But if the brain can analyze the individual notes that make up the chord, why do these note intervals make up a chord? Perhaps it has to do with the ratio then? – ZAR Mar 13 '16 at 16:38
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    You might need to edit your question, taking a step back, removing the incorrect assumptions, and ask something more direct to help you understand more about chords or maybe just single notes and why they sound the way they do. Right now it's hard to write any kind of answer to this question since the question doesn't really make sense as it is. Not all chords use thirds and fifths. The most popular chords do, but that's not all chords. – Todd Wilcox Mar 13 '16 at 16:39
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    If A2 is 110, why would A3 be 220.5 rather than 220.0? Also, C# and E would be 138.6 and 164.8 respectively, calculated using the twelfth root of 2 (that is, in 12-tone equal temperament). – phoog Apr 4 '19 at 15:09
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As Todd noted in a comment, frequencies combine rather than average. Remember that sound is simply moving air (or other medium) being pushed around by, for example, a vibrating string. If you push on something 3 times a second and someone else pushes on it 2 times a second, you wouldn't see it move only 2.5 times per second! You also wouldn't see it move exactly the same amount 5 times, either, since some of your pushes are going to line up at the same time and result in one bigger movement. And when it comes to sound, you're not just pushing but also pulling — so some of your pushes will cancel out.

This graphic (credit to Alex Basson) illustrates how waveforms combine:

x wave y=2x wave x+y wave

As you can see from the final image, both of the original frequencies are represented — the overall frequency is still that of x, but it has a clear sub-frequency that matches y (which is double that of x, in this case). This allows your brain to "pick apart" the sound and hear all of the notes in a chord — different combinations will make different shapes, all representing their components.

Related post that the above was taken from: How do harmonics work?

As for why notes are "chosen" for a chord — well, any two or more notes form a chord. But the most common are structured around a note and its fifth (seven semitones above) because they form a 2:3 frequency ratio. Other than 1:1, 1:2, or 1:3 ratios (two of the same note separate by 0-2 octaves), this is the simplest ratio (the one with the smallest sides) — which means that, while it has bit of intermingling to sound interesting, the frequencies regularly "line up" and form peaks and troughs together, as above. This is called constructive interference, and when it works well and sounds pleasing it's called consonance.

If you want your sound to be more "interesting", you add in notes that line up less often, giving more twists and turns to the waveform. When this becomes too much and just a jumble of frequencies it tends to be unpleasant, and is called dissonance.

For a mathematical representation of consonance/dissonance, see my answer here.

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A good question to ask!

As said by Tood Wilcox and Matthew Read, combined sounds don't just average their frequency; instead they superimpose. To get a better idea, it might help to consider water waves: you can have big shallow ocean waves and much smaller excitations on top at the same time. Clearly, the result is not just a single wave (monochromatic, to use the proper term) with the average frequency, but, well, a distinguishable combination. It works just the same way for sound waves.

That is, except when it doesn't. Your assumption actually holds sometimes, if only under special circumstances.
When you hear two violinists playing a single note unisono, you hear in fact not two individual pitches which just happen to be the same frequency (almost the same: intonation is never completely accurate). Instead, you do hear a single note with something like the average frequency, but also a slightly different tonal character. The reason is quite a remarkable mathematical fact, namely that the additive superposition of two sine oscillations is equal to the multiplicative superposition of two other oscillations.

sin(θ) + sin(φ) = 2 · sin(θ+φ/2) · cos(θφ/2)

Here, θ+φ/2 is your average frequency! θφ/2 is the beat frequency; if θ and φ are very similar then the difference will be much smaller than the individual frequency. In fact, if one violinist plays 439 and the other 442 Hz, then the difference frequency is inaudible infrasound! The cos(θφ/2) factor is nevertheless not irrelevant. Multiplication by a constant like 2 means modifying the amplitude and thus also the loudness. cos(θφ/2) is almost constant because the frequency is so low, but nevertheless it changes in time, making the sound louder and quieter periodically. This is the phenomenon called beat.

Now, it only works quite so simple for two pure sine (or cosine, doesn't matter) waves of the same amplitude. Real instruments do not cause sine oscillations, they have more complicated waveforms, and somewhat different between any two instruments. Nevertheless, beat is audible when you tune e.g. the empty strings of two violins to match, together with other effects that make the sound generally more “smooth / silky” – I won't discuss that here.

I said special circumstances. Well, when the frequency is not almost the same, then the sine beat will be too fast to actually be recognised as loudness modulation. However, there are other special cases which lead to a similar phenomen: instead of almost the same, φθ consider now twice the frequency, i.e. φ=2·θ. (Again the following will also work with almost twice, you just get an additional beat term.) Then two oscillations of φ will exactly fit into one oscillation of θ. So the result has the same periodicity as a θ-oscillation alone – by definition, this means it has the same frequency! Does this then mean that superimposing a double frequency doesn't change anything? Well, not quite. The waveform will sure be different. In fact, one useful way to interpret the different waveforms of various instruments is that they all consist of a superposition of many integer-factor sine waves, the overtones, mixed with different amplitudes. Some instruments actively exploit that: an organ creates its different sounds by mixing pipes whose frequency is a factor 2, 4, 6 or 8 apart. Each combination sounds different, but they're all perceived as one tone with a “shared frequency”.

Except when they aren't. This illusion does work for an organ passage where every note uses the same combination. If you have actual independent instrument voices, then each chord has a slightly different mixture. Our ears notice this and the brain is able to deduce back the individual frequencies. There can be other cues as well, e.g. in a strummed guitar chord the notes don't begin quite at the same time. Nevertheless, the sound of (almost) integer-related frequencies is perceived as, well, consonant, harmonious. That's how chords work!


Actually, sound waves are simpler than water waves – these involve funny circular movements around imaginary surface points. Sound waves behave more like a queue of people (≈ air molecules) in which everybody waits to get pushed from behind, and then transmits the push to the guy standing in front (but nobody actually steps forward). You could then imagine an ongoing sequence of shoulder-taps every few seconds, interspersed by two-hand pushes every minute or so. Again, superposition of two frequencies. Of course that's also not really how sound waves behave, but you get the idea.

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    Are "imaginary surface points" points on an imaginary surface, or imaginary points on a surface? Or something else? – phoog Apr 4 '19 at 15:13

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