What is a chord in terms of frequencies?

Question

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?

Answer 1

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:

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.

Answer 2

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(^θ−φ/₂)

Here, ^θ+φ/₂ is your average frequency! ^θ−φ/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(^θ−φ/₂) factor is nevertheless not irrelevant. Multiplication by a constant like 2 means modifying the amplitude and thus also the loudness. cos(^θ−φ/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, φ≈θ 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!

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^†_{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.}

Answer 3

They do not "blend" in any way shape or form. To understand this better you need to delve into a few disciplines; physics, math, biology, and psychology. I will come from the math-physics angle. First of all when you play the 110Hz that is the lowest note in your chord you are NOT playing or experiencing 110Hz alone. Any instrument that is "attacked" to produce a note produces a whole spectrum of notes based on the harmonics of the system. Within that spectrum you actually have an octave, a 5th and the maj 3rd (actually 2 octaves and a 3rd above the fundamental). This is also true of the other notes you play in the chord, the C# has harmonics that correspond to E# and G#, etc. The mechanical wave and the acoustic wave produced by the instrument is a fairly complex entity but it can be described as the linear superposition of a bunch of pure sine curves, each with three pieces of information in them; the frequency, amplitude, and phase. A single plucked note on the guitar for example would, in theory, have an infinite number of harmonics in it.

wave = sum over all harmonics n = 1...Infinity of (Ansin(2pi * fn * t - kn*x + phi_n)).

kn and fn are related.

When you play a chord you play three or more of these infinite combinations of harmonics.

The reason this is important is because your ear creates harmonics too. Even if one could build a sophisticated machine that can create a pure sine wave (and we have such machines in physics labs, maybe not perfect or ideal but close) you ear will create the harmonic sequence. So, it is not possible (as far as I am aware) for a human to NOT hear the harmonics.

What does it even mean to "combine" frequencies? That depends on how you used the term. It seems based on your post that you were thinking that something like the average might be present in the wave or heard by the listener. So lets discuss the wave first. It is true that the superposition of multiple sine curves can be massaged into a single function that has an effective average frequency and something called a beat frequency that is experienced as amplitude modulation. This can be demonstrated with trig identities. Consider the wave given by

w = Asin(w1t) + Asin(w2t) = 2Acos((w2 - w1)*t/2) * sin((w2 + w1)*t/2)

The severity of the envelope beats depends on how close the two pitches are to each other. If they are very close then the resulting wave looks like a single frequency wave with a long amplitude modulation. We use this to tune instruments but comparing a note with a standard and listening for when the beats go away.

Just looking at the equation you might get the impression that there are 4 frequencies present, f1, f2, |f1-f2|/2, and (f1+f2)/2. w = 2pif in the equations. But this is not the case. There are only 2 frequencies present and they are f1, and f2. If one were to capture the wave form described by the right hand side of the equation and analyze it with a spectrum analyzer you would find f1, and f2 and NOT the other two.

This is why I ask, as part of my answer to you, what does it mean to combine frequencies. The idea that a complex wave form can be described by an infinite series of sines is due to a mathematical theorem by Fourier and is the foundation of a lot of engineering and scientific analysis. The fact that this can also be used to describe acoustic phenomenon is due to the approximate fact that acoustics and mechanical vibrations in musical instrument is to very good degree linear. The phenomenon are described by linear equations and this allows the solutions to be described completely by a Fourier series with appropriate boundary conditions applied. So the individual sines do mix up in the sense that they add together to make a complicated looking function. But the only frequencies present are the ones in the sequence! They do not really "MIX" in any way. When we say a frequency is present in a spectrum this is because we've done a Fourier transform on it and found the individual parts of the sum on the left hand side of the equation. You will NEVER find the sum and difference tones in a spectrum. Unless, ... the analysis or the system used to capture the sound is itself non-linear. This is rare but an example if such a system is your ear. This non-linearity is what creates aural harmonics and prevents people from every experiencing a sine tone. Once the information is passed to our brain we are doing some sort of spectral analysis on it (I am not sure how as this is far from my training). The specific nonlinearity of the ear does not (as far as I know) inject sum and difference tones into the signal, just harmonics. So there is no "mixing" or blending of frequencies.

There is however a limit to human pitch discrimination and at some point we cannot tell the difference between two pitches. I do not know if this is due to a non-linearity or a simple resolution threshold. But when two pitches are very close to each other you will tend to hear the average and the beats as I mentioned before regarding the tuning of an instrument. But this is a property of us as signal detectors and spectrum analyzers, and NOT necessarily an intrinsic property of the wave propagating in space. That acoustic wave however complicated it looks only has the components of the harmonic sequence in it, and not the average tones.

I also wanted to state that the notes in a chord are typically far enough apart so that they are not below human pitch discrimination threshold and are not perceived as an average. However that also depends on voicing! In fact common practices in chord voicing probably evolved to avoid the issue of pitch discrimination.