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My question is based on what I read here: https://music.stackexchange.com/a/56316/23919.

When I follow the link to 'fundamental frequency' it seems to be about how the partials of a single voice become perceived as a single, fundamental pitch. Basically, how I perceive B flat on a trumpet as B flat only and not a jumble of partials.

But, this sentence in the answer - "Putting the 3rd in the base can cause confusion about which frequency the fundamental is." - seems to be talking about the chord root and inversions. My assumption is that the 'confusion' is what enables me to perceive 1st inversion chords and different from root position chords.

Am I misunderstanding the answer?

Is the psychoacoustic phenomena of 'fundamental frequency' the same thing that results in the perception of a chord root and chord inversions?

Obviously, I don't understand acoustics. Please try to answer in a way a non-specialist can understand.

EDIT: I understand some basics of acoustics. I know about the harmonic series and its relation to a major triad. I understand complex waveforms are a composite of simpler wave forms. I will add two images which I hope will make my question clearer.

Complex waveforms can share fundamental frequencies. They have different timbre, but the same pitch. This graphic shows it clearly: enter image description here

A triad is three fundamentals which combine to make a complex wave form: enter image description here

When I look at the complex waveform for the triad it looks like there is a repeating pattern. Does that mean the chord has a fundamental frequency? If yes, is it the same as the chord root?

I couldn't find any graphics comparing the waveforms for various inversions of a triad.

I was thinking of Rameau when I put fundamental bass in my question, because I'm asking - in part - about the chord root.

  • Thanks for the clarification. I'm not sure I understand what you mean by the term fundamental frequency. You've restated your question as, "does that mean the chord has a fundamental frequency?" But the "fundamental frequency" isn't a property of a chord. It's a property of a single vibrating source (e.g., a single vibrating string or a single vibrating air column). You can play any three notes together and the waveform will be a repeating pattern. – jdjazz Jun 14 '17 at 22:50
  • So when you strike a C maj triad, you're playing 3 fundamental frequencies: C4, E4, and G4. Similarly, you're producing 3 harmonic series: the harmonic series with C4 as the fundamental, the harmonic series with E4 as the fundamental, and the harmonic series with G4 as the fundamental. Despite there being 3, we only ever talk about the harmonic series associated with the lowest note in the chord. This is where my answer comes in: if the higher notes in the triad "fit" into the lowest note's harmonic series, then we achieve the euphony I described below and that @AlphonsoBalvenie referenced. – jdjazz Jun 14 '17 at 23:04
  • I can rephrase my answer in terms of the root: There is only one way for the higher notes of a chord to "fit in" the lowest note's harmonic series. To get this matching, the lowest note must be the root. When a chord meets this condition, it achieves the extra euphony described in the post by @AlphonsoBalvenie that you've linked to. Would it be useful to edit this statement into my answer? – jdjazz Jun 14 '17 at 23:13
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    I know of Fourier, but can't do the math. If I'm following you the repeating shape can be seen in both of my graphics, right? The violin shape in graphic 1 repeats and the chord shape in graphic 2 repeats. Am I right so far? – Michael Curtis Jun 15 '17 at 15:45
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    @jdjazz, your last comment hits the nail on the head. I can't say why exactly it works that way, but I don't doubt it, and it makes common sense to me. Would you be willing to add this last comment into your answer? – Michael Curtis Jun 15 '17 at 16:44
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I can offer the physical explanation, which helps me understand all of this. In my answer, I'll briefly describe how hearing is produced so that the notion of the fundamental frequency and harmonic frequencies make good sense too. This will set up the explanation for why major chords in root position have a natural euphony--a euphony that inverted chords lack. Long story short, here's why major chords in root position are special: the lowest note of the chord (e.g., C for C maj) has harmonic frequencies that match the higher notes of the chord (e.g., E and G).

How is Hearing Produced?

Everything you hear is the result of tiny swaying movements by cilia. Cilia are incredibly small hairs that line the walls of your inner ear:

http://www.ciliopathyalliance.org/images/cilia/cilia-sem.jpg

This inner chamber of your ear is closed off from the air and is filled with a fluid. That fluid is literally flowing around your inner ear to the beat. When the fluid flows, those tiny cilia hairs move with the flow:

http://www.ciliopathyalliance.org/images/cilia/cilia.gif

But what causes that cochlear fluid to slosh around your inner ear in rhythm? The answer: air particles bumping against the eardrum. The eardrum is a thin membrane stretched across the end of the ear canal. The eardrum blocks air from passing any farther into the ear. When air particles bump against the eardrum, this causes the fluid on the other side to move.1

So the process of hearing is: air particles bump into the eardrum, which vibrates the cochlear fluid. The fluid flows around the inner ear in rhythm, blowing the cilia back and forth like blades of grass swaying with the wind. The cilia are extremely special: when they sway back and forth, they generate electrical signals that travel to the brain via nerves. If the air particles bump into your ear canal very frequently, the cilia move back and forth faster, and your brain receives the signals closer together in time. Perceptually, we "hear" this as a higher-pitched sound. By contrast, if the air particles bump against your ear less frequently, the cilia move back and forth slower, and your brain receives the signals less frequently. This creates the perception of a lower-pitched sound.

The crucial fact here is that the pitch of the sound is controlled by the frequency with which the air vibrates and bumps into the eardrum.

What is Sound?

The following statement should now make sense: sound is vibrations in air. Here's what the air inside your ear canal looks like when you're hearing a pure tone:

www.acs.psu.edu/drussell/Demos/waves/wavemotion.html

Each dot is an air particle. Take away the red line and instead imagine that your eardrum is stretched across the right edge of the canal. If you watch the animation above, you'll see that the air particles will bump into/vibrate your eardrum every time a compression reaches the rightmost edge. To summarize: the compressions travel through the air and initiate the physical process of hearing when they reach the eardrum.

The animation shown above perhaps could represent a high-frequency (high-pitched) sound wave. Why? Because there are so many compressions--they are very frequent. Every time a compression reaches the eardrum, the air particles are hurled against the eardrum, causing that stretched-out membrane to vibrate. So the wave shown above--with its many compressions--will vibrate your eardrum very often. We could instead imagine that the compressions (the denser/darker spots) are half as frequent as the animation shows. What we're imagining is a lower-frequency (lower-pitched) sound wave.

When you put your finger on the guitar string and pluck it, the string vibrates back and forth like this:

http://waiferx.blogspot.com/2014/11/physics-presentation-waves.html

And if we keep the vibration going, it would look like this:

http://waiferx.blogspot.com/2014/11/physics-presentation-waves.html

Here's the shape of the string as it vibrates:

http://hep.physics.indiana.edu/~rickv/sw_1.gif

This produces sound because as the string vibrates up, it bumps into air particles and compresses those air particles. Then, those compressed air particles bump against their neighbors, compressing them. The compressed neighbors then bump against their neighbors, and the compression is sent through the room in a domino-effect-like fashion. Finally, the compression reaches your ear canal, travels through the air in your ear canal, and strikes your eardrum. This happens every time the string vibrates up: each upward vibration produces a compression that strikes your eardrum. So the frequency with which the string vibrates is exactly equal to the frequency with which your eardrum vibrates.

What is a Fundamental Frequency?

We give a name to that big loop in the string (shown above and also again below): we call it the fundamental frequency of the string. Why do we need a fancy name for this big-looped vibration? Because it turns out that there are other vibrations with different shapes that simultaneously occur in the string. You can't see these other vibrations because they're so small, but the string is actually vibrating with all of these shapes:

https://cnx.org/resources/aa67fb155f2f0a86a459138ef2af6da96c5b86d0/StandingWaveHarmonics.png

To make the string look like the bottommost shape (the 4th harmonic), you have to vibrate it more frequently than you would to produce the top shape (the 1st harmonic). I won't prove that this is true, but instead will offer this intuitive explanation: if you hold a jump rope and wiggle the end of the string up and down slowly, you only get one big loop/crest. But if you wiggle your hand up and down very quickly/frequently, you get several crests. So the bottommost shape (the 4th harmonic) has a higher frequency and thus produces a higher-pitched sound.

While it's true that all of these vibrations are produced at once when you pluck the string, and while it's also true that all of these different pitches are produced at once, we don't hear them all. We predominantly hear the fundamental frequency because it's the loudest: the farther you go down the list of harmonics, the smaller/fainter/quieter the vibrations become. The fundamental frequency is by far the largest and loudest vibration--in fact, when we watch the video of the string moving up and down, we can't even see the other vibration shapes at all. So when you pluck the guitar string, those higher harmonic frequencies are produced, but we really just hear one note--the fundamental frequency.

As an aside, you might be wondering what the string would look like if the higher harmonics were louder. To illustrate this without getting too complicated, here's what the string would look like with just the fundamental frequency and a really loud 8th harmonic. The two waves shapes add together, as the bottom image shows:

superposition of harmonics on a string

(I've created this image using PhET's really outstanding Fourier Transform animation.)

What Does the Fundamental Frequency Have to Do With Hearing Chords?

So when you pluck a guitar string, you primarily hear just one note--the fundamental frequency associated with the big single-loop-looking vibration. But other notes (the "harmonics") appear too. The harmonic frequencies are higher in pitch, and they are extremely faint--barely audible at all.

Here's why the harmonics are so important to your question: those harmonics spell out chords. When you press down on middle C on the piano (C4), here are the extra notes: C4 are C5, G5, C6, and E6.

Harmonics of C4

In other words, pressing down just C4 causes other higher-frequency notes to ring out, collectively spelling out a C maj chord. Richard's analogy of the harmonics being like branches higher up on a tree is extremely on point.

The exact same effect of producing harmonics occurs in wind instruments (though the particular harmonics that are produced depend on properties of the instrument). Here's the conclusion we can come to: our ear is accustomed to hearing faint, higher-frequency groups of notes that are produced along with the much-more-audible fundamental frequency. Very rarely do we hear pure tones consisting of just the fundamental with no harmonics, and because such pure tones are unfamiliar they don't sound as good to our ears. (Click here to test this out.)

Major chords in root position have a certain euphony because the higher notes in the chord match the natural harmonics of the lowest note in the chord. The pattern established by the harmonics is crucial and isn't random: the higher harmonics are all integer multiples of the fundamental frequency, as seen by the fact that the 2nd harmonic has 2x as many crests as the fundamental, the 3rd harmonic has 3x as many crests as the fundamental, etc. If you break this pattern, you lose the natural euphony of the chord.

When you invert a C maj chord once, you establish a new lowest note: E4. But the harmonics of E4 don't match up with the notes of the C maj chord, and so we've broken the natural euphony of a C maj root position chord. The effect on our ear is subtle, and I think Alphonso Balvenie has described it perfectly by saying that "the chord doesn't sound as 'C ish'."

In summary: root position chords set up a natural euphony because the lowest note of the chord has harmonics which match the higher notes of the chord. Inverted chords break this pattern. Our ear can pick up on this lack of euphony when we hear inverted chords, but this isn't the primary mechanism by which our ear detects inverted chords. The notes themselves--and their ordering/intervals--are much more audible to us than the subtle, faint harmonics.

Distinguishing Waves of Chords from Waves of a Single Oscillating Source

So if given a waveform of an oboe playing a "single" note, we'll actually see a superposition (combination) of many different wave shapes. And if we look at the waveform of a trumpet playing the same note, we'll again see a superposition of many different wave shapes. But if the instruments play the same note, then the waveforms should all reflect the same fundamental frequency. The image you added in the edit section of your question shows this really well.

But the question might arise: how are those waveforms different from a waveform we produce when playing multiple notes together (as in a C maj/E)? Indeed, there will be many similarities between the two: they'll both have more complex shapes formed by combining the underlying frequencies. And they'll both show repetition. That repetition alone, though, can't be taken as an indication about whether a waveform has a single fundamental frequency. There is a crucial fact which distinguishes the top 4 waveforms (violin, trumpet, flute, oboe) from the bottom waveform (200 Hz + 250 Hz + 300 Hz): the top 4 waveforms were each produced from a single oscillating source, whereas the bottom waveform was produced from three different oscillating sources. This fact dictates that we can talk about a single fundamental frequency for the top 4 waves and can't talk about a single fundamental frequency for the bottom wave.

There's a really interesting follow-up question: is it possible to look at the bottom waveform and see that it wasn't produced by a single oscillating source? The answer is yes, and here's how we would do it. Using a Fourier Transform, we would decompose the complex waveform into its constituent/underlying pure tones/frequencies. Then we would look at the lowest frequency and do a check: are all of the higher harmonic frequencies integer multiplies of that lowest frequency we see? In the case of an inverted chord, the answer would be "no" and we would be able to see that there must be more than one oscillating source.2, 3


1In my explanation of the ear, I'm cutting out the function of the middle ear, which acts as an amplifier much in the same way that high heels amplify the pressure on the pointy top piece.

2As you'll appreciate, I'm ignoring other overtones that are produced besides the harmonic frequencies. If we could eliminate all other vibrations in the wood, etc., maybe then we would see just the harmonic frequencies produced in the oscillating air.

3The next logical question would be "if four violins play the notes in the harmonic series for C4, then would this waveform look any different from a single violin playing C4 alone?" I would expect the physics to dictate certain amplitudes/intensities for each of the harmonics, but I don't know enough about the physics to say this definitively. Assuming that's correct, we could decompose the waveform for the four violins and say "this must have been produced from four instruments, because the higher harmonic frequencies are far too loud to have been the result of a single violin."

  • Holy cow, this is an amazing answer. There is so much in here! – Ben I. Jun 14 '17 at 17:45
  • @jdjazz, thank you so much for staying with me on this question! This last part really answers my question. Your follow up questions and answers are exactly the nagging questions that were in my head. This Fourier decomposing is absolutely fascinating. – Michael Curtis Jun 15 '17 at 18:00
  • @jdjazz, thanks for including the term 'oscillating source' I knew there must be such a term, but didn't know what it was. – Michael Curtis Jun 15 '17 at 18:02
  • You're welcome! And thanks for the clarification on this really interesting question. In addition to the term 'oscillating source', an equally common term is 'vibrating source'. – jdjazz Jun 15 '17 at 18:16
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Here's an attempt at an ELI5 answer:

The overtone series of a given pitch is what helps define that pitch.

For a crude metaphor, imagine the overtone series like a tree. You have your fundamental pitch at the very bottom, then a long bit of space (the trunk) before the overtones (the branches) start to appear, and before you know it the overtones (the branches) are so closely spaced together it's tough to really tell them apart.

Keep in mind that this "tree" of overtones helps define a given pitch. Thus these trees need to be properly spaced so that we can clearly discern a given musical environment.

But now imagine that we have two trees planted right next to each other (let's say they're planted "a third apart"). When you look at the two trees, there are going to be moments where it's next to impossible to determine if a given branch belongs to Tree 1 or Tree 2, and it all starts to become a jumbled mess of branches (or overtones).

That's basically what's happening when you have these small intervals down low in the bass; you end up with a mesh of overtones up top and we can't really discern what belongs to what, and it clouds up the acoustics of the given musical excerpt.


It's tough to answer whether "fundamental frequency" is the same as "fundamental bass," because it depends on how someone is using the latter term. Make sure you're not confusing it with Rameau's notion of fundamental bass, which is quite different (though a bit related).

And yes, it is this interaction of fundamental frequencies and their overtones that help inform us of chords, chord qualities, and inversions.

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Fundamental bass is a very old term that predates modern harmony theory, in the days of figured bass. The fundamental bass is exactly equivalent to the root of the chord.

A fundamental frequency is a mathematical concept referring to the frequency from which overtones can be derived via whole number multiplication. So it is somewhat different.

That being said, many tonal theorists (Schenker, Schoenberg) construe all harmony theory as deriving from the acoustic properties of harmonics and fundamentals existent in nature, a notion known as Naturklang. If you believe these folks, the concepts are indeed closely related.

  • can you elaborate on "if you believe these folks?" Is there good reading from the non-believers? – Michael Curtis Jun 14 '17 at 13:55

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