Music Theory 000.1 – Part 5: Tonics and Intervals

This is part of my series on Understanding How Music Works. If you haven’t yet, you may wish to start at the beginning. This post, finally, is about how two or more notes interact within a piece of music — when there is a “home base” note and some interval relationships between that note and the others in the piece.

The most succinct and accurate definition I’ve encountered for “music” is “organized sounds and silences.” That broad definition encompasses just about everything that’s seriously regarded as music, and even most things that many people don’t regard as music. Things like John Cage, Daft Punk, Schoenberg, Stomp, turntablism, and Ke$ha. But we’re not going to deal with these outliers yet; we’re going to start by looking at one way that we humans organize sounds (and silences) into music that is so common that we might reasonably conclude that it’s physiologically hard-wired into us.

This organizing principle I’m referring to is called “tone-centeredness.” For most pieces of music, there is one particular pitch or tone that is central, a main or “home base” pitch that all the other pitches relate to. Again, it’s not quite universal, but across cultures and time periods, around the world and throughout history (as far as we can tell), people make music that somehow establishes one tone as the main tone, and any other tones in the piece have a meaning that is based on some kind of relationship they have to that base tone. We’re going to need a name for this home-base tone, so let’s call it what others have: “the tonic.”

(I feel compelled to address an objection that some of my uber-schooled colleagues will raise at this point. Yes, around the world and in many cultures there are musical traditions that are completely divorced from organizing music around a central pitch. Clearly, tone-centeredness it is not a necessary feature of music. My point, however, is that it is very common. Consider the posibility that it is as common as it is because it represents a default way that humans make and experience music. We don’t have to do music this way, but we usually do, and that means something.)

Remember that there are an infinite number of pitches available. But remember “Doug’s Garage” as we called it. For every pitch, there are other ones “higher” and “lower” that are somehow also “the same.” So in practice, we’re limited to the infinite number of pitches that are between one note and that same note an octave higher (or lower) — the infinite number of points around one loop of that spiral. And again, in practice we’ve got far few than “infinity,” because the human capacity to distinguish between two notes that are very close in pitch has a finite limit. When two pitches are close enough, our ears (or our brains) say “meh, close enough” and conflate the two into the same “note” or “pitch class.”

So what we see again and again is that one pitch (from the infinity available) gets selected as the “tonic” pitch that a piece of music is centered around and, once it’s picked, a reasonably limited number of other pitches are used, each with some kind of relationship to the tonic and to each other.

That’s pretty much it; that’s the whole gig. All that remains to be decided, described, and understood is: how many pitches? which ones, and why? and, what is their relationship to one another?

Some of the numbers seem like magic

I want to keep the focus on observing music “in its natural habitat,” because I believe Music Theory should be post-game commentary — a description of what happens in the real world — and not a rulebook or set of proscriptions. But before continuing that, we should note of some interesting things that have been measured in musical tones.

Starting with the octave, that mysterious thing that happens when two pitches are “the same (pitch class) but different,” directly up or down an elevator at some point around the spiral of Doug’s Garage. Say that a pitch was selected to be the tonic for a piece, and we measured how fast the air was vibrating to make that pitch, and found it to be 200 times per second (Hz). Now someone produces the note one octave above it. If we measure the air vibrations we’ll find that note to be 400Hz. Then someone produces the note one octave above that, and we find that the air is vibrating at 800Hz. Finally someone produces the note one octave below the original note; we measure the air vibrations and find the note to be 100Hz.

If we continue up and down, we find that the speed of the vibration in the air doubles for every octave we go up, and halves for every octave we go down, to the practical limits of the human hearing range. It’s a simple ratio; when the frequencies of two pitches are in a 2:1 ratio, we hear that as an octave. Every time.

(Don’t worry: there will be no need for you to remember what these ratios are. All that’s important is that you see that notes have real physical properties that have something to do with the relationship that they have to one another. You’ll gain an intuitive grasp of the relationships as you do some ear training; it’s not necessary to know the ratios to hear the relationships.)

Say that this imaginary piece we’re considering were to continue, and we were able to make a perfect recording and analysis of each and every pitch used in the piece. Say, also, that it sounded like a “simple” piece (whatever that means — just go with it for now) and had a certain “sing-song” quality to it (again, without saying what that means). Here are a few notes that will commonly appear in such a piece, and their frequencies, and their relationships to other notes in the piece.

One quite common note in the piece is measured and found to be air vibrating at 300Hz, plus its octave counterparts at 600Hz, 1200Hz et cetera. Your ears will notice that this note is definitely different than the tonic of the piece, but sounds quite nice when played at the same time as it. Now let’s figure out the ratio relationship this note has to the tonic note below it — 300:200, or 3:2. Or, maybe we should figure out the ratio relationship between the tonic note above and this one — 400:300, or 4:3. Hmm, what does 3:2 have to do with 4:3? Let’s look at another note first.

Another common note in the piece is measured and found to be air vibrating at 266.67Hz, plus its octave counterparts. Again your ears will notice that this note is definitely different than the tonic of the piece and the other note we looked at. This one sounds quite nice when played with the tonic, but not nearly as nice when played with Mr. 300Hz. Now let’s figure out some ratio relationships. With the tonic below it, this note is 266.67:200, which works out to 4:3. With the tonic above it, this note is 400:266.67, or 3:2.

Musical ratio-relationships are, in general, referred to as “intervals.” An interval is “the distance between two notes,” and to say that is to describe what kind of relationship those notes have with one another. So, somehow, the two additional notes that we’ve looked at in this piece seem to be inverses of one another. The one is 3:2 in relation to the tonic, and the tonic an octave up is 4:3 up from there. The other is 4:3 in relation to the tonic, and the the tonic an octave up is 3:2 up from there.

And, indeed, it even sounds like that. (But I suppose you’d like an audio example, to hear that for yourself. I’ll try to produce one; if I do, I’ll link it here.)

Non-math geek friends may wish to skip to the next section, because right now my math geek friends are pulling out their sliderules and starting to compute some of the implications of this, multiplying ratios together, plotting logarithmic functions, and drawing conclusions. I did, and it drove me nuts for some time. If this is you, then to you I say: get down with your bad self. You’ll notice lots of interesting relationships, and the conclusions that you might draw from those relationships seem to be borne out in what we observe in most good pieces of music. Have fun. But beware: at a certain point, the human ear is less fussy, less exacting than your plotted graph. At a certain point of complexity, our ears (or brains) say “meh, close enough,” and instead of adhering to your pristine theories, maddeningly round things off to some simpler ratio relationship.

Just to give you an example of the real-world phenomenon that may add a hopeless complexity to your model, I’ll tell you the two other frequencies that appear in our simple, sing-song piece. One of them is smack-dab on 250Hz. This one sounds great when played with “home base” 200Hz, and with Mr. 300, and even together with both of them at the same time. It doesn’t sound very nice at all along with Mr. 266.67. So right there we’ve created ratios of 5:4 (and inverse 8:5), 6:5 (and inverse 5:3), and a compound ratio of 6:5:4 — plus an unpleasant-sounding ratio of 16:15. Then there’s one last one at 333.33Hz. Go ahead and draw a five-pointed star and plot the 10 ratio relationships (7 unique) that exist between these 5 notes. The rest of us will sing ourselves a song while we wait.

(But I’ll tell you right now: the interval of a 9:8 ratio sounds too much like the interval of a 10:9 ratio for your ears to be very good at telling them apart!)

The numbers don’t matter — the relationships do

The whole point of that discussion was not to baffle you with numbers, but to point out the kind of relationships that the notes in a piece of music have with one another. The relationships — the intervals — seem to be based on some kind of simple ratio in the frequencies of the notes. As such, the intervals themselves seem to behave in much the way ratios do: they can be added together to make other intervals (multiplied actually, but let’s not split hairs on that point), and they can be inverted. And when there are more than 4 or 5 pitches (or “pitch classes”) in a piece, they form a complex system of relationships with the tonic and with each other.

So we seem to have addressed the question of “what is their relationship to one another?” in broad terms, leaving us the questions “which notes, how many, and why?” And in broad terms, the answer is “it depends.” Alot. On a lot of things. And that’s not merely an evasion. There is such a wide variety of criteria used in the selection of notes for a piece that it quite literally depends on the individual piece, and sometimes on which cultural context the piece is in.

Watch Bobby McFerrin hack your brain. In this piece, there are 5 notes used (a different 5 than in the example I created above). The same kind of scale — just 5 notes — is used for the melody of this tune, although you’ll hear a few additional notes in the accompaniment. Then there are a bunch of pieces that just use 7 notes; some of these pieces would be described as “major,” or “minor,” or in a “mode” of some kind like “mixolydian” or “dorian” — all of which just describes one kind of “recipe” or another for which 7 notes (in relation to the tonic of the piece) are “allowed” in the piece.

Then there’s pieces like Midnight Sun or this etude by Debussy which take advantage of the 12-tone equal-tempered tuning system that Western music standardized on by around 1850. That deserves a column all to itself, mostly because I’ve previously advised you to throw this particular abstraction out. It will be useful, in a future post, to discuss why we might consider adopting it again. But for now, just enjoy Ella singing. Because, mmmmmm.

Then there’s Middle Eastern and Eastern European music. This, for most readers, should really bend your ears around for a loop. As acclimated as we are to the scale that we’re used to, a lot of this is going to sound “out of tune” to you. Trust me, these people are playing exactly the notes they intend — they’re the “right notes.” They’re simply correct for a different tuning system than you or I are familiar with — one that defies performance on a keyboard that is “in tune” for 12-tone equal temperament.

So it’s really about building up your personal catalog of scales and modes that you’re familiar with, and about building your personal understanding of how these different scales and modes might relate to each other. If you can pick up two different pentatonic (i.e., 5-note) scales, four different 7-note modes, a concept known as “secondary dominants,” and what we in the West call a “chromatic” (i.e., 12-note) scale, and understand which of these things might be sub- and super-sets of which others, you’ll have a pretty complete tonal toolbox. So, let’s say we’ll cover all of those things in future posts.

Next time, I plan to take up rhythm. Music takes place in time, no? So, how do we organize chunks of musical time? Let’s get into it next time, and get funky, get our groove on.

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