Lesson 2
The harmonic series
Lesson 1 talked about a sound as though it had one frequency. Almost nothing you will ever record does. This is the lesson that everything after it leans on, and by the end of it you will have found the consonant intervals hiding inside a single note.
A string cannot vibrate at just one frequency
Pin a string at both ends and pluck it. It swings as a whole, which gives you the lowest frequency it can produce, the fundamental. But it also swings in halves, with a stationary point in the middle. And in thirds. And quarters. All at the same time, all on the same string.
The halves complete a cycle in half the time, so they sound at exactly twice the frequency. The thirds at three times, the quarters at four. Not approximately: the string is one length, and the only waves that fit on it with both ends pinned are the ones that divide it into a whole number of parts.
So a plucked string emits a fundamental plus a stack of partials at integer multiples above it. Blown air columns do much the same thing for the same reason. Almost every instrument you own is doing this, right now, on every note.
Timbre is the recipe, not the ingredients
Every pitched instrument produces roughly the same set of frequencies. What differs is how loud each one is. A clarinet leans on odd-numbered partials, a bowed string spreads energy across all of them, a flute is dominated by its fundamental.
That distribution is what your ear reads as the difference between a clarinet and a violin playing the same written note. Timbre is the amplitude envelope over the harmonic series. Pull the faders around and listen: the character changes completely and the pitch does not move at all.
Twelve harmonics, one note
The pitch never changes. Only the recipe above it does.
Click a number to hear that harmonic on its own. Harmonic 2 is an octave above the fundamental, 3 is a fifth above 2, 5 is a major third above 4. The consonant intervals are already inside a single note, which is where lesson 3 begins.
The intervals were already in there
Now look at what the harmonics are, relative to each other, rather than relative to the fundamental. Solo them and listen in pairs.
Harmonic 2 is twice the fundamental: an octave. Harmonic 3 is three times the fundamental, which makes it 3:2 against harmonic 2, and that is a perfect fifth. Harmonic 4 against harmonic 3 is 4:3, a fourth. Harmonic 5 against 4 is 5:4, a major third. Harmonic 6 against 5 is 6:5, a minor third.
Stack harmonics 4, 5 and 6 and you have 4:5:6, which is a major triad. Nobody invented that chord. It is sitting inside every single note any string has ever played, and it has been there the whole time you have been dragging faders.
Two honest caveats. Real instruments are not ideal strings: stiffness pushes upper partials slightly sharp of exact integers, which is audible on piano and is why piano tuning stretches octaves. And plenty of sounds are not harmonic at all. Bells, drums and most metal produce partials at non-integer ratios, which is exactly why they do not carry a clear pitch. Lesson 3 shows what happens to consonance when the partials stop being integers, and it is not subtle.
Which changes what playing two notes means
If a single note is a stack of partials, then two notes are two stacks. Playing an interval is not comparing two frequencies. It is overlaying two combs of frequencies and listening to what happens where the teeth land near each other.
Sometimes they coincide exactly. Sometimes they miss by a little. The difference between those two cases turns out to be the entire explanation of why some intervals sound smooth and others do not.