Note: This post is a branch off of the Baroque period lesson
WARNING: MATH CONTENT AHEAD
So I hope we all understand the basics of what a pitch is, and what music is. A sound wave is simply a pressure differential, basically, in whatever medium it's going through. It takes the form of a longitudinal wave. Basically, the air particles don't move that much to make sound happen, but there's a wave of pressure that does. If it's just a single disturbance, or any non-periodic pressure change, it's noise. However, when we have a series of waves at a specific frequency, that is to say, quasi-periodic or the theoretical periodic changes, we get a pitch. An example is if the air is changing pressure at 440Hz, or cycles per second, we hear an A. And a specific A. We refer to it as A440. Guess why.
Now, onto the music part:
Intervals are definable as ratios of frequencies. If you play A440 and another note at 880Hz, you will get a perfect octave. If you play A440 and another note at 660Hz, you will get a perfect fifth. A440 and 586.66...? Perfect Fourth. Ok, so that's simple enough, right? Each interval is a ratio, and since this is all based on relatively simple math I'm sure it ends up lining up all prim and proper, especially since the Greeks were all "These intervals are perfect due to their fitting in with math correctly", right?
Let's try something. Let's take A220, an octave below A440, and let's go a fifth up from that, which is the ratio 3:2. 330, which is an E. Cool. A fifth up from that? 495, a B. Now, for the purpose of this example, every time we go above A440, let's adjust for octave, so that we're between 220 and 440. So we divide by 2 and get: 247.5(B). Now let's keep doing this to see where we end up: 371.25(F#), 278.4375(C#), 417.65625(G#). Ok. We're starting to get some serious decimal shit all up ins. I'm no math expert, but I'm pretty sure it'd take some awesome to bring us to the nice clean 440 going at a ratio of 3:2 all the time. Let's see though. 313.2421875(Eb), 234.931640625(Bb), 352.3974609375(F), 264.298095703125(C), 396.4471435546875(G), 297.335357666015625(D), and finally: 446.0030364990234375(A).
Again, no math expert, but I'm fairly sure that 446.0030364990234375 is in fact a different number than 440. So we end up with a different frequency than expected if we just loop fifths around. Uh oh. How different of a pitch? Well, the Bb one half step above A440 is approximately 465Hz. So we're sharp by a little under a quarter of a half step when we loop fifths. We've got a frequency for every pitch there too, so what's going on?
Well, it turns out that we get a little fudged here, simply on account of the math(There's some Pythagorean therum thing that explains this, but I couldn't for the life of me remember what it was). In order to make all the intervals work and fit in an octave, we have to have something be out of tune. The trick is figuring out what should be out of tune. For non-fixed-pitch instruments, this isn't actually a problem. Vocalists for instance, can fudge their pitches a little bit so that every interval is always in tune with itself, regardless of the intervals or where it lies in the tuning system. But on a keyboard, you can't really fudge any notes, since every time you hit the key the exact same frequency is going to come out. Unless there's something wrong with your keyboard. Well, the way that tuning was accomplished historically was dependent on what exactly they needed to do with the music.
The first tuning here is a form of Just intonation, where ratios are defined by whole numbers, and traditionally by small prime whole numbers. It contrasts to the Equal Temperment we'll see later. Just intonation is how we often express intervals, such as a fifth being 3:2, but as we've seen it doesn't really line up too well, so we have to throw a few things out of whack. Just intonation is also what we contrast other tuning with, but we use its theoretical form, where it actually works out.
Pythagorean tuning is the first tuning system we see in the western tradition. In Pythagorean, we tune the fifths. Well... that means the octave isn't right, so the Pythagorean tuning way of handling this is the beautifully simple way of "Call the out of tune one a different note". That's right, in Pythagorean tuning, let's say based off of D, where we go both ways to get the tuning, so that fifths and fourths are the most in tune(So D is the middle based pitch, we then tune G below and A above, then C below and E above, etc), G#/Ab are two different pitches, separated by what we call the "Pythagorean comma". This is fine-ish if we're playing in D major or any key that doesn't have G# or Ab anywhere in it ever, but that means that we'd have to retune our instrument to play in other keys. This also means that any fifth from C# to Ab or G# to Eb will be outrageously wide, and is referred to as a "wolf interval", which is essentially a noticeable out of tune interval due to a tuning system. This tuning sounds super-great when dealing with fifths and fourths, because that's what we tune to, and we have a nice simple interval of 3:2 for fifths that makes them sound all nice and consonant. But it makes thirds really complex intervals like 81:64(Major) or 32:27(minor). This makes thirds sound not as cool and a little out of tune. This is part of the reason that in early music the third was considered a dissonance, because they were dissonances in Pythagorean tuning.
Now, to get to the next system, it's important to note the difference between "tuning" and "Temperament" Tuning is accomplished by tuning just intervals, where Temperament attempts to correct the single super-out-of-tune interval by adjusting an interval by a small amount off of its just interval to get it to fit better.
Meantone Temperament comes next. In meantone, we, basically, tune the thirds. The most common Meantone Temperament, and the one we're dealing with around this era, is Quarter-comma meantone. In Quarter-comma, we technically tune the fifths, but then we shrink the fifths by one-quarter of a Syntonic comma. What is a Syntonic comma? A syntonic comma is the ratio of 81:80, and is a tiny bit more than 1/5th of a semitone. It's barely off from the Pythagorean comma. It's much simpler to think about how we derive it though. A syntonic comma is the difference between a Major third in Pythagorean tuning(referred to as a "Ditone") and a Major third of the interval 4:3. And now Meantone starts to make sense. Stack four Perfect fifths on top of one another: C-G, G-D, D-A, A-E. C-E is a major third. So we shrink each fifth there by a quarter of a syntonic comma, which is the difference between the E derived from stacking fifths and the E derived just as a major third. Oh look, now the Major third is in tune. Awesome. Well, we'll notice a problem here too. Try stacking major thirds like we did with fifths and see if this time we loop back into a perfect octave. FUCK! Basically, we've just replaced one wolf interval with another. Now the good news is we can play more in a single key while still sounding in tune, since the fifths are shrunk by a very small amount(about One quarter of one fifth of a half-step. essentially cutting the out-of-tune sound of thirds from Pythagorean tuning into 4), they don't sound too bad, and now thirds sound awesome. But we still have wolf intervals. Luckily, they're between really awkward intervals that probably won't show up like the #1 and 4s M3, or the #5 and b3 P5(this one was the worst, the true wolf fifth). Unfortunately, if we meantone tune to C this means that the P5 between root and fifth in Ab would be a wolf interval. And thus anything in the key of Ab would sound like shit. So you couldn't play a suite or a group of pieces that covered certain key relations on a single keyboard without having to take a break to retune. You'll notice we don't retune our pianos between each piece of music now a days, so what gives?
Well, next up is Well-temperament. Well-temperament is an attempt to make intervals a closed circle. That is to say, if you stack all the fifth intervals, they'll get you a perfect octave. Well-temperament is slightly irregular though, which differentiates it from tuning we use today. The good part about Well-temperament though is that since each interval was slightly off, most or all keys could be played without needing to re-tune. Technically, Well-temperament isn't a set tuning structure like quater-comma meantone, but a range of temperaments with irregular intervals. The great thing about Well-temperament was that there was no wolf interval, because it was so distributed around the different notes. Keys excessively away from the base would still often sound a little wonky due to their thirds... since the distribution wasn't entirely regular, but for the most part, it opened up all keys on a single, 12-tone keyboard.
Equal Temperament is next, and is what we use today. Specifically 12-tone Equal Temperament, but mostly we just say Equal Temperament, because the others are rare outside of the western tradition(24-tone Equal temperament is in use in very contemporary music, but the 12 tones of 12-tone equal temperament are the same in 24-tone). Equal Temperament divides the octave into 12 equal intervals, basically distributing the comma over all notes and intervals, so it can't be heard, and allowing the most ability to move around keys, since there is literally no difference in keys.