Wednesday, December 16, 2009

Aside: Tuning and Temperament.

Note: This post is a branch off of the Baroque period lesson 


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.

Lesson 6: Baroque, as well as an explanation of tuning systems.

Hey everyone.  I want to start off by apologizing for the lack of update recently.  I'm trying to keep somewhat regular with this and trying to clip along at a reasonable pace since unless something goes horribly wrong, I'm going to be unable to really have any contact with the outside world starting mid-May for like 8 months, and I'd like to give at least a pretty good outline before then even if I don't get to the more advanced stuff before then.  I have a studio recital later today and I was playing for a bunch of Juries and had some concerts, which were keeping me pretty busy, plus I've got a pretty big audition coming up in a few weeks, and I'm recording an album next week, so all together I just didn't have time to update.

Also, I'd like to apologize in advance for a probable lack of updates in the next 2 weeks and possibly the next month or so.  I'll be out of town starting this Friday and that'll possibly extend straight through to January, and then I'm not entirely sure exactly how much but I'll have a very limited time to practice for my audition again so will probably be locked in a room with a piano for most of the day.

Ok, so we're ending the renaissance period today and moving towards the music that most people start knowing, and that probably sounds a lot closer to stuff you've heard of now.  The biggest difference into the Baroque period is the rise of functional tonality instead of relying on modes.  Also, in the mid-late Baroque we get the "rules" of counterpoint, which are essentially guidelines for writing parts in a way that doesn't suck.  We'll cover counterpoint when we're done with history.

This is also a period that I have much less familiarity with than the early music, and it'll probably show.  Also, as a brief note, while most of this is coming from my head, I'm fact checking with Norton Anthology of Western Music, and wiki for some specific subjects.  While earlier music I knew more about personally, with Baroque and on I'm leaning a lot more on my sources.  I'm obviously rewording like, everything to make it less stuffy and boring, also since it'd be super lame and pointless to have this blog if I was just plagiarizing everything.  And illegal, but I did want to cite that my biggest sources are the Norton Anthology and Wiki.

I'm kicking myself a little because one thing I meant to mention before was formes fixes.  Luckily for me, it doesn't really matter too much until now, so I might as well talk about it now.  Formes fixes was a form of poetry for songs, originating in France in about the 14th century.  So we're stepping back a century or two from where we ended last update, but basically there were three formes fixes: The ballade, rondeau, and virelai.  Basically, this was used for songs around the post-Notre Dame School, or ars nova period.  And they were specific in the form of the way the poem was put together.  Essentially they were like tablets.  Fill in the words in the right places and you have a song.

Ok, now that that's out of the way, let's jump back forwards to 1600 or so.  We often place the beginning of the Baroque period around 1650, where Purcell and Scarlatti start entering the ring, and often times this period of 50-100 years right before the Baroque starts all proper like is referred to as the "bridge period", since it's really pretty obvious that music kind of morphs from what is clearly Renaissance to what is clearly Baroque.
The Baroque period's beginnings are often cited as a council of Florence meeting, where they discussed arts, among other things.  They specifically looked back at the Grecian texts for inspiration, deciding to resurrect the musical drama, where text reigned supreme.  You may have noticed that in the renaissance stuff, sometimes text was impossible to understand, because there was so much going on over itself and words were so elongated that it took an hour or to to get to the second goddamn syllable.  The wiki article on Baroque music even says that this is the "conventional dividing line" for the Baroque period, but I take a bit of issue with that.  While it was a designed meeting that set specific ideas down, since there was still a pretty damn big delay in communication ability, and since if you do things like listen to the late 16th century stuff, you still find many things that the Baroque period had.  Either way, it's a nice dividing line if you're looking for one, just keep in mind that it wasn't all prim and proper with one style suddenly no longer happening and another taking over immediately.

What we're going to see over this period of time is a movement towards more structure in music, and the emergence of what today is a refrain in art music.  Madrigals, which we looked at, often had a repeating chorus and sort of song structure approximating todays song structures, except again, those were for a bunch of people to get together and sing for fun and to be all dirty and talk about sex.  Since the Ars Antiqua stuff started using secular music though, the secular and sacred parts of music sort of started to come a little closer together.  In the Baroque period, we have an emergence of much more commonplace concert music.  A lot of this is due to economic sorts of factors, with the Age of absolutism roughly overlapping.  We had rich, powerful people who wanted entertainment, and could now hire musicians to write an opera for them on the holophoner.  In fact, my first example of bridge period music is from Claudio Monteverdi, at the beginning of the 17th century, and was one of the first Operas.

So we hear that returning refrain in the orchestra between verses... it's quite a change compared to the flow-y, melt-y lines of the renaissance... it's a lot closer to standard song forms today.
We also see the arrival of Opera.  As I mentioned, Music was always linked to drama in some way, such as the early greek texts that note that their works of drama would have musicians, to heighten the emotion, as music can do.  The modes often are associated with different feelings that they evoke as well.  Opera was pretty much an advancement of this, melding the two.  In opera, we see basically two types of song.  First, we have the recitative, often shortened to recit.  This is the exposition song style.  Often there are a bunch of words over a chord, maybe on one note.  It's the equivalent of speech, pretty much.  Just sung.  The aria, on the other hand is the long florid song.  It's really the fuller song, and most people think of the aria when they're thinking of any specific work from any song.

We also see multi-movement works coming on scene more.  Similar to masses, but as concert/public music and not just sacred music.  Rather than list and talk about each style here, I'm going to link to wiki, which has a list of the parts of the suite as well as a few other style listings.  I'll talk about some of the more important ones in terms of theory and specific style.

First I want to talk about something called "Basso continuo".  Basso continuo was a type of figured bass, sort of an early way to give a bass note and a chord over it without writing all of the notes out.  The continuo was semi-improvised, and even the group of instruments playing it wasn't specified most of the time.  Think today if you went to a rock concert and instead of playing guitar, bass, keyboard, and drums instead they decided to pull out violins and brass and play the songs on those instead.  And that was completely normal and par for the course.  In terms of the notation, it was pretty similar to the way a jazz chart is written now, mixed with theory analysis of today.

We also see using "Ground bass", which is essentially a bass ostinato.  For those of you who don't know, an ostinato is a constant, repeated figure over changing music.  So ground bass is a bass ostinato... it's a bass line as we think of it today, essentially.  I'm skipping a few years for this example, and I'll go back to cover temperment and general style, but this is the example that is used in every theory class for Ground bass:

skip to about 0:45 ish to get to the end of the recit and into the aria, and you'll hear that descending chromatic bass line, which then repeats like a billion times while the aria is going on.

What we hear here too is a movement towards vertical thinking.  While lines in the renaissance tradition would construct what could be analyzed as chords today when they were overlapping, the thinking was all about the horizontal nature of the lines on the page.  The idea of "vertical" and "horizontal" here makes a lot of sense if you look at a multi-part score... the vertical nature is how the parts interact with each other, the horizontal aspect is how a single line behaves in time.

While this was going on we also find that now music is taught as a specific art.  Pedagogy emerges, things are structured, and there are now musicians instead of either people getting together to sing, or monks singing.  This means two things basically.  First, all of a sudden we have musicians that are musicians all the time.  People can now devote pretty much their entire lives to composition, performance, and study of music.  And the second thing, which grows off the first, is that musicians in general are a lot better at their jobs.  Now, what this means is that with the emergence of more ability, we have an emergence of showing off that ability, which is why we have the Baroque style emerging.  And what is the Baroque style?  Simple:

Baroque music is technical, virtuosic, and structured.  Baroque music is very much show-off music.  Baroque music uses heavy ornamentation and is very flowery sounding.  It's also very technical about it.  In many examples of Baroque music, you're basically just talking about playing chords.  But fast.  This makes Baroque music super-awesome for teaching.  There's a reason a lot of kids learning instruments learn Baroque music, because it was either about teaching good technique, or showing off that the performer had good technique.  Also, it's important to keep in mind here that pedagogy has improved over the past few centuries.  While today 12-year-olds can play Bach pieces, back then this was the height of performance ability.  There's a story with I believe Aaron Copeland, who was getting a piece of his performed by a modern orchestra, and the violinist played a certain lick he had written as written, and he said that it was all wrong, because even in the short time of a single composers lifetime, when he wrote the piece, violinists couldn't play that lick together and correctly, so the end result was a lot mushier than what was played with the more modern orchestra.  Consider with that sort of improvement over a relatively short period of time, how difficult this music must have been at the time.

Anyways, let's listen to some good examples that show off... well, showing off.

Oh my, that last one.
You may notice the last piece is from "The Well-tempered Clavier"  Allright, so other than an awesome band name, what does "Well-tempered" mean?  For this, we unfortunately have to use math.  I know, I'm as angry as you are.  Math?!  in Music?!

NOTE: My attempted "simple" explanation of tuning and temperament ended up being... well, super awesomely long and full of really boring math shit, so I'm going to post it separately, and just summarize here. 

 Essentially, before the mid-Baroque, we were using tuning systems that gave "Wolf intervals", which were wildly out of tune, and certain intervals just didn't sound right, because of some math stuff.  Also, we would be tuned to a specific key, and moving around to other keys required retuning any fixed-pitch instrument like a harpsichord, or, much harder to retune, an organ.  In the mid-Baroque a system of tuning started to arise called "Well-Temperament", which was a great advancement that allowed every key to be played(though some still sounded a touch off)

This was what Bach was using in that last example, he was so excited that he could do shit in every key that he wrote a suite, Das Wohltemperierte Klavier, or The Well-tempered Clavier, which contains a piece in every one of the 24 major and minor keys.  The original suite, what is now "book 1", was composed in 1722.  Later, in 1742, he wrote "24 preludes and fuges", which is basically the same idea, and is now considered "book 2" of the full Well-tempered Clavier.

A final style thing I want to talk about in the Baroque period is the "fugue", or "Fuga". The fugue is a texture of music that is the closest thing to the direct following of the tradition of prima practica Renaissance music.  Technically, I believe "fuga" comes from Latin related to both the terms "to chase" and "to flee".  Basically, in a fugue, a musical idea gets passed around parts and played with a bit.

The traditional fugue opens with the entire ensemble playing a single theme, and then a single voice starts playing that theme, which is referred to as the "subject".  Then, a second voice enters with the subject but transposed, often to the dominant.  Once the subject is restated, the first voice takes up counterpoint to it.  If the counterpoint is used later and is a constant companion to the subjects restatement, it is referred to as the "countersubject", otherwise, it's just called "free counterpoint".   Pretty simple, right?  Well, when this gets super-layered it can be pretty intense.  Bach is probably the most famous for these, but then again he's probably the most famous Baroque composer so.

This stuff sounds like this:

One final thing to mention, though I'll go into more detail after the history series, is counterpoint.  Bach measured statistics and wrote, in texts like "The art of fugue", "rules" for counterpoint, which is essentially the means to write moving voices.  This is another part of the structured nature of the music.

And that's all for Baroque styles and characteristics for now.  Next update will be Classical, though again, there might be a bit of a delay here since I'll be out of town for the next two weeks.

As always, any questions?  Feel free to leave a comment and I'll try to get back to you.