What's Wrong with Western Music? Part III. "Passacaglia in Cm"

By: Bernard Chazelle

A few remarks first:
Don't waste your time "disagreeing" with me because, so far, I've only been stating established, known facts. Well, maybe not known to everyone and, hopefully, a few of you will find in these postings food for thought. But please try to dial the hostility down and the civility up. I find some of the aggressivity in the comments frankly baffling. This blog is all about pointing out weird aspects of things we love: art, politics, humor, etc. Those of you who still think I am trying to prove the superiority of music A over music B are missing my point entirely, and -- if I may add -- that of this blog, in general.

But many of the comments are from genuine lovers of music (you know who you are) who, like me, are perpetually puzzled by the mystery behind it. Yours are the voices I want to hear above all. Please share your thoughts and experiences, especially what and why music moves you.

Again, sounds odd to say it, but yes these are just hastily written blog posts (like all my blog posts). I discuss only a tiny corner of the musical world, the acoustic implications of harmony and the social choices they imply, and I greatly oversimplify. Music has so many other facets. One topic that fascinates even more than Western harmony is African rhythm, which is extremely intricate (much more so than in Jazz). It's a world onto itself. Anyway, let's get cracking. (If you spot technical mistakes, please let me know so I can correct them. I have no time to proofread this and I apologize in advance for the sloppiness.) Thanks.

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So it's the small fractions (2:1, 3:2, 4:3...) that make music the physical art form that it is. Any true music lover knows exactly what it means to be overpowered with ecstasy while listening to [FILL IN YOUR FAVORITE MUSICAL PIECE]. For me, that would be Bach's Passacaille in Cm. (French spelling.) Amazing what Bach could do with 3:2! It's in a minor key. I still remember when I learned music theory and I discovered with amazement that minor and major "modes," which sounded so emotionally different to me, were in fact the exact same notes played in the exact same sequence, just starting from a different place! If you read Romeo and Juliet from the middle to the end, and then from the beginning to the middle, do you get a happy play? (Well, maybe you do!)

When it's all said and done, I cannot think of any music that has had more power on me.... (6:06-6:30 is scary). It's personal. My grandfather was an organist. The sound of the organ in a big cathedral is one of my earliest childhood memories, and so the effect of that piece on me has a context. Before he passed away, my grandfather confided to my mom that if he gets to sit between God and Bach, the afterlife shouldn't be so bad. Opera lovers will tell you similar stories of ecstasy. As will rock, blues, folk, Jazz buffs. I don't know enough about the visual arts to know if there's an equivalent. Can a painting cause people to stop breathing? I am curious to hear your experience. I played guitar in local rock & blues bands for many years. I can't count the number of times we'd look at each other while playing and think the same thought "How can living be so pleasurable?" Music buffs know exactly what I am talking about. If you haven't experienced the overwhelming physical power of music, you just don't know what you're missing.

This post is long, so you might want to let Bach make it more tolerable. Some oppose on principle the concept of playing music and doing work at the same time. (Not that reading this is work.) They're idiots. Life is finite and I have thousands of Jazz CDs to listen to before I die. (And I am very lucky to have a job that allows me "music at work.")

What Bach can do with small fractions is staggering. But he was the ultimate music genius (sad to think it peaked just as it got really started) and to draw from his example the lesson that pretty intervals properly placed makes great music would be a serious mistake. Humans are complex beasts. Sometimes pleasure is enhanced when it follows pain. In the end, make no mistake, music is about pleasure. But pleasure is a tricky thing. Too much of 3:2 and 4:3 will numb you. Muzak is "pretty." So what? And so perhaps preceding a 3:2 with a dissonance will sometimes enhance it. For example, take the tritone C-F#.

It's a fascinating interval: its ratio is sqrt(2):1, which means that, if you take the tritone of a tritone, you get an octave. The math is simple:

(sqrt(2):1)*(sqrt(2):1) = 2:1.

Millions of years of evolution have made your ear into a giant logarithm table (no one knows why for sure), so when air goes sqrt(2) of the way through its natural period, your ear thinks it goes log(sqrt(2))/log(2) = one-half of the way. So the tritone is very natural. Trouble is, as a small fraction, it sucks. First of all, sqrt(2) is irrational, which means? Well, which means precisely that it cannot be expressed as a fraction. A close approximation might be 45:32. But, hey, that's awfully close to the subdominant 4:3! Remember my chocolate sundae metaphor. The fraction 45:32 sounds horrible because it's so close to 4:3 yet so far! But then why is the tritone used all the time, not just in jazz and rock, but also in 19-c and 20-c classical music?

Being half an octave means that your brain treats it as not just a horrible dissonance but a very special one! So if you're going to use dissonance to prepare the grounds for higher pleasures, then a tritone might be the ideal candidate. Renaissance musicians hated it and called it a "Satanic interval," which of course implied their recognition of its special status. (Not every random asshole gets to be Satan!) Bach used it to create tension that only a motion to the root chord could resolve. In the 19th century it was used to modulate (more on that below). The idea was this: I am tired of meat so I'm going to yank that steak from your plate and replace it by a piece of fish. But I don't want you to scream WTF when I do that, so I'll distract you by yelling: Oh my God, did you see the flying pig over your head? And when you look up, pronto, I switch your dish. Then you'll eat your fish all happy without even realizing the change. The tritone is the flying pig. Wagner was a fanatic flying pig farmer, an obsessive modulator: he could easily change your dish every two measures for 10 minutes. (Trust me, with too much of that, you would invade Poland, too!)

Think of dissonance as the word "fuck" in comedy. It can be used to transgress, to liberate, to ridicule, to humanize, to bring down to earth, to change topics. In "Curb Your Enthusiasm," Elaine has a cameo appearance where she says fuck all the time. People ask her: Why are you saying fuck all the time? She replies: "Because I'm on HBO." By that she means to expose the hypocrisy of network TV. (As though American kids don't hear 'fuck' at school all the time that they have to be "protected" on TV.)

In "Last Tango in Paris," this is how Marlon Brando asks for forgiveness for the love he failed to give his now-deceased wife. He addresses her in her open coffin:

"You cheap goddamn fucking God-forsaken whore, I hope you rot in hell, you lying cunt!"

These are the necessary dissonances, if you will, of a scene that captures despair as poignantly and powerfully as film ever has. "You, lying cunt" is an eloquent affirmation of love. You have to see it to believe it, but this scene will leave you in tears. The dissonance works. But it works because it's great art. In any other context, it stinks. Art can make the stinky sublime. And, in music, bad sounds help you make it happen.

Life is a bitch, and sooner or later, believe you me, you'll need your tritones to get by, you'll need your Satanic intervals to make it through the day. But here's the thing: you also need a purpose. A stand-up comic who thinks of "fuck" as a "comedic enhancer" does not get it: "2+2 is 4" is not funny, but "2+2 is fucking 4" is not funny either. Dissonances should be in your music and, hopefully in your life, only where and when you need them. The artist has to feel the necessity of them, else it's manipulation.

Good, but now, if deep inside you really wished you had 3:2 but your technology or your culture imposed, say, 31:20, then it would be wrong. How can denying an artist her creative need be called right? Keep this in mind. I'll get back to it.

Change of scenery: Scales. You need them. Between 1:1 and 2:1 you need to specify special points (ie, notes) that you favor. You need them for many reasons. One of them is technological: many instruments cannot be built otherwise. But there's a more fundamental reason: writing down music on paper. Before the age of recording, people could learn music by oral transmission (but that does not work well for complex instruments) or by reading it. But to read it, you have to write it first. And to write it, you need an alphabet. Western music chose an alphabet of 12 letters (the 7 white keys on your piano between two Cs plus the 5 black keys). English has 26; Western music has 12.

Why 12? It's a cool number. The best way to see how cool is to draw a regular polygon (like the Pentagon in DC but with 12 sides). Then draw the diagonals. If you blink hard enough, you will see all sorts of beautiful interlacing patterns. If you have the time, do it. Take a circle and draw 12 equally-spaced dots on it. Number them 0,1,...,11 (with 0 at the North Pole and 6 at the South Pole). Now connect 0 to 7. There is exactly one other diagonal of the same length starting from 7. It goes to 2. Draw it. Then repeat. Lo and behold, this will take you back to 0 after 12 diagonals, and you won't even have to lift your pen! Exactly two diagonals connect 0: they are 0-7 and 0-5. Guess what? These correspond to 3:2 and 4:3! (Don't try to read the numbers 3:2 and 4:3 from the polygon: you can't!)

If you draw all the diagonals, you end up with a grand total of 66 lines. Every piece of Western music can be interpreted by looking at the symmetries among these lines! Neat, huh! This picture bends the diagonals for effect.

People weaving Persian, Chinese, or Indian rugs will not be impressed. Islamic artists will shrug their shoulders. The symmetries of the 12-sided polygon are, shall we say, babyish! To give you an idea, physicists sometimes use generalized polygons (called Lie groups) with one trillion "diagonals"! Not just 66...

There's one serious human limitation on music, though: one must be able to hear it, to play it, to memorize it. So big numbers are out. Certainly, 12 is on the low side. And that will come back to haunt Western musicians. (Much of Indian music uses 22 tones: 12 is to 22 what a bicycle is to a Mercedes, because the number of possibilities grows exponentially in the size of the scale.)

Anyway, we have our 12 notes, we have our diagonals, symmetries, and all that. We're ready to go. Except for one thing: what should these 12 notes sound like?

You need 12 notes between 1:1 and 2:1 ? That's easy:

1. Throw in 1:1 (our root)
2. Throw in 3:2 (the dominant, our favorite)
3. Throw in 4:3 (the subdominant, our second favorite)
4. Now what ?

Going 3:2 from anywhere sounds good, so let's take our dominant and move up by 3:2, which lands us at (3:2)*(3:2), which is 9:4. But we want a number between 1:1 and 2:1, so we raise the denominator 4 of (9:4) by an octave to get 9:8.

That gives us 4 notes:

(1:1) (3:2) (4:3) (9:8)

Keep on doing this (multiplying by 3:2 and bringing the numbers back to the range between 1 and 2) until you get 12 notes and that'll get you a scale.

Done.

Only one problem: the closest fraction to 5:4 (the major third) you get in this scale is 81:64, which is the difference between 1.25 and 1.27. Close but no dice. Can we just say we don't care about thirds? No, we can't! You need 5:4 or something very close to it because all chords in Western music are formed by stacking thirds, so if you get those guys badly wrong you're out of luck. Especially that by stacking them together the error gets compounded.

Let's fix it this way:

Keep (1:1) (3:2) (4:3) (9:8) as before but now add in the major thirds (5:4) above and below each of these 4 pitches, for a total of 12. This gives you:

(1:1) (16/15) (9/8) (6/5) (5/4) (4/3) (45/32) (3/2) (8/5) (5/3) (9/5) (15/8)

Wunderbar!

This is a great scale. You can still hear funny dissonances if you're not careful (but, remember, you want them to say "fuck" and to fly pigs!) For a measly 12 notes, you got yourself a good deal. OK, if you're dying to hear all those intervals you can't get in Western music, well, tough. Go get yourself a 22-sided polygon and learn to play Indian ragas. The trouble is that to be really good at Indian music takes enormous skill AND a lifetime of learning. So if you want young children to sing in a choir or music to dance to in the town square, 12 is better than 22.

Medieval European music was based on this principle. But then it all changed in the 17th century! (Yes, nitpickers, I know, nothing ever happens all of a sudden!)

Why? How?

First, why? Even though a mere 12-pitch scale is already asked to do way more than it can (ie, accommodate all these good intervals), the Western man asked for more and more. When the Western man sees a mule loaded with 1,000 pounds of bricks on its back, what does the Western man say? The Western man says: "Let's add another 1,000 pounds of bricks and see what happens!"

Why is the Western man (with an Italian/French/German accent -- for once the "Anglo-Saxons" are off the hook!) is being so demanding? Because of their sisters. You see, the way we defined the scale will get us good dominants, subdominants, and thirds, but only when we start from certain lucky notes. So the Western man looked at the 12-scale and said:

My sister doesn't like you! When I sing my song, I begin in C (ie, 1:1) and go up to a dominant and that sounds great. But C is too low for my sister, so she starts at the third note, which is labeled (9:8). We call it D. And then she goes up to the dominant, which is (9:8)*(3:2) = 27:16 But 27:16 is not in your damn scale. The closest you have to offer is (5:3), which is so way off it makes my sister's ears bleed. Fix yourself!

The answer could have been:

"Hey, buster, that scale is already overtaxed. It can't be changed. Ask your sister to get her own piano tuned to D and she'll be OK."

But for the Western man, you see, no mule is ever overloaded and devotion to one's sister knows no bound. Here is what the Western man will do. He'll find a number, call it R, and form this scale:

(1:1) (R:1) (R*R:1) (R*R*R:1) (R*R*R:1) (R*R*R*R:1) ..... (R*R*R*R*R*R*R*R*R*R*R*R*R*R*R*R:1)

If R were an integer you would get the harmonics, which would be, like, really stupid. You want all these ratios to be between 1 and 2. So, make R very tiny: so small that when you multiply it 12 times with itself you get 2. That means P is the twelth root of 2. So the last note in the list above is actually

(R*R*R*R*R*R*R*R*R*R*R*R*R*R*R*R:1) = (2:1)

The neat thing is that the ratio corresponding to any interval is the same. For example, take a third like (R:1) (R*R*R:1). [It's of length 3 because in our new scale, there is exactly one note in between.] See, you go from one note to the next by multiplying the numerator by R*R. But, now, consider the interval of length 3 from (R*R*R:1) to (R*R*R*R*R:1). Ah, you go from one note to the next by multiplying the numerator by R*R. Voila! All intervals of a given length like, say, all fourths, will sound exactly the same. Remember that your ear is a logarithm table. This means that it will perceive exactly the same increment as you sing an interval of a given length. This new scale is essentially a 12-step ladder where each step takes you up by exactly the same amount (or so your ear thinks). We'll call it equal-tempered (don't ask why).

So, the Western man and his sister will sing exactly the same music (only transposed higher). You can play from any note and as long as you go up and down as you should it does not matter a bit where you start. There's only one trouble. Yes, the music will sound the same wherever you start, but unfortunately it won't be the one you had in mind! All intervals have now ratios that are powers of R. But these powers can never be what we want, ie, things like 3:2, 4:3, 5:4. Never !!!!!!!!!!!!!!!!

But do we get close?

No!

The scale is wrong in at least 3 ways:

1. It is an assault on the ear. You don't have to be Mozart to hear that it's all wrong. Every chord Maurizio Pollini plays on his piano at Carnegie Hall is off. It's off by at least 10% of a semitone, which most people can hear if they try. Worse, we're so used to hearing wrong music we don't let it bother us too much. But a classical Indian musician will be horrified by the sound of a piano. You've heard a sibling or a cousin butcher a tune so badly you wanted to get a lifetime membership to the NRA. Be honest, you have no qualms calling that singing "wrong." Well, then Ravi Shankar should have no qualms calling an equal-tempered piano "wrong."

2. The piano teaches singers to sing out of tune. To sing out of tune is hard. It takes practice. In fact opera singers will naturally revert back to form and sing in tune, with perfect thirds, dominants, etc, as soon as the piano shuts up! Barbershop quartets sing in tune naturally. In fact, humans who can carry a tune are pretty much incapable of singing out of tune as pianos do even if they try. The only way they can do it is when they're accompanied by a fucking piano! (Do appreciate the judicious use of a dissonance in the last sentence.) Piano accompaniment is, of course, how all singers practice! This is wrong because it's not wanted. It has no purpose, except convenience. This is wrong because not even the piano's biggest fan would keep that system if it could be fixed. But it can't. Unless, that is, you throw away equal temperament.

3. Mathematically you're trying to do something that, in better worlds, gets people shot. I won't go into the math, but only mention a relevant anecdote. One of the world's greatest number theorists once told me,

"Bernard, do you realize that all of the world's mysteries can be traced back to the fact that addition and multiplication don't go together."

This is (in all seriousness) one of the most profound truths I've ever heard. "Equal temperament," which is what this new scale is called, pretends that addition and multiplication are compatible, when it is the very essence of everything that's beautiful in life that they are not.

Bach is often accused of having forced equal temperament down our throats. That is a lie! He wrote pieces for well-temperament, which is not the same. In equal temperament, all consecutive keys are equidistant. But piano tuners, especially before electronics, used their ears and their ears were still accurate so that they tweaked the tuning to "fix" the thirds and fifths, etc. In fact, Bach wrote his famous "Well-Tempered Clavier" as a set of pieces in all 12 keys, major and minor. Why did Bach bother with all these different keys? Because when you don't do equal temperament different keys will sound different. That too is lost in post-Bach music (OK, not quite true, because pianos get tuned in certain ways sometimes for certain pieces, like Beethoven's amazing Waldstein sonata being a good example). In equal temperament you get the famous joke:

"You say potato, I say potato; you say tomato, I say tomato." Hmm, I don't get it!

Now our 12-pitch mule has 2,000 pounds of brick on its back. Its sounds are all audibly wrong. The wrongness is undesired. So it's not like the dissonances we love to throw in. It's plain wrong all across the board. Is the cost huge? Yes, absolutely. Many musicians (including Western ones) have tried to break away from it.

But do we get a reward for our sins?

Yes, a huge bundle of prizes.

Transposition is one benefit. But the more important is modulation: changing tonal center within the same melody. When you start singing the note C, chances are you'll soon be hitting E, and F, and G. But playing F# would sound weird. (I haven't talked about the diatonic scale so I'll do it quickly: the white keys on the piano. Remember our first attempt to build a scale, get the root, the subdominant, the dominant, and then keep multiplying by (3:2). If you stop once you have 7 notes, that's your diatonic scale right there: C,D,E,F,G,A,B. Well not in that order. The cycle of fifths goes: F,C,G,D,A,E,B, etc. So your 7-note scale is built out of the 12 tones by cycling through the dominants. There are other ways of justifying its construction but this one will do. The famous pentatonic is built out of the first 5 notes from the cycle of 5ths (with its "minor" variant). I don't want to digress, but the way the pentatonic is understood in rock and blues is completely different: it's downright absurd to think a rock minor pentatonic as being extracted from a Western scale. It's no longer a matter of debate among musical ethnographers that the origin of that scale, as used in blues/rock/Jazz is non-Western. We simply define the minor pentatonic as its closest match in the Western scale, but it is a poor approximation.

Equal temperament allows you to do the same thing starting from anywhere, say, F#. This adds great flexibility to music writing. It allows you to break from the hierarchy imposed by tonality (which favors certain notes over others.) Our long diagonals in the polygon allow us to modulate to any key we like. So we gain great "syntactic" power at the price of poorer sounds. We bloggers know that well: the more we bullshit the more we can say.

Equal temperament was tremendously liberating for musicians. By sticking to 12 pitches, the language was still very simple, making innovations easier. Modulation allowed for the sort of versatility that is difficult to get in other musics.

Some have argued that if you want equal temperament, then 20 pitches should be the minimum acceptable; or 30. They say that 12 makes the music so wrong as to be offensive. I wouldn't go that far. For one thing, we Westerners are too conditioned to it by now to get so upset. But some non-Western musicians are horrified to see their compatriots abandon their much richer native musical vocabularies for the Western "approximation." Bollywood has a lot of Westernized Indian music which many natives regard as destructive. I think we can all agree that, in an ideal world, all musical traditions should enrich each other but not suffocate each other.

Equal-temperament costs a lot (wrong sounds: again, I use the word wrong because it violates the rules of acoustics in a way not even fans of equal-temperament like). The wrongness is accepted only because it opens new doors, transposition and especially modulation.

No other music cheats that much. But all do to some extent. Even Indian music has wrong intervals that it tries to hide with drones and harmonics, etc. But Westerners do it so much more.... Well, like this: Take pi=3.14159.... No one uses the exact value of pi numerically. It's impossible. But the great state of Indiana tried to pass a bill in the late 19c decreeing that pi would be officially 3.2. (Funny they didn't choose 3.1.) Many musicians, including American ones (but not Hoosiers), think of the equal-tempered rule of Western music as the pi=3.2 of music. Some will say: But imagine, as a thought experiment, that, without a literary equivalent of this ridiculous approximation, writing King Lear had proven impossible, wouldn't it had been all worthwhile then. Well, yes!

The choices made by Western music were without a doubt worthwhile. To say that Western music would have been better off if it had opted to be less wrong and, say, stick to "just intonation" (the scale I discussed before equal temperament) is silly because such a statement is entirely unfalsifiable.

The interesting question is why Westerners made a drastic choice that no other big music traditions seem to have made. (I could be wrong about that, but certainly not among the "major" ones). It was drastic because, when it was made, for many people, music was still a medium to communicate with God. It takes guts to offer God imperfect dominants just so that little Sis can sing Mary Had a Little Lamb! Why did it happen? Was it:

1. Technology?

2. Expediency?

3. Curiosity about modulation: much of Western harmony is really the art of modulating and substituting.

4. Dance music? Don't forget that Bach borrowed melodies from everyone, including local villagers singing folk tunes. In some ways, music was much more democratic then. I don't see modern classical musicians borrowing much from hip hop. (OK, Radiohead borrowed 1 measly sample from Lansky.) But Schubert spent his evenings in local bars to hear people sing folk songs. Also, remember that classical music often incorporated dance elements.

5. Church music? Bach was both a court musician and later a church composer. He'd compose music from one sunday Mass to the next. Local parishioners knew his music. Again, there was a smooth continuity between high art and vernacular music in the 17-19th c. that seems to have been lost. Which villagers today listen to Philip Glass?

6. A desire for abstraction? Equal temperament decouples harmony from acoustics. Since you don't care how things will really sound in the end, you're allowed much more freedom to explore. Composition acquires an abstract flavor whereby mathematical patterns matter more than aural perception. At the same time, political theory became more abstract; empires grew. The enlightenment made it easier to be seduced by the sure rationality of the new rules. (Just to clarify, Western harmony is perfectly fine mathematically as an abstract model. It only breaks down when you connect its impeccable logic to the logic of acoustics. It violates the math of acoustics, not the math of harmony.) So was it part of a general philosophical trend toward abstraction and universality?

Who knows? Take it away. I am done.

— Bernard Chazelle

Posted at August 11, 2008 07:22 PM