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August 11, 2008

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.


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.


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)


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?


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

I found this to be a very accessible discussion of temperament. You have my congratulations.

What did you mean by calling Lie groups "generalized polygons"? I've never heard them referred to that way before.

What did you mean in the last post, when you made that comment about N*(N+1)?

What did your number-theorist associate mean by his comment? They surely "go together" in the sense that the integers form a ring, but that obviously is not what (s)he meant.

Whence would you say the pentatonicism of blues originates?

Posted by: Save the Oocytes at August 12, 2008 05:08 AM

Prof Chazelle, I did not understand a single letter of this blog ( my ignorance and loss ). But I LOVE music, could not live without it.
And yes, though I have not had a moment when I thought my breathing was going to stop, I have had an experience, only once in my lifetime and will never forget it. I LOVE the song "Ave Maria" and have heard different versions. I was attending a friend's wedding and suddenly it was all quiet and I heard a voice from behind sing "Ave Maria". I literally froze and I thought my heart was going to jump out of my chest and and it scared me a bit because I really thought, my heart would stop but it was an unbelievable physical experience of pure joy and ecstacy. The voice (to this day, I have not heard a voice like that), the acoustics in the small church and the accompanying music and of course the song did it for me.

Posted by: Rupa Shah at August 12, 2008 10:23 AM

I think one thing to keep in mind is that the human ear “adjusts.” For individuals the sense of pitch can change over time. Some musicians play sharper than others but to them it sounds perfectly in tune as their ear has adjusted to it but for other musicians it sounds sharp. Of course some play a little sharp purposefully because for some reason playing flat sounds worse than playing sharp also some believe it projects better than when you are perfectly (theoretically) in tune. At any rate if there are “imperfections” over time that would become the norm as the ear adjusts to it as it does to dissonance, well at least if the music is well written.

It seems to me that you really have in part perhaps answered your own question as to why when you say that Western music is all about modulation and substitution. Western ears are used to hearing modulations and expect to hear it. In most popular music a tune is broken down into 8 bar phrases and one of those phrases is called a bridge where the tune modulates to another key and then back to the original key and people expect to hear that even if they don’t realize it or know anything about music. The purpose of modulation is to keep the music interesting or that is my impression anyway. The same thing is true about substitution because it adds another color to the music keeping the listener interested.

I really have no idea why Western music went in the direction that it did. I would hazard a guess that it was the influence of a few individuals which usually seems to be the case when a new style or type of music evolves. For an abstract language music has an amazing ability to covey information. Watching movies or TV one device I have seen often is the camera is zoomed in on an actor’s feet as the actor walks along. Obviously the sight of someone walking is not unusual and since you are zoomed in on the feet it does not tell the viewer much about the movie but when you add the movie music it gives you a clue. If the music is sinister you know it is probably a bad guy about to do something bad for example. But why does the listener know that the music is sinister or more accurately perhaps is why do we associate certain musical constructions with sinister? I’m not talking about the conventions of Western music but the basic question of why do certain combinations of tones convey emotion and information in the first place?

This is totally interesting Bernard, thanks again for taking time to write about it.

Posted by: Rob Payne at August 12, 2008 11:50 AM

Thank you. I know have a much better understanding of why I have such "weird" musical tastes. I listen to electronic music, but I love "broken beats" best of all. I listen to hip hop, but my focus is on hearing new African and Middle Eastern hip hop bands.

Across the spectrum I see now that my conventional genres are heavily loaded with reinterpretations and remixes that borrow heavily from non-Western influences.

Now, if you would just help me to pick the right drugs to go with all this new music....

Posted by: Mark Gisleson at August 12, 2008 12:20 PM

I had no idea about singers naturally singing properly proportioned 3rds unless they have a piano there to muck them up. As a pianist, I actually find that pretty disturbing. And it makes me wonder if the vogue for that Sonny Rollins set off of firing the rhythm player in the quartet may not have been only about the rhythmic and harmonic freedom it gave, but also-- on an unconscious level-- the fact that the tuning could be freed as well. (Tenor sax isn't really made for just intonation, but it certainly has more freedom in that arena than a piano or fretted guitar. And of course, acoustic bass-- like any fretless string instruments-- is as perfect as a human voice for "non-wrong" tuning. Guess I'll go give another listen to "Way Out West" soon and see if I can identify any extra sonorousness compared to usual jazz.)

I remember when I first discovered that the whole apparently astonishingly elegant 12-tone system was just a sham approximation imposed by human hands. It was very disillusioning at first. In what may be a fairly common story among other musicians, I mucked around for a few days with a synthesizer, adjusting it to various systems of just intonation. But it just was no good for jazz. I gave up quickly. At least as we traditionally know and love it, jazz is a skin that grew almost entirely over the skeleton of equal temperament.

With non-equal temperament, upper extensions on dominant seventh chords usually sound appalling. And even if you limit the chord voicings to completely vanilla triads, the movement of the chords themselves over nearly any jazz standard out there is nearly always too adventurous. Too much modulation. Though maybe instruments that can change their tuning on the fly can pull it off better. But a justly tempered piano, no way.

Very simple modal stuff can work fine, but it's pretty limited. If you tune it so you can play in D dorian, you can forget about shifting to, for instance, Eb dorian. ("So what?", most people are probably thinking right now.)

So quitter that I am, I gave up and promptly forgot all about it until you rekindled my memories with this fine post.

I do know that finer jazz musicians than I have experimented more rigorously with alternate tunings. Don Ellis had that special quarter tone trumpet. Steve Lacy intentionally developed an extremely flexible tone to imitate the human voice, and I seem to recall that imitating the way it finds pitches might have been part of it.

Rob, you point out that the ear "adjusts". The question is, how much does this matter? Does listening to music played closer to "not wrong" tuning give some extra sense of well-being, or anything like that? In the same way that, for instance, recordings on vinyl just fill you with so much more life than recordings on standard 44.1 kHz audio CD. If so, then it's something I should seriously start thinking about messing around with more.

Posted by: Quin at August 12, 2008 01:24 PM

When I mentioned how Sonny Rollins famously did without a "rhythm player", I meant to say "comper"-- that is, without a pianist or guitarist accompanying with rhythmic chords. Maybe jazz fans knew what I meant, but I thought it might not be clear to people who don't know jazz. Then again, probably the rest of what I wrote wasn't, either.

Posted by: Quin at August 12, 2008 01:43 PM

Bernard, you've stimulated me and pissed me off a lot over the course of this blog, but every sin you have committed or may commit is hereby absolved with these sentences:

"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."

That is my experience, too. In fact, having worked with people with a variety of physical and sensorial difficulties, I'll tell you this: if an unjust God, or an interrogator in Gitmo offers you the choice of losing your sight or losing your hearing, choose to your sight.

And thank you for the discussion. Your posts are well-appreciated (even if I think you did know how people would react when you told them that they and the music they loved was "wrong").

I'd love to hear more of your opinions about music, and your own preferences. (I myself have a pretty broad taste, and one of my favourite albums of all time is one by the great singer Nusrat Fateh Aly Khan. I heartily recommend him to anyone wanting to expand their musical tastes a bit.)

Posted by: Hairhead at August 12, 2008 03:05 PM

Prof Chazelle, I never studied 'MUSIC' but your posts have got me hooked and I am going to try to be a self-taught student ( with the help of your posts ).
I saw a movie some years ago, "La double vie de Véronique" and it had some real beautiful haunting music. I could never have enough of it.

Zbigniew Preisner - Van den Budenmayer Concerto en Mi Mineur

Van den Budenmayer Concerto 1798

Posted by: Rupa Shah at August 12, 2008 06:30 PM

>Which villagers today listen to Philip Glass?

Haven't found any so far. When I play Glass (quietly) at my desk, some of my fellow cube rats will ask "Are you _trying_ to annoy me?"
Not euphonious with Toby Keith, I guess.

I know from nothing about music theory. (well, more now than a week ago) I DO know that one of the reasons I enjoy Philip Glass is that his music often comes at me just a little faster than I can comfortably process it.

Posted by: George Lowry at August 12, 2008 07:03 PM

Your point being it all boils down to: "Can you dance to it"?

That makes sense. Apologies for my earlier snark.

Posted by: J Lumlum at August 12, 2008 07:22 PM

GREAT post.

As a Lutheran PK choirboy who eventually would up singing in a Bach society, I have been thinking about these matters for literally decades.

Mostly I think you got it mostly right--especially the part about sitting between Bach and God. But I would add a friendly amendment. Bach would have never gained fame, and more importantly employment, without the cultural manifestations of Luther's Reformation--which are infinitely more interesting than his theology. As I see it they are:

1) Luther substituted music for statuary are THE approved religious art form. In many ways, the organists and choir directors are MUCH more important in a Lutheran church than the clergy--ask any preacher--and no matter how tiny the congregation, the organist get paid!

2) While the Lutherans never built churches like, say, the Vatican, they spent BIG for pipe organs. Because organs require such high levels of precision, Lutheran areas of the world have consistently "punched above their weight" when it comes to precision manufacture--and have for centuries.

3) Luther's payback to the printers that made him famous was that he insisted that good Lutherans HAD to be literate (so they would buy a Bible and catechism) and also that they must sing--in four-part harmony (so they would own a fat hymnal.)

So when Bach came along, pipe organs were already spectacular and tuning was already pretty much settled. When they recently rebuilt the organ for the Lutheran cathedral in Dresden, there was a furious debate about how it would be built. Of course, the purists wanted it to be built EXACTLY like the one JS Bach had tested when it was new.

The point is that while Bach was an incredible genius who gave us the well-tempered clavier along with his magnificent music, he had a superb supporting infrastructure--including congregations who thought what he did was worth supporting financially AND what he wrote was worth singing.

It should noted that Lutes still love their Bach. My aunt sang the St. Matthew's Passion when she was 13 (1936) and my sister was such a pipe organ prodigy she played Bach's Jesu Joy Man´s desiring, BWV 147 for a wedding while she was still in fourth grade. And in Scandinavia where most folks are indifferent agnostics and stopped going to church generations ago, there are probably 12 live performances in a city like Stockholm of the St. Matthew's Passion on any given Good Friday.

Posted by: Jonathan Larson at August 12, 2008 07:42 PM


Well I remember listening to George Shearing talking (during a concert) about how when he performed with a group of singers that they began the tune in one key and at the end of the tune they were in a key up a whole step apparently because the singers unknowingly modulated up throughout the performance of that song.

I wasn’t really trying to make a point about the importance of intonation rather was trying to illustrate how a person’s sense of pitch can change over time trying to relate it to what Bernard is talking about though probably not very well. I think what I was trying to say is that if a person becomes more exposed to the use of say dissonance (within reason) then it becomes accepted as normal with time if that makes any sense.

A couple months ago I watched an old 40’s mystery movie and the sound track for the beginning was truly awful, endless diminished chords played mercilessly with trumpets, you know, really bad writing it was so bad it was actually comical and corny. It was a case of just too much sameness with no relief anywhere. I don’t mean there is anything wrong with diminished chords just this particular piece of music.

I noticed your comment about tenor sax and I don’t think it is that they are not made for intonation since all good horns have basically very good intonation. If you are arranging for a big band you can actually get away with close voicing with the sax section more than you can with the brass section the reason being that saxes have a bit of a spread sound and they blend very easily so dissonance created by close voicing sort of has the edge taken off of it.

Posted by: Rob Payne at August 12, 2008 08:06 PM

"generalized polygon" = group polytope like the 8-dimensional beast you get from the Lie group E8 (whose matrix has close to 1 trillion entries IIRC). you may have read about it in the news recently.
On string theory blogs (which i stay away from), people slaughter each other over such matters! Those temperamental physicists....

N(N+1) is what you need for the standard Conservatory exam question "How do you modulate from C to C#?" Answer: run your cycle of fifths one extra step. This is a crucial property to have in Western harmony. N(N+1) is easy to prove with a tiny bit of group theory but it does require a proof.

Why adds and multiplies don't go together? This is HUGE! That's the reason Riemann's hypothesis is believed to be true, ie, the primes really are random! Because addition and multiplication are essentially alien to each other. Not sure how much math you know so I might be wasting everyone's time but Fermat's Last Theorem is exactly about that. The proof is so hard because you need to define two objects (called Fourier transforms): one is standard and is what math majors learn in college (Fourier transforms over the characters of a finite group); the other one is much more exotic: it's essentially a multiplicative version of Fourier analysis. Then you realize that these two beasts (additive and multiplicative) completely refuse to cooperate, ie when you put them together you get random-like behavior.

Of course what made my friend's remark particularly witty is that, on their own, addition and multiplication are the same (the logarithm is the function that proves that). So species of animals, one thinking in addition, the other in multiplication, would think exactly the same things. But one you get that ring you're talking about, the interaction between the two operations produces mysterious patterns. But elementary math doesn't really allow one to see why. Takes some work.

Thanks, all, for the great comments. A pleasure to read you!

Posted by: Bernard Chazelle at August 12, 2008 08:18 PM

Thank you Bernard for this post,

I once studied piano tuning (about 30 years ago). I learned about the circle of 5ths. My memory on this period of my life is vague, but I do remember being told that a change in the manufacturing of the harpsichord meant that the instrument could no longer be tuned with a diatonic scale. Knowing a bit about math, I thought it was due to the increased tensions in the strings that introduce non-linearities into the scales. Obviously, I didn't know what I was talking about. Perhaps the daunting task of retuning a harpsichord or piano to play in certain keys might have had something to do with the introduction of the well tempered scale. I'm hoping you can shed more light on this issue.

Also, there is one issue I had with your previous post which you seemed to resolve for me in this latest one. You likened music to masterbation for the ears. While I love both materbation and music, and often enjoy both together, music does server other purposes. Your enjoyment of music at work is no mystery to me. Songs and rhythms can motivate and intensify the performance of physical labor, so why not intellectual labor as well.

Anyway, great post,

Posted by: IronButterfly at August 12, 2008 09:28 PM

I'd like to drop a quick note to let everyone know how happy I am to see the words "tenor sax" keep popping up, because just recently I used far too great a proportion of my limited funds to buy myself one, which I've been wanting to do for years.

Posted by: ethan at August 13, 2008 12:53 AM

Wrong? Broken? Cheating? All this heat? For what??

It seems to me there's only one problem here, and it can be represented quite easily in simple mathematical terms.

ALL we need to do is find two integers, N and K, such that 3 to the power of K equals 2 to the power of N, like this:

3^K = 2^N

The answer(s) will tell us how many perfect fifths must be stacked to create how many perfect octaves. After that everything else is easy.

The solution to this simple equation is left as an exercise for the reader.



Thanks, Bernard, for a fascinating series.

Posted by: Warren Jones at August 13, 2008 03:18 AM


Thanks for the interesting link. I see what you meant about intonation and tenor saxes. Almost everything in the world is a compromise of something. For birds it is a compromise between weight and muscle. Almost all good designs require compromise of some kind and perhaps the same thing can be said about music in regards to this intonation thing between just intonation and well tempered. As it is it’s a compromise and evidently it works well enough. Giant Steps is a favorite of mine too. To me it comes off as a declaration of the sheer joy of being alive and an astonishing performance.

That’s an interesting thought about not coming back to quite the same key. I would think that as long as the band members were in tune with each other that a modulation without returning to the original key would work fine or perhaps depending on the listening skills of individuals in the audience, hey it might even become a new musical convention like 32 bar songs.

Yes, your Red Rodney story is what I was thinking. At first more dissonance would sound strange or off at first but with familiarity they really sound great to the listener.

I can still recall the first lecture given to me by my best teacher and it was one of the most useful things I learned. Here it is…

Music is a language and like language it is logical. Musical notes are like words and like words they need to be imbued with logic in order to convey meaning. A single word really has little or no meaning by itself rather it is where and how the word appears in a sentence that gives that word meaning. If I walked up to another person and said the word “tree” and then walked away I would not have conveyed any meaning to that person who would conclude I was probably crazy. But if I said “That tree over there is starting to lose its leaves” or something of that nature I would have conveyed meaning as the words that come before and after the word “tree” have imbued it with logic that is understandable. Likewise if I said “Orange ocean begs down bags church keypad if up sooner pig cloud telescope” I would have conveyed no meaning or logic because it is also the order in which words are used that gives them intelligence and this same concept holds true for musical notes since music is a language.

Today many so-called jazz educators give their students a list of scale choices that accompany a given chord usually beginning with the most consonant choice followed by less consonant scale choices. What I intend to show you is how you can make use of all twelve tones with any given chord, this is called the twelve tone system. To do this I will be discussing how the musician lends logic and intelligence to their improvising by understanding the importance of where a given note is placed in a melodic line, in other words, like words, notes are dumb and it is the job of the musician to lend intelligence to them.

Posted by: Rob Payne at August 13, 2008 01:48 PM

What is a "group polytope"? Does it have something to do with the convex hull of a root diagram?

I still don't understand what you are talking about with N(N+1). It may be that I would fail the conservatory exam, but the most obvious way of getting from C to C# with circle of fifths would be modulating, in turn, through C G D A E B F# C#. My understanding for 12, as I said before, is simply that (3/2)^{12} is about 2^7, or as Warren Jones would put it, 3^{12} is kind of near 2^{19}. What am I missing?

What number theorist made your remark, out of curiosity? I'm a mathematician, but FLT is far out of the scope of my knowledge, and I only know the standard L^2(R^n) Fourier transform. What you are referring to as the "multiplicative version"?

Posted by: StO at August 13, 2008 03:09 PM

As to your comment that people naturally sing in tune: Well, guy, you've obviously never heard ME sing.

Posted by: Green Eagle at August 13, 2008 05:49 PM

StO: You're absolutely right about the C to C# and the rest!

Andre Weil.

Re. the multiplicative business, here's the thing. Fermat's Last Theorem follows from the fact that elliptic curves (ie, cubic curves) are modular, meaning that the Fourier series whose coefficients a_p are given by the number of points on the elliptic curve in the finite field GF(p) is a modular form (ie, satisfy a whole bunch of amazing symmetries in the complex plane). Now if you define something called the L function (Dirichlet series sum_{n>0} a_n/n^s) then by studying this L function you get all the additive/multiplicative interactions of the system. The way you do that is to define two types of Fourier transform: one like the one you mention, ie, over an additive group (unlike R^n you take only finite groups but it's the same idea); then you do the same over multiplicative groups. You get characters that are multiplicative, ie, they are homomorphisms from the group to the unit complex circle, in other words they respect multiplication (as in f(pq)= f(p)f(q)), whereas in the standard Fourier transforms of engineering and physics, only addition is respected, ie the characters satisfy f(p+q)=f(p)f(q).

Posted by: Bernard Chazelle at August 13, 2008 06:03 PM

If I understand part of the discussion, here's a dry rehash, and then some "experimental" results.

We like to hear pairs of sounds whose frequency ratios are 3/2, so a way to create a "nice" family of notes is to create a collection whose frequencies are (3/2)^i times that of some starting note A: then we'll get lots of those ratios.

Since notes whose frequency ratios are a power a two fit together so well that we might as well call them the same note, we could just as well look at frequency ratios of the form
3^k/2^{k + k'}, where k and k' are integers, and k' is chosen just to make the ratio between 1 and 2.

But, we can't afford an infinite family of notes, so we want to stop when we have
3^k/2^{k + k'}
close enough to one that we can hardly tell the difference. Since we hear pitch ratios logarithmically, we could express this as saying that we want
log_2(3^k/2^{k + k'}) = k+k'+k log_2(3)
close to zero, which can be true for some k' when
k log_2(3)
is close to an integer.

I wrote a little python program to look at the difference between k log_2(3) and the nearest integer to it, and (with the major assumption of no errors in my 3 line python program), k=12 is among the best 5% of integers less than 1000 with respect to this difference, and among those less than 100, only 53, 94, and 41 are better. Among the k less than 12, the next best is k=5, and the distance of 5 log_2(3) to an integer is four times larger than the distance for k=12.

So, for this particular idea of what makes a good scale, it's not like there's some pretty good second choice: twelve is the hands-down choice. (Well, unless maybe you're the second coming of Harry Partch.)

Posted by: Ken Clarkson at August 13, 2008 07:36 PM

Ken: Interesting. I wonder how much better you get if you throw in log5 (or even log7?) into your lattice basis. I seem to recall some Just Intonation folks work with 5 and 7, too.

Posted by: Bernard Chazelle at August 14, 2008 12:43 AM

Someone mentioned Nusrat Fateh Ali Khan. He was behind one of my forgetting to breathe moments. I was driving my brother's car, and I turned on the CD player, it was NFAK singing "Allah Muhammad Char Yaar". I had to pull over and let the CD finish. It was 1989 i think, and this was my first time hearing him.

Most of NFAK recordings that are famous or available in the West (heck, even in Pakistan) are in Urdu. Yes his more secular stuff in Punjabi is a whole cut above.

Posted by: doosra at August 15, 2008 01:59 PM