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

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