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from the department of musical mythology:
Breaking news! (from the middle ages) Music is Math!
posted Feb. 1, 2009
Recently I came across an article in which a mathematics professor who happens to just love the Beatles has decided to use math to ‘decode’ their music. In other words, he is going to deconstruct their songs and find out that math is the reason for their success. Buried in the article, he summarizes excitedly "Music is basically just math."
No—no it ain’t, professor. Sorry. But let me be more eloquent upon the matter at hand.
Imagine if I could copyright that little phrase (hey, the Donald tried to copyright "you're fired" so don't assume too much) and got royalties from everybody who made that observation. Although I would rather people didn't say it at all. I don't think people really know what they are saying.
Suppose you are a person who only knows a little about math, and less about music. Many people use this expression as an unsubstantiated aphorism at parties, without getting at all specific about what they mean. If you can memorize this sentence as is, and not go into detail you can apparently appear to know a lot without actually knowing anything. Taking one huge field of human endeavor and equating it with another one seems like exactly the kind of sweeping pronouncement you would make if you had so much knowledge in your head you could authoritatively make these large-scale comparisons. Because you have such command of the mass of details, you can see things whole. Only I’ve never heard anyone in my personal experience give any reasons for their assertion other than that both fields use numbers.
However, this professor apparently does know quite a bit about math, since he chairs the math department at a university in Nova Scotia. His musical knowledge is suspect, although the woman who wrote the article probably didn’t help his cause any since she apparently knows about as much as most journalists know about music, which is next to nothing. Certain obvious errors in the article led me to that idea, and I’ll point them out later.
Now I have no problem with the idea that music involves math in various ways we already know about, and that the two disciplines might be related in interesting ways we haven’t yet thought of—some composers have used this idea to write music, though I don’t think it is always successful. But to simply and baldly state that music IS math, or music is JUST math, as if knowing math were the key to knowing music, presents more than a few problems.
For one thing, if they are identical, why not eliminate one of them, or make the head of the math department chair the music department as well?
I probably shouldn’t try to break up this convivial marriage, since there may be some benefits for both sides. For one thing, math has always been considered an important part of the curriculum in school. If your child is struggling in math, he’d better get a tutor. Nobody says, 'just have fun honey, and if you aren’t having fun you don’t have to do it.' He has to do it. For many years, like it or not. Music, on the other hand, is often regarded as a fun but essentially frivolous extra. It tends to get its funding cut quite a lot when budgets get tight and kids quit studying it without penalty when they get too busy. Now that we know it is just another form of math, it is suddenly important again. It matters. Somebody tell the Illinois legislature before funding for the Arts Council gets cut out of this year’s budget! whooops! too late!
If we’d known this in the 80s, when, every time a new survey came out showing Japanese kids were beating us in math, instead of just requiring an extra year of math (more is better) and ignoring music (unlike the Japanese) we might have taken a different tack. Alas.
But there are other benefits for aligning the two—for the mathematicians. For one thing, if music can be reduced to math (like a complex fraction), than math is the supreme discipline. It largely was, in the Middle Ages, when philosophers saw math, by which they usually meant simple ratios and convivial integers, strewn throughout the universe. Why stop with music? The connection between math and the sciences is obvious, but what about the humanities? Does the effective use of language involve ratios in the structure of sentences or the rhythm of syllables? Than English is math. Are there patterns in rhyme and meter? Poetry is definitely math. What about philosophy? If it is well ordered, it too, like math, must be based on just proportions. Philosophy is math. What about sports? Can we track the arc of a football through the air using charts and graphs, collecting data? Football is math.
Now if we stopped by saying there are certain things in these activities that can involve mathematical operations of one kind of another, we’d have something interesting. If we go a step farther and make a metaphor out of the conflation, then we have domination by math. Or, as the Beatles would put it, all ya need is math. Everything else just wants to be math. Math is math. How gratifying. (This kind of departmental snobbism reminds me of the quote by somebody or other that all the arts aspire to the condition of music. Yeah, we rock! You other guys are just pale imitations!)
On the other hand, there is something kind of sad about aligning math with things like music and sports. Even if these are not considered important the way math is, people are not likely to line up for a math lecture, and they will definitely go to a football game. Another part of the human parade might gain in respectability by close contact with math, but math gets to be fun for people who are not the punching bags of the rest of us—those ‘math nerds.’ It is application to things that more people care about that gets math its notice. Math isn’t just a bully, trying to one-up everything, it is a lonely fellow saying, "I want to be fun, too!"
This kind of intra-disciplinary rivalry is as old as the hills. Musicians, particularly in the Baroque era, have tried to tie music to rhetoric and geometry and logic and all sorts of things to deflect the idea that it was an idle activity, but something that very bright people can productively engage in. Seeing a mathematician come calling now is just an old reverse on an old marriage of convenience. When it is convenient.
But it’s really not all that convenient in this context, because, as I said, the matter is so often oversimplified. The professor’s claims in this article are often based on faulty musical knowledge or exaggerated notions about his favorite band’s tendencies in musical creation. But I’ll try to keep in mind that these findings were also filtered through the mind of a journalist. Even though the article appeared in no less than the Wall Street Journal, it is pretty clear that this particular journalist is not a musician, either. The professor, who can play the guitar, and apparently write songs, does have some working knowledge of music. But not a lot of a sense of musical history.
One reason this is evident is in a video that accompanies the article where Mr. Brown points out some of the features that make the Beatles’ songs great works of genius. One of these occurs at the beginning of the song "I Wanna hold your hand." Mr. Brown has observed that there are three notes that occur before the downbeat of the first measure. He illustrates this using a puzzle, and referencing the "principle of the hidden assumption." In the puzzle, which has nine dots, the only way to connect all of the dots using only four lines is to extend some of the lines outside the space of the box. Most people assume that the lines have to remain within the space taken up by the dots and thus cannot solve the puzzle. Our professor cites this as evidence that the Beatles were thinking outside the box when they came up with that introduction.
Actually, we have a term for those three extra notes the Beetles chose as the beginning of their song. It is called a pickup—a note or notes occurring before the start of the first full measure of a piece. The reason we have a term for this is that musicians have done it before—lots and lots of times. It is standard issue; it has been for a long time. Composers great and small have used pickups to begin pieces long before the Beetles thought of it, and it is hardly a guarantor of great music. Lots of mediocre compositions begin with pickups, as well as employing the tricks I'll describe below. Professor Brown does not seem to realize what an ordinary thing this is. Apparently the "hidden assumption" is his—that it is something special.
Besides, if this is a mathematical principle it is sure a strange one. Starting in the middle of a measure means you have a stray fraction of time that doesn’t fit into your convenient containers of rhythm. Your piece is now 44 and 2/3 measures long, say. Now ‘mathematical’ minds fixed this problem years ago by a strange rule that the last measure of any piece with a pickup must contain whatever portion of beats was missing out of the first measure, thus completing the opening partial measure at the end of the piece. That way we’d have no incomplete measures (even if it took eight pages to finish the first one!). Being able to contain your data in convenient whole number packets has frequently been a hallmark of minds looking for math in music, or the universe at large.
I don’t know what makes that any more mathematical. Is the number 44 more ‘mathematical’ than the number 44 2/3? If you are looking for the presence of numbers you can find them anywhere. If you are looking for patterns, and Mr. Brown thinks he is (he defines math as the ‘systematic study of patterns’), are you only looking for simple ones? It is a good thing he is restricting himself to the music of the Beetles. The music of Chopin might come across like differential calculus, at least. (Not that he couldn't handle that)
Another item of interest to our professor/hero is in a kind of syncopation where groups of four notes in a beat are regrouped by accenting every third one. This produces a situation where the accented notes are not on the beat for a while and is a neat, somewhat disorienting effect. For some reason, he thinks that the reason this works is because 3 and 4 are common multiples of 12, even though there were 16 eighth notes needed to fill out in the two measures in his example. The effect occurs during that famous lick in "Here Comes the Sun" and four more (unaccented) notes are required to restore balance to the metric order.
This is also a very old trick, and I agree with him that it is a neat thing for a composer to do. Brown points out that three and four are factors of twelve, but I don't see how having a group of twelve (which translates into a measure and a half) will make the effect more magical. For instance, if you’d rather do this in larger groups, the effect can be kept up for several more measures. It is true if you want things to come out evenly, getting the accents to line up again, you will need a group of 12 notes for this particular kind of effect, but many times composers do not want things to be even. Sometimes when you divide numbers you wind up with a remainder, no?
Apparently not. This is one of the old convictions I’ve had about people who want music to be math: that they are oversimplifying both music and math. Math gets to be messy and complex—but not in this scheme. Instead, Mr. Brown asserts that math is all about "a systematic study of patterns" and music, he assures us, is all about following patterns.
I happen to think that music is as much about breaking out of patterns as it is about establishing them. The best music I know is full of surprises. Some of them seem in retrospect to be well placed, and some succeed on their own terms. One ‘mathematical trick’ that Brown didn’t invoke was the golden section—the idea that 61.6% of the way into a musical composition is the best place to put the climax. An article I came across recently had the decency to admit that some of the musical masterpieces they tested didn’t adhere to this little formula. Some did. Some were intentionally crafted to do so by 20th century composers who knew what the golden mean was and planned their compositions accordingly. That doesn’t always yield great music, though it might help—or hinder. Still, you can find several articles on the internet that swear up and down that the principle of the golden section must be observed in all great music, and there can be no exceptions. One of those articles wondered whether Mozart knew about this golden section or if his "great musical instincts" made use of it without knowing (which I suppose would make it better in the mythology of so many). Unfortunately it turns out many of his pieces didn’t fit this scheme anyway. But you’ll never convince a starry-eyed mathematician who is convinced that it is this kind of predictable pattern that unerringly leads to the most satisfying music.
And that is what Professor Brown is, really, a dewy romantic, a kid in the blushes of first love. Every typical musical practice is evidence of something special, and it is evidence of a beautiful order in things. Everything that does not fit the pattern or patterns doesn’t count. His sweetheart has a nose! And eyebrows! And the most beautiful teeth of anyone in the world! Sigh. Who would want to take that away? On the video he comes across with such enthusiasm and joy--obviously here is a guy who loves what he is doing. So it feels a little mean challenging him. But music has had to be emancipated from so many things, so many boxes that it is put into over the centuries by various people who are usually not musicians, and this is one of them.
I don’t want to give the impression that everything our professor does is simple—he is after all a professor of higher math. But the methodology often seems that way. One of his projects is to determine whether Lennon or McCartney wrote ‘In My Life’ since they both claimed the lion’s share of its authorship. From the article it sounds like Brown is going to plug the chord progressions into a computer model of the known output of both men to see which is best conforms to. Not only does this not sound like it needs a computer to check on, he seems to be ignoring things like rhythmic patterns and melody and song construction. But it might be the journalist crunching the stuff out of it who is to blame. She’s on a deadline, right?
Speaking of journalistic gee-whiz, I’m going to take a little side trip to discuss Brown’s take on the guitar solo in a Hard Day’s Night, which is one of those places I get the impression a lot of the people involved don’t know what they are saying.
Here’s the journalist: "Other problems have since yielded to Mr. Brown's mathematics. Fans have always marveled at Mr. Harrison's guitar solo in 'A Hard Day's Night,' a rapid-fire sequence of 1/16th notes, accompanied on piano, that seemed to require superhuman dexterity."
Ok. two problems here. One is that 16th notes are not fractions—they are simply 16th notes, not 1/16th of anything. They may have originated that way back in the middle ages when thinkers were trying to connect music, math, and the cosmos, and also trying to find a why to explain these new, fast little notes. One whole note is 16 times as long as a 16th note, as it happens. But unless you are always writing your piece in 4/4 time (a meter signature which is not reducible to 1, since it is not a fraction, but rather two numbers, top and bottom, which mean different things) you can’t refer to your 16th note as being a fraction of anything since the entirety of which it is a fraction changes depending on the musical context. (Whew!) For instance, a waltz in ¾ time does not mean each whole measure is only three-quarters full. There is only room for 12 16th notes in a complete measure of three-four time, and in math that does not equal one whole, but in music it does. Similarly in 6/4 time, a whole note is only 2/3 of a measure. Time signatures like ¾ time and 6/4 time aren’t fractions. They only look like fractions when you have to write them in an article like this one, and then only for convenience’s sake. In an actual piece of music, there is no line between the two numbers. One of the first things we had to get kids who didn’t know their theory in the remedial class at music school to do was to stop writing time signatures as fractions.
If your eyes glazed over during that paragraph the gist was that somebody made a goof that shows that they don’t know basic musical terminology.
That little error is probably the journalist’s fault. She heard the words "sixteenth note" and assumed it had to be a fraction. The other one has more to do with the fans. "a rapid-fire sequence…that seemed to require superhuman dexterity." You’re kidding, right? I just tried it out on the piano. Nothing remotely hard about it. It helps if you are a pianist, of course. Maybe the guy on the recording wasn’t, because apparently, they had to record the thing at half-speed and then speed up the tape, whereas I could have gone in and sight-read it at full speed without breaking a sweat. And it’s not because I’m the only pianist on earth that can play like that. There are thousands of pianists around the world who could do the same thing. Still, it sounds fast to most people, and that kind of thing always drops jaws. Piano showmen make whole careers out of doing things that people think are hard and making sure to tell them how hard it is.
Anyway, this is the sort of thing that makes musicians who really know what they are doing chuckle when they read about it. And also frustrated, because they realize how simple it is to get people to fall all over musicians who can barely play or sing or have much of a musical imagination and how difficult it is for the more talented ones to get the attention of that same audience. But before you go hanging me in effigy, I think the Beatles did some pretty good stuff, at least later on. All I said was that guitar/piano solo is not exactly ‘superhuman.’ It still works, though. But it is not revolutionary.
Brown gets to try out his really fancy equipment and fine sense of higher math when he tries to analyze the famous first chord of "A Hard Day’s Night." His doings are probably out of my league at this point, though he made it sound simple enough. By analyzing various groups of harmonic frequencies he determined who was playing what. The explanation in the article is full of holes but I’ll leave him to deal with this on his own. To me the thing just sounds like a guitarist striking all the open strings of his 6-string guitar with some distortion. Disclosure: I don’t get too worked up about the Petrushka chord or the Tristan chord, either. In doing some research on this phenomenon, though, I found that like the famous chords from classical literature mentioned above, the ‘hard day’s night’ chord has gotten a lot of complicated explanations from theorists who like to explain every note as a stroke of genius, and, quite frankly, seriously overcomplicate the construction of the melody and the choice of chords to boot. Of course, if by the phrase music is math you mean music is 4th grade math a lot of bright minds are going to get bored, so it is fascinating to see in the age of ‘intelligent design’ that musical phenomenon are now being made the subjects of bewildering complexity and ferocious arguments about those theories. After a while, though, I just want to go practice. Despite having been a theory professor, there is a point when I weary of all this. I want to go compose something and stop arguing about what someone else’s intentions or instincts were. If I spent all day explaining in detail the construction of someone else’s song that would make me a theorist. It is obvious from what is on the web that we don’t have a shortage. We don’t have a shortage of people telling them to just shut up and enjoy the music either. Is there any room in between?
The Beetles had a quote about people who were analyzing the music to death. I used to think it displayed an instinctive musician’s disdain for musical knowledge, but given the kind of ridiculous lengths people have gone to in justifying their favorite songs by making them academically acceptable, I now have sympathy for it as well. Knowing they couldn’t do anything about the wave of critical analysis at the molecular level of every song, they took refuge in sarcasm:
"Do you remember when everyone began analyzing Beatles songs-I don't think I ever understood what some of them were supposed to be about."
That would have packed a little more punch if it had been said by John Lennon or Paul McCartney, rather than Ringo Starr, But here's Paul with a similarly frustrating quote for people who want in the deep secrets of the creative process:
"There are two things that John and I do when we write a song. One is we sit down. Then we write the song."
Whether there is bemusement or resignation in those words, it won’t stop the analyses, or the argument. And yet Brown thinks he’s found a way. "You can’t argue with math," he says. Not his math, anyway.
And that is also the point. When you say music is math you generally assume that math is something and music is something and that they are equal. You basically take some phrases from the language of math and apply them to some phrases from the language of music. It yields interesting insights, some of the time, as interdisciplinary projects tend to do. But sometimes it is just plain dumb. People who are idol-worshipped with the kind of fervor that the Beetles generate bring that out of people, including some otherwise smart ones.
One of the things that is just plain dumb is the unmoving conviction of many of the people involved in this millennia-long music-is-math project that their ideas are the right ideas and there should be no further argument. That is supposed to be the beauty of math. It can't be interpreted differently by different people. And then, when you aren't looking, a mathematician inserts his own ideas about how these things work and says they too are just math. All is either true or it is false. Just like religion. And we all agree on that subject too, don't we? No matter what a mathematician or anyone else thinks about the elegance of their theory getting everyone to buy into a universal field theory of music isn’t going to happen. Arguments will continue to rage around us, fascinating, productive, and just dumb. There isn’t any point in trying to quell them. Some people need something to do. I, on the other hand, have to get to the grocery store. And then go practice.
So I’m under no illusions (well, few, anyway): This little critique of mine won’t stem the torrent of people who want to think of music as just a pleasant play of simple numbers, or patterns, or 'math.' Those who want to see evidence that everything fits together a certain way will simply ignore evidence that runs contrary, or will adapt contradictory reasons for appreciating it. Some will do just the opposite and show the complex genius that lurks behind every C-Major chord the master chose to put on paper. (If Tolstoy wrote it, is the word "the" still just a definite article? Or is it groundbreaking?) Some will have it both ways. For Professor Brown, every set of jostling cross-rhythms or standard chord progressions that shows a simple numerical congruence will be the music of math and harmony. Every instance where things don’t quite add up will be ‘thinking outside the box.’ The box is wonderful, and not being in it is wonderful. So much of what is normal in music is glorious and what is abnormal is also glorious. And you know what? On that we agree.