The Pythagorean Comma

One of the advantages of learning music on a piano as opposed to other instruments is the amount of music theory one picks up without realizing it.

However, one of the disadvantages of learning music on a piano is that one never learns that most of that theory is arbitrary and, depending on how you look at it, wrong.

I’ve been a pianist since I was 10, but have not played for a very long time. During the time I played, I had a habit of regarding the music I was producing more as a by-product that would tell me if I had moved my fingers correctly than as an artistic experience. Obviously, I have
been moved by the artistry of music — but not when I was producing it. If I was producing the music, I was obsessed almost entirely with doing it correctly. It was an experience of great stress and only rarely great beauty. The best I could hope for when I was on that stage was to not fail.

Only lately have I begun to try to combine the two experiences of music: rigor and beauty. And hence only lately have I begun to actually think about and investigate music as a mathematical, theoretical thing instead of simply either an experience of emotional transport (when I wasn’t responsible for it) or an experience of great stress to get it right (when I was).

One of the most interesting artifacts that I’ve run into in this new fascination is something that I was permitted to remain ignorant of for decades as a pianist, but which any guitarist or violinist probably knows like the back of their hand: the Pythagorean comma, the discrepancy between twelve perfect fifths and seven perfect octaves.

It’s not mysterious or mystical to me since I’ve forgotten more hard mathematics than most people have ever learned, and this truly is dirt-simple. It’s simply the difference between iterating a 1:2 ratio seven times versus a 2:3 ratio twelve times. *shrug* Easy as pie. Since these numbers don’t work out exactly, if you hop up by perfect fifths twelve times, you don’t wind up exactly seven octaves away from where you started. You overshoot by a tiny bit.

According to the numbers, that makes perfect sen– wait a minute.

What about the piano keyboard? The circle of fifths closes.

Except it doesn’t. Or rather, it closes on a piano only because each fifth has been flattened by just a bit, enough to allow the dissonance to ride under the radar and allow twelve not-quite-perfect fifths to fit neatly into seven octaves.

And the only reason a piano can force this is because each note has its own set of strings (a “choir,” as it’s called) that is dedicated to producing only that note. Unlike a guitar, where each string must do multiple duty to produce many notes as it is “cut” by a finger on the fretboard, a piano produces each note independently of all others.

I got a lot of music theory learning to play a piano, but the guitarists and players of other stringed instruments picked up a lot more acoustic physics. I recall reading an interview with rock guitarist Eddie Van Halen wherein he bemoaned the strangeness of the guitar, where tuning perfectly toward one consonance will knock another out of whack. In reality, it’s the piano that’s strange, where each note is slightly out of tune by exactly the same amount, and only because Western music demands that twelve fifths equal seven octaves, by hook or by crook. Bringing together one instrument with arbitrary tuning (the piano) and another that is constrained to obey the laws of acoustics (the guitar, violin, or any wind instrument) has, since the dawn of humanity, been a dicey proposition.

We expect to be able to transpose any piece of music to any other key with impunity, and we expect to be able to produce the same music (more or less) on instruments that operate in vastly different ways. Over the history of Western music, a menagerie of solutions has been developed by dedicated musicians to make this possible by addressing the problem of the Pythagorean comma, only one of which is the current “equally out of whack everywhere” solution called equal temperament. Meantone temperament, just temperament, well temperament … all are attempts to address this oddity in the mathematics which is inherently unsolvable. One can no more force twelve perfect fifths to fit into seven octaves than one can force pi to be a rational number.

At any rate, I’ve become fascinated with this new discovery of mine and have ordered several books from Amazon.com to learn more about this topic — all of which began with my diving into “A History of Western Music” and realizing that, for some murky reason, the authors seemed to imply odd things now and again, like G# and Ab might not be the same note.

This is why the best rock musicians start out in classical and jazz.

Enjoy:

Jazz/Rock drummer Steve Smith having a bit of fun!

:-)

“The Castrati in Opera” by Angus Heriot

This is a good enough book for anyone interested in early and Baroque opera and the florid music (and people) involved in it, but it’s a bit strange at times. It does include a useful list of the most well-known castrati and functions nicely as a quick catalogue of the better known names, along with their birth and death dates, places of birth and activity, and best known operas and roles, which does earn the book a spot on the shelf of any early opera devotee. But there are definitely some oddities that render it much less useful than Barbier’s later book.

The first thing that struck me is the familiar highbrow habit of refusing to translate foreign passages. For me, this was not a problem, but it always strikes me as a bit elitist. Contemporary books in this vein, that quote extensively from foreign sources, usually include translations alongside the original language in order to expand the reach of the book for any interested amateurs.

The second thing that struck me, and surprisingly noticeably, was the way the man never lost an opportunity to describe women singers in terms of contempt. Even when two men — a castrati and a well-known impresario, for example — were revealed as having acted like asses to one another, Heriot always made sure to interject a completely irrelevant comment to the effect that some imbecilic diva, surely above her station, acted in a much worse manner. Such examples are not always entirely out of place, but Heriot’s insistence on ensuring that the reader remains aware at all times that women singers were never as good as their castrated contemporaries and were always far more unstable makes one wonder.

In a modern book, such an attitude would stand out like the proverbial sore thumb, and this book was written in the 1950s, where such attitudes were taken more as a matter of course. Yet, in a book that purports to be an authority on male performers with feminine qualities, such undisguised contempt for the feminine does render Heriot’s judgments suspect. He spares no opportunity to print the most slanderous reviews of Cuzzoni and Bordoni’s performances and personalities, for example — and yet both women were deified in their lifetimes and the target of much ink spilled in their praise. I doubt they and the other prime donne of their times were quite the shrill harpies of pantomime comedy that Heriot would believe.

Now, I’m not incapable of rolling my eyes and moving on, and an occasional foolish remark can be overlooked. Unfortunately, Heriot’s remarks are frequent and predictable enough to become tedious. One settles into the book repeatedly only to be constantly brought up short by some shaded, Freudian agenda on the part of the author.

Again, as a reference book on the best known of the castrati, “The Castrati in Opera” does have a place on the bookshelf of the opera lover. However, the same person would probably find Barbier’s book a superior choice for its less oddly canted tone, greater psychological understanding of the castrati themselves, better treatment of the Neapolitan conservatories and understanding of the physiological results of castration, more thorough description of the history of opera up to that point, and greater awareness of both the features and faults of these extraordinary men.

Tamerlano tix acquired!

Tamerlano with Bejun Mehta, Sarah Coburn and the indefatigable Placido Domingo

Now, it’s time to hop back to Petrucci and print/bind the score, and grab a good version from Amazon.com to get up to speed on it. There’s one that I’ve found with Michael Chance, who is a bit flutey and definitely has that falsettist “buzz” to his voice but is a wonderfully creative singer who can have fun with a da capo, so I imagine it will be a good listen. I’ve also found another version with Domingo, but I’d rather be surprised by him since he’s quite good. Mehta, from what little I’ve heard of him, is glittering in vocal quality but occasionally goes off the center of the note by a shade, so I’m not sure what to expect. I may find him extremely frustrating, or I may come away very impressed.