by Adam Johnston
It was with the Vienna Boys Choir that I first sang Dona Nobis Pacem — Grant Us Peace.
Okay, I was never really a member, but I did sing with the Vienna Boys Choir. We were in the same concert hall and we were directed to sing at the same time. It just so happened that I was shoulder to shoulder with the boys of my own elementary school choir, and the Austrians were on the stage. We sang together, as directed, and it was beautiful. It was a big deal at the time. It didn’t create world peace, obviously, but it didn’t hurt, either.
I’ve been thinking about that canon lately. At Christmas, especially, I find myself poking it out of the keys of the piano, the pedal heavy on sustain. The tune, the progression of the chords, the simplicity of the sequences of notes, it all draws me.
Music is one of those things I try to teach in classes. “The Music of Physics,” I call it, rather than the converse, not simply to be witty but to make that point that what we study can be both about the beauty of nature and the nature of beauty. That’s a heavy load to bear, but one I think we regularly try to carry in physics class.
The physics of “good” music is a tricky thing. We would like to believe and even teach that the beauty of the music comes from the relationships between the frequencies. An octave is defined by a factor of two in frequency between two notes. Thirds, fourths, fifths, and all other intervals you hear in chords and scales are each defined by factors, so that each major key may be made up of distinct notes, but the same proportions from one note to the next. When played together, the difference between any two notes becomes another unique “beat frequency.” So, for example, the difference between two notes separated by an octave is the same as the lower note (delta = y-x where y=2x, so delta = 2x-x=x). We think of an octave’s difference as really being the same note simply shifted up or down — so the baritones can sing in unison with the sopranos. On a piano keyboard this just shows up as a different place on the array of keys, but the same note within the pattern of repeating white and black. This is wonderful consequence of this geometry.
What’s really splendid is that this doesn’t have to be. It’s amazing, come to think of it, that this could work at all. Why would two notes with completely different frequencies, ever have any chance to be the “same,” or in “tune” with one another? I suppose the brain hears each note, and hears the difference between these notes, forms the beats in some physiological and mental way. Hearing two notes an octave apart, there’s a pattern that’s detected that creates that beat frequency that’s exactly equal to the base, bass note. Other differences between notes (as well as differences between harmonics heard on a singular note of a singular instrument) that sound good together are created by similar geometry. The difference in frequencies is itself a frequency that is in phase with the notes themselves.
That’s one thing I think about when I’m playing Dona Nobis Pacem on the piano. This chord progression is almost the least complicated one can imagine. A garage band of 17-year-olds could be cranking away on the three chords and we’d scoff or grin at the simplicity of it. And, because the three lines can be sung in unison or in round, they are each modeled after the same pattern. (Pull out your dusty guitar and strum G-D-G-D-C-G-D-G; repeat two more times and you can accompany my colleagues from Vienna.) It’s all so fulfilling that we go to the trouble to re-create it in contraptions like tubas and cellos and pianos. Considering all the design work put into the inner workings of my piano, I realize that these simple patterns are something we are driven to re-create. Not only are the patterns there in nature, but we’re hard-wired and compelled to construct them over and over.
So, I think understand how the difference between two notes produces a beat that itself is in tune with either of the notes; and this geometry of the physics of the music makes it pleasant, interesting, coherent. And then we play with patterns in this, both in the sequences in time as well as the audial space. In class, during that “music of physics” unit, I teach this with a tuning fork, a keyboard, a microphone, and an oscilloscope. I swing a tube over my head to play a series of harmonics that emulates “Taps” and they all laugh at my cleverness, or silliness — or perhaps out of sympathy for the bizarre instructor. I can get an organ tube to resonate with a piece of wire mesh and a blowtorch; and this phenomenon can be used to calculate the temperature of the room. I know where to stroke a violin bow on a piece of aluminum in order to play that octave’s difference, and we can see the standing wave set up by sprinkling some corn meal on the surface of the metal. I’m delighted with how much we can understand, as well as what we can do with it.
But then I throw my hands up and turn up the music. I’ve been known to tear up when, for example, Brandi Carlile sings Hallelujah. When Clapton has his hands on a guitar I understand why anyone called him God, and I believe. When Bach is played on a church organ I forgive him for all the pieces I had to learn in his name (though I still curse him, the asshole, when my fingers return to those pieces). Or when I find the right notes for peace, for prayer, on a piano. My left hand finds a way to march up a scale and meet my right hand in the middle, and I think, maybe, this is what I was meant to learn from it all. The physics of the harmonics and the physicality of sound is all beautiful, but it’s only as beautiful as we choose to make it. Dona Nobis Pacem. It’s stunning, and maybe my own version and belief in a miracle, that such a sentiment can come out of three chords. Grant us peace.