Okay, people, here’s a New Year’s resolution for you: V - I! V7 effing I! Isn’t it satisfying when things do resolve? And don’t you hate that other kind of New Year’s resolution, the kind which, in hindsight, turns out to have been nothing but a recipe for failure and a mandate to feel bad about yourself? Or maybe you are someone who actually keeps your New Year’s resolutions. If that’s the case, I don’t believe we’ve met. Hi, I’m Rachel.
Here’s another New Year’s resolution: IV - I. That one is more gentle, more subtle, softer. Maybe it shouldn’t be a New Year’s resolution, but a resolution for some other, more understated day of the year. Not for a milestone, not your birthday—unless your hope is to treat that day as you would any other day, with a minimum of fuss. Still, that is a little bit weird, because you don’t hear IV - I every day, at least not in most of the pieces you play most of the time—whereas that über-satisfying, turning-of-the-new-century-worthy V - I is everywhere! What do you think, about celebrating the new year as opposed to thinking to oneself, today is today, yesterday was today yesterday, tomorrow will be today tomorrow—what’s the difference anyhow? Just another today. I don’t know! But I did like how at my grandmother’s retirement community, where (for all their spunk) those octo- and nonagenarians did not wish to stay up til midnight, they celebrated “New Year’s in Reykjavik” from North Carolina, simply by turning the clocks ahead.
What does resolution mean, anyway? In music we talk about resolutions a lot… we fuss over them (like birthdays!), give them lots of special care. And they deserve it! What is it about resolutions? First of all: what we call resolutions are moments where musical tension is released. And in most of the music that most of us play, most of the time: that means harmonic tension, and THAT means the use of certain pitches and combinations of pitches, as opposed to other pitches or combinations of pitches. And that means: sensitivity to the different notes of the scale. Off at music camp one summer, my cabin-mates and I dreamed up what we thought was the most marvelous prank. It involved sneaking over to our rival cabin before the break of dawn, instruments in hand (I played the violin too in those days; Ingrid was there, and as I recall there was at least one trombone student among us), and we played absolutely as loud as we could: DO… RE… MI…. FA…. SOL….. LA….. TI……….. and then running back to our own cabin and jumping back into our beds. We knew that our rivals (you know, those other insufferable 13 year olds) would never be able to go back to sleep. Ha, ha, ha, very funny, says the tired and grouchy grownup as she thinks back on this. But the point is, musical tension exists, and it acts on us, whether we know it or want it to or believe in it or not! That ti really, really wants its do.
And what do the different roles of the different notes of the scale really consist of? MATH. Let’s dip our toes into this, just ever so gingerly, so as to be able to make some New Year’s resolutions we can keep. A pitch is a certain frequency, a certain number of vibrations per second (hertz). Different pitches have different frequencies. So far, so good.
What is interesting, musically, is the relationship between these pitches: their relative frequencies. Different numbers have different relationships. The relationship between 3 and 9 is one thing (or 4 and 8, or 10 and 2); the relationship between 4 and 11, (or 6 and 7, or 23 and 5) is something very different. We don’t have to get too math-y to see that, right?
An interval is the distance between two pitches, and what that distance is determines the relationship between those two notes. Whether they blend, balance against one another, or antagonize one another. Whether we want them to resolve or not! What we call perfect intervals (we might find them to sound clean, pure, empty, hollow, strong…) have the closest mathematical relationship between their frequencies. Consonant intervals (warm, blended, friendly, familiar, textured…) are the next most closely related. Dissonant intervals (ones that carry tension!) have ratios between their two frequencies that are more difficult to reconcile.
To be more specific: the ratios of the frequencies of the two notes in perfect intervals are: 1:1 (unison); 2:1 (octave); 3:2 (perfect 5th); and 4:3 (perfect 4th). Isn’t that nice? For consonant intervals, we have 5:4 (major 3rd); 6:5 (minor third)….. Dissonant intervals: the major second has a ratio of 9:8; the minor second is 16:15…. For the tritone, it’s 45:32!
What is happening, as we move from perfect and consonant intervals to dissonant ones, is that these overall waveforms are getting much more complex and harder for our brains to analyze. We don’t like that! When we’re presented with sounds like this, what do we want? We want these sounds to RESOLVE, or to settle back into relationships we can more easily make sense of. We want simple math, as manifested in sound. We want larger numbers to collapse into smaller ones. We want to move in the direction of simplicity.
It seems we don’t want things to be too simple, though! Resolutions tend to satisfy us when they bring us to the tonic triad, whose intervallic relationships (encompassing now three different frequencies) are simple enough to be understandable, but still offer some level of mathematical complexity or interest. Maybe we are frightened of resolutions that get too close to the mark! It would be unsettling, for example, if a Brahms symphony, after all that, concluded with an open fifth—or (even more confronting to our psyche) a unison. Wouldn’t it? We seem to want closure, but also to leave ourselves something to chew on. Our tonic triad gives us that needed resolution, but still carries just enough tension & activity in its mathematical relationships to remind us that we’re still alive.
A certain frequency is established as tonic, as home. It doesn’t matter so much what note it is. And it doesn’t matter whether our “A” is 440 hz, or 415 as was used in the Baroque period (today we call that A flat), or the 432 for which Verdi passionately lobbied the Italian Parliament. (They went with 440. The whole episode was not without its bitter political machinations. Are American opera singers suffering from pitch too high? Maybe there is a comparable bill languishing on Mitch McConnell’s desk right now, but somehow I doubt it.) Just as it doesn’t matter whether we celebrate New Year’s in North Carolina or Iceland, it doesn’t matter what frequency we decide is “home.” We set the clock, or the tonic, somewhere, and we relate to everything from that.
We understand this home note to be free of tension. The chord built above that note we know as our harmonic home. Then, because our brains are in the business of relating things to other things, every other pitch of the scale relates to that tonic, and creates a particular interval with it. Chords built upon all these different notes of the scale take on complex characters, created through all the relationships of the component pitches not just to each other, but to the tonic. Individual pitches resolve at a melodic level (move towards do, or towards the frequency of least mathematical tension against the frequency of do). Intervals resolve, from dissonant ones towards consonant or perfect ones—collapsing the frequency ratio between the two pitches to something smaller. Whole chord structures make their way inexorably towards the tonic. This movement towards home: these are the kinds of resolutions I am interested in today! Not I Will Take 3,650,000 Steps in 2022, or No Cookies for a Year!
I’m fascinated that in musical cultures all around the world, including many that divide the octave in completely different ways than we do in the West, there is a primary role for the sound we call the “perfect fifth.” We can feel the special quality of this sound in our bones. It’s not the invention of some dead, white male in the 18th century—it is math, it is nature. The perfect fifth outlines that tonic triad, to which we return over and over and over again (as long as we all shall live); it generates all those ubiquitous V - I’s (be they triumphant or tragic, simple or grandiose, elegant or tacky…), even as it serves also as the constant drone background layer on the tanpura in Indian classical music, or defines the tuning of the Chinese erhu, to name just a few non-Western examples. Closer to the heart of things, in the resolution department, we find only the octave (two different frequencies, but mathematically so closely related in the ratio of their wavelengths that we perceive them as “the same note”)…. and the unison. We have come down to one note. Dear readers, we have now RESOLVED—not to do anything, not anything at all! We’ve just resolved, and can now slip into silence for a while.
But don’t worry! We’ll start practicing again soon. Happy new year, dear readers! And happy practicing, too, and thanks for reading. If you enjoy Pride & Practicing, please share it with your friends.
Thank you for such a beautiful and perfectly (!) illuminating article, Rachel! I will distill some of it for my studio's group theory classes!
What a great article Rachel. It's a different perspective that helps clarify some of my confusion over intervals and resolutions. I'll revisit this again and again.