Perfect Harmony?
Pythagorus said, "There is geometry in the humming of the strings ... there is music in the spacing of the spheres." He and Plato are among those who have theorized that the same simple mathematical ratios generating the purest musical harmonies also govern planetary spacing and motion. And that as the planets whirl through their orbit, each makes a sound, these sounds blending to create a gorgeous cosmic harmony.
I'm not sure that I believe in The Creator, in the standard theological sense, but I do take comfort and find beauty in these theories that point to some sort of higher order of things. Now, I'm no expert on solar system harmony, but I can tell you something about the math of music here on earth.
To start off, a little audience participation...Does anyone here have a birthday today? Okay, let's all sing "Happy Birthday" to ____________. Shirley, can you get us started? (Sing.) Thank you.
I'm sorry, was that a little high for some of you? Maybe we should have sung in a different key. Shirley, can you play just the first phrase of Happy Birthday for us in a lower key?
We could have started on any note. We recognize a sequence of notes as a particular melody because of the distances, or intervals, between the successive notes.
Intervals result from frequency ratios. Shirley, can you you play an A, and then play the A above it? That's called an octave. You hear it in the first two notes of "Somewhere Over the Rainbow." The lower note was vibrating at 220 cycles per second, and the upper one at 440. Shirley, let's have a different starting note - and now go up an octave from that. No matter what pitch we start on, the pitch an octave higher will be vibrating at twice the speed - a frequency ratio of 1 to 2.
To take one more example, Shirley, will you please play the first phrase of Twinkle, Twinkle Little Star? The distance between the notes of the first "twinkle" and the second "twinkle" is an interval of a fifth. (Shirley, can you play the two notes together?) The frequency ratio between those pitches is 2 to 3. Again, we can start on any note. (Shirley, can you play just Twinkle Twinkle in a different key?) The frequency ratio stays the same. Each musical interval has a corresponding frequency ratio. The simplest ratios - 1 to 2, 2:3, 3:4, 4:5, and 5:6 - produce the musical intervals that are most harmonious to our ears, the building blocks of our basic chords.
I have to say, my confidence in the coherence of the universe was somewhat shattered when I learned that it's actually not quite that simple. Actually, as soon as we add more than two notes to each octave, the system breaks down. It turns out that frequency ratios that we hear as perfectly-tuned intervals are mathematically inconsistent with each other! The bottom line is that it's impossible to get all of the keys of a piano in tune with each other at once. You have to make room for dissonance someplace.
How this unavoidable dissonance gets handled has changed throughout keyboard history. Apparently, in some of the early tuning methods, some horribly out-of-tune intervals created overtones reminiscent of wolves howling. So, this necessary dissonance has come to be called "the wolf". The question is, then, how do we deal with the wolf?
In the early baroque period, the meantone tuning method was employed. The choice was made to have certain notes maximally in tune with each other. But this meant that the other notes were so out of tune as to be unusable. "The wolf", then, was concentrated into a few notes of each octave. The resulting music was extremely harmonious, but limited. You might think of being able to use only the white keys. If you stuck with the key of C, your music would actually be much more in-tune than on a keyboard of today. But venturing onto the black keys, modulating into another key signature, would create unbearable dissonances.
Compare this with today's equal-tempered method of tuning. In equal-tempered tuning, "the wolf" is evenly scattered throughout all 12 notes of the octave by compromising each interval just a little bit. This makes all notes equally usuable and all key signatures musically equivalent. A practical solution which maximizes versatility. But we have given up the purity of every interval. Our ears are accustomed to these slight imperfections. Nevertheless, we have lost the ability to create, and thus perhaps to discern, the purest of harmonies.
Now, in between the meantone era and the introduction of equal-tempered tuning, a method called well-tempered tuning was utilized. In well-tempered tuning, the wolf is not concentrated into just a few keys, but neither is it equally distributed throughout all of the keys. Well-tempered tuning assigns varying amounts of dissonance to the keys. The result is that all key signatures are usable, but they are no longer equivalent. Each has a distinctive tonal character that contributes to the emotional quality of music played in that key.
With well-tempered tuning, at least in theory, Happy Birthday played in the key of G will feel different than Happy Birthday in the key of A or F#, because of the subtle differences in the relationships between the notes. These affective differences are referred to as "key characters".
Listen to a couple of these rather sophisticated key character descriptions from the early 1800's: F # major: Triumph over difficulty, free sigh of relief uttered when hurdles are surmounted; the echo of a soul which has fiercely struggled and finally conquered lies in all uses of this key.
Compare that with, say, A major:
This key includes declarations of innocent love, satisfaction with one's state of affairs; hope of seeing one's beloved again when parting; youthful cheerfulness and trust in God.
Bach wrote the prelude that Shirley played earlier to complement the specific key character of E-flat minor. One description of E-flat minor reads in part: If ghosts could speak, their speech would approximate this key. Beautiful as it sounded to us just now, please appreciate that Bach intended for it to be played on a well-tempered keyboard, and that something is lost by playing it on an equal-tempered piano.
Well-tempered tuning is remarkable to me in that this fundamental flaw in the system is turned into an asset. The mathematical imperfections are exploited, rather than hidden, to add a dimension to the music. There are numerous practical reasons to use the equal-tempered approach of today. However, musical "special effects" of sorts could be achieved with well-tempered tuning that are no longer possible.
"The wolf" in this musical context represents the unavoidable dissonance resulting from our inability to perfectly tune every note. Life is full of unavoidable dissonance. Might our tuning methods offer us a glimpse of self-awareness regarding how we deal with "the wolf" in our lives?
Recall that in the meantone method, a couple of key signatures were actually far more harmonious than in today's tuning, while the rest were rendered unusable. The strategy? Get rid of the wolf by avoiding it.
My family and friends know that I am a pack rat and a magnet for clutter. Our house always seems to have too much stuff in it. We used to take a "meantone" approach to dealing with our clutter. Before our 3rd child was born, we had a spare bedroom - if you could call it a bedroom. Bascially, all of our mess went in that room. Everything we couldn't part with, but didn't have a place for found its way to that room -- papers, outgrown clothes and toys, idle exercise equipment, hardware store doodads, etc. The rest of the house looked good, but that one room was totally off-limits as living space.
And that's the big downside to the meantone method of tuning - or housekeeping. The "wolf" is highly concentrated. Some utility gets completely sacrificed; you are constrained. And if you are forced to go to that off-limits place, it's really ugly.
Since Kurt was born, we've found another way to handle our mess. It's basically an "equal-tempered" approach, in accordance with today's most accepted tuning method. Instead of throwing everything in one place, we evenly distribute it throughout the house. Now every room is functional, but each contains a pile of stuff. For better or for worse, we have become pretty non-discerning about this. And that's where equal-temperedness leaves you - with a kind of uninspired acceptance of imperfection.
Now the saga of my failure to triumph over clutter doesn't rank up there with problems like war or world hunger, and the best solution is far more obvious: I need to deal with the wolf! Acknowledge it, not shove it in a corner or sprinkle it around, but confront it with intelligence and creativity. This would not only result in a nicer-looking home, but would give the junk a purpose! Photos, recipes, receipts, and children's schoolwork are annoying miscellanea when piled around the house. But most of it has the potential to add to my life in a positive way. This is the beauty of the well-tempered tuning approach: manage the wolf resourcefully and make him work for you, not against you.
How often we fall back on a meantone approach to life, staying out of the wolf's way entirely. Living the meantone way is playing on the white keys, playing it safe. It means avoiding conflict and risk, an intolerance for venturing outside of one's comfort zone.
My son refuses to try new foods. I have been limited for years by my fear of flying. People remain in jobs that are secure, even though they are no longer rewarding. We may dodge former friends with whom we've had a falling out. We can eat our meat and forget about the cow, wear our designer clothes and put the sweatshop out of our minds. How many ways we find to keep our lives simple and our minds uncluttered. It's easier than processing the complications.
And then sometimes we dwell in a state of equal-temperedness. We cope with the wolf not by evading him, but by hiding him just well enough to become undiscerning and complacent. We use the knowledge that perfection is impossible as an excuse to accept the barely okay. We justify unhealthy relationships and situations by minimizing the issues. We compromise too easily. We give up on excellence. We decide that the world is way too flawed for us to try to make a difference. We sell ourselves short when we tolerate what we have the capacity to fix. We do ourselves and others no favors by repressing, rather than addressing, problems. And yet, this is often the path of least resistance.
A well-tempered approach to tuning your life may not be the easiest, but I say it's the most honest and ultimately the most satisfying. For in a well-tempered life, flaws and inconsistencies are turned to your advantage. You're not trying to hide the wolf, or hide from him, but rather to weave him into your life in a way that adds purpose. You may take a risk and in so doing, discover a new passion. You may find that someone who vexes you eventually gives you a precious gift. Your life's work may help to solve the problems of the world, or at least a corner of it. Face challenges head-on. Conflict is creative tension, waiting for your energy. Your limitations are fuel for personal growth.
That musical intervals cannot be made simultaneously perfect to our ears illustrates the global truth that you just can't get everything right at once. The harmonious theories of Pythagorus notwithstanding, paradox, conflict, and error are hard-wired into the universe. Our thoughts, our choices, and our actions represent our attempts to make sense of and optimize our tiny individual pieces of an infinitely complex and imperfect world.
We can limit ourselves by steering clear of the things we don't like or aren't good at or don't want to think about. Or, we can feed mediocrity by masking imperfections and simply resigning ourselves to them.
OR, we can view the world's inadequacies and our own shortcomings not as things to be shunned or obscured, but as substance to be present with. It's all part of the richness of being, providing us with opportunities for reflection and growth. As we head into the new year, let us call upon our personal treasuries of integrity, ingenuity, humor, and love to create from life's dissonances not a perfectly harmonious song, but one of authenticity and meaning.
December 26, 2004