Well? Tempered

Catching up with the Roderick on the Line podcast, episode 187 includes a discussion of the temperament of musical instruments. In particular of the piano, but also a nod by John Roderick to some B-string-flattening guitar folklore espoused by Eddie Van Halen.

In my travels, it has become clear that most ordinary people do not understand what it means for an instrument to be tempered. Anybody with even a pop-culture level of exposure to classical music has probably heard of Bach's "Well Tempered Clavier." But what does it mean to be well-tempered? Is the keyboard in a good mood?

I'm not a music expert, but I did earn a B.A. in Music from San Francisco State University, and I remember enjoying a lesson specifically on the subject of tempering instruments. Here I will try to explain as succinctly as I can what it is and why it's done.

In a nutshell: tempering an instrument involves deliberately detuning some of the notes, so that the whole instrument will sound in tune regardless of the key you are playing in.

To understand why you would need to detune an instrument in order to make it sound more in tune, appreciate that all our western music names and notations are elaborations on some pretty low-level physical phenomena:

1. Sounds are perceived from vibrations in the air.
2. A sound with a consistent frequency is perceived as having a corresponding "pitch."
3. Sounds with frequencies that are relatable by simple ratios are more consonant sounding when heard together than those relatable by complex ratios.

For example, a sound vibrating at 440 Hz is considered in western music to be a standard "A" note. Another sound vibrating at 880 Hz is perceived as very consonant with the first, so much that it is also called A. This simple 2:1 ratio defines what we think of as an "octave" in musical scale. So notes at 220 Hz or 1760 Hz are also considered A, and you will produce (roughly) these frequencies if you locate an A on a piano and play octaves up and down the keyboard.

The rest of the intervals on the piano, related to the same A, will have less consonant frequency ratios, ranging from the still pleasant perfect 5th of A to E, which represents a clean 3:2 frequency ratio, to the jarring minor 2nd from A to Bb, the black key right above A, which is related by a ratio of 1.0595:1. The simpler the ratio of frequencies between two notes, the more their oscillations overlap, causing a sense of harmony and belonging together. The more complex? The more they clash and are perceived as dissonant.

So if you imagine how a piano tuner might set to work perfecting the sound of the piano for songs in A Major, they could start by tuning one A string precisely to 440 Hz, the octave above to 880 Hz, and the perfect 5th E between them to 660 Hz. Then, applying the same process to each of the other keys on the piano, tuning them to the precise ratios for each interval in the chromatic scale, the piano would be perfectly tuned. This approach to tuning is called just intonation, and it very just indeed for A Major (in this case), but not so fair to music written for any other key.

To understand the problem, consider that although every key on the piano is now tuned perfectly to one of the ratios relating it to the pitch called A, it is not necessarily tuned perfectly in relation to some other pitch. For example, the interval from A to C, a minor third, has an interval ratio of 9:8. So on our justly tuned piano, C is (440 * 9/8) or 495 Hz. OK, let's switch to C Major. The interval from C to E, a major third, has a 5:4 ratio, so given a C key tuned to 495 Hz, the major third above it should be (495 * 5/4) or 618.75 Hz. But we already tuned the E key on this piano to be a perfect 5th above A, so it's (440 * 3/2) or 660 Hz. On our justly tuned piano, the E is perfectly tuned for A Major, but a little flat for C Major. There's no getting around this problem. It's pure science, I mean music, I mean math. I mean all of the above.

So the notion of tempering an instrument is to compromise the tuning of the instrument so that, while it's not perfectly tuned in any key, it is at least consistently and predictably tuned for every key. By deviating by a few cents (a standard measure of fractions of a semitone) from perfection for each of the tunings on an instrument, it trades pure tuning for the versatility of being playable in any key.

How much to deviate, and where to focus those deviations, depends on the particular form of tempering being applied. An instrument tuned with equal temperament aims for consistency between the keys, such that the ratio between any key on the piano and its immediate neighbor, is exactly the same. This effectively "divides up the keyboard" in a manner that averages out the rounding errors alluded to before when using perfect ratios for every key.

Was Bach's keyboard in a good mood? Possibly. But the Well Tempered Clavier consists of works written in a variety of keys that would be unsuitable for a keyboard that was not tuned with some amount of temperament. I guess in this sense the title is a celebration of the way tempering opens up the possibilities of composing in multitudes of keys.

I hope this helps you appreciate the meaning of temperament in the context of music. On the other hand, if you've understood it quite well, better than myself, and have corrections or elucidations to offer, please do share your thoughts!

Runner's World

I've described my running habit before, wherein I run "every other day," more or less without fail.

Today Boston was hit with a significant snow storm. Schools are shut down. The city is blanketed in white. It's a sight to behold.

And it's my running day.

That's just how I roll!

Hipster Shit

I finally got around to reading the widely-celebrated New Yorker article about Megan Phelps-Roper and her departure from the Westboro Baptist Church, a hate group based in Topeka, Kansas.

The article is a fascinating profile of a young woman who transitioned from a committed, utterly brainwashed member of the "church," to an empathetic, thoughtful member of society at large. It's a valuable reminder that such transformations are possible even among people you would assume are the least capable of changing.

I was amused by a passage that describes Phelps-Roper's tentative online relationship with an anonymous "C.G.", who sparked her interest in a variety of finer things:

He introduced her [...] to the novels of David Foster Wallace and Marilynne Robinson. “Hipster shit,” Phelps-Roper said. He turned her on to the Field Notes brand of notebooks. He poked fun at the inelegant fonts that Westboro used for its press releases. After C.G. complimented her on her grammar, she took pains to make sure that her tweets were free of clunky text-message abbreviations.

C.G.? The David Foster Wallace and Field Notes references were enough to draw my thoughts to John Gruber of Daring Fireball. But the additional emphasis on typefaces and correct use of grammar? Come on! It's J.G., not C.G. Right?

She and C.G. connected as strongly in person as they had online, and they now live together.

OK, maybe C.G. is not J.G. But I bet C.G. also reads Daring Fireball.

My Running Habit

I was a sprinter as a kid: I counted myself among the fastest runners in my grade-school classes, and would routinely challenge friends to ad hoc dashes across arbitrary stretches of playground grass or asphalt.

As much as I enjoyed pushing my body to its limits for these explosive, 30-second jaunts, the idea of running any longer distance filled me, as it did most kids my age, with utter dread.

In Junior High (around age 13), we were compelled to "run the mile" at least a couple times a year. We anticipated such events for up to a week in advance. "We have to run the mile on Friday," was the empty refrain of doom shared by all except the weird cross-country kids, who seemed to somehow derive pleasure from the challenge.

25 years later, I am no longer a sprinter. Running a mile sounds like a mere warm-up. I typically cover distances that my younger self would have seen as impossible. And somehow, I too derive pleasure from it.

I started flirting with long-distance running in my early 20s. I worked at Apple, and the company fitness center had treadmills with TV. In an era before podcasting had taken off, what better way to pass the time than to watch endless episodes of Judge Judy and reruns of Matlock? I gained an extremely dubious understanding of the legal system while running in place at a very comfortable pace.

I carried the treadmill habit with me after I quit Apple. I ran through my second college career at SFSU, and made a priority of finding a suitable gym every place I lived for many years. I enjoyed the grittiness of the Cambridge YMCA, where I recalled an old iPod's near death experience while running on the treadmill. I upgraded to the lush MIT Zesiger Center, where the impressive olympic swimming pool even got me cross-training for a stretch. In Brooklyn, I stuck to the academic plan and took advantage of Brooklyn College's modest yet reliable athletic center.

By this point I had been running "long distance" for 15 years or more, but had scarcely run a mile off of treadmills. I had come to lean on the predictability and relative ease of treadmill running: it's easy to do in any weather, at any time of day, and requires little motivation beyond stepping onto the platform. You simply set the machine to the speed you think you should run, then do your best to keep up.

Running off-treadmill, as people have more typically run for millennia, was a fairly foreign concept to me until a few years ago. I had occasionally gone for jogs around Boston or Brooklyn, but found it frustrating to keep my pace, to plan the route for the desired distance, and to keep my momentum in urban areas where pedestrians and vehicle traffic were constantly interfering. If I ran 100 times in a year, at least 98 of those times would be on a treadmill and only 1 or 2 in the great outdoors.

That ratio inverted completely when I moved from Brooklyn back to the Boston area, settling in to Arlington, MA. I was too far from my beloved MIT fitness center, and frankly all the local gyms seemed bleak to me. I didn't know what to do, but it was September and the weather was nice. I committed to sucking it up and running outdoors at least until I found a better long-term solution. Surely I would need a gym for treadmill running by the time winter came.

That was the fall of 2011, and I have rarely stepped foot on a treadmill since. The only times that come to mind are when I've been traveling and found for whatever reason that a hotel's fitness center made for a more expeditious workout than venturing out into the surrounding city or town.

I am very lucky to live in a town with a rail trail: a recreational path recovered from former railroad right-of-ways. A typical outdoor run for me consists of a few blocks jog through suburban streets, a few miles run up "the bike path," and a few miles back home. Occasionally I take more adventurous routes, especially if I want to explore an area of town I haven't seen much of, or if I'm trying to diversify the distance or terrain of my workouts.

It turned out I didn't need a gym and I didn't need a treadmill. Not even in the dead of winter. That first season of running outdoors was relatively mild, so I gradually invested in the necessary gear: thermal underwear, gloves, and a beanie cap. In subsequent winters I've made good use of extra-insulated running shirts, even a balaclava! There were days when I ran in 5F weather and practically froze my face off, but I got, and get some kind of perverse thrill running in all weather conditions.

What's kept me going all these years, on treadmills and then on roads and paths, has been the establishment of some kind of rule system, and then sticking to those rules as if they were not negotiable. Eventually, this becomes a habit. Back in the early Apple days, I made a sort of pact with two co-workers to go to the gym nearly every afternoon (yes, the flexible hours were a god-send). More often than not it would be one of us getting motivated, and dragging the other two away from their screens kicking and screaming. But it worked. Committing to the habit together made it easier for all of us to start the habit, but eventually it became easy for each of us to preserve the habit on our own as well.

In more recent years, I've had to piece together new rules and conditions. Especially after switching to outdoor running and inevitably upping my mileage, I struggled with some moderate injuries that threatened to put an end to my running. I countered this by taking a hardline stance that was rigid and unforgiving, but also gave my body time to rest: I must run every other day, but must not run two days in a row. To me it's important in establishing any habit to have a fairly easy to understand system. Will I run today? If I didn't run yesterday, then barring a very good excuse, the answer is yes.

To give the habit some teeth, I have also imposed certain minimums on my running. At first it was just "2 miles or more." Then, as I dabbled in my first efforts at running in official races, I upped my minimum to 5K. Eventually I worked up to a point where I was running at least 5 miles, and for the past several months I am running to a landmark that logs me 6.5 miles in total. I'm free to run longer than the minimum, and in extenuating circumstances I run less, but as a general rule the decisions are all made for me.

Eliminating questions about when and how far I run has made it far easier to keep the habit going. Is it a running day? If so, put on clothes, load up a podcast, and go. Meet the minimum distance, then turn around, or keep going. I still have choices, but they are delightfully few.

When people ask me about starting a running habit of their own, I try to emphasize how gradually you can grow one. Start by jogging around the block once or twice. Set a standard, and then stick to it. If your experience is anything like mine, you'll find that as your body gets used to the standard regimen, it will eventually yearn for something more. Add another block. Make it an even mile. Shoot for 5K. You'll feel better, be stronger, and develop a hopefully lifelong habit that makes it easy to rest assured you are getting enough exercise.