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!