Bach and Equal Temperament

Bach’s influence on keyboard instruments is unrivaled. But before getting into his works for solo keyboard, it is important to talk about Equal Temperament, which solved for all time the problem of the Pythagorean Comma, which we discussed at the beginning of this blog. To recap, Western music was based on a natural scale with each successive note having a 2/3 relationship to the next. But the spacing between each note was slightly off, and increasingly so as you went up the scale. By the time you got to the next octave, the pitch was noticeably off. The historical compromise had been to discard 5 of the 12 tones. To use more tones, composers developed keys—but each key required a unique tuning. So switching between keys became difficult, if not impossible in many cases. Equal temperament was developed to enable instruments to play in all keys in a single, uniform, tuning. To do so required music to make a fundamental shift from nature—correcting the Pythagorean Comma so that each note was the exact same distance from the next changed the fundamental mathematics that linked music to the natural world of sound. As Howard Goodell is fond of saying, due to Equal Temperment, every note you hear today is a monstrous lie. Here, again, is the math: https://pages.mtu.edu/~suits/scales.html

Bach wrote the two volumes of the Well-Tempered Klavier to prove that a single keyboard could play all the tones in each key without being re-tuned. Bach did not write this to perform; he may not even have written it for teaching his students. It is just possible that Bach wrote this as the proof of Equal Temperament. But, oh, what a proof! The reclusive pianist Glenn Gould brought the Well-Tempered Klavier into the concert hall and his recording of them became one of the true landmarks of recorded music. Along with his classic recording of the Goldberg Variations, Gould is the modern master of Bach. Only one problem—these pieces were not composed for the piano.

In Bach’s day, the dominant form of keyboard was the harpsichord, where strings are plucked rather than struck. Harpsichords, however beautiful, cannot vary their volume. Every note, no matter how forcefully struck, will be just as loud as any other. As noted earlier in this blog, a harpsichord maker had invented a keyboard that could play both softly (piano, in Italian) and loudly (forte, in Italian). So was born the “fortepiano”, the forerunner of our modern piano.

These new fortepianos made their way to Germany and, naturally, one was presented to Bach. If they were looking for an endorsement, however, they went away disappointed. Bach was not impressed. Thus, to really understand these pieces, you must hear them on the harpsichord. Here, we have the Kenneth Gilbert recording, who plays a harpsichord from 1671! From the very lengthy set across two books, I have selected Fugue No. 4 in C-Sharp Minor from the first book. This fugue has the familiar three subjects, one inverted. An inverted subject “retains the rhythm and the basic contour of the material, but flips it upside down: where the original moves up, the inversion moves down just as down changes to up.” It is a mirror image of the original. There is no break in this fugue as the subjects gather one on top of the other. It is a brilliant composition. I’ve also chosen the B minor Fugue from the first book. It is particularly notable as the subject uses all twelve notes in the chromatic scale. I’ve been looking for the earliest example of this, but I believe this is the first time that feat was achieved. Conversation alert for the 20^th century.

J.S. Bach, The Well-Tempered Klavier

Book 1, Fugue No. 4 in C-Sharp Minor (on harpsichord):