04 December 2009

Glenn Gould - Prelude in C minor, BWV 847

The incomparable Glenn Gould plays a prelude from Das Wohltemperierte Klavier (The Well-Tempered Clavier).

The liner notes of my excellent CBS Masterworks recording explain the significance of Bach's use of the "well-tempered" tuning system in this set:

Johann Sebastian Bach was probably the most important, albeit not the first, avant-garde Baroque musician to recognize immediately the advantages of tempered tuning and the expanded scope of new harmonic frontiers thereby opened to the musical imagination. In 1722, he finished a collection of twenty-four preludes and fugues in which he systematically utilize every single major and minor key, ascending and so forth until reaching B Major and B Minor.[Note 1.]

The well-tempered tuning was an innovation of a 17th century German music theorist, Andreas Werckmeister. He figured out a practical way to tune instruments so that one could transpose any tune into any key. Up until that time, the notes of scales were not evenly spaced, which is to say, the frequency ratio from one note in the scale to the next was not constant. Werckmeister's system defines twelve evenly-spaced chromatic tones in the scale, the frequency of each tone being related to the previous one by a factor of the twelfth root of two.

What's so special about that? Well, for one thing, since every half-tone is available, every tune is entirely transposable; if the key is shifted up or down, all the relative notes can still be played accurately. (In other tuning systems with irregularly spaced notes, shifting the tonic tone of a song could make some of the notes fall noticeably sharp or flat.)

But what is special about the twelfth root of two? This is the neat part! If you construct such a scale in a spreadsheet, you can notice some interesting relationships between the frequencies of the chromatic tones. For instance, since everything is relative to the (tonic) tone I, let's start with 1.000 for it, then increase each subsequent row by the factor 2^(1/12). I've done this below, labeling the notes I, II, III, etc. corresponding to the conventions for do, re, mi, etc.

I 1.000
II 1.122
III 1.260
IV 1.335
V 1.498
VI 1.682
VII 1.888
I 2.000

I remember being stunned when I first did this many years ago. Some of the numbers in the list, particularly the cluster from III through V, absolutely jump off the page! For instance, the ratio of V:I is almost exactly 3:2. Similarly, the ratio of IV:I is very nearly 4:3, and the ratio of III:I is closely approximated by 5:4. In other words, the significant intervals in the scale are related by simple integer ratios, which must account for the pleasing consonance to the human ear. The factor of 2^(1/12) works well because its evenly-spaced tones just happen to fall near these "golden" ratios!

These observations can be extended. A major chord, I-III-V, can be constructed with the ratios 5:4 and 3:2 (III:I and V:I). A minor chord, I-IIIb-V can be constructed with intervals 5:4 and 4:3 (V:IIIb and 2I:V). And so on. Very cool!


1. Liner notes from "Glenn Gould: Bach, The Well-Tempered Clavier," CBS Records Masterworks, M3K 42266, 1986.

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