The Fifth Element #40

One of the engrossing and rewarding things about the study of history is that, every now and then, someone comes up with a plausible solution to a historical mystery—or some aspect of a historical mystery—that is decades, or even centuries, old. It often is the case that all the evidence was right there under everyone's noses all along. It's just that no one had yet managed to put all the pieces together properly and look at them from the right angle.

One excellent example of what I'm talking about comes from the disappearance of aviatrix Amelia Earhart. For decades, historians, aviation buffs, and conspiracy theorists have been troubled by the apparent discrepancy between the line of position Earhart radioed she was following on the morning she and her navigator disappeared, and what their position most likely was.

Earhart's navigator, Fred Noonan, had an excellent reputation. There is no indication that either their compass or their chronometer was off. So either something caused Noonan to miscalculate their position, or they really had suddenly gone off course without realizing it.

The key to the riddle, as recently worked out by historian Roy Nesbit, is that whereas Noonan navigated through the night using the stars, at dawn he doubtless would have sought to confirm their east-west position by sighting and timing the sunrise. For this purpose, he would have used navigation tables prepared for use aboard ships. These tables correlate the time of sunrise to one's east-west position.

Problem was, the tables assumed that one was on the surface of the ocean, and Earhart and Noonan were up in the air. Had Noonan and Earhart not been up all night, and were they not becoming increasingly anxious about their predicament, they would have realized that, without a trigonometric correction to account for the height from which they saw the sun rise, the sunrise table would be off. Nesbit calculated that correction—a difference of about 35 miles—and solved that aspect of that puzzle.

However, one of my other favorite "solutions," to "a Table of greene fields," from Henry V, II, iii, these days seems to be under attack or even discredited. So that might serve as a cautionary example.

Be that as it may, organist and harpsichordist Peter Sykes, who performed Bach's Goldberg Variations at Home Entertainment 2004, has alerted me to a proposed solution to a centuries-old musical mystery. To me, as to Peter, it looks and sounds like a winner. But before I explain the solution, a little background on the problem:

Since the time of the ancient Greeks, specifically our old chum Pythagoras, it has been known that several perplexities lurk within the conventional Western musical scale. A most significant problem is, if you tune the intervals of the scale so that the various pitches are derived purely by whole-number ratios of the starting pitch (eg, the fifth note of the scale, such as E in the key of A, has a frequency of three halves, or 150%, of the key note: in this case, 660Hz to A's 440Hz), the farther up the keyboard (or around the "circle of fifths") you go, the more "off-tune" the purely derived pitches will be.

Counting on the keys of the keyboard, it would seem that going up by leaps of fifths 12 times in the sequence F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E# should land you in the same place as going up by leaps of octaves seven times. And while you do end up on the same keyboard key (F), if you tune the 12 leaps of fifths so that each is in a pure, or just, ratio—that is, the higher note's frequency is exactly 150% of the lower note's—the frequency you end up on will be noticeably sharp compared to the frequency you get going up by octaves. Although they take up the same space on a keyboard, 12 pure fifths don't exactly equal seven pure octaves. Hmmm.

There is the temptation to emulate your average teenager and say, "Get over it." But this incongruity, known as the Pythagorean Comma, has important ramifications for writing and playing music, especially music in which the tonal center moves from one key to another. Tuning all the notes within the octave to "just" or whole-number-ratio tuning in one key—such as C major (no sharps or flats)—results in pitches that don't line up with just intonation in a remote key such as B major (five sharps).

The modern solution is to distribute the necessary correction equally among all intervals of the octave. Changing scalar intervals from their whole-number-ratio–derived pure or just positions is called tempering. So, modern tuning is called equal temperament. The flattening of the pure fifth required by equal temperament is why a guitar tuning fork pitches E at 329.63Hz and not the pure pitch of 330Hz. Equal temperament allows you to move the tonal center of your music between any keys without encountering any noticeably out-of-tune intervals.

The drawbacks to equal temperament are several. First, by distributing the adjustment equally among all the intervals, all the intervals are, to an equal extent, out of tune. No two given notes sound really out of tune, but no two given notes are ever perfectly in tune, either. You may not notice this when hearing music played using equal temperament, but you would notice it compared to the clarity and sustain of pure intervals.

Second, equal temperament makes all the key signatures sound essentially similar, except for starting pitch: some higher, some lower. But G-sharp in one key and A-flat in another are really different notes; it is a compromise of convenience that on most keyboard instruments they occupy the same keyboard key. Tuning G-sharp as a perfect major third above E is off by nearly a quarter of a half-step from A-flat tuned a perfect major third below C. That difference is clearly audible to just about anybody, and splitting it just makes both intervals wrong.

In an ideal world, G-sharp and A-flat would sound different from each other (and so on); as a result, the different key signatures would each have its own distinct character. In historical times, the workarounds employed to serve these values ranged from avoiding certain modulations, retuning instruments between pieces in contrasting keys, and even building instruments with staggered or split keyboard keys: G-sharp on one side, A-flat on the other.

Until recently, the biggest unsolved mystery in the history of tuning has been this: While everyone has always known that Johann Sebastian Bach employed a specific and perhaps unique tuning system for his The Well-Tempered Clavier, for the last 200 years or so no one has known exactly what that system was. The assumption has been that it was lost forever.

To the rescue came harpsichordist and organist Bradley Lehman. In a startling intellectual coup I find more dazzling than anything in The Da Vinci Code, Lehman makes a compelling case that the solution has been right there under our noses all these years, on the autograph title page of The Well-Tempered Clavier itself. Lehman's thesis is that the calligraphic doodle above the title is not just a space-filler, but a graphical memory aid that would remind students to whom Bach had personally taught his tuning method how to go about tuning a keyboard instrument (such as a harpsichord). Specifically, the doodle illustrates Bach's uneven, unequal distribution of the correction. Bach, it seems, did not equally flat those fifths; some were left pure, some were flatted by one-twelfth of the Pythagorean Comma, and others by one-sixth of the comma (footnote 1).

Here's how the graphic works: From right to left, it goes through the circle of fifths, beginning with F, indicating the amount of correction by the number of nested loops: two, one, or none. Starting with F allows you to tune all of the white keys before you tune any of the black keys.

Lehman's solution jibes with much of what we know of J.S. Bach. A practical musician as well as an indisputable genius, Bach was not one to put words on paper often, and was always more comfortable with practice than with theory. A succinct graphical mnemonic fits the historical evidence of Bach's being a great teacher but no great shakes as a prose stylist, let alone a music theorist for its own sake.

It was said that Bach could tune a harpsichord in 15 minutes, and also that his contemporaries were taken aback at the fluidity and flexibility of the temperament he used. It is axiomatic that the entirety of The Well-Tempered Clavier could be played without retuning, using Bach's own temperament. It all fits.

Lehman has put up a huge website, at which you'll find far more involved explanations than I have room for here, as well as downloadable MP3 musical examples. As far as I'm concerned, one listen to Prelude 1 of WTC clinches the case. The "home" keys sound more in tune, while outlying keys (more sharps or flats) sound more distinctive: spicier for the sharps, mellower for the flats.

Lehman is not out there on the Grassy Knoll all by his lonesome, to be sure. Goshen College, in Goshen, Indiana, site of some of John Atkinson's recording exploits with Cantus, commissioned organ builders Taylor and Boody to build an instrument to the "Bach Spiral" temperament. It, too, is represented by downloadable musical examples.

Lehman's work, in and of itself, does not immanentize the eschaton. But it's a very satisfying start. Well done.

Back to Basics
If your music-reading skills are stuck at the level of being just about able to follow "A Mighty Fortress Is Our God" in the hymnal, an inexpensive and relatively painless way to gain greater proficiency is Terence Ashley's Learn to Read Music in 10 Lessons. But don't delude yourself—it's 10 lessons, not 10 minutes! Budgeting a week or two's worth of spare time and odd moments for each lesson is probably realistic. The lessons cover such subjects as "Pitch," "Timing and Rhythm," "Scales and Keys," "Harmony," and so on. I advocate photocopying the test sheets and retaking each test a few times. A companion CD includes musical examples and listening tests. At $9.98 for both book and CD, you can't beat it with a stick.

The Right Way to Play the Violin
Here's a real treat. Video Artists International (VAI) has released on DVD a duo recital by violinist David Oistrakh and pianist Sviatoslav Richter, recorded in Alice Tully Hall in 1970. The program is Beethoven's Violin Sonata 6 in A Major, Op.30 No.1, paired with Brahms' Sonata 3 in D Minor, Op.108. The single encore is the Scherzo movement of Beethoven's Sonata 5 in F Major, Op. 24, "Spring." The DVD's liner notes are silent on the source of the program, but it probably was originally produced as a local or national public-television broadcast, in that it lasts slightly under an hour.

Footnote 1: Lehman acknowledges that previous investigators have suggested that the calligraphic doodle holds the key, but he claims with some justification that the previously proposed solutions are unworkable.