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The Circle of Fifths and the Comma of Pythagoras


Imagine a violin tuned in pure fifths to G-D-A-E, and a viola similarly to C-G-D-A. Compared to the viola's C-string, the violin's E-string will come out a little too sharp to make a pure major third (plus two octaves). The difference is about a quarter of a semitone, and is called the syntonic comma.

Now imagine a sequence of thirteen strings tuned in pure fifths, as

C   G   D   A   E   B   F♯   C♯   G♯   E♭   B♭   F   C

Here the top C will be a little too sharp to match the lower C (modulo seven-octaves). The mismatch, comparable in size to the syntonic comma, is called the Pythagorean comma.

Whether Pythagoras discovered the Pythagorean comma we do not know, but he or his disciples certainly did discover the underlying reason for these commas, which is that an octave is a factor of 2 in pitch, a pure fifth a factor of 3/2, a major third 5/4, and so on. Hence, we can think of these commas as simply arithmetic statements: the Pythagorean comma is the difference between (3/2)12 and 27, and the syntonic comma is the difference between (3/2)4 and 4×(5/4). The musician, however, who contributed the most to understanding harmony as a consequence of natural principles was Rameau.

In music with a single voice, commas are probably not important. But in music with harmony based on fifths and thirds, a quarter-semitone comma produces a dissonance so awful it is called a wolf. Even a fraction of a comma is easily audible. Have a listen to this capriccio cromatico by the little-known (but clearly unjustly so) early-Baroque composer Tarquino Merula. More on this piece later, but meanwhile, enjoy the expressive effect of the out-of-tune chromatic scale.

The applet provides a visual representation of commas. Seeing is no substitute for listening, obviously, but maybe these diagrams can help provide some insight.

     

You should see a button to the left saying Show Commas window.   (If not, see here.)   To download, see below.

The circle represents an octave. Pitch rises clockwise, with the angle around the circle proportional to the logarithm of the frequency. (In other words, 30° degrees around the circle is 100 cents of pitch.) The white dots represent special intervals: they can all be reached from the top of the circle in no more than four steps of 210° or 120° (that is, pure fifths or pure major thirds). There are more intervals that could be built in this way. I have simply chosen the ones that seem reasonably identifiable as named notes, as labelled.

On the chromatic circle, you can now compare different tunings.

Pythagorean

Now we add lines indicating pure or Pythagorean fifths: C-G and so on upwards, and C-F and so on downwards. This gives us a Circle of Fifths.

Except that it isn't quite a circle. The ascending and descending fifths do not meet, instead they collide at F♯/G♭ with a Comma of Pythagoras. Digital pianos often have a Pythagorean-tuning option. This makes all the fifths pure, except one that is too narrow by a Pythagorean comma. This wolf fifth will probably try to hide among the black notes, but you can hear it by its howl.

The syntonic comma appears in the figure as well, as the mismatch between the red vertex and the white dot at E.

Equal temperament

What to do about the Pythagorean comma? One elegant solution is to make all the fifths too flat by 1/12 of a Pythagorean comma, which is barely audible. This automatically spreads out the syntonic comma as well, making all major thirds a little sharp. Through the twentieth century, equal temperament was almost universally regarded as the ideal system, and the Gospel of Sebastian was invariably cited in its support.

The problem with equal temperament is   .   .   .   well, its equalness. All fifths are slightly flat, and more noticeably, major thirds are always a little sharp and minor thirds always a little flat, and there is no escape. If this is really what Bach meant by wohltemperirt, why does he tend to give individual keys their own character? For that matter, why is Mozart at his most buoyant in A major, and why does Beethoven become almost placid in F major?

Mean-tone

If you find equal temperament boring (and the sandal-wearing early-music types will tell you it was the worst thing for music, ever) you are free to spread the commas in another way. In the Baroque era, you had to have designed a tuning system at least once or you were a nobody (kind of like writing a completion of Mozart's Requiem today). Bach's pupil Kirnberger managed three. The general idea is to make some keys more accurate at the expense of others. Another consideration, at least in the pre-electronic era, was that there needed to be some practical algorithm to tune by ear.

Mean-tone tuning is the best known of the unequal temperaments. Broadly speaking, it makes the simpler key signatures more accurate than in equal temperament, and in particular C-E is a pure major third. But in the process the black notes become quite unevenly spaced, as we can see in the figure. The interval A♭-E♯ is another wolf fifth, this time too wide. This makes A♭ major an unusable key in mean-tone. Perhaps not insignificantly, A♭ major does not appear in the two and three-part inventions, even though F minor does.

The unevenly-spaced black notes in mean-tone are what give the Merula piece its peculiar quality. The opening chromatic scale at the beginning is mean-tone, starting from D. Whether Merula intended it to sound like that, or whether the harpsichordist Jean-Marc Aymes simply seized an opportunity for dramatic effect when he saw one, I do not know. Either way, the piece is quite memorable.

Lehman-Bach

The last example shown is a tuning system (re-)discovered in the 21st century. Bradley Lehman tells that in a moment of idle speculation he tried reading the ornamental curlicue on the title-page of the Well-Tempered Clavier as a set of instructions for tuning, and tuned a harpsichord accordingly, but when he started to play, his ears told him he had the answer. Lehman was not the first to suggest that this ornament, sitting so prominently on the title-page of a book about tuning, might explain what Bach really meant by wohltemperirt, but he was probably the first harpsichordist with an expert knowledge of tuning to do so in modern times.

If we read Bach's curlicue as Lehman suggests, the resulting tuning system is remarkable. The white notes are very close to mean-tone, but the black notes are much more even. The triads vary in character: C major and F major are almost perfect, E major sounds bright because of that sharp G♯ while B♭ major sounds rather dark. But there are no lurking wolfs anywhere, and all 24 keys are usable. In the figure (as more importantly, in sound) the differences from equal temperament are small, barely perceptible.

Was this the tuning Bach used? Quite a few musicians are convinced. Decide for yourself. Have a listen to Lehman playing this familiar and deceptively simple music and pause to enjoy Bach gently taking you on a tour of chords all around the keyboard, including each black and white note.


The Commas program is written in literate Java and is free software. For source and Java byte code, download the Commas.jar file.