Pythagorean Comma
Recently my friend Paul Suino away out in California asked me what a "Pythagorean Comma" was. This rang a faint bell. I knew it had something to do with the tempering of a musical scale, for example when tuning a harpsichord, which I had tried years ago. But the details remained murky in my dim gourd. Only one thing to do: Google.
Here are three definitions for starters. Forgive me if they're too simplistic, too complex, or just too generally murky:
FREQUENCY: In music, frequency refers to the number of complete oscillations of a sound wave that hits your ear every second. In other words, it is the number of vibrations per second of the air in your ear, or the string on your uke. This determines the pitch. Frequency is indicated by the word "Hertz" (Hz). One vibration per second is one Hz. The higher the number, the higher the pitch. The piano goes from a low of 27.5 Hz to a high of 4186 Hz.
OCTAVE: An octave is what you get when you take a note and multiply its frequency by 2. For example, middle C's frequency is 261.63 Hz. If you multiply this by 2, you get 523.26 Hz, which is the next higher C on the keyboard. An octave is usually described as containing a scale of 7 tones -- some of which are separated by a half-step, and some of which are separated by a whole step. These tones are often represented in the familiar system called "Solfa" as do, re, mi, fa, so (or "sol"), la, and ti.
Actually, if you count every note in an octave -- every black key and white key -- you'll find that every octave contains 12 half-steps (or "semitones"). When you go up the keyboard note by note, and include all white and black keys, you are progressing up the keyboard by half-steps or semitones.
FIFTH: A fifth is the fifth note of a scale. In other words, in Solfa, the "so" of "do, re, mi, fa, so." Or, in the key of C, it's the first G you hit, going up. Technically, it is what you get when you multiply a note by one and one-half. Take middle C's frequency of 261.63 and multiply it times 1.5, and you get 392.445, which is the Hz of the G note mentioned above.
The term "fifth" also applies to the interval between the tonic, or base note of the scale, and its fifth note. In cultures throughout the world, the fifth (middle C played with the G above it for instance) is acknowledged to be the most consonant (least dissonant) of all intervals, except for the octave interval itself (middle C played with C-above-middle-C, for instance).
To make things a little more complicated again, just as an octave actually contains 12 half-steps or semitones, a fifth actually contains 7 of them. There are 7 semitones between C and G; 7 half-steps, in other words, if you play all the white and black keys from C to G.
The Rub
Now, here's the rub. There is a very annoying discrepancy between the mathematics for the octave (multiply a note times 2 to get the octave) and the mathematics for the fifth (multiply a note times 1.5 to get the fifth). This discrepancy makes it literally impossible to tune a piano (or anything else) perfectly for every conceivable use. You have to put up with some fudging to get it to sound halfway decent all the time, which means it never sounds perfect.
Here's an example of why this is true. Let's start with the lowest note on the piano, the A of 27.5 Hz. To move up from this A to the next A, we multiply 27.5 times 2, then THAT times 2, and so forth. We do that 7 times and we get this column of A's, each one an octave higher than the last:
27.5
55
110
220
440
880
1760
3520
This last number, 3520 Hz, is the frequency of the A note 7 octaves above the original A note. Since an octave has 12 semitones, and we have gone up 7 octaves, we have gone up the keyboard by a distance of 12 times 7, or 84 semitones.
Now, if we start with that same low A and multiply it times 1.5 -- to get to the FIFTH of that A -- then multiply THAT number times 1.5, and so on and so on, we end up with this column:
27.5 Hz · A
41.25 · E
61.875 · B
92.8125 · F#
139.21875 · C#
208.82812 · G#
313.24218 · D#
469.86327 · A#
704.7949 · F
1057.1923 · C
1585.7884 · G
2378.6826 · D
3568.0239 · A
This last number, 3568.0239, is the frequency of the A note 12 fifths above the original A note. Since a fifth has 7 semitones, and we have gone up 12 fifths, we have gone up the keyboard by a distance of 7 times 12, or 84 semitones. Same as with the octave column. 7 times 12 should be exactly the same as 12 times 7, one would think.
But the end result is not the same number, exactly. Going up octave by octave, the highest note we hit is 3520 Hz. Going up fifth by fifth, the highest note we hit is 3568.0239. These are close, obviously, but ARE NOT THE SAME. I tried it on my calculator, and it's true!
Naming Rights
Because this discrepancy was studied by Pythagoras and his bunch, and because the word "comma" has a stem meaning "segment," the discrepancy between these two numbers is called the Pythagorean Comma.
This weirdness not only can be observed mathematically, but can be heard by your own two ears. Which makes it a problem for piano tuners. Who should they answer to, the fifths or the octaves?
Many tuning systems, or "temperings," have been devised over the years to deal with this issue. These days, it's usually handled by dividing the problem evenly between all the intervals on the whole piano. This means that every interval on the entire piano is a tiny bit out of tune. Apparently my guitar has used a similar system for decades ha ha.
Next episode, The Aristotelian Semicolon. Just kidding. For more info about the Pythagorean Comma, and for thousands of web sites about peculiarities of music theory in general, just Google using the main words in this episode: semitone, Pythagorean Comma, Solfa, tempering, octave... You'll be delightfully swamped.