Twelfth Root of Two

Today is Pi Day. Maybe someday I will write some music based on Pi, but today it is just a happy coincidence that I have finished a new music piece that is also math-based and uses an irrational number: the 12th root of 2 (1.05946309...). This algebraic number is most important for setting the frequency ratios to fine-tune the twelve tones of Western music into an arrangement known as equal temperament (12-TET). Starting three centuries ago, this tuning has made possible the awesome beauty, power and sophistication of great orchestral symphonies, and today has become the most widespread system for tuning, also used in most rock and jazz and contemporary music styles almost everywhere.

Due to years of piano lessons as a youth, I have been influenced by the genius melodic and harmonic development in the Well-Tempered Clavier. The WTC are preludes and fugues for solo keyboard, set in all 12 major and minor keys, and written by J. S. Bach about 300 years ago. The first set was published when he was 37, and an even richer second set was created 20 years later. I play some of these pieces on piano, slowly and in a more romantic style than the usual practice of using a higher-speed baroque interpretation. The WTC are vitally important compositions, having influenced later famous composers like Mozart, Beethoven and Chopin, among others. Their publication helped promote and popularize the new (at the time) equal temperament system of tuning.

My instrumental piece entitled Twelfth Root of Two uses do-re-mi-fa-so-la-ti-do-re-mi to map out 1-2-3-4-5-6-7-8-9-0. Hence the 1.05946309 (the 12th root of 2) number then becomes a sequence of notes. Set in the key of B flat major, the first six notes of this slow-moving melody are Bb and high D (the first and third of the tonic chord), then F and high C (the first and fifth of the dominant chord), and then Eb and G (the first and third of the subdominant chord). I was instantly quite taken that this 'outlined' the Primary Triads of I, V, and IV very nicely! This theme doesn't appear until about 2 and a half minutes in, lifting the tonality from minor into major, but then it repeats constantly until the end.

Not that it matters at all, but recalling the melody now is how I have memorized the actual numerals for the twelfth root of two . . . so, a bit of a math nerd here.