senior project quilt: while we may fear the day that physics explains aesthetics in full...

Fourier Analysis of Frequencies, Ascertaining Anharmonicity, and Explorations of Equal Temperament in The Well-Tempered Clavier. Abstract: After centuries of deliberation, equal temperament (a system of piano tuning wherein frequencies progress by a factor of the twelfth root of two) subverted Pythagorean conceptions of acoustical purity involving whole-number ratios. Soon after, Bach’s Well-Tempered Clavier—a work rightly deemed the “Old Testament,” or firm foundation, of piano repertoire—configured twenty-four pairs of preludes of fugues most ideally to the unequal temperament of his time. Interpreters ascribe synesthetic characteristics to each key (D major = burnished brassy gold, C = pure innocence of white, etc.), spurring this attempt to either dispel or affirm such arbitrary associations by verifying each piece’s optimal suitability to its given tonality. Following Fourier analysis of natural frequencies, I outline the extents to which stretched tuning and anharmonicity affect aural perception, and also provide preliminary confirmation that these pieces are indeed imperceptibly transposable amongst all keys on equally-tempered keyboards.

Piano temperament, or “the adjustment of intervals in tuning a piano or other musical instruments so as to fit the scale for use in different keys,” is analogous to ikebana (Japanese flower arrangement) in its dependence on balance and taste

Guys who did the groundwork, from upper then lower left: Pythagoras (b. 570 BC), Euclid (b. 325 BC), Gioseffo Zarlino (b. 1517), Zhu Zaiyu (b. 1536), Simon Stevin (b. 1548), Descartes (b. 1596), Daniel Bernoulli (b. 1700 AD), and Rameau (b. 1683)

Infinite overtones can be derived from the fundamental, providing endless harmonic color.

Sustaining the fundamental (1/1) on a piano at a loud dynamic permits one to hear the overtones more clearly. Notice how the higher overtones include discordant notes. The “cents” are equal to one-hundredth of a semitone.

Musical frequency, pitch, keyboard, interval, and notation relation chart.

The table above corresponds to the intervals named right above it.

Railsback piano stretch, which is an average deviation from equal temperament amongst 16 different pianos (Martin and Ward, 1961). These deviations are tiny but perceptible!

Mathematica does all Fourier transforms for me, which is lovely because I'm still not quite sure how to do them by hand... :) Here, its squeezes out a dominant frequency (in Hz) from 100,000 sample points collected using Audacity.

The black lines illustrate the deviations of Mead Chapel's piano's "A"s from what they are predicted to be by the twelfth root of two. Notice that A3, A4, and A5— which are near the center of the keyboard—align fairly well with their nominal frequencies.

Determining consistent values for anharmonicity (non-Hookean motion in an otherwise simple harmonic system) for the fundamental frequency and a few of its overtones.

some visualizations of what it means to transpose WTC Prelude No. 1 in C Major to other major keys. The interval sizes do indeed contract and expand in slightly significant ways, suggesting that yes, C Major is the best "home" for this piece!

Another juxtaposition of interval-progressions in twelve major keys. See how some in the back are taller than those in the front. Though we can see the height disparity, our ears likely can't hear the difference.

Another attempt to point out differences in interval progression-shape. G Major, the key located "farthest" from C on the keyboard, displays the greatest deviation. Again, we probably can't hear it, so the implications of this finding are nil, I'd say.

While we may fear the day that physics explains aesthetics in full, we have much to glean from the aspects explained in part.

In summary.