A Horological and Mathematical Defense of Philosophical Pitch
By Brendan Bombaci
All Rights Reserved
Copyright 2013 Lulu Press
I propose an alteration of the concert pitch standard outlined in ISO 16. As of now, it is set to A440 (A=440Hz), which has been chosen subjectively (rather than empirically as based upon the mathematical or geometrical values of art composition), as most all other concert pitch standards have been chosen throughout history. I have sought out various ways to make a compositionally cogent concert pitch standard, and I have succeeded at finding one which is perfectly tailored to synchronize with both the sexagesimal timekeeping system upon which all music is measured, and the 5 Limit Tuning system. It is well-known that this form of just intonation is the most consonant of all tuning systems, including equal temperament (whether or not equal temperament mostly corrects for the near-Wolf fifth of just intonation). In as much, it is perfectly suited to be the model tuning system for this innovative new pitch standard, especially when one considers its fractional values for deriving each note of the chromatic scale. I will now explain both of my justifications in detail with some corroborative horological references.
It should be imagined that music, which hinges upon the second and minute hands of the clock for metering rhythm, should have a pitch frequency which is similarly correlated. When tuning music to A440, most of the pitch frequencies are not whole numbers; the first octave of B (B1), for example, is 61.74Hz. If this were set to 60Hz instead, being the only note of the chromatic scale which comes close to synchronizing with the clock as a fractal continuance of the sexagesimal system, we would find the middle C note, C256, at the “scientific” or “philosophical” pitch of Joseph Sauveur  and Ernst Chladni [1, 2]. At the first octave of C, we would have the value of 1Hz, perfectly matching the second hand of the clock.
Using 5 Limit Tuning set to C256, the frequencies of notes C4 (256), G4 (384), E4 (320), D4 (288), and B4 (240) are reducible to, respectively: 1, 3, 5, 9, and 15, and you may notice that these notes rearrange to a set of “stacking thirds,” in perfect chordal harmony, when denoted as C, E, G, B, and D. With the lowest C also standing in for its multiples of 2, 4, 8, 16, and 32, all of the numbers which are member to that set of stacking thirds are the very same numbers which comprise the numerators and denominators by which every chromatic note is derived (except 45, which is a harmonic of 15). This makes for more mellifluous tonicity. In addition, the numbers 1, 2, 3, 4 and 5 represent the most commonly used values for meter in classical and modern music. Any “brighter” compositional sound, such is desired by proponents of A440, can be manifested by simply transposing a song. There are important historical implications to this system as well, making it quite geometrically and astronomically intrinsic.
The helakim is an ancient and still used unit of time in Hebrew horology . The second hand of Western timekeeping was extrapolated from it, and it is a duplicate measurement of the further preceding Babylonian “barleycorn” or “she” unit of time, effectively marking the passage of 1/72nd of one degree of celestial rotation in a day. There are 1080 helakim per hour, and therefore 25920 helakim per day (and that many years in one astronomical Precession of the Equinoxes). This gives a discrete measurement unit that relates each minute to a visibly interesting astronomical cycle that has captured the imaginations of many cultures worldwide. Half of a day is akin to half of a precession of equinoxes, thereby; and likewise, periods of 2160 helakim are similar to the 2160 years of one astrological Age, meaning there are 12 Signs that pass in one day. This is historically and modernly observed in various parts of Asia as well. The conversion between helakim and seconds is this: 1 helakim = 3.333 seconds, or 60 seconds to every 18 helakim. 72 helakim, like the 72 years that pass in one degree of celestial precession, are equal to 4 minutes. 4 minutes multiplied by the whole 360 degrees equals 1440, the amount of minutes in one day. This number is equivalent to the amount of years that it takes for a Sothic Cycle to pass*, and the frequency in Hz of the F# when tuned with this method.
The usefulness of tuning to C256 is inarguably better than any other standard, for the sake of remaining true to horology in sonic form, harking back to but making better sense than the “Music of the Spheres,” and also because it becomes far more intuitive to explain, in simple math, how various notes interact with one another. There are likely many more astronomical references to be made through this system, and I will leave those discoveries to the curious. In the meanwhile, it is safe to say that a standardization of musical frequencies ought to be tailored to the very horological science which tone and rhythm are measured by to begin with.
* The Sothic Cycle calendar that Eduard Meyer invented in 1904, in order to calibrate historical recordings of ancient Egypt to modern time tables using Leap Years, is based upon a miscalculation. Meyer derived the duration of the cycle by multiplying the number of days in one year by the 4 years it takes to see one degree of Equinox Precession. What he should have done is recognize, first, that 72 years pass for each one degree of celestial rotation – 72 years by 360 degrees = 25,920 years. As such, he would have to note that the 360 degree system is used to divide the Great Year, and not an equivalent to 365.25 degrees. Thusly, 360 degrees x 4 years = 1,440 years. If his system was correct, then the Great Year (Precession of Equinoxes) would be 26,208 years long, which is off by approximately three centuries compared to modern calculations. There are indeed more modern criticisms of his original version of the Sothic Cycle, such as a lack of synchrony between Egyptian historical records and those of other nations whom Egypt had contact with, a lack of definitely incontrovertible markers of time, and a lack of precision in the original astronomical dates used to reference his conclusion by .
1. Bruce Haynes, History of Performing Pitch: The Story of “A,” pp 42,53 (Lanham, Maryland: Scarecrow Press, 2002).
2. Ernst Florens Friedrich Chladni, Traitéd’acoustique, pp 363 (Paris, France: Chez Courcier, 1809)
4. Hebra, Alex, Measure for Measure: The Story of Imperial, Metric, and Other Units, pp 53 (The John Hopkins University Press, 2003)
5. Mackey, Damien F, The Sothic Star Theory of the Egyptian Calendar: A Critical Evaluation, abr. ed. (Sydney, New South Wales, Australia: University of Sydney, 1995).