A Horological and Mathematical Defense of Philosophical Pitch

By Brendan Bombaci

All Rights Reserved

Copyright 2013 Lulu Press

ISBN: 978-1-304-36230-8

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Introduction

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 that 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 that of equal temperament (whether or not equal temperament mostly corrects for the arguably noticeable near-Wolf fifths 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.

Time In

It should be imagined that Western music – with an original meter basis of 4/4 that originally hinged upon the second hand of the clock for metering rhythm (a la the 120bpm Roman standard for marches) even before the second was academically identified[5] – should have a pitch frequency that 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 (a mathematician, physicist, and music theorist) [1] and Ernst Chladni [1, 2], “the father of acoustics.”  At the first octave of C, we would have the value of 1Hz, perfectly matching the second hand complication (movement).

Using 5 Limit Tuning with the root set toA (at 216) rather than C, 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.  You may notice that these notes, C, E, G, B, and D respectively, rearrange to a set of “stacking thirds,” in perfect chordal harmony.  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 from the root (except 45, but this is still a harmonic of 15).  This makes for more mellifluous tonal vibrations.  In addition, the numbers 1, 2, 3, 4, 5, and 6 represent the most commonly used values for meter in classical and modern music.  There are also important historical implications to this system, making it more geometrically and even astronomically intrinsic.

The helek (helakim, pl.) is an ancient and still used unit of time in Hebrew horology [3], from which the second of modern global timekeeping was extrapolated.  Further preceding helakim were units of the same duration but with the Babylonian names barleycornor she, but, no matter which name is used, all effectively mark the passage of 1/72nd of one degree of celestial rotation in a day.  There are 1080 helakim per hour, and therefore 25,920 helakim per day, which, in total, represent the number of years in one astronomical Precession of the Equinoxes.  This gives a discrete measurement unit that relates each “moment” 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.  Many historical European clocktowers, such as the Torre dell’Orologio in Venice, graphically purvey this along with the 24 hour segments.  The conversion between helakim and seconds is this: 1 Helen = 3.^–33 seconds, or 18 helakim = 60 seconds.  72 helakim, like the 72 years that pass in one of 360 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 is also the frequency in Hertz of the F# (the 7^thinterval, or perfect chromatic center) when the chromatic scale is tuned as proposed herein.

Making the transition from helakim to seconds would only be a matter of deciding that the sexagesimal “navigational” system should also be standard for the measurement of time, for better precision.  That is, 25,920 helakim multiplied by 3.^–33 is 86,400 seconds.  Musicians of the Middle Ages would have noticed that the twelve divisional factors of that navigation system (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60) are attractively coherent with four of the five stacking thirds frequencies of the new 5 Limit Tuning system which was designed to fix Pythagorean tuning dissonance in thirds intervals.  With the addition of the fifth stacking third (9/18/36hz, etc.: the 2^ndinterval D note), they altogether cross-correlate with all aforementioned time measurement references within the Precession of Equinoxes, paying ultimate homage to the more prolific origins of timekeeping (remember: 18 helakim per 60 seconds).  It seems that the timekeeping conversion has eluded historians of music, as the concert pitch value has wavered over time and place for hundreds of years, and now has nothing to do with horological frequencyor periodicity at all – it has been reduced to arbitrary ‘pitch,’ where tone becomes subjective.  See the below frequency conversion table, which calculates differentials in cents between current frequency values of certain notes and their proposed frequency values.  Because alternate tuning in software usually requires in-octave cent differentials, I have provided all note values from C4 to C5 even though the root for deriving the frequencies was A3.  That root value was decided upon because 108 multiplied by 3.^–33 is 360 cycles per helek (representing therefore 360 nautical degrees of observed celestial motion per year).  216 is the next octave up for that root note.

Root of ‘A,’1:1= 216

16:15(Bb) = 230.4

10:9(B) = 240

*C, 32:27= 256 (binary; divisible to 1) [-37.661 cents]

• optionally 6:5 = 259.2 (power of 25920)[-16.154 cents]

5:4(C#) = 270 [-45.436 cents]

4:3(D) = 288 (divisible to 9) [-33.694 cents]

64:45(Eb) = 307.2 (higheroctave of 0.3hz or 1 cycle per helakim) …[-22.063 cents]

40:27(E) = 320 (divisible to 5) [-51.331 cents]

• optionally 3:2 = 324 (multiple of 2592) [-29.824]

8:5(F) = 345.6 [-18.089 cents]

5:3(F#) = 360 [-47.434 cents]

16:9(G) = 384 (divisible to 3) [-35.697 cents]

50:27 (Ab) = 400 (divisible to 25; supplementary low denominator for stacking thirds)

• optionally 15:8 = 405 (multiple of 25,920) [-43.478 cents]

2:1(A) = 432 [-31.767 cents]

16:15(Bb) = 460.8 [-20.021 cents]

10:9(B) = 480 (divisible to 15) [-16.883 cents]

• optionally 9:8 = 486 [-27.845 cents]

32:27 (C) = 512 (again, binary; divisible to 1)

On a more esoteric note, my proposed system also corresponds in some cases to culturally relevant “sacred” geometrical figures, whether or not any ancient musicians played note values that represented the same cosmic motions their timing system held to.  Some of the latter include the conversions: 1440(F#)/3.^–33 = 432 (a value considered by some to be the “spiritually” correct concert pitch)… 720(F#)/3.33 = 216 (purported by some to be the “Number of the Beast” in Biblical Revelations)… 360(F#)/3.^–33 = 108 (a value of great importance in Buddhist and Hindu tradition, and the number of beads on Christian rosaries)… 180(F#)/3.^–33 = 54 (the number of years in one Exeligmos or Triple Saros eclipse prediction cycle discovered by the ancient Greeks)… and 240(B)/3.^–33 = 72 (the number of years in one degree of equinox precession, and the “tetragrammaton” value given to the Hebrew name for God, i.e., YHWH).  No less, these numbers appear as pentagram angle degrees, which reference the phi ratio and Fibonacci sequence (a fundamentally common pattern which all biological matter utilizes for efficient growth, and one that has been venerated throughout history via great artworks), and the faces of the dodecahedral (soccer ball shaped) Cosmic Microwave Background itself [4] that shows how our cosmos expanded from the Big Bang.  Interesting as they are, these solution values are not the note values we should make standard in our chromatic tuning system while using seconds rather than helakim, but rather intriguing sign posts that show the astro-horological bases for certain compositional conventions in both secular and religious visual (including architectural) and sonic art.

For the sake of remaining true to horology in sonic form, harking back to but making better sense than the “Music of the Spheres,” the usefulness and the intricate aesthetics of tuning to C256 is inarguably better than any other standard.  It also becomes far more intuitive to explain, due to whole number relationships, how various notes interact with one another and with tempo bases.  Any “brighter” compositional sound, such as desired by proponents of A440, can be manifested by simply transposing a song.  Although utilizing this proposed standard alters interval relationships (because just intonation is not equally tempered), it offers a new way to experience music, with the same revolutionary impact that modes within a key provided for mood and depth when choral music was innovated.  Now, slight dissonances will be heard in a number of modes stemming from any key.  Many Western composers prefer this and use just intonation specifically to achieve enhanced dramatic effect; some people who do so are: John Luther Adams, Glenn Branca, Martin Bresnick, Wendy Carlos, Lawrence Chandler, Tony Conrad, Fabio Costa, Stuart Dempster, David B. Doty, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Michael Harrison, Ben Johnston, Elodie Lauten, György Ligeti, Douglas Leedy, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, Marc Sabat, Wolfgang von Schweinitz, Adam Silverman, James Tenney, Michael Waller, Daniel James Wolf, and La Monte Young.  Perhaps, with the rationality I provide in this article, many more yet will.

References

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)
3.  Hebra, Alex. Measure for Measure: The Story of Imperial, Metric, and Other Units, pp 53 (The John Hopkins University Press, 2003)
4. Luminet, Jean-Pierre, Jeffrey R. Weeks, Alain Riazuelo, Roland Lehoucq, and Jean-Phillipe Uzan.Dodecahedral Space Typology as an Explanation for Weak Wide-Angle Temperature Correlations in the Cosmic Microwave Background.  Nature 425:593-595.
5. Sachau, Edward C.The Chronology of Ancient Nations. (Kessinger Publishing, 2004).