Summary:
Interlude B, the second “extra-musical” discussion, explores the relationship between musical taste and the scientific realms of math and physics. The interlude begins by relaying a famed trope within music history: the fictional account of Pythagoras and musical hammers—which introduces readers to the vital, empirical nexus of intervals (perfect and imperfect) and mathematical ratios. From there, the author commences an extended and historically-grounded discussion of the headier, physics-based concepts of tuning, frequency (pitch), and the overtones series—culminating in the assertion that our collective love of perfect intervals and triads (not to mention jazz) is grounded in the very laws of nature.
Supplements:
( More on the strength of perfect intervals ):
It should not be surprising, then, that the “sonic strength” of these intervals can be seen as reflecting not just our vocal anatomy (as in the case of the parallel octaves), but also our neurobiology; as Patel put it, the special perceptual status of these intervals “reflects the neurophysiology of the auditory system.” (Patel, Music, Language, and the Brain , 13).
( More on non-human preference for “small integer” intervals ):
Octave recognition, for example, has been demonstrated via experiment with birds (e.g., the European starling), cats, and several species of monkeys—such as a study showing that a group of rhesus monkeys was able to consistently recognize octave equivalency (transposition) of simple, tonal children’s songs.
( More on deducing the Pythagorean ratios of other notes of the chromatic scale ):
There are, in fact, several means by which one can deduce the ratios of the various intervals. To cite one example: for the major 2nd (e.g., F to G), one arrives the ratio of 9:8 by calculating the difference between the 5th and the 4th, which indeed is a major 2nd. Specifically, this amounts to dividing the ratio of 3:2 by 4:3 (3/2 : 4/3 = 9/6 : 8/6 = 9:8). See n. 196 for bibliographic sources on this topic.
( More on the “sour”-sounding intervals arising from Pythagorean tuning ):
To get a bit technical: by virtue of the most traditional schema to derive ratios—starting at a given “base” note (e.g., Middle C) and moving by 3:2 (perfect 5th) up and down by circle of 5ths, and then reconciling the notes within a single octave—the notes at the extreme end of the circle (F# and Db if starting at C) are extremely out-of-tune. So much so that they are called “wolf” tones, since the “beating” interference of their frequencies resembles the howling of a wolf.
( More on the English roots of “just” intonation ):
Specifically, the English theorist Walter Odington (c. 1330) was the first to propose these “just” versions of the major and minor 3rd (as well as 6ths)—technically, by reducing each interval by a tiny distance known as a “syntonic comma” (81:80, about 1/4 of a half step). Interestingly, Odington notes that singers were already adjusting to these “just” ratios intuitively, by which “they steer them into a sweet blend and wholly into consonance.” Cited in Koster, John. “Questions of Keyboard Temperament in the Sixteenth Century.” In Interpreting Historical Keyboard Music: Sources, Contexts, and Performance , ed. Woolley, Andrew. London: Routledge, 2013: 117.
( More on the relegation of 5ths and 4ths to structural intervals ):
In fact, by the late 15th century, the perfect 4th was considered a kind of “dissonance”—unless supported by a 5th or 3rd below (as in a 1st inversion triad); the first such designation was by the famed Renaissance theorist Johannes Tinctoris, in his Dictionary of Musical Terms (1473); see Drabkin, William. “Fourth.” Grove Music Online (2001).
( More on the limitations of just intonation ):
For example, by tweaking the 3rds, some “perfect” 5ths (e.g., E to B) were suddenly out of tune.
( More on other experimental tuning systems in 17th-18th centuries ):
The most popular system during the 17th and early 18th centuries is a temperament known as quarter-comma meantone, where each 5th is lowered by a quarter of the “syntonic comma” (about 1/20th of a half step). It maintained the just intonation of 3rds and 6ths, while likewise keeping most 5ths in tune; this was particularly beneficial for keyboard instruments, whose pitches were fixed.
( More on equal temperament ):
To get technical: when an octave (2:1) is divided into 12 half steps (as on the piano), each half step is the 12th root of
2 ( 12√ 2), whose ratio is somewhere between 18:17 and 107:101. To avoid such convolutions, a logarithmic system of “cents” was adopted (from the 1830s), were each half step is 100 cents—such that a 5th is 700 cents (compared to 701.96 cents in Pythagorean tuning), a major 3rd is 400 cents (compared to 386 cents in just intonation), etc. These differences are too small for human perception, and yet they enable any note to be sounded simultaneously with any others—including via adventurous harmonic modulations—without “wolf” tones or other indelicacies. Equal temperament, interestingly, has roots in the early years of the “scientific revolution”: when in 1584 composer / theorist Vincenzo Galilei (father of Galileo) proposed dividing the octave into 12 tones at a ratio of 18:17 per half step.
( More on Vincenzo Galilei’s experiments ):
For example, in the 1590s Galilei demonstrated that multiple ratios could generate the same interval—for example, the octave could be produced by a ratio of 1:4 if tension was measured instead of length.
( More on Mersenne’s law of frequency ):
Mersenne’s 3-part law (in his Traité de l’harmonie universelle , 1637) states that a pitch’s fundamental frequency is proportional to the square root of string tension, and inversely proportional to string length as well as to the square root of the string’s thickness.
( More on experiments by Hooke, Sauveur, Bernoulli, and Euler ):
In brief, Hooke was among first (1681) to make a rough direct measurement of frequency, using a spinning wheel of brass teeth—where the pitch varied with speed, thus demonstrating how pitch relates to frequency; Sauveur (early 18th century) was the first to propose a distinct field of “acoustics”, and the first to accurately determine (near) absolute frequency by counting beat frequencies on low-pitched organ pipes; Bernoulli (1727) was able to calculate the fundamental frequency of pitches generated by strings; and Euler made numerous contributions to acoustics in the early 18th century, including defining three distinct sources of sound—the tremblings of solid bodies; the sudden release of compressed or rarefied air; and oscillations of air, either freely or confined—as well as producing frequency-based theory of consonance and dissonance.
( More on Mersenne’s perception of those “petits sons” ):
Indeed, Mersenne recognized that most musicians fail to hear these “petit sons”, given the strength of the fundamental, but that he had succeeded over 100 times, “on the viol, the theorbo [a kind of lute], as well as on two monochords.”
( More on “upper partials” ):
To get a bit technical, each individual frequency (each sine wave that combines into a “complex tone”) is called a “partial”—including the fundamental; those above the fundamental are called “upper partials”—numbered 2nd, 3rd, etc. As such, the term “overtone” is any “partial” except the fundamental—since it is an “upper partial”.
( More on additional intervals arising from the overtone series ):
The next four partials (8:7, 9:8, 10:9, and 11:10), interestingly, produce four distinct “versions” of the major 2nd (e.g., from C to D); as you may recall, 9:8 was identified via Pythagorean tuning and cited in Plato’s Timaeus ; the other three then produce 2nds that are either larger (8:7) or smaller (10:9, 11:10) than that produced by 9:8—being nearly identical (only 4 cents different) to that on a modern piano. From there, the successive partials produce multiple, varying sizes of half steps (e.g., C-Db)—that on a modern piano being somewhere around 17:16. As we continue up, we find every conceivable interval above the fundamental, including dissonances like the tritone (e.g., C-F#—via the 11th partial), both in and out of tune.
( More on Rameau’s interpretation of the overtone series ):
In the wake of discoveries by Sauveur, Euler, and others, Rameau seized upon the overtone series as a means to identify a rational and single-principled “source” for all aspects of music, not least harmony. In writings from the 1720s onward, he labeled this source the corps sonore (sounding body)—the vibrating systems (strings and otherwise) that produced all the overtones necessary to understand the basics of harmony and the relationship of chords to one another. Indeed, not only harmony was beholden to it:
The corps sonore— which I rightfully call the fundamental sound—this single source, generator, and master of all music, this immediate cause of all its effects, the corps sonore , I say, does not resonate without producing at the same time all the continuous proportions from which are born harmony, melody, modes, and genres, and even the least rules necessary to practice…. It is the pure and simple operation of nature and the foundation of all harmony and all succession.
As noted, Rameau made numerous inaccurate or untenable assumptions—not least arising from the actual inability of a single fundamental to produce a minor triad above it, nor any of the other 7th chords (e.g., the diminished 7th) common to music. This led him to devise various byzantine arguments (e.g., relying on an “undertone”, two fundamentals, etc.), or to simply bypass specifics. It also led to a number of heated debates with his contemporaries, including Rousseau, who preferred to see melody as the “source” of all music. Rousseau, Jean-Jacques. “Essay on the origin of languages” (1789). In Essays on the origin of languages and writings related to music, trans. J.T. Scott. London: J.M. Dent & Sons, 1990: 289-332.
Principal Bibliography:
Kitty Ferguson, e Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space (New York: Walker & Company, 2008)
Murray Barbour, Tuning and Temperament: A Historical Survey (Mineola, NY: Dover, 2004)
William A. Sethares, Tuning, Timbre, Spectrum, Scale (Madison, WI: Springer, 2005) Mark Lindley, “Temperaments”. Grove Music Online (2001)
H. D. Cohen, Quantifying Music: e Science of Music at the First Stage of the Scientific Revolution, 1580–1650 (Ontario: Springer Science, 1984)
Peter Pesic, Music and the Making of Modern Science (Cambridge, MA: MIT Press, 2014)
External Links:
Music & Noise (The Physics Hypertextbook)
Sound Waves and Music (The Physics Classroom)