In 500-something BC, the Greek philosopher/mathematician/astronomer/ musician/etc. etc. Pythagoras noticed interesting sounds coming from some hammersmiths. This didn’t actually happen,1 but: Pythagoras observed that different pitches resulted from hammers’ different sizes/weights. Smaller hammers sounded higher, larger ones lower. Examining their relative weights in detail, he discovered ratios that corresponded with the hammers’ chords/intervals (2:1 resulted in an octave, 4:3 in a fourth, 3:2 a fifth, etc.). Applying these ratios to a monochord (1-string instrument with a movable bridge – and this did actually happen), he showed that the ratios applied universally. Then he used the ratios to construct scales2 that sound familiar, even beautiful to us still today. Here we see the beginnings of Western pitch structure, arising from observation of nature, analysis of its mathematical foundations, and application of that math experimentally. This connection between math and the physical world became a fundamental point for Pythagoras, of religious-level importance. Math was a bridge between the conceptual and the physical, and music arose from math, so music was showing us something about physical reality. Understanding this offered a way to project our reasoning even into faraway space, to understand the universe.
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1. That or the details of the story got jumbled by the time it was written down – the oldest account we have was written about 700 years after Pythagoras, and I don’t think anyone takes it literally – but anyway somehow he discovered these ratios.
2. Diatonic ones. [This is slightly technical, feel free to skip!] The focus was on 2 tetrachords within an octave, so 2 perfect fourths (4:3) separated by a whole tone (9:8), like C-F, G-C, and then there were options on how to fill in the tetrachords with 2 other notes each. That’s 7 pitches to the octave, just like our modes (any white-key piano octave). What we now call “Pythagorean Tuning” came much later, but it is a chromatic scale: multiply a root by 3:2 to get a fifth, multiply that pitch by 3:2, repeat until you have 12 pitches across 7 octaves, then divide the higher notes by 2:1 repeatedly as needed (i.e. transpose them down) to bring them all into one octave. The intervals sound fishy to the modern ear, chords get a bit wild, the last interval has to be adjusted to preserve the perfect octave, and countless adjustments would be made over the next 2000+ years before we got Bach’s well-tuned chromatic scale... but it’s incredible how familiar it sounds.
He ran a secretive operation as he developed his ideas3 and attracted followers, and we have no writings directly from him. But his following outlived him, and thankfully they eventually started talking. Around 390 BC, the Greek philosopher/mathematician/educator Plato went on a long journey to study with the Pythagoreans, and his writings after that point reflect their influence. In the deeply weird dialogue Timaeus, Plato describes a divine artisan (dēmiourgos = demiurge) using combinations of ratios to bring different aspects of physical reality into being. A certain Pythagorean musical scale is the tool with which the artisan creates the soul of the world. It’s a creation myth based on that idea about math/music bridging the conceptual and the physical. Our reality is an expression of underlying mathematical structures. Timaeus includes a beautiful description of the planets’ orbits set in place according to ratios, and just as music arises via ratios from hammers or monochords, sure why not, it arises from planets. Plato’s Republic, ends with “the legend of Er,” rich with Pythagorean outlook, in which a soldier named Er is killed and returns from the afterlife with knowledge of the grand workings of the universe. He describes having seen the 8 orbits of the heavenly bodies and the music they make by moving in perfect ratios in relation to one another: “The eight together form one harmony.”
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3. Including ethical vegetarianism, acceptance of women as students and possibly as teachers (both very much against the norm), the idea of the soul being separate from the body (a popular idea among later Greeks and then Christians), transmigration of those souls into humans, animals, and even plants, lots of strict rules on behavior/etiquette, the triangle thing of course, the belief that Earth is spherical, something about beans (it’s a bizarre tangent, you’ve been warned), and more, but I’m sticking to the music of numbers here.
Such ideas kept stimulating thinkers through the centuries. Around 150 AD, Greco-Roman mathematician/astronomer/geographer/ music theorist Ptolemy wrote his treatise Harmonics, using the already ancient Pythagoreans as a foil, highlighting problems with Pythagorean tuning, but in the process also obliquely admitting an inescapability of Pythagorean thought, offering his own tweaks to it and expanding on the idea of musical ratios paralleling those of the human soul, and of the heavenly bodies (this in a music theory treatise!). Ptolemy’s astronomical writings, which became the standard for the next 1400 years, describe the heavenly bodies as being fixed on nested spheres. In 500-something AD, Roman senator/philosopher/music theorist Boethius codified Pythagorean ideas in his De Musica, dividing music into 3 types: Musica instrumentalis (literally “instrumental music” but just meaning “audible music”), Musica humana, which was the “music” of the human body and spirit, basically about healthy balance (ratios!), and Musica mundana, the “music of the cosmos.” The latter included the music of Earth’s seasons, and the music of the elements (earth, wind, fire, air), which arose from proportional harmonic relationships among each. And then there were the heavenly bodies fixed on distant invisible spheres, each spinning at its own speed, in specific ratio-relations to one another, their music arising from moving in those ratios.4 In time, this would be called “the music of the spheres,” 5 which concept became popular, even among hard-nosed astronomers, even a thousand years later.
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4. Incidentally there’s a whole thing about can you hear this music, can you not, is it inaudible because it’s so ever-present from birth to death that we can’t distinguish it, is it meant to be perceived not with the ear but with the mind (this is what Aristotle argued), on and on.
5. In German it’s Sphärenklänge, and you know that 1868 Josef Strauss waltz!
In 1619 AD, the German astronomer/mathematician/philosopher/ musical thinker/“first astrophysicist” 6 Johannes Kepler wrote his Harmonices Mvndi, the “Harmony of the World.” 7 He had adopted the heliocentric planetary model from Copernicus,8 and improved its accuracy via better planetary observation data. His calculations of the planets’ maximum/minimum speeds in their newly demonstrated elliptical orbits looked to him like musical ratios, and because it was the 1600s and music notation was great, he could write them down as actual notes. He transcribed notes for each planet, the Earth’s, for example, being a sad little half-step up, then back (he says this “mi-fa-mi” is an appropriate representation of Earth, where MIsery and FAmine prevail, and I don’t know if it’s a joke). It’s no surprise that humans make scales and organize sounds into music, he says, as we are simply imitating the Creator and acting out “a certain drama of the ordination of the celestial movements.” Pythagoras looms large here. While Kepler too uses him as a foil in his writing, making improvements to those old ideas, he also speaks of being “thoroughly warmed by taking a fairly liberal draft from that bowl of Pythagoras.” I mean, Kepler has now literally notated some music of the spheres, and describes the celestial choir formed by the planets, assigning vocal ranges to each (Mercury sings soprano, Mars is a tenor, etc.). Imagine this choir from the viewpoint of Kepler’s musical milieu – that lush, sophisticated 17th century polyphony. His contemporary Shakespeare must have imagined similar celestial polyphony:
How sweet the moonlight sleeps upon this bank!
Here will we sit and let the sounds of music
Creep in our ears; soft stillness and the night
Become the touches of sweet harmony.
Sit, Jessica. Look how the floor of heaven
Is thick inlaid with patens of bright gold.
There’s not the smallest orb which thou beholdest
But in his motion like an angel sings,
Still quiring to the young-eyed cherubins;
Such harmony is in immortal souls
But whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it.
– Lorenzo, wooing his girl with Pythagorean poetry (Merchant of Venice)
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6. –Carl Sagan!
7. Which incidentally 330+ years later inspired Hindemith’s powerful yet underperformed late opera Die Harmonie der Welt, which uses a planetary model to structure its scales and harmonic relations. He wrote a 3 movement symphony from the same material, and maybe you could guess the titles: Musica instrumentalis, Musica humana, and Musica mundana
8. Pythagoreans believed that instead of a stationary Earth around which other planets revolved, Earth revolved like other planets, around a “central fire.” Copernicus cited this when writing about his new heliocentric model, upending Ptolemy’s longstanding geocentric model (1400 years after Ptolemy and 2000 years after Pythagoras).
Looked at individually, most Pythagorean ideas feel like fascinating history, or whatever a² + b² = c² feels like to you. But because he believed in crossing disciplines freely, in enriching each area of investigation with the other, and in knowing about things even beyond direct reach via application of universal principles, all of which also describes everyone else I’ve mentioned here too, he managed to charge this one particular concept in a way that has kept it fresh across the millennia, growing and changing as it stimulated more and different minds. I think of it like this (and get a little tingle): all is number, number is music = All is music.
But Pythagoras himself was primarily a natural scientist/philosopher, in whose thought-periphery music was usefully swept up. What happens when such a thinker is primarily a composer?
My next post will be on Xenakis.