Diabolus in Musica: Harmonic profiles of two Tibetan singing bowls


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I have two Tibetan singing bowls that I acquired during my travels. Each bowl is enjoyable on its own, but when the two bowls are struck or rubbed simultaneously, their composite sound is truly blissful.


Bowl 1:
  Handmade, and the rim is ever so slightly irregular in shape, causing some of its resonant frequencies to “split,” causing a curious sensation to the ear. The bowl is made of (presumably) an alloy of copper and tin that has been formed in such a way that the thin metal releases a lower frequency than one would expect from a bowl its size. It belonged to some old hippies that I met at the 2004 national rainbow gathering in northern California, and after becoming quite mesmerized by its sound, I traded them some interesting rhyolite specimens that I had collected. (It was the best kind of musical “score.”)

Bowl 2:  Much heavier than Bowl 1, and may not be hand-forged. It is a rounded “Lotus”-style singing bowl. I acquired it in Bylakuppe, Karnataka, India, where I visited the acclaimed Sera and Namdroling monasteries, which, except for Dharamsala in the north, is the second largest Tibetan settlement outside of Tibet, with approximately 70,000 Tibetans living there. I bought it from a store that donates all their proceeds to the local Tibetan primary school. The bowl is inscribed with the eight auspicious symbols around its perimeter and an endless knot inside the bowl. On the exterior bottom of the bowl is the “Om” symbol written in Tibetan script (ༀ).

SPECTROGRAM2

The fundamental frequency of Bowl 2 is about 20 cents below a Bb (461 Hz), and the fundamental of Bowl 1 is about 20 cents above Ab, but an octave lower.  The blend of frequencies is especially beautiful because the first harmonic above the fundamental of Bowl 2 (which in a circular bowl like this is a tritone above the fundamental; alpha in the figure), harmonizes with the fundamental frequency of Bowl 1, producing a lovely Major 3rd interval — Bb and its upper neighbor D.

In the figure, the fundamental frequencies (f) are followed by the following series of harmonics: (α) tritone, (β) perfect 4th, (γ) octave, (δ) tritone, (ε) minor 7th, (ζ) octave, (η) major 2nd, (θ) perfect 4th. Splitting of frequencies is evident in Bowl 1, presumably due to imperfections in its shape.

I have often wondered if the prominence of the tritone in singing bowl harmonics explains why that auspicious interval was more readily embraced by eastern musical cultures (e.g., Indonesian Gamelan), whereas western cultures that did not use singing bowls and relied more heavily on stringed and woodwind instruments (which follow the traditional harmonic series), developed a musical theory that reflected that bias. Pythagoras of Samos had no problem wrapping his mind around the ratios of string lengths, but the complex non-linear vibrations of singing bowls (e.g., Terwagne & Bush 2011) were apparently not widely encountered in the west, where the tritone and other dissonances (e.g., minor 2nd, mi contra fa) became known as diabolus in musica, “the Devil in music” (Smith 1979).

Literature Cited

Smith, F. J. 1979. Some Aspects of the Tritone and the Semitritone in the Speculum Musicae: The Non-Emergence of the Diabolus in Music. Journal of Musicological Research 3:63–74.

Terwagne D., and J. M. W. Bush. 2011. Tibetan singing bowls. Nonlinearity 24(8):R51-R66(16).

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Namdroling Monastery, AKA “the Golden Temple” — Bylakuppe, India — 2009. Photo by MR Halley

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Categories: History of Music, Natural HistoryTags: , , , , ,

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