Music Of The Spheres Examples

[Cherie Trojak]

Pythagoras (569-490 BC) established that the octave (the ratio 1:2 of a string's length) was the fundamental musical interval and saw it as a sign that nature itself, including the planetary orbits ("the spheres"), is governed by numerical ratios. "There is geometry in the humming of the strings and there is music in the spacing of the spheres," he concluded. In "Mysterium Cosmographycum and Harmonices Mundi," Kepler (1571-1630) further developed Pythagoras' ideas and suggested that the planets of the solar system are spaced according to specific geometries and produce tones as they orbit the sun.

We discovered an equation that relates the spacing of the orbits of the solar system. It is depicted in Figure 1 and emphasizes both mirror and scaling features. We obtained it by using mathematical metrics derived from the popular 12-tone music tuning systems, which divides the octave into 12 notes.

We focused on the most distinctive feature of the solar system: its mirror-reflection architecture with respect to the asteroid belt located between Mars and Jupiter. This feature was first studied by Geddes and King-Hele in 1983.

The solar system is composed of four inner terrestrial planets (Mercury, Venus, Earth and Mars) and four outer gas giant planets (Jupiter, Saturn, Uranus and Neptune) separated precisely by the asteroid belt. Four mirrored planetary pairs can be readily identified using the asteroid belt as the mirror point: Jupiter-Mars, Saturn-Earth, Uranus-Venus, and Neptune-Mercury.

We also found that the same relation relates the orbital radii that correspond to the two main Kirkwood asteroid gaps that are generated by the 3:1 and 7:3 resonances of Jupiter, which indicates that the planetary self-organization of our solar system is due to Jupiter. Finally, we can postulate two additional pairs of gravitational rings that complete the structure of the solar disk: They include, on the outer side, the Kuiper Belt and the scattered disk, respectively, and, on the inner side toward the sun, the hypothesized Vulcanoid asteroid belt and the empty region surrounding the sun.

By taking a as the length of the semi-major axis of an orbit, the found scaling equation linking together all planetary pairs is depicted in Figure 1. For each ratio, the figure also shows the value (val.) and its accuracy (acc.) relative to the ratio 9/8 = 1.125. We found that each planetary-pair ratio differs from the ratio 9/8 by no more than 1 percent, which suggests that the found equation is quite robust. The proposed equation appears "elegant" and is clearly suggestive of a planetary "order," that is, a "kosmos," as Pythagoras would have called it.

The fundamental ratio of the planetary scale is 9:8. This ratio is important in the history of mathematics and music because it is the Pythagorean epogdoon, which is displayed by Raphael in a famous detail of his masterpiece "The School of Athens."

The musical interpretation of our model and its robustness in predicting the relative spacing among the planets of the solar system is further detailed in our published paper. For example, the scaling involving consecutive powers of 2 musically corresponds to tones on subsequent octaves. We also found that by using the same metrics, the ratios of the orbital radii of neighboring planetary pairs elevated to the 2/3 power correspond to four musical "consonances" having frequency ratios of 5/4 (Major Third), 4/3 (Perfect Fourth), 3/2 (Perfect Fifth) and 8/5 (Minor Sixth). The probability of obtaining this result randomly has a p < 0.001.

Musical consonances are "pleasing" tones that harmoniously interrelate when sounded together, which further suggests that the orbits of the planets of the solar system could form a gravitationally optimized and coordinated structure mimicking musical harmony. On the contrary, dissonant ratios were found when the Kirkwood asteroid gaps, the scattered disc (characterized by Eris), and the region closest to the sun were considered: Indeed, these are gravitationally divergent regions.

Thus, the discovered equation (depicted in Figure 1) appears to fully characterize the solar system and supports the idea that a harmony described by typical music tone ratios characterizes the solar system, as Pythagoras and Kepler conjectured. It performs better than any other model proposed in the literature.

This story is part of Science X Dialog, where researchers can report findings from their published research articles. Visit this page for information about ScienceX Dialog and how to participate.

As a sound designer, musician and filmmaker, much of my creative work is based on personal experience in the world, based on my own senses. I have spent a great deal of time alone in the wilderness listening to unknown animal calls and finely sculpted natural soundscapes, as well as in foreign countries that offer unexpected sonic reflections of human culture. Through the simple act of listening and observing my own physical, mental and emotional reactions to the surrounding sounds, the stories of these places, people, creatures and events began to coalesce into a pattern. This pattern was drawn from the previous theoretical structures I had learned from studying and creating films (traditional models mentioned above), but extended beyond into this dynamic model that I now call Sound Spheres.

If we consider the human experience of our environment from its most intimate to most external, a model of six concentric spheres serves to describe the various levels of sonic information available. Like the layers of an onion, an outer sphere may encompass some of the properties of inner spheres, but not as an absolute rule. As a perceptual construct of our world, this is a model to be to be explored, debated and expanded upon in relation to other audiovisual theories, as well as psychoacoustic and philosophical approaches.

Our bodies are organic factories full of vibration, friction and impacts that create sound. Many of these can only be heard by ourselves, if we even notice, as normally we are habituated to constant low level rhythms of breathing, heartbeat and even neurologically based auditory stimuli like tinnitus. However, slightly louder bodily functions become audible to us and to those around us, sometimes with unintended, embarrassing results. Speaking and clapping are more obvious sounds we make for the purpose of communicating with others. This sphere represents the interface between the very personal, private and personalized arena of sound making and that of interaction with others.

When we make contact with the outside world, manifesting our willpower through our bodily movements, this action sets up sonic vibrations. Often it is initiated by our hands, the major anatomical marvel that distinguishes us from most other animals. We have the capacity to smash materials with heavy objects, delicately finger minute particles and complex musical instruments, and communicate through sophisticated symbols on electric devices. Our whole body plays the environment like a drum set, slamming doors, pounding up stairs, sweeping the floor and turning the pages of a newspaper.

The purpose of the assignment is to have the students experience their own sound spheres, then to apply this in a possible dramatic film scene with an evolving plot. An extra benefit that frequently derives from this exercise is the opportunity to explore related audio theory and applications, which are noted below. Here are some of the results of this assignment.

NOTE: This experience highlighted the overlap between music, sound effects and dialogue in our everyday experience. Although these areas of audio are frequently separated in the production and postproduction processes of filmmaking, the application of the Sound Spheres model applies equally to all types of sounds.

Remember to leave your comments below (before Friday 5/13/11) to have them included in the discussion with David. Again, for the full article, check out Volume 1.1 of The New Soundtrack, available through Edinburgh University Press in both print and electronic document. Reprinted with permission.

Gerhard Sonnert, a research associate at the Harvard-Smithsonian Center for Astrophysics, has published a new website that allows listeners to literally hear the music of the stars. He worked with Wanda Diaz-Merced, a postdoctoral student at the University of Glasgow whose blindness led her into the field of sonification (turning astrophysical data into sound); and with composer Volkmar Studtrucker, who turned the sound into music.

Diaz-Merced lost her sight in her early 20s while studying physics. When she visited an astronomy lab and heard the hiss of a signal from a radio telescope, she realized that she might be able to continue doing the science she loved. She now works with a program called xSonify, which allows users to present numerical data as sound and use pitch, volume, or rhythm to distinguish between different data values.

Diaz-Merced plugged the Chandra X-ray data into xSonify and converted it into musical notes. The results sound random, but Sonnert sensed that they could become something more pleasing to the ear. He contacted Studtrucker who chose short passages from the sonified notes, perhaps 70 bars in total, and added harmonies in different musical styles. Sound files that began as atonal compositions transformed into blues jams and jazz ballads, to name just two examples of the nine songs produced.

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