Researchers at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam have discovered a universal property of musical scales. Until now it was assumed that the only thing scales throughout the world have in common is the octave.
The many hundreds of scales, however, seem to possess a deeper commonality: if their tones are compared in a two- or three-dimensional way by means of a coordinate system, they form convex or star-convex structures. Convex structures are patterns without indentations or holes, such as a circle, square or oval.
Almost all music in the world is based on an underlying scale from which compositions are built. In Western music, the major scale (do-re-mi-fa-sol-la-ti-do) is the best known scale. However, there are many other scales in use, such as the minor and the chromatic scale. Besides these 'traditional' scales there are also artificial scales created by modern composers. At a superficial level, scales consist of an ascending or descending sequence of tones where the initial and final tones are separated by an octave, which means the frequency of the final tone is twice that of the initial tone (the fundamental).
By placing scales in a coordinate system (an 'Euler lattice') they can be studied as multidimensional objects. Dr. Aline Honingh and Prof. Rens Bod from the ILLC did this for nearly 1,000 scales from all over the world, from Japan to Indonesia and from China to Greece. To their surprise, they discovered that all traditional scales produced star-convex patterns. This was also the case with almost 97% of non-traditional, scales conceived by contemporary composers, even though contemporary composers often state they have designed unconventional scales. This percentage is very high, because the probability that a random series of notes will produce a star-convex pattern is very small. Honingh and Bod try to explain this phenomenon by using the notion of consonance (harmony of sounds). They connect their research results with language and visual perception where convex patterns have also been detected, possibly indicating a cognitive universal (a general cognitive property).
The research results were recently published in the scientific Journal of New Music Research. The research is part of the Vici programme 'Integrating Cognition' of the Netherlands Organisation for Scientific Research (NWO) led by Rens Bod.
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