Pitch, Pi. and Other Musical Paradoxes (a practical guide to natural microtonality)
by Charles E. H. Lucy
Ever wondered??????? *
* How many notes there should be in an octave?
* Why the black note between G and A has two or more names?
* Why some keys are called sharp whilst others are flat?
* Why the music of some other cultures use different scales and tuning systems?
* Why conventional guitars may sound in tune for some keys, in some positions, but very out of tune in others, and how to refret them?
* Whether the musical circle of fourths and fifths should really be a spiral, a torus, or a cylinder?
* How to microtune the Yamaha DX7 MkII, TX81Z, Korg M1, Synclaviers and many Roland and Ensoniqs and other instruments have microtonal capability? (Includes MIDI tuning dump and pitchbend data)
* How pitch and colour are connected?
* How music is related to quantum physics, longitude, and cosmology?
* Why musical tuning was of paramount importance to Chinese emperors?
* Whether what the music colleges taught about harmony is based on some fundamentally false premises?
* Why some foreign music sounds out of tune?
* How musical scales are related to the only irrational and transcendental number which occurs in all cultures naturally ( pi )?
There are many mythological references to the music of the spheres, and countless learned attempts to construct a unifying theory to explain the relationship between music, physics, astronomy, and mathematics. No single theory has become generally accepted, yet the search continues to find patterns in music which, in some mysterious way, reflect patterns fundamental to the nature of the universe. Recent discoveries in quantum physics and mathematics suggest that this link is not so tenuous as it has seemed. The musical aspects of this puzzle are particularly paradoxical; for although contemporary Western society divides the musical octave into twelve equal parts (semitones), this is merely a convenient compromise to represent an underlying organisation of frequencies which are only now fully understood. Thousands of ways had been devised to split the octave into discrete intervals. Other temperaments persist in the diverse musical traditions found today as ethnic music or revivals of older scales and tunings.
Many of the alternative temperaments are well documented. One system, proposed by the British horologist John Harrison (1693-1776), is unique in that he uses the ratio of the diameter to the circumference of the circle (i.e. pi = 3.14159 26535 etc.) as the basis of his 'natural' scale.
This tuning system uses a Large interval and a small interval. By selecting permutations and combinations of these two intervals, an infinite number of notes may be computed, which can represent any possible scale and hence produce a universal musical notation, and harmonic mapping.
Using this tuning system, fretted instruments have been built with nineteen, twenty-five and thirty-one frets per octave, and synthesisers programmed, to play it. Scales and harmonic structures have been analysed to create a 'Musical Esperanto', on which any instrument may be build or adapted to play in any key or modality, using conventional Western musical notation. Using LucyTuning, the circle of fourths and fifths which equal temperament uses as its harmonic basis, have been found to be a spiral of fourths and fifths expanding octaves in fourths for flat keys, and contracting in fifths for sharp keys. The use of this scale, opens musical possibilities for the re-interpretation of existing music, and unlimited potential for new composition. Adjacent sharps and flats which are assumed to be of the same frequency in conventional harmony, may now be treated as separate pitches. This increases the tonal vocabulary as the extra altered notes may also modulate into double, triple, or more sharps or flats, giving greater pitch choice and precision, which matches natural harmonics.
ISBN 0 - 9512879 - 0 - 7. 1987 and ISBN 0 - 9512879 - 1 - 5. 1998. updated, expanded professional edition.
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