Magic temperament
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In microtonal music, Magic temperament is a regular temperament whose period is an octave and whose generator is an approximation to the 5/4 just major third. In 12-tone equal temperament, three major thirds equal an octave, but this is only because its tempering replaces 128/125 with a unison. In Magic temperament, this comma is not tempered away, and the sequence of notes separated by major thirds continues infinitely.
The defining feature of Magic temperament is the magic comma of 3125/3072, which is tempered away by tempering each major third 5/4 slightly narrow (to about 380.4 cents) so that five of them are equal to 3/1 (an octave plus a perfect fifth). This cycle of thirds can, for instance, generate a 7-tone scale with the following interval distribution (given in cents):
0 322 381 703 762 1084 1142 1201
Note that this is only one possible concrete tuning of Magic temperament. The important property is that the major third is tempered slightly flatter than its just value of 386.3 cents so that five of them minus an octave yield a good approximation to the perfect fifth of 702.0 cents.
If the sequence of major thirds is continued, this is most logically followed by 10-, 13-, and 16-tone scales. Magic temperament is compatible with divisions of the octave into nineteen, twenty-two, and forty-one equal parts, which is to say that those equal temperaments contain good approximations of Magic temperament, and therefore a piece written in Magic temperament can be performed in any of them.
Magic temperament divides the 3:1 ratio into 5 equal divisions. Similarly, the Bohlen-Pierce scale divides the 3:1 ratio (a tritave) into 13 equal divisions.
[edit] References
Smith, Gene Ward. Tuning-math mailing list, message 10917, July 17 2004.
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