microtonal music Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of tw ...

, magic temperament is a

regular temperament Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most c ...

whose period is an

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

and whose

generator Generator may refer to: * Signal generator, electronic devices that generate repeating or non-repeating electronic signals * Electric generator, a device that converts mechanical energy to electrical energy. * Generator (circuit theory), an eleme ...

is an approximation to the 5/4

just Just or JUST may refer to: __NOTOC__ People * Just (surname) * Just (given name) Arts and entertainment * ''Just'', a 1998 album by Dave Lindholm * "Just" (song), a song by Radiohead * "Just", a song from the album ''Lost and Found'' by Mudvayne ...

major third In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...

. In 12-tone

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

, three major thirds add up to an octave, since it tempers the interval 128/125 to a unison. In magic temperament, this comma is not tempered away, and the sequence of notes separated by major thirds continues indefinitely. Instead of 128/125, 3125/3072 vanishes in magic temperament, where each 5/4 ''major third'' is made slightly narrow (about 380 cents ()), so that five of them add up to an approximate 3/1 (an octave plus a ''perfect fifth''). A chain of these thirds can be used to generate a 7-tone scale with the following interval distribution (given in cents): :0 322 381 703 762 1084 1142 1201 Note that this represents only one possible tuning of magic temperament. The important property is that the major third is tempered slightly flatter than its just value of 386 cents, so that five of them less an octave yield a good approximation to the perfect fifth (702 cents). If the sequence of major thirds is continued, the next moments of symmetry are at 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 these equal temperaments make reasonable tunings for magic temperament, and therefore a piece written in magic temperament can be performed in any of them.

References

Further reading