Miracle temperament

Miracle temperament

In music, miracle temperament is a regular temperament discovered by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones. Because 15:14 and 16:15 are equated, their ratio 225:224 \left(\tfrac{15}{14}\div\tfrac{16}{15} = \tfrac{225}{224}\right) is tempered out, and two secors give an 8:7 interval, a septimal whole tone. Three of these 8:7 intervals, or six secors, make up a fifth, so that 1029:1024 \left(\tfrac{3}{2}\div\left(\tfrac{8}{7}\right)^3 = \tfrac{1029}{1024}\right) is also tempered out. This gives the 7-limit version of miracle.

A septimal whole tone of 8:7 as we have seen is approximated by two secors, and a neutral third of 11:9 by three secors. In miracle, a minor third plus a septimal whole tone is also equated with the 11th harmonic. This means that the gap between a minor third plus a septimal whole tone \left(\tfrac{8}{7} \times \tfrac{6}{5} = \tfrac{48}{35}\right) and the 11th harmonic (an 11:8 ratio), 385:384 \left(\tfrac{11}{8}\div\tfrac{48}{35} = \tfrac{385}{384}\right)., is also tempered out. Miracle, therefore, is the temperament tempering out 225:224, 1029:1024 and 385:384 at the same time.

For tuning purposes, a secor of seven steps of 72 equal temperament can be used. While this also tempers out 4375:4374 (the ragisma), doing this is not regarded as a part of the definition of miracle temperament.

Miracle temperament, particularly in the distributionally even scale known as Blackjack, a scale of twenty-one notes derived from twenty successive secors, has been used by several composers, including New York composer Joseph Pehrson.

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