“Music is subordinate to arithmetics, and if one knows several fundamental experiments with harmonies and dissonances, then one knows that all its other principles depend on numbers. Music is a hidden arithmetic exercise of the mind that is not aware that it is counting. (…) In music we count only to five (…) If we were capable of a little more subtlety then we could also include the interval with the prime number 7. And I believe that it does truly exist. But there will hardly be anyone who can also use the intervals from the next prime numbers 11 and 13.” (Leibniz in a letter to Christian Goldbach from 17 April 1712, Leibniz, Opera Omnia, vol. III, ed. Dutens, Geneva 1768)
Explanations
By means of de-tuning a number of the instruments, this orchestral piece goes beyond Leibniz’s ideal of employing up to the thirteenth overtone. It intends to reach an approximate just intonation, up to the thirty-second partial of a fundamental E:
As you see above, in this piece, the partial which deviate from the tones in the twelve-tone equal temperament are rounded up to the closest 1/6-tone or 1/4-tone higher or lower, to achieve the approximate just intonation.
The instruments do not have microtones to play. They play their normal tones. But, due to de-tuning, the sounding result of the instruments will be different than normal.
The musicians should concentrate on their own intonation and rhythms. They should not pay attention and merge with the intonation and rhythms of the other instruments. This is necessary to achieve the subtle fine-tuning and rhythmical character of the piece.
The conductor should feel free to balance the dynamics to achieve the polyphonic texture of the first section (m.1-142), which is a canon. The same point extends to the homophonic texture of the middle section (m.143-187), and again the polyphonic texture of the last section (m.188-273).