TEMPERAMENT AND JUST INTONATION Even in pure i 6th century polyphony the ideal diatonic scale implies distinctions of intonation beyond the capacity of any mechanical instrument with a limited number of notes. In the Ionian mode or major scale of C the interval C—D is not the same kind of whole tone as the interval D—E, but differs as 8:9 from 9 : i o.
The normal position for the supertonic is a "major tone" (8:9) above the tonic ; but even so common a discord as the dominant 7th will set up a conflict, the dominant requiring its 5th to be as 9:8 above the tonic, while the 7th will want to make a true minor 3rd from a supertonic in the position of io:9. Such con flicts are about very minute distinctions, but every discord pro duces them if it is dwelt upon. Nevertheless, the 12 notes that human hands can negotiate within a span-stretched octave suffice to express the most chromatic harmony with less average inaccu racy than is cheerfully permitted in human singing and violin playing. Singers and violinists can and do constantly achieve a purer intonation than that of keyed instruments ; but the only aesthetic issue between free voices and tempered instruments is the difference between a human intonation liable to human error and an instrumental intonation with an inherent systematic error. The human error is often not only accidentally, but deliberately in excess of the systematic error, for the slightest vibrato is larger than the quantities involved.
The subject of just intonation is fatally fascinating to people whose mathematical insight has not attained to the notion of ap proximation. In art, as in mathematics, accuracy lies in estimat ing the relevant degree of approximation rather than in unrolling interminable decimals. Music is no more to be heard through Helmholtz resonators than pictures are to be enjoyed through microscopes. The true musical ear will recognize the real meaning of harmonies though the practical intonation confounds them with homonyms. Bach introduced no new musical thought when he arranged Das Wohltemperirte Klavier to stimulate the adoption of equal temperament by providing music in every major and minor key for which the keyboard had notes. Systems of un equal temperament tuned the commoner keys as well as possible, in the hope that remoter keys would never be visited. Bach decided that it was better to have all keys equally out of tune than to have some keys intolerable. The miraculous modulations of his Chromatic Fantasia deliberately emphasize all the chords that were "wolves" in unequal temperament, and thus Bach devoted his highest efforts of imagination to a humble practical purpose. But Marenzio had modulated as far in madrigals written in the purest golden-age polyphony. No true harmonic ideas are based on equal temperament, any more than a true geometry is based on exclusively rational quantities.