Tri-tone - Size in Different Tuning Systems

Size in Different Tuning Systems

In 12-tone equal temperament, the A4 is exactly half an octave (i.e., a ratio of √2:1 or 600 cents; play). The inverse of 600 cents is 600 cents. Thus, in this tuning system, the A4 and its inverse (d5) are equivalent.

The half-octave or equal tempered A4 and d5 are unique in being equal to their own inverse (each to the other). In other meantone tuning systems, besides 12-tone equal temperament, A4 and d5 are distinct intervals because neither is exactly half an octave. In any meantone tuning near to 2⁄9-comma meantone the A4 will be near to the ratio 7⁄5 (582.51) and the d5 to 10⁄7 (617.49), which is what these intervals are taken to be in septimal meantone temperament. In 31 equal temperament, for example, the A4, or tritone proper, is 619.35 cents, whereas the D5 is 580.65 cents. This is perceptually indistinguishable from septimal meantone temperament.

Since they are the inverse of each other, by definition A4 and d5 always add up to exactly one perfect octave:

A4 + d5 = P8.

On the other hand, two A4 (proper tritones) add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:

A4 + A4 = P8.

In quarter-comma meantone temperament, this is a diesis (128/125) less than a perfect octave:

A4 + A4 = P8 − diesis.

In just intonation several different sizes can be chosen both for the A4 and the d5. For instance, in 5-limit tuning, the A4 is either 45/32 or 25/18, and the d5 is either 64/45 Play or 36/25, or 1024:729 Play. The 64:45 just diminished fifth arises in the C major scale between B and F, consequently the 45:32 augmented fourth arises between F and B.

These ratios are not in all contexts regarded as strictly just but they are the justest possible in 5-limit tuning. 7-limit tuning allows for the justest possible ratios (ratios with the smallest numerator and denominator), namely 7/5 for the A4 (about 582.5 cents, also known as septimal tritone) and 10/7 for the d5 (about 617.5 cents, also known as Euler's tritone). These ratios are more consonant than 17/12 (about 603.0 cents) and 24/17 (about 597.0 cents), which can be obtained in 17-limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600.0 cents.