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 A4 is unique in being equal to its own inverse. 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 and the d5 to 10⁄7, which is what these intervals are taken to be in septimal meantone temperament. In 31 equal temperament, for example, the A4, a 10⁄7, or tritone proper, is 617.49 cents, whereas a 7⁄5 is 582.51 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, 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.
Read more about this topic: Tritone
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