# Ditonus

In harmony, the ditonus (Latin for double tone — plural ditoni) is the ratio 5:4 (sesquiquartum) between a pair of frequencies or, equivalently, the ratio 4:5 between a pair of wavelengths. It is the arithmetic mean of unison and diapente.

It is 1.01 in binary — 1 + 2−2 — and it is the inversion of the exadem minus (minor sixth) (8:5), which means that it is equal to diapason divided by the exadem minus:

[itex] {2:1 \over 8:5} = {2 \cdot 5 \over 8 \cdot 1} = 5:4 \ .[itex]

The exadem minus is the sum of the first four reciprocals of triangular numbers:

[itex] {1 \over 1} + {1 \over 3} + {1 \over 6} + {1 \over 10} = {8 \over 5}. [itex]

The ditonus is approximately equal to 4 semitones of equal temperament:

[itex] \lg (5:4) = 0.3219 = {3.8631 \over 12} = {386.31 \over 1200} [itex]

Four semitones is equal to 400 cents in equal temperament, but the ditonus in just intonation is 386.31 cents, which makes a error of −13.68 cents with respect 4 semitones of equal temperament.

Three ditoni are slightly less (−41.05 cents) than a diapason, since

[itex] \left( {5 \over 4} \right)^3 = {125 \over 64} [itex]

is three ditoni, and is comparable to a diapason, which is 128/64. The ratio between three ditoni and diapason is 128/125, which is called diesis. Notice that three ditoni should equal a diapason exactly: ditonus has four semitoni, whereas diapason has twelve semitoni, and four goes into twelve three times. In a constructed scale the discrepancy of the diesis would be solved by letting the three ditoni which subtend the entire diapason be slightly unequal, then one or two of the three ditoni would have a ratio of 5:4, but not all three.

A ditonus is also called a major third, but notice that here it has been defined as a simple ratio, such as is used in just intonation.

Meantone temperament emphasizes the use of ditoni perfectly tuned in the ratio 5:4 = 386.31 cents. Meantone temperament also shifts the emphasis slightly away from perfectly tuned diapentes. A ditonus interval, and its corresponding major trichord, sounds better in meantone than in equal temperament, but only for a given set of possible root notes.

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