Semiditonus

In harmony, the semiditonus is the ratio 6:5 (sesquiquintum) between a pair of frequencies or, equivalently, the ratio 5:6 between a pair of wavelengths. It is the harmonic mean of unison and diapente:

<math> {2 \over {1 \over 1:1} + {1 \over 3:2}} = {2 \over 1:1 + 2:3} = {2 \over 5:3} = 6:5 \ .<math>

It is equal to diapente divided by ditonus:

<math> {3:2 \over 5:4} = {3 \cdot 4 \over 5 \cdot 2} = {3 \cdot 2 \over 5} = 6:5 \ .<math>

This means that diapente is equal to a ditonus and a semiditonus, put together.

The semiditonus is 1.00110011001100110011... in binary, and it is the inversion of the exadem maius (major sixth) (5:3), viz.

<math> {2:1 \over 5:3} = {2 \cdot 3 \over 5 \cdot 1} = 6:5 \ . <math>

The exadem maius is the sum of the first five reciprocals of triangular numbers:

<math> {1 \over 1} + {1 \over 3} + {1 \over 6} + {1 \over 10} + {1 \over 15} = {5 \over 3} . <math>

The semiditonus is also called the minor third.

See also: unison, diapason, diapente, diatessaron, ditonus, tonus, semitonium.