# Tonus

In harmony, the tonus is the ratio 9:8 between a pair of frequencies or, equivalently, the ratio 8:9 between a pair of wavelengths. It is the arithmetic mean of unison and ditono:

[itex] {5:4 + 1:1 \over 2} = {5:4 + 4:4 \over 2} = {9:4 \over 2} = 9:8 .[itex]

It is equal to diapente divided by diatessaron:

[itex] {3:2 \over 4:3} = {3 \cdot 3 \over 4 \cdot 2} = 9:8 .[itex]

This means that a diapente is equal to a diatessaron and a tonus, put together.

The tonus is 1.001 in binary — 1 + 2−3 — and it is the inversion of the eptadem minus (minor seventh) (16:9),

[itex] {2:1 \over 16:9} = {2 \cdot 9 \over 16} = {18 \over 16} = 9:8. [itex]

Notice also that the eptadem minus is a pair of diatessarons put together:

[itex] (4:3)^2 = 16:9 . \ [itex]

The eptadem minus is the sum of the first eight reciprocals of triangular numbers:

[itex] {1 \over 1} + {1 \over 3} + {1 \over 6} + {1 \over 10} + {1 \over 15} + {1 \over 21} + {1 \over 28} + {1 \over 36} = {16 \over 9}. [itex]

The tonus is also called major second.

## Major and Minor

The tonus comes in two slightly different versions: the tuono maggiore (9:8) and the tuono minore (10:9).

The tuono minore is the harmonic mean of unison and ditono:

[itex] {2 \over {1 \over 1:1} + {1 \over 5:4}} = {2 \over 1:1 + 4:5} = {2 \over 5:5 + 4:5} = {2 \over 9:5} = {2 \cdot 5 \over 9} = 10:9 . [itex]

In binary, the tuono minore is equal to 1.000111000111000111000111...

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