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:

${5:4+1:1 \over 2}={5:4+4:4 \over 2}={9:4 \over 2}=9:8.$

It is equal to diapente divided by diatessaron:

${3:2 \over 4:3}={3\cdot 3 \over 4\cdot 2}=9:8.$

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 minor seventh (16:9),

${2:1 \over 16:9}={2\cdot 9 \over 16}={18 \over 16}=9:8.$

Notice also that the minor seventh is a pair of diatessarons put together:

$(4:3)^{2}=16:9.\$

The minor seventh is the sum of the first eight reciprocals of triangular numbers:

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

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:

${2 \over {1 \over 1:1}+{1 \over 5:4}}={2 \over 1:1+4:5}={2 \over 5:5+4:5}={2 \over 9:5}={2\cdot 5 \over 9}=10:9.$

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

See also: semiditonus, semitonium.