# Semitonium

In harmony, the semitonium is the ratio 17:16 — or 18:17 — between a pair of frequencies or, equivalently, the ratio 16:17 — or 17:18 — between a pair of wavelengths (or lengths of a monochord). It is the mean between unison and ditono.

The arithmetic mean between unison and ditono is

[itex] {1:1 + 9:8 \over 2} = {8:8 + 9:8 \over 2} = {17:8 \over 2} = 17:16, [itex]

which is equal to 1.0001 in binary, or 1 + 2−4.

The harmonic mean between unison and ditono is

[itex] {2 \over {1 \over 1:1} + {1 \over 9:8}} = {2 \over 1:1 + 8:9} = {2 \over 9:9 + 8:9} = {2 \over 17:9} = {2 \cdot 9 \over 17} = 18:17 , [itex]

which is equal to 1.0000111100001111000011110000111100001111... in binary.

The ratio 18:17 is the inversion of the eptadem maius (major seventh) (17:9), viz.

[itex] {2 \over 17:9} = {2 \cdot 9 \over 17} = 18:17 . [itex]

In Pythagorean tuning, the semitonium is equal to the ratio 256:243 (which is specifically called limma), i.e.

[itex] {2^8 \over 3^5} = {256 \over 243} [itex].

The Pythagorean diatonic scale has five toni, each of ratio 9:8, and two semitonia, each of ratio 256:243. Multiplying all of these together yields

[itex] \left( {9 \over 8} \right)^5 \times \left( {256 \over 243} \right)^2 = {3^{2 \times 5} \times 2^{8 \times 2} \over 2^{3 \times 5} \times 3^{5 \times 2}} = {2^{16} \over 2^{15}} = 2 [itex]

which is diapason exactly.

The semitonium is also called minor second, or semitone.

A tone is equal to a pair of semitones. That is, a tonus can be composed by joining together a pair of semitonia:

[itex] {18 \over 17} \times {17 \over 16} = {18 \over 16} = 9:8 [itex],

but notice that the semitonia are slightly unequal.

Of the two ratios given above for the semitonium, the ratio 18:17 is closer to the minor second of equal temperament. The reason is that, given that an octave should equal twelve semitones, then both
(17/16)12 and (18/17)12 should be close to 2, but (18/17)12 is closer:

[itex] \left( {18 \over 17} \right)^{12} = 1.98555995207 [itex]
[itex] \left( {17 \over 16} \right)^{12} = 2.06988999178 [itex]
[itex] {2 \over (18/17)^{12}} = 1.00727253182 [itex]
[itex] {(17/16)^{12} \over 2} = 1.03494499589, [itex]

and 1.00727 < 1.03494, so that the ratio 18:17 better approximates the ideal semitone.

It is possible to combine 18:17 and 17:16, so that there are ten 18:17 semitones and two 17:16 semitones:

[itex] \left( {18 \over 17} \right)^{10} \times \left( {17 \over 16} \right)^2 = 1.9993725014 [itex]

which is extremely close to perfect diapason: the result is equal to 1199.4567 cents, less than one cent from a perfect octave. Also,

[itex] {18 \over 17} < 2^{1/12} < {17 \over 16} [itex]

where 21/12 is exactly 100 cents: the semitone of equal temperament.

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