In ancient Greek music there were three genera (singular: genus) for classifying musical scales:

1. diatonic
2. chromatic
3. enharmonic,

diatonic being the simplest and enharmonic the most complex. The chromatic scale is an extension of the diatonic scale: it can be generated by combining the diatonic scale with a complementary pentatonic scale. The enharmonic scale is an extension of the chromatic scale, in which pairs of enharmonic notes are distinguished from each other.

The Greeks analzyed genera using various terms, including diatonic, enharmonic, and chromatic, the latter being the color between the two other types of modes which were seen as being black and white. Scales are constructed from conjunct or disjunct tetrachords: the tetrachords of the chromatic genus contained a minor third on top and two semitones at the bottom, the diatonic contained a minor second at top with two major seconds at the bottom, and the enharmonic contained a major third on top with two quarter tones at the bottom, all filling in the perfect fourth (Miller and Lieberman, 1998) of the fixed outer strings. However, the closest term used by the Greeks to our modern usage of chromatic is pyknon or the density ("condensation") of chromatic or enharmonic genera.

(ibid)

The pentatonic scale is one subset of the diatonic scale, its complementary subset being the trivial 2-tone scale, in which the octave is divided into a perfect fifth and a perfect fourth.

It is possible to generalize this concept of genus by establishing a hierarchy of genera G₁, G₂, G₃, et cetera, such that either

G_n = G_n−1 ∪ G_n−2

or

G_n = G_n−1 ∪ (G_n−1 − G_n−2).

So let G₁ be a 1-tone scale, then

G₂ = G₁ ∪ G'₁

is a 2-tone scale,

G₃ = G₂ ∪ G'₁

is a 3-tone scale,

G₄ = G₃ ∪ G'₂

is a pentatonic scale,

G₅ = G₄ ∪ G'₂

is a diatonic scale,

G₆ = G₅ ∪ G'₄

is a chromatic scale, and

G₇ = G₆ ∪ G'₄

is an enharmonic scale, or, alternatively,

G₇ = G₆ ∪ G'₅

could be a microtonal scale with 19 tones in the octave.

This microtonal 19-tone scale could be followed by

G₈ = G₇ ∪ G'₆

which would be a microtonal 31-tone scale (19 + 12 = 31),

G₉ = G₈ ∪ G'₆

which would be a microtonal 43-tone scale (31 + 12 = 43).

Examples:

• G₁ = {C}
• G₂ = {C,G} = {C} ∪ {G}
• G₃ = {C,F,G} = {C,G} ∪ {F}
• G₄ = {C,D,F,G,A} = {C,F,G} ∪ {D,A}
• G₅ = {C,D,E,F,G,A,B} = {C,D,F,G,A} ∪ {E,B}
• G₆ = {C,C#,D,D#,E,F,F#,G,G#,A,A#,B} = {C,D,E,F,G,A,B} ∪ {C#,D#,F#,G#,A#}
• G₇ = {C,C#,Db,D,D#,Eb,E,F,F#,Gb,G,G#,Ab,A,A#,Bb,B} = {C,C#,D,D#,E,F,F#,G,G#,A,A#,B} ∪ {Db,Eb,Gb,Ab,Bb}

Source

• Miller, Leta E. and Lieberman, Frederic (1998). Lou Harrison: Composing a World. Oxford University Press. ISBN 0195110226.