Octave species various available intervals could be combined in any order, even restricting species to just the diatonic genus would result in twelve ways of dividing the octave (and his 17th-century editor, Marcus Meibom, pointed out that the actual number is 21), but "only seven species or forms are melodic and symphonic" . Those octave species that cannot be mapped onto the system are therefore rejected . In chant theory beginning in the 9th century, the "New Exposition" of the composite treatise called "Alia musica" developed an eightfold modal system from the seven diatonic octave species of ancient Greek theory, transmitted to
Holdrian comma fact that the Ancient Greeks (or at least Boethius) believed that in the Pythagorean tuning the tone could be divided in nine commas, four of which forming the diatonic semitone and five the chromatic semitone. If all these commas are exactly of the same size, there results an octave of 5 tones + 2 diatonic semitones, 5 × 9 + 2 × 4 = 53 equal commas. Holder attributes the division of the octave in 53 equal parts to Nicolas Mercator, who would have named the 1/53 part of the octave the "Artificial Comma". Mercator's comma is a name often