Diversity and Semiconvergents in Pythagorean Tuning

Several approaches to building generalized Pythagorean scales provide interpretation for the rational approximation of an irrational number. Generally, attention is paid to the convergents of the continued fraction expansions. The present paper focuses on the sequences of semiconvergents corresponding to the alternating best one-sided approximations. These sequences are interpreted as scale lineages organized as a kinship. Their properties are studied in terms of the two types of tones and elementary intervals, since each scale contains the tones of the previous scale plus the newly added tones, i.e., the generic diatones and accidentals. For the last scale of a lineage, the octave is regularly subdivided by sections, separated by a single elementary interval of the other type. Lineages are therefore related to the scale diversity with regard to their generic diatones and accidentals, which is analyzed from the Shannon diversity index, either for tone abundance or interval occupancy.