Extended Expressive Intonation: An Application of the Convergents and Semiconvergents in Pythagorean Tuning

Cyclic scales are associated with convergents and semiconvergents of the continued fraction expansions of the generator tone. After each convergent, a scale lineage ends and another begins. Along a lineage, a constant number of generic accidentals are successively added to its first scale, becoming regularly interspersed. In this way, it is easier to know where each note is to go. This process, applied to the lineage of the 7-, 12-, and 17-tone scales, is related to expressive intonation. Such a concept is extended to larger scales with added microtones and it is described how they can be chosen in terms of the starting index of the scale. An automorphism in terms of the step and co-step indices associated with the two elementary intervals provides a two-dimensional representation that shares some common features with the musical staff.