Formal Diatonic Intervallic Notation

Abstract

Numbers called quality modifiers are used to identify interval qualities: 0 numerically represents perfect, 1/2 represents major, –1/2 represents minor, and so on. These modifiers are linked with diatonic class intervals as ordered pairs that mimic common interval notation. For example, a minor third is represented by (–1/2, 2). A binary operator is constructed that allows these ordered pairs to be added consistent with our expectations. Similarly, accidental modifiers numerically identify the number of sharps or flats attached to a given note: 0 indicates no attached accidentals, negative integers indicate the number of flats attached, and positive integers indicate the number of sharps attached. These modifiers are linked with diatonic classes as ordered pairs that mimic common note names. For example, the note G$\flat$ is represented by (–1,4) and Gx by (2,4). Intervals and notes represented by these ordered pairs are said to be in MD-notation (MD for modifier-diatonic). A group action and generalized interval system are defined for intervals and notes in MD-notation. An implied quartertone system is also discussed.