Formal Diatonic Intervallic Notation

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.