Intervals

An interval in the musical sense, strictly speaking, is some quantification of the distance between any number of tones to one another. There are many ways in which one could go about naming intervals (e.g. frets on a guitar, frequency, cents). The predominant method used in music is to quantify intervals is based on the diatonic scale. Given a scale spelled, for example, C D E F G A B, we represent these members as the set of roman numerals I II III IV V VI VII, respectively. So the distance from C to E, or I to III, would be spoken as a third (e.g. E is the third of C). Define + and – as new symbols, and + and – as the familiar symbols for addition and subtraction. Then T2 + (T1– 1) = T2 + T1, and T2 – (T1– 1) = T2 – T1 , for any two scale tones. So when one says add two fourths together they mean to perform this alternate operation, such that IV + IV = VII. This last non-italicized Roman numeral is an interval. It is not necessarily the seventh of the scale it is a member of, it is the distance across two consecutive, stacked, fourths. It may also be convenient to use the Arabic numerals we are familiar with, so then in terms of addition/subtraction we would use 0 through 6 one to one for the roman numerals I through VII. So, 2 + 4 = III+V = 6 = VII, and I+I = I = 0 + 0 = 0. It’s important to remember that 0 corresponds to one I , 1→ II, 2→II, and so on to 6→7. There are two or three species of each interval, either minor and major, or diminished perfect and augmented(4ths and 5ths). Major intervals are larger than minor intervals and augmented intervals are greater than perfect intervals, which are greater than diminished intervals. In terms of a twelve-tone chromatic scale, whose elements are named 0 through 11, we can define diatonic intervals thusly: I = 0; II = 1,2; III= 3,4; IV=4,5,6; V=6,7,8; VI= 8,9; VII= 10, 11.