Interval arithmetic

Interval arithmetic systems are sets of rules regarding the names and qualities of stacked intervals. The most common rule implied by most interval naming and notation systems says that:

• an a-step interval stacked on a b-step interval is an (a + b)-step interval, and
• an a-step interval stacked n times is an na-step interval.

Diatonic interval arithmetic

Diatonic interval arithmetic is a set of rules governing diatonic notation systems, which differs in that the degrees are 1-indexed. By 1-indexed is meant that the first degree is the unison, a single step makes a second, two steps make a third, and so on. Another way to derive them from note names is to write down all the note names between the interval and count the total number of appearing note names. For example, C–E is a third because "C", "D", and "E" are three letters.

The same arithmetic rules follow if 1 is subtracted from all degree numbers. For example, a stack of two thirds is always a fifth, since (3 - 1) + (3 - 1) = 5 - 1. 1-indexed degrees are used today mostly for maintaining the consistency with the rich historical body of work.

Todo: expand, clarify Add general rules for interval quality

Figure 1: diatonic generator step markers