Musical interval

The successive execution of intervals can be described by addition or subtraction. The corresponding frequency ratios are multiplied or divided.

For example:

• Addition: minor third + major third = fifth or subtraction: fifth - minor third = major third.
• In cents: 316 cents + 386 cents = 702 cents or 702 cents - 316 cents = 386 cents.
• Frequency ratios: ^6/5-5/4=3/2 or ^3/2:^6/5=5/4.

The frequency ratios of the intervals behave exponentially. Therefore, the size of an interval is calculated logarithmically.

Main article → Music theory in ancient Greece → The tonal families

According to the legend of Pythagoras in the forge, he defined the intervals central to tonality as integer frequency proportions of lengths of vibrating strings of a monochord:

• Octave (frequency): 2:1 (octave up when length is halved)
• Fifth (frequency): 3:2 (fifth up at two-thirds of the length)
• Fourth (frequency): 4:3 (octave 2:1 up, then fifth 3:2 down, so: ^2⁄1 : ^3⁄2 = ^4⁄3).
• Whole tone (frequency): 9:8 (fifth 3:2 up, then fourth 4:3 down, so: ^3⁄2 : ^4⁄3 = ^9⁄8).

He did not consider the major third (5:4), but an interval consisting of two major whole tones larger by the syntonic comma (81:80): the ditone (81:64). If one subtracted the ditonus from a pure fourth, the leimma remained (256:243). With these intervals it was not possible to form a stable harmonic triad, so that ancient Greek music did not yet develop harmonics in the later European sense. Only Archytas and Didymos determined the major third (5:4), Eratosthenes the minor third (6:5).

The Pythagoreans only allowed intervals that could be calculated as whole-number ratios. They found no quotient whose doubling yielded 9:8, so that they could not divide the whole tone into two equal semitones, but only into a smaller (diesis) and a larger (apotome) semitone. Also, for them, an octave was not mathematically exactly the same as the sum of six whole tones or twelve semitones, because twelve pure fifths strung together yield a slightly higher target tone than the seventh octave of the original tone. The difference is called the Pythagorean comma.

Philolaos first converted added musical intervals into multiplied acoustic proportions. This method was optimized after 1585 by Simon Stevin with an exponential function and around 1640 by Bonaventura Francesco Cavalieri and Juan Caramuel y Lobkowitz with the logarithmic inverse function. Euclid hypothetically understood interval proportions as frequency ratios without being able to measure them yet.

In contrast to the Pythagoreans, Aristoxenos defined intervals not mathematically, but acoustically as an audible "space" (diastema) between two tones of a continuous melody, in accordance with Greek musical practice. Accordingly, he assigned to each interval a certain number of fixed pitches (tones) that it comprises. Thus the fourth contained four successive tones, a so-called tetrachord. Its outer tones were later also briefly referred to as intervals, so that the term henceforth meant the distance from the first to the last tone of such a sequence of tones.

Aristoxenos practically divided the whole tone into two, three or four equal subintervals. The different combination of semitones and whole tones within a tetrachord resulted in its genus (tonal gender: diatonic, chromatic or enharmonic). Two tetrachords following one another at intervals of a whole tone produced different scales (modes) within the framework of an octave.