Periods and generators: Difference between revisions

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A '''generator''' or '''generating interval''' is an interval which is stacked repeatedly on top of itself to form a scale, an EDO, or part or whole of a tempered structure. The resulting chain of stacked generators is often called a '''generator chain'''. Although they may seem somewhat abstract at first, generator chains are fairly simple to understand, and are important in developing an intuitive understanding of regular temperament theory.

The simplest and most intuitive example of a chain of generators is an [[Equal_Temperaments|equal temperament]], such as the familiar [[12edo|12EDO]]. An equal temperament, regardless of whether it subdivides the octave, the tritave, or anything else, is itself a single generator chain. For example, 12EDO is generated by the 100 cent interval stacked on top of (and below) itself ad infinitum. Likewise, 5EDO is generated by creating a generator chain of 240 cent intervals, which generates the entire temperament. Nonoctave equal temperaments, such as [[13edt|13EDT]] (or the "Bohlen-Pierce" scale) are also formed by a single generating interval, which in the case of BP is 146 cents.

As is made apparent by the above example, 7 consecutive notes out of this chain will yield all of the pitches of the familiar diatonic scale. However, if we stick to the Pythagorean chain of fifths, these pitches will be spread out as a stack of fifths, rather than being octave-reduced and forming the familiar LLsLLLs diatonic scale pattern. If we want to octave-reduce the notes in this chain, we need to be able to move around by 1200 cents. Since 1200 cents doesn't exist in the Pythagorean circle of fifths, we need to create it as a new "prime interval" and hence create a second generator chain of octaves.

In the case of the above example, the addition of 2/1 as a second prime interval will allow us the additional degree of freedom needed to turn the Pythagorean chain of fifths into the Pythagorean diatonic scale. The operation to do so is trivial: simply take seven consecutive fifths out of the chain, and then reduce each note to the octave. In this case, we obtain 0 = 204 = 408 = 498 = 702 = 906 = 1110 cents, which in diatonic notation spells C-D-E-F-G-A-B.

Hence, in the above example, the Pythagorean diatonic scale is generated by two different intervals, which are the 3/2 and the 2/1, and the 2/1 is the period.

It is possible to construct scales in which the period isn't the octave. Of these, the most common are scales in which the period subdivides the octave, which are often called "symmetric scales" in 12-equal music theory.

It should be noted that in the above example, 300 cents is itself generated by 100 cents, which means that strictly speaking, it isn't a second "prime" generator interval. The choice of 100 cent generator was chosen for simplicity, but this isn't true of all tunings for this scale - for example if our generator were 91 cents instead of 100, then the 300 cent period wouldn't consist of 3 stacked generators anymore.