If ignoring extreme registers and counting only the augmented series with tempered primary intervals, one can count 144 augmentations before all the notes of the series are the octave transpositions of the fundamental (the intervals are augmented twelve times). There are 156 augmentations before the order of a given series repeats (the intervals are augmented thirteen times). The purpose of the augmentations is twofold: it proportionally diminishes the microtonal deviations when applied to equal temperament and, most importantly, creates various series and sub-series each founded on a primary interval other than the octave thus offering new consonant hierarchies.
Since one can choose to modulate from one series to another, it is important to mention exponential relationships between various augmentations. When a given series is multiplied by a positive integer, the new augmented series consists of notes ordered in the given series by the exponent of the same integer. It means that the integral augmentations result in the series consisting exclusively of the notes found in the initial series and might be, as such, viewed as sub-series of the initial series rather than new augmentations. Since each note of the series is a fundamental of a new series that is an exact transposition of the given series, and since any of these new series relates to the initial series by the order of n, 2n, 3n, 4n, 5n, etc., it follows that the smaller the n, the larger the number of “pivot” notes there is between a transposed and a given series.