As a way of considering intervallic relationships and musical scales, furthermore, the tetrachord has a broad, elemental utility, since it is not derived from a chordal framework (as the major scale is conventionally regarded), nor are a single key-center or other hierarchical pitch-relationships implied within the tetrachordal building blocks.
After a brief survey of the use of alternative tuning systems in twentieth-century music, Chalmers begins his elucidation and expansion of tetrachordal theories with the arithmetic approach of Pythagoras and Ptolemy, in which intervals are expressed as numerical ratios, or as harmonic divisions - 1/2, 1/3, 1/4, and so on - of a string, and the complementary approach of Aristoxenus and his followers, who thought of musical intervals as spatial distances divisible into parts in the same way that a line can be divided with a ruler (and much as on a piano keyboard, where equal intervals span constant distances).
Chalmers retains the ancient Greek tetrachordal classifications, or genera, of enharmonic (two approximate quarter tones and a major third, ascending), chromatic (two semitones and a minor third), and diatonic (a semitone and two whole tones).
Chalmers then considers systems of scales and modes as if they evolved outward from tetrachordal modules, beginning with a thorough and succinct presentation of the ancient Greek theoretical system in its entirety, along with hexachordal, heptachordal, and pentatonic scales, and the theorist Ervin Wilson's methods of scale construction by largely symmetrical permutations and modulations of tetrachordal intervals.
Thus far, the author has treated the tetrachord as did the ancient theorists, primarily as a melodic structure; he now presents a number of systematic methods of harmonizing tetrachordal scales.