28, emphasizing its tetrachordal
design (BACH motive).
The mapping of tetrachordal lines below maintains the reflection between a row and its 3 x 4 cross-partition:
In this case, we can preserve Qs tetrachordal pitch-class sets under [I.
Example 16(b) lists the tetrachordal set classes found in the horizontal lines of the cross-partitions.
The pervasiveness of (0145) throughout the work gives validity to these tetrachordal segmentations, and their usage calls attention to the symmetrical relationships inherent in their registrations.
Example 13 lists the  earlier in the network not as an indication of chronology but as a demonstration of how the tetrachordal relationships at the end of the piece reflect those at the beginning: a concluding [T.
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).
voicings of the twenty-nine tetrachordal
set classes is an intimidating prospect.
5] These two writers disagree principally over the question of whether the piece is constructed from a chromatic tetrachord or an (0,1,2,3,6) pentachord, and von Blumroder's analysis demonstrates conclusively that Maconie is correct on this point, even if on other matters he falls short: "Robin Maconie has indeed deciphered the three tetrachordal
groups of the first metrical section, but not, however, their subsequent modification, on account of which he describes the notes C and D-flat as 'free radicals' and the pitch composition from measure 8 onward as 'difficult to follow'" (page 126).
0]P in aggregate 1 can be recomposed from the partial ordering of aggregate 7, where the disposition of pc 1 in lyne 5 and <95> in lyne 2 also admits pitch contour repetition; (2) only aggregate 6, not aggregate 8, contains tetrachordal
row segments that embed a member of sc 3-12 , thus enabling the mirrored repetition of sc's 3-4  and 3-12  across the barline.