In my contemporary work, I treat these tetrachords
as found artifacts, rather as I have approached al-Ma'arri's translated words.
The major and melodic minor scales are printed as whole notes, eighth and sixteenth notes, tetrachords
, modes and in thirds.
And it's not too difficult to see his frequent use of tetrachords
containing all intervals and hexachords containing all triads--and most recently, twelve-note chords--repeated without transposition and in key positions as an acknowledgment of tonality.
Most of the pitch analysis focuses on set theory, beginning with the three primary sets that pervade all of Carter's music from this time period (the two all-interval tetrachords
and the all-trichord hexachord).
No significant relationships regarding tetrachords
or hexachords appear in the associative harmony, but the dyadic content offers even more diatonic references.
His atonal language of the time relied heavily on two pitch-class sets, the all-interval tetrachords
[0,1,3,7] and [0,1,4,6].
In it he discusses the problem of acoustics and elementary theory about tetrachords
At this point I show the student on the keyboard how to move around the circle of fifths using the tetrachords
before drawing it for them, relating the key signatures as I go.
It is a sort of iterative operator, which starts from the lower category of tetrachords
and their derivatives, the pentachord and the octochord, and builds up a chain of more complex organisms, in the same manner as chromosomes based on genes.
This group works by marked points within the twinned totals of 108 that represent -- but again proportionally -- perfect musical intervals seen either as "a group of poems, differentiated at each end by some structural mark (the ends of a sequence, say, or the ends of a series of poems in the same stanzaic pattern)" (71) along a monochord or, inversely, as harmonic divisions of the whole such as "an octave that could be divided by similar means into tetrachords
, or tones, by counting equal divisions of the whole -- under this scheme, the fifth would occur seven-twelfths of the way through the group" (71).
The pitch materials of paragraph 2 are all pentachords or tetrachords
, and the interval contents of all the pentachords are identical, as are those of the hexachords.
The fourth and fifth octaves were obviously tuned, and also intervals deriving from non-musical methods (equidistant measuring) were considered to form probably tetrachords
similar to those known from ancient as well as early medieval sources (e.