Documentary evidence exists in the famous postcard Carter wrote to a Boston music critic; both tetrachords
are notated, their "all-interval" properties demonstrated, and Carter states that both tetrachords
were used in both the First and Second Quartets.
I like to explain the intervals by using the tetrachord
system (Figure 1).
It was necessary to have free circulation between the notes and their subdivisions, between the kinds of tetrachords
, between the genera, between the systems, and between the echoi--hence the need for a sketch of the in-time structure .
The two anonymous treatises that it translates have become celebrated for a variety of reasons: the `poor Greek and worse Latin' (Palisca's phrase) of their titles, their system of disjunct tetrachords
and its expression in `dasian' (or `daseian') characters, and their account of what was evidently a contemporary form of polyphonic practice.
6) Some justification for a division into groups of four can be gleaned from the Greek practice of analysing extended runs of notes (`scales') as sequences of tetrachords
The five full tones of the octave are articulated by two overlapping subsystems, the triad (I, III, V) and two tetrachords
(I, IV and V, VIII).
Patterns include scales with arpeggios; scales in tetrachords
, modes and thirds; arpeggiated diatonic chords and chromatics; blues; pentatonic and whole-tone scales.
I therefore subdivide the hexachord into three tetrachords
(or succession of four notes) in order to secure purity of intonation.
3 (1971), for instance, Mead has shown how Carter extracts all of the tetrachords
formed by registrally adjacent pitches in that work's governing twelve-note chord and uses transformations of them in passages in which the twelve-note chord is not otherwise present.
Nevertheless, Byrd's 'Ave rerum corpus' provides a well-known early example, for its opening phrase uses g'-f[sharp]' (in the topmost part), then f[natural]-e[flat]-d in the bass; only e[natural] is missing from the chromatic fourth, and Byrd continues to use tetrachords
(though not chromatic ones) at various points later in the piece (indeed, the opening bass line is only one stage removed from quoting Dowland's Lachrimae theme, and the rest of the piece is virtually a meditation on that shape).
The word (one unconvertible word in Greek Character) appears the papyrus remnants provide little intelligible context for it we can be confident that musical technicalities are still in play, in view of the indubitable reference to a tetrachord
in the next line (the surviving letters are (one unconvertible word in Gre
Exercises in this book cover long tones, scales, arpeggios, ascending and descending tetrachords
, modes, scales in thirds, and arpeggiated diatonic chords for all major and minor keys followed by pairs of exercises for chromatic, major blues, minor blues, major pentatonic, minor pentatonic and whole-tone scales.