Pitch Systems

Combinatorial Tetrachords

Speculating on Milton Babbit’s “all-combinatorial hexachords” based on set theory, and on the idea of the division of the chromatic range into four mutually exclusive triads, I thought I’d come up with quadriads instead, as they represent a much used vehicle in improvisation (at least as used as the aforementioned formations).

Examples starting from G:

[G,B,D,F] – [A,C#,E,G#] – [C,Eb,Gb,Bb]

[G,B,Db,F] – [A,C,Eb,F#] – [Bb,D,Fb,Ab]

[G,B,D,F] – [F#,A#,C#,E] – [A,C,Eb,G#]

[G,Bb,Db,E] – [F,Ab,B,Eb] – [D,F#,A,C]

Nested Tonal Signatures in Messiaen’s 6th Mode

Expanding on Gilbert Nouno’s and Malik Mezzadri’s article found here:


which deals with Malik’s system of tonal signatures, I thought I’d post the ones that are found in Messiaen’s sixth mode of limited transposition (tone,tone,semitone,semitone,tone,tone,semitone,semitone – in other words the first tetrachord of a major scale repeated with a tritone steer).

In C, these are:

m3Eb (F,Gb,C,D)
m3A (B,C,F#,G#)

h3Eb (F,Bb,B,D)
h3A (B,E,F,G#)

The original article deals only with nested tonal signatures in regards to Messiaen’s third and seventh mode, so I thought this would be a nice addition as I find the sixth quite useful too…

The missing MLT + Another combinatorial formation

Well…not really, but it is a nice one. Augmented fourth steer, 8 notes, 6 transpositions:

C Db E F F# G A# B

And loads of conventional harmonies in there…(if that’s what you like)

As for the combinatorial formation, here it is:

C G Ab Eb D F# A F E B C# A#

Expressed according to set theory, the prime forms for the above would be:

0378, 0347, 0239