Such simplicial sub-melodies have much more information for melody composition, as they contain a pitch direction (up, down , steady) as pitch moves for each chord. They have also the advantage that they utilize the 66% of the chord, thus there is less indeterminacy in the relation melody-chord.
These pairs of notes are usually even number odd odd number notes of the full melody, in other words they need extension with interpolated notes.
To compose the full melody we interpolate notes of the 6-notes scale Celtic minor between the notes of the simplicial sub-melody so as to have intervals of 2nds, and so that the interpolated note (usually outside the chord) has duration 1/2. or 1/3 of the corresponding note of the simplicial submelody. In this way we are certain the the derived melody fits to the underlying chord.
The pitch direction (up, dowm , steady) shows the topological pattern of the simplicial melodic themes. Of course a full melodic theme may involve more than one chord, e.g. 2 chords with the characterization of the transition relation as chromatic , melodic and harmonic.
We give an example here with the Irish folk song Frost in Celtic minor (6 notes scale g,a,b,c,d,e)
The chord progression and structure of repetitions is 4 times the
Am Am
Am G
Am G
Am Am
And then 2 times the
Am G
Am G
Am G
Am Am
G Am
Am G
Am G
Am Am
And then all these 2 parts 3 times
The arpeggio interval simplicial submelody for the first part and for each chord is the
Am(e5->c5) Am (c5->a4)
Am(a4->e4) G(g4->b4)
Am(c5-c5) G(b4->e4)
Am(a4->c5) Am (c5->e5)
And for the 2nd part is
Am(a4->c5) G(e5->b4)
Am(e5->a5) G(g5->d5)
Am(e5->a5) G(b5->g5)
Am(a5->e5) Am(e5->a5)
G(g5->d5) Am(c5->a4)
Am(e4->a4) G(b4->d5)
Am(a4->a4) G(d4->d4)
Am(a4->a4) Am(a4->a4)
To compose the full melody we interpolate notes of the 6-notes scale Celtic minor between the notes of the simplicial sub-melody so as to have intervals of 2nds, and so that the interpolated note (usually outside the chord) has duration 1/2. or 1/3 of the corresponding note of the simplicial submelody.
