Wound Honey is my new piece for the wonderful trio terra invisus. Honey has been used to dress wounds for millenia. This new piece investigates its sugar sweetness with the astringency of iron-rich blood; its amber and red and bandange; its crystalline wax and clotting stasis.
The piece is an experiment in two different types of harmony: modal harmony and combination tone harmony. I thought I’d dig into the details. (I hope you find hand-drawn examples charming rather than exasperating: I didn't have time for the typesetting today!)
The types of modes that I’m using are symmetrical: they extend a series of intervals out from a central point to generate notes, replicating those notes in every octave. This is just how major and minor scales are formed by the circle of fifths, but here, instead of using one ‘generator interval’ (like a fifth), I’m using two. (That’s because in Just Intonation, you get more consonant intervals this way, as you can preserve the chains of 4:3 fourths and 3:2 fifths that run through the scales.) These scales are what Erv Wilson calls ‘constant structures’ and they are rather useful: they create modes with consistent intervals that can still generate complex harmonies.
This piece uses four such modes, in pairs separated by an equal-tempered tritone. This tempered interval allows for complex modulations and tonality flux, as it generates intervals that are very close to another but have radically distinct tonal implications. In the example on the left, the top note shifts by 18¢ in two pairs of chords. These small shifts support a dramatic reharmonisation around the tritone axis. Shifts of this type occur several times at critical moments of Wound Honey.
Contrasting with this cyclical, symmetrical modal approach is a use of combination tones and the harmonic series. Combination tones are the notes caused by the combination of two frequencies. They are acoustic phenomena audible when two notes are played with sufficient clarity and volume. Any dyad has two combination tones, its sum and difference tones. For example, two notes in a ratio 9:8 have combination tones of 1 and 17. Dyads are thus projected into four note chords that share a harmonic fundamental. A key aspect of writing this piece was hunting for good difference tone chords. You can hear one of the files I created to explore this here:
This piece features two difference-tone chorales as its second and fourth movements, in which dyads are presented in two ways: first with a modal harmonisation, then with their difference tones. (Though sometimes, for voice leading reasons, the modal harmonisation is omitted.)
One challenge in all this was to integrate the piano. The piano, of course, is constrained to twelve-tone equal temperament. To allow it to interact with the harmonic world of the piece, I researched which harmonics were closest to equal temperament. These were mainly derivations of 17- and 19-limit tunings. As such, I started to integrate these minor harmonies alongside the modal harmonies I use elsewhere. In particular, I noted that the 11-limit harmonies in the neutral mode I was using stack well with the 19-limit harmonics of the same fundamental. This resulted in a hybrid modal system, where the neutral mode on D was combined with a minor scale on C.
This had implications for chorale sections. The final chorale section primarily uses 17- and 19-limit difference tones, allowing the piano to take a more prominent role in the harmony. In addition to the dyads and difference tones, this section uses the 10 harmonics closest to equal temperament in the harmonic series of the fundamental of the chord in question. This allows the piano to fill out the harmonic series in elaborate arpeggios, creating an increasingly dense sound world towards the end of the piece.
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