Blackwood's diatonic behaviour of xenharmonic equal temperament

 Blackwood's Modes and Chord Progressions in Equal Tunings (1991) put the traditional functional harmony and chord progression into a new context, non-12 equal temperament, in other equal temperament numbers, nineteen, seventeen, sixteen, and fifteen. His article describes what if the subdominants, dominants, or tonic chords are applied in these ET.

Blackwood proposes two major concepts of interval property, Dissonance versus Discordance. Dissonance describes the interval property between, it is not based on the sonic experience to our ears, pure or harsh. If the interval is defined as major 3rd, perfect 5th, major 6th or octave etc. It is always described as a consonant; if verse versa, dissonant. A chord that contains a combination of tendency notes, such as a dominant seventh, a diminished seventh, or an augmented sixth; these latter are dissonant intervals/chords whatever the tuning, disregarding the how it sounds. Consonant and dissonant are paired to describe in terms the harmonic intervals. 

It is contrasting to the elaboration of Sethares (2010), measuring sonic experience by the dissonant curve. The perfect fifth can sound dissonant of the inharmonic nature of an instrument (see inharmonicity). On the contrary, Blackwood's discordance is the word that is regarded as Sethares's dissonant, describing our hearing experience. In Blackwood's example, major and minor thirds are consonant intervals described as "jangling discord" in non-diatonic behaviour with equal temperament.

Diatonic behaviour of equal temperaments

The formation of a diatonic scale contains 5 whole tones and 2 halftones in an octave. In ET, the smallest step (s) always stays at the same interval. The octave equals the number of steps of a TET in the octave. Thus, the relationship of a diatonic scale can be written as 5w+2h=n-TET. w, h and n must be interger and n>w>h>0. If an n-TET satisfies this relationship, for example, 12-TET=5(2)+2(1) and 19-TET=5(3)+2(2), this n-TET satisfies the diatonic behaviour. In 19-TET example, it can be considered as there are 3 microsteps in a whole step, while 2 microsteps in a half step. The traditional naming of a chord still can be used, but sound different.

On the contrary, 10-TET and 15-TET do not satisfy this relationship. By the sounding, we can still find traditional chords by pitch, but the pattern and notation are hugely different from what we used in the diatonic scale. It will sound extremely contrasting compared to the n-TET that satisfies the diatonic behaviour. However, 5-TET is used in gamelan music and sound consonant. It is about how to use the tool in a suitable context. It is interesting to listen to Blackwood's Suite for Guitar in 15-Note Equal Tuning using harmonic serious spectrum guitar.

In Blackwood's thesis discussing diatonic behaviour, he does not mention the sound to judge the sonic quality of the equal temperaments.

  Blackwood, E. (1991) Modes and Chord Progressions in Equal Tunings. Perspectives of new music. [Online] 29 (2), 166–200.