Quarter-Tone Harmony/Modulations

With twelve new notes comes twenty-four new keys, twelve major and twelve minor. Key signatures can be made to accomodate these new keys, but they will not be used for the sake of brevity.

Importantly, the keys made by the new notes don't share any notes with any conventional key. This means that if we want to modulate from a conventional key to a new key (or vice versa), in the words of Wyschnegradsky, "a progression of modulation passages by means of transitory chords will not bring us any closer to the key to which we want to modulate. To do such a modulation, the progresion must move suddenly from the chord of a semitone scale to a chord of the quarter-tone scale (or vice versa)."

Basic Modulations

Modulations from a conventional key to a new key are smoothest when all voices move by a single quarter tone, rather than three or five:

$\fixed c' { \partial 2 <c e g>2^"C major" <bqs, eqs gqs>2 <bqs, dtqs ftqs>2 <eqs eqs gqs>1^\markup{"E"\semisharp" minor"} }$

All notes in the C major chord individually move up or down in single quartertones: the C to B, the E to E, and the G to G.

Here are some promising examples by Wyschnegradsky in four parts, filling in the gaps using conventional theory:

$\new PianoStaff << \new Staff { \clef treble << {g'2^"C maj." a' aqs' aqf' gqs'1^\markup{"E"\semiflat "maj."} \bar "||" c''2^"C maj." cqs'' cqs'' bqs' bqs'1^\markup{"E"\semisharp "maj."}} \\ {e'2 e' eqf' dqs' eqf'1 e'2 eqs' eqs' dtqs' eqs'1} >> } \new Staff { \clef bass << {g2 fs fqs fqs eqf1 g2 gqs ftqs ftqs gtqs1} \\ {c2 c cqf bqf, eqf,1 c2 bqs, aqs, bqs, eqs1} >> } >>$