End Notes

1 Jean-Philippe Rameau, Treatise on Harmony, translated, with introduction and notes, by Philip Gossett, (New York: Dover Publications, Inc., 1971).
2 Hugo Riemann, Musikalische Logik (Leipzig, 1873; reprint, Präludien und Studien: Gesammelte Aufsätze zur Asthetik, Theorie und Geschichte der Musik, III, Leipzig: Hermann Seeman Nachfolger, 1900-1901); idem, Skizze einer neuen Methode der Harmonielehre (Leipzig: Breitkopf & Härtel, 1880; reprint, Handbuch der Harmonielehre, Leipzig, 1887 and Leipzig, 1912).
3 Linear does not necessarily equate to stepwise, and neither linear nor stepwise equates to non-functional. For example, IV-V involves stepwise root movement-as well as potential stepwise motion between other chord members-and yet is functional, when acting as pre-dominant followed by dominant; a tritone substitute dominant chord followed by tonic replaces harmonic root movement of a fifth with stepwise root movement (e.g., G7-C becomes Db7-C) so that although stepwise root movement is involved, the relationship between the chords is functional.
4 Quartal voicing of harmonies is frequently used for non-functional chords. Although they are of non-tertian construction (thereby contradicting the earlier claim that non-functional harmony generally consists of traditional chords built in thirds), such structures may be aptly named "non-functional" because the chord symbols supplied by the composer continue to indicate familiar tertian structures: quartal voicing generally is an artistic choice made by the performer. (When used for a functional chord, voicing the harmony in fourths lends ambiguity to its role, taking a step away from typical functional harmony by departing from tertian construction.)