1. Gioseffo Zarlino, Le Istitutioni Harmoniche, part 3, translated in Guy A. Marco and Claude V. Palisca The Art of Counterpoint: Gioseffo Zarlino. Part Three of 'Le Istitutioni Harmoniche', 1558 (New York, 1983). Full texts and graphics are available online at http://www.music.indiana.edu/smi/cinquecento.

2. Though writing somewhat earlier, both Prosdocimus and Ugolino give very accurate specifications for the monochord positions of the notes of musica ficta. Equal weight is given to the power these ficta pitches must possess both in "perfecting the consonance" and in "coloring the dissonance". The pitch "F#" therefore not only perfects the consonance with a lower B, but also colors the dissonance with a lower D. In this regard, the major third D-F# is described as a "colored dissonance". By this we must understand the "Pythagorean ditone" whose ratio is 81:64.

3. This was first extensively discussed in Margaret Bent, "Diatonic Ficta", Early Music History 4 (1984), pp. 16. The version used here is the one proposed by Bent.

4. I am now introducing a new method of notating comma inflection: each pitch is indicated by its letter name followed by a suffix consisting of an operand and a number. The notation "G" indicates the pitch of "Pythagorean G" (in any octave); while the notation "G-1" indicates "Pythagorean G lowered by a Syntonic comma". In other cases the notation "C+1" will indicate "Pythagorean C raised by a Syntonic comma", while the notation "E-2" will indicate "Pythagorean E lowered by two Syntonic commas". The purpose in using this notation is both analytic and functional. Not only does it enable each pitch to be understood in relation to the Pythagorean default against which it was deemed to exist in the world of JI, but it also instructs the computer program that generates the Audio files to provide exactly the correct pitch for the particular note employed. These pitches have been calibrated with mathematical precision, and when a comma analysis is undertaken, using this method of annotation, the results are given automatic and accurate playback through the computer program. A technical account of this process will be offered in a future article, together with a tool that will enable readers to generate accurate tuning through their own MIDI systems.

5. Jonathan Walker, "Intonational Injustice: A Defense of Just Intonation in the Performance of Renaissance Polyphony", Music Theory Online 2.6 (1996), I.10.

6. While, for example, B-flat/F forms a pure Pythagorean fifth, Walker's use of B-flat/F+ (in measure 48) injects an unacceptable and severe "wolf". The same combination occurs also in measures 49 and 51. In measure 49, Walker not only places an F+ against a B-flat, but also adds a D- in voices 1 and 2. While the use of D- creates a pure 5:4 third with the B-flat, the also-present F+ is wildly dissonant against both. While the combination of F/D- would have provided a pure 5:3 major sixth (being narrower by a Syntonic comma than its dissonant Pythagorean equivalent) the combination F+/D- actually prescribed narrows the interval by a further Syntonic comma creating thereby a double dissonance with the surrounding voices.

7. No theorist even from the ancient world ever considered an interval as small as a comma (of any kind) to be melodic, and even Aristoxenus considered the smallest melodic interval to be the diesis.

8. In order to perform according to Walker's prescription, the singers would need to engage in a game of "Deaf Man's Bluff". In this game, the singer of one voice would need to know in advance against which notes in another voice he should--in order to remain in tune with them--deliberately avoid singing in tune, because he will have been persuaded (in rehearsal) that the singer of the other voice will actually be intending to change the pitch of the note heard in such a way that if he does sing in tune against it he will then be "out of tune". All parties will also need to have been convinced that one (or more) of them was actively capable of singing a melodic Syntonic comma, and that the others were able to achieve the even more complex feat of imagining in advance what the effect of this will be so that they can then pitch their own notes accordingly in order to ensure that they all end up in tune. Since the last person to be considered during this entertainment is the composer, I would suggest that such anarchy has no connection whatsoever with the proper conceptual creation and execution of Syntonic tuning.

9. I mean nothing more by this term than would reasonably be understood by a reader who can hear the sound of a major key and understand the ways in which it differs from that of a minor key.

10. Hermann Zenck and Walter Gerstenberg, "Adrian Willaert, Opera Omnia", Corpus Mensurabilis Musicae vol. 3, 8.

11. Willaert might even have felt some gratification that the music of other composers--especially when juxtaposed so closely in performance with his own--sounded less harmonious and more dissonant. Such differences would certainly have been noticed by informed listeners like Zarlino and Ganassi, both of whom remarked upon Willaert's uniqueness (the former noting his correction of "numerous errors" and the latter asserting his position as "the new Prometheus of celestial harmony").

12. Such comma control shows that Willaert was aware throughout the composition of a new piece of exactly what, at any given moment, was the current "comma position" in relation to the base pitch. Careful comma analysis reveals that the base pitch will frequently but temporarily rise or fall by a comma during the unfolding of a composition, but that Willaert will generally hold this "comma drift" under tight control by taking appropriate evasive action to restore the pitch.

13.  I view these comma cross relations as "comma control points" and shall henceforward use this term to describe them. In this case the audible cross relation occurs between the new "F+1" in voice 1 and the previous "C" in voices 2 and 3. There is a perceptible "wolf", but it is absolutely indirect and the listener's awareness of its presence is only via a retrospective cognition of the previously-heard "C".

14. In measure 7, beat 3, the "G+1" applied to voice 1 signals another audible "comma control point" by virtue of its juxtaposition with the previous pitch "D" in voice 3 against which there is another indirect "wolf".

End of footnotes