Saturday, June 04, 2005

The Art of Counterpoint, Part One

I made it through the first nine chapters of Zarlino's "The Art of Counterpoint" today (Which is Part III of Le istitutione harmoniche), and after the introductory nicities, Zarlino plunges right into interval classification. That will be the main topic of today's post, but first I want to mention something I found in Claude Palisca's introduction.

Zarlino, it seems, was a big believer in the rational purity of the just intervals found in the Natural Harmonic Overtone Series, and he believed singers approximated these simple ratios instinctively in their singing. Evidently, he got into some heated debates over the matter, one series of which was with Vincenzo Gallilei, father of the now immortalized Gallileo. I obviously take Zarlino's side on this issue, and history and experiments have proven us right. In any event, Zarlino divided his monochord into six divisions as per Ptolomy versus the previous four of Pythagoras to arrive at a 5-limit Just Intonation Tuning Scheme versus the older 3-limit Pythagorean tuning. Here we meet a man who is taking his lessons from nature, and in fact Zarlino mentons the naturalness of musical concord often (I've been through this book twice previously). Nature and the natural aspects of music are a strong undercurrent of his thought, just as they are mine. Now, on to the interval classifications.

By this time, Zarlino has it just about perfectly right, but due to his point in history, he still hasn't reduced the issue of consonance and dissonance to it's simplest terms so far as explaining the phenomenon is concerned. In order to do that, we shall consult the following example:



The entire essence of musical acoustics is built on the naturally obvious phenomenon of octave equivalence, meaning - for example - that a C is a C no matter which octave it appears in. This is defined by the very first overtone in the Natural Harmonic Overtone Series. Using the octave in either the form of a displacement or the form an inversion (Or both), I have already shown how perfect consonances are superparticular ratios in both of their octave exchange positions but imperfect consonances are only superparticular in one of the two positions. Now, I want to take that further to show how few consonances and dissonances there really are by using this very same device.

At the beginning of the example above I have the Perfect Unison and it's octave displacement, the Perfect Octave: These are really just the same interval duplicated at the octave, as I have indicated by labelling them 1a and 1b. Again, the octave is the first harmonic overtone. The same principle reveals, obviously, that the Perfect Fourth is essentially the same as the Perfect Fifth through the type of octave displacement we know as the principle of inversion. So, essentially, there are only two perfect consonances: The perfect octave and the perfect fifth, the other two being only octave displacements or octave inversions of these (I have moved fron the unison to the octave because the unison isn't an interval per se).

The same holds true for the Imperfect Consonances: there are really only two, those being the major and minor thirds; the sixths are simply octave inversions of those thirds. Again, the same process reveals that there are only two major/minor dissonances, the major and minor second, with the major and minor sevenths resulting from octave displacement inversions. Finally, the Tritone is in a unique category of it's own, but it too appears in two forms - the Diminished Fifth and the Augmented Fourth - one simply being the octave displacement inversion of the other. Needless to say, the tritone is treated as a dissonance. These are all seven of the natural intervals then: Perfect Octave, Perfect Fifth, Major Third, Minor Third, Major Second, Minor Second, and Tritone (Again, I used the octave versus the unison in this list because a unison isn't an interval, strictly speaking). All other natural intervals result from the principle of octave displacement or octave displacement inversion of these seven intervals and are essentially their equavelents. Intervals beyond the octave are simply octave duplicates of these. All augmented and diminished intervals are equavelents of these seven or their octave displacement or octave duplicate equivalents as well, but they are notated enharmonically.

Note that all consonances are what we would call harmonic intervals: Those that appear when we make triads out of stacks of thirds and invert them or duplicate them at the octave. Note also that the major/minor dissonances are the purely melodic intervals of the half step and the whole step with the sevents being simply octave inversions of these essentially melodic seconds. The tritone is the exception in that it is a dissonance that is a harmonic interval, it being the fifth of the diminished triad. We need not concern ourselves with seventh chords until we get to harmony proper.

Reducing the number of natural intervals from fourteen to seven greatly simplifies coming to terms with countrapuntal writing, as we shall see.

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