[Smt-talk] rationalizing the octenary system

Richard Porterfield porterfr at hotmail.com
Fri Apr 17 18:56:26 PDT 2009

Dear Nicolas Meeus, 
Thank you for your valuable paper! I will have more to say about it later. Below are a few comments on your latest reply to Eyton Agmon. 
Richard Porterfield
Instructor, Mannes College 
Ph.D. candidate in Music Theory, CUNY GC
porterfr at hotmail.com
Date: Fri, 17 Apr 2009 22:39:12 +0200
From: nicolas.meeus at paris-sorbonne.fr
To: agmonz at 012.net.il
CC: smt-talk at societymusictheory.org
Subject: Re: [Smt-talk] rationalizing the octenary system

... Hucbald's readers were familiar with the chant repertory, but they had no idea of the diatonic system, nor of ways of comparing the melodies between themselves from an intervallic point of view; they did not even have note names, nor any clear notion of the modal final. Hucbald's purpose, therefore, cannot have been to discuss which degrees of the system could serve as finals, as none of these notions was available.
It seems possible that the notion of the modal final was in circulation already (Aurelian is uncomfortable with chants that seem to begin in one mode and end in another: with this we have at least the final phrase being of a mode, if not the final note). Even if in oral pedagogy and/or a treatise now lost these finals had already been recognized as discrete tones occupying successively higher modal degrees -- and their names protus, deuterus, tritus, tetrardus being in tonal order cannot have been accidental -- it would then have been Hucbald's theoretical contribution to locate those finals in a tone-system for western chant adapted from ancient Greek instrumental models, itself no small feat. 
What he tried to show is that, if the melodies were considered in terms of their inner intervals (to the description of which he devotes a lot of space), it is possible to align them all along one single general scale (the diatonic system) ...
Two scales, that is, including that of the synemmenon allowing for soft b. 

A and C were not rejected as possible finals, they merely were not deemed necessary in Hucbald's highly pedagogical description because modes on D and F respectively could be written with a B flat. As a matter of fact, melodies ending on F almost always have a B flat, so that strictly speaking it are the modes on F that do not exist, not those on C. 
Many F-mode melodies have both hard and soft b (Gradual Viderunt, e.g.); many melodies in protus, deuterus, and tetrardus do as well. 
It would theoretically have been possible to write melodies on E with B flat (corresponding to modes on B), but these merely do not exist. 
They certainly do exist. Credo 1, for example (LU 64-66). 
As to melodies on G with B flat, they can as easily be written on D without.
Some alternate between a ditone and semiditone (M3 and m3) above the final, such as the antiphon Missus est gabriel (which Hucbald cites, by the way); unless this chant is notated on G it goes outside the notational system. 

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