[Smt-talk] Gravity (Was: Car names)

[Thomas Noll]

Dear Nicolas,
thank you very much for your very helpful comments, questions and the link to your paper. The quotation from Taruskin which we use as a teaser to our article may indeed raise misleading expectations. It is not clear whether the concept of height/width-duality may be made productive for oriental traditions of modality. As you suggest, the beginnings of polyphony may have played a catalytic role for the medieval approaches to the Pythagorean diatonic scale.
We investigate properties of and interdependencies between mathematical concepts, which are motivated by historical concepts. The success of such a mathematization project should perhaps be measured by the amount and utility of conceptual cross-relations, which are natural consequences of the theory. 
For example, to systematize those discrepancies in the counting of heights and widths, which you mention, is a central part of our investigation. Mathematicians nowadays communicate the properties of and interdependencies between central words, standard words, conjugates of standard words, etc through definitions and proofs on a suitable level of generality. The particular achievement of Aldo de Luca and other mathematicians is that these definitions are quite well adopted to the knowledge that can be derived from them. It is therefore remarkable that the medieval theorists came up with a closely related conceptual network.               

There are obstacles to interpret the Greek chromatic system in terms of height/width-duality. I will nevertheless make an ad-hoc attempt, just for brain storming. It relates the divisions of the folding with the position of the "de-clutched" fifth. Let us first consider a descending diatonic genus of the ancient "Dorian" species of the octave. The tripartition of the descending step pattern TTS / T / TTS  (descending TTS tetrachords  separated by a disjunctive descending T) corresponds under height-wifth duality to a sharpward folding XY | XY | XYY into two conjunct species of the major second (X = fifth, Y = fourth) and a species of the minor third (= the width analogue of the diazeuxis). The tones in width direction are thus divided as F-C-(G) | G-D-(A) | A-E-B-(F#). It appears that the "de-clutched" fifth G-D of the chromatic mode is separated from the other tones. In the corresponding enharmonic mode one could analogously and separately "de-clutch" two fifths, both of which correspond to different factors of the division of the folding. 

Sincerely
Thomas Noll


> Dear Thomas,
> 
> Your MTO article begins with a quotation of Taruskin,
> Where actual musical practice is concerned, the relevant historical fact is that people have evidently internalized the diatonic pitch set—carried it around in their heads as a means of organizing, receiving, and reproducing meaningful sound patterns—as far back as what is as of now the very beginning of recorded musical history, some three and a half millennia ago.
> a statement that seems highly questionable. I suppose that 'the diatonic pitch set' refers to what I'd call 'Pythagorean' diatonicism, a scale that can be generated by a cycle of (pure) fifths, and producing a pattern of tones (T) and semitones (S) of the type TST T TST (two TST tetrachords separated by a disjunctive T). There are other (Antique) definitions of diatonicism, but let's agree that this is the strict one.
>     Oriental ('Arabic') modality makes an extensive use of another scale, the 'scale of Zalzal', with major (T) and neutral (N) seconds, for instance in the arrangement NNT T NNT (NNT tetrachords and a T of disjunction). This scale, which may be the chromatic system of the Greek, seems to have been used also in early Christian chant. It cannot be fully produced by a cycle of fifths and the degrees that are not in fifth-relation with the others often are of imprecise intonation.
>     The link between Pythagorean diatonic and the 'actual musical practice' mentioned by Taruskin is unclear, to say the least. The earliest descriptions of the Pythagorian diatonic scale in the West (after Boethius who may not have been much concerned with practice) are those by Hucbald and the Enchiriadis group of treatises, c900. This is too close to the beginnings of polyphony to be a mere coincidence.
> 
> Handschin's idea of the character of tones is convincing, but he may have been mislead (by ideas common in the earlier 20th century) in believing that it depended on the cycle of fifths. Medieval theorists described 'qualities' of tones, the 'modi vocum', at least from Hucbald, c900, to Hermannus contractus in the later 11th century. They described four qualities, corresponding to the four degrees of the tetrachord and determining four pairs of modes; the modal final shared the same quality as the fifth above and the fourth below (which is the origin of the theory of species of fifths and fourths); etc. See my paper on Modi vocum, available on http://paris-sorbonne.academia.edu/NicolasMee%C3%B9s/Papers. What you call 'Guidonian affinity' (why Guidonian?) obviously has to do with this (but this may not always have been properly understood).
> 
> Medieval theory was torn between a tetrachordal (or hexachordal) and a heptachordal conceptions of the diatonic system. The medieval equivalents to your height and width cannot be dimensions of the same degrees, as there are so to say seven 'heights' and four 'widths'. And neither can be assimilated with pitch, as pitch is not a relevant category in medieval theory – as the pseudo Odo of Cluny clearly stated, modes do not differ from each other by their pitch. The 'quality' of the notes (your 'width') depends exclusively of their intervallic surrounding: they may be described as systemic functions. What you describe as 'height' concerns what might be described as a modal function, without necessary link with 'pitch' properly speaking.
> 
> Your description of the height-width duality forms and interesting modern view of diatonic scales. I have no objection against your dubbing these scales 'modes'. I am not sure, however, that it does explain medieval (or oriental) modality... To say it in other words: your expression "Guidonian modes" may make sense today, but what does it mean for medieval theory?
> 
> Nicolas Meeùs
> Université Paris-Sorbonne
> 
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Thomas Noll
http://user.cs.tu-berlin.de/~noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona 
Departament de Teoria i Composició 

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