<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Dear Nicolas,<div>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.</div><div>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. </div><div>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. </div><div><br></div><div>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. </div><div><br></div><div>Sincerely</div><div>Thomas Noll</div><div><div><br><div><br class="Apple-interchange-newline"><blockquote type="cite"><div bgcolor="#FFFFFF" text="#000000">
<font face="Calibri">Dear Thomas,<br>
<br>
Your MTO article begins with a quotation of Taruskin,<br>
</font>
<blockquote>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.<br>
</blockquote>
<font face="Calibri"></font>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.<br>
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.<br>
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.<br>
<br>
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 <i>Modi vocum</i>, available on
<a class="moz-txt-link-freetext" href="http://paris-sorbonne.academia.edu/NicolasMee%C3%B9s/Papers">http://paris-sorbonne.academia.edu/NicolasMee%C3%B9s/Papers</a>. What
you call 'Guidonian affinity' (why Guidonian?) obviously has to do
with this (but this may not always have been properly understood).<br>
<br>
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.<br>
<br>
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?<br>
<br>
Nicolas Meeůs<br>
Université Paris-Sorbonne<br>
<br>
<br>
<font face="Calibri"><br>
<br>
</font><br>
</div>
</blockquote></div><br><div>
<span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div><br class="Apple-interchange-newline"><br><div><div>*********************************************************</div><div>Thomas Noll</div><div><a href="http://user.cs.tu-berlin.de/~noll">http://user.cs.tu-berlin.de/~noll</a></div><div><a href="mailto:noll@cs.tu-berlin.de">noll@cs.tu-berlin.de</a></div><div>Escola Superior de Musica de Catalunya, Barcelona </div><div>Departament de Teoria i Composició </div><div><br class="khtml-block-placeholder"></div><div>*********************************************************</div></div></div></div><div><br></div></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"><br class="Apple-interchange-newline">
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