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Le 1/08/2012 23:21, Thomas Noll a écrit :<br>
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<div bgcolor="#FFFFFF" text="#000000"><font face="Calibri">The
fact is that a sharpward series of fifths in Pythagorean
intonation does raise in pitch, by one Pythagorean comma
after twelve steps.<br>
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<div>That's a sophisticated perspective. With the same line of
argument one could say that five sharpward fifths lower the
pitch by a semitone, and that seven sharpward fifths raise it
by an augmented prime. But the pitch height direction of those
"commata" is not directly concerned with the pitch height
directions of the scale steps themselves. It only measures
their difference. My concern about directionality is more
elementary.<br>
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My view has to do with enharmonic equivalence. What you describe
would presuppose diatonic semitone (limma) equivalence (five fifths)
or chromatic semitone (apotome) equivalence (seven fifths). I have
some sympathy for this view, as these two limits are those of the
pentatonic and the diatonic scales. It makes sense if one considers
a full, unlimited and non hierarchized cycle of fifths. But the
medieval cycle of fifths was highly hierarchized, with seven main
degrees and five (or twice five) secundary ones belonging to musica
ficta. My own reading of Handschin always was that he was fully
aware of this hierarchy, which resulted in each degree having its
own, unique character – at the level of the system itself and
independently of any particular mode. <br>
My own view is that we lost this hierarchized reading of the
system. To us, hierarchies belong to modes or keys exclusively, not
to the overarching system. And I believe that this overarching
hierarchy is what may best define modality at large (by which I
mean, the modality of all modal music in the world, that which
Harold Powers said is not real). What I mean is, for instance, that
the mode of <u>mi</u> owes its characteristics, in any culture,
from the character of the <u>mi</u> (a weak note) in the system.
This belief originated in my former reading of *Der Toncharacter* –
which I admit it is time that I should reread.<br>
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In short, a mode is not a scale.<br>
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A mode relates the perfect fifth and perfect fourth to a species
of the fifth and a species of the fourth. It further relates the
perfect octave to the concatenations of the species of the fifth
and the fourth. In the dorian mode, for example, the species of
the fourth is Y = TST (Tone, Semitone, Tone) and the species of
the fifth is X = TSTT, wherein the species of the fourth is a
prefix, i.e. X = YT, with T playing the role the major step as
well as the role of the diazeuxis. <br>
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This is a late conception, that of Hermannus Contractus and, more
generally, of the St. Emmeran monastery in the later part of the
11th century. As you say, the fifth really is a fourth+diazeuxis.
The novel idea of St. Emmeran is that the octave species can place
the diazeuxis anywhere in the species of fifth (and therefore also
of the fourth). But this conception nevertheless originates in a
tetrachordal one that, again, organizes the hierarchy of the system,
not of the individual scales. Indeed, there is only one model of the
tetrachord which defines the system ever since Hucbald first
described it c900. The novel idea of St. Emmeran is that the
tetrachord is degraded into the Dorian (i.e. first species) fourth.<br>
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<div>The point where I might be making too much <span
class="Apple-style-span" style="font-family: Calibri; ">of
Jacques Handschin's ideas </span>is the construction of a
"species of the major step" T = Y^(-1) X and a "species of the
minor step" S = T^(-1)YT^(-1) = X^(-1)YYX^(-1)Y. Handschin
doesn't deliberately distinguish between an ascending fifth and
a descending fourth. These concepts seem to be algebraically
motivated neologisms. But is this historically true? Schenker's
"ausgeworfener Grundton" is quite close to that construction and
maybe other theorists have also considered the five species of
the semitone?</div>
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One should never mix logic and history. The cycle of fifths makes
logical sense; but it would be definitively wrong to believe (as
some did) that the history of music developed along this cycle.
There is an algebraic logic that very much nourishes our theories,
but which did not play the same role in former times. The medieval
logic was one of analogy ("there are seven degrees in the scale <u>because</u>
there are seven days in the week"), one which we hardly understand
any more.<br>
<br>
Nicolas Meeùs<br>
Université Paris-Sorbonne<br>
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