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

Nicolas Meeùs nicolas.meeus at paris-sorbonne.fr
Thu Aug 2 00:23:37 PDT 2012

Le 1/08/2012 23:21, Thomas Noll a écrit :
>> The fact is that a sharpward series of fifths in Pythagorean 
>> intonation does raise in pitch, by one Pythagorean comma after twelve 
>> steps.
> 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.
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.
     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 _mi_ owes its 
characteristics, in any culture, from the character of the _mi_ (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.
>>     In short, a mode is not a scale.
> 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.
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.
> The point where I might be making too much of Jacques Handschin's 
> ideas 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?
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 _because_ there are seven 
days in the week"), one which we hardly understand any more.

Nicolas Meeùs
Université Paris-Sorbonne
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