[Smt-talk] Soudominante versus Sous-dominante (was: Subdominant versus Predominant)

[Thomas Noll]

Dear Nicolas, dear Colleagues,
With the two meanings of subdominant (scale step below the dominant vs. fith below the tonic) you thematize another interesting distinction, which perhaps deserves to be given its own subject in the discussion. [NB: Is it possible that Rameau in "Nouveau système de musique théorique" not only adopts but also slightly changes Dandrieu's term "soudominant" towards "sous-dominant" just in order to leave a trace of the shift in the meaning, which he undertakes?]

In accomplishment to the historical and geographical subtleties in the currency of both concepts it is insightful to look at these alternatives from a mode-theoretical point of view. On the one hand the distinction yields different derviations of the same scale degree. On the other hand, - and that is the focus of my posting - one may ask for the circumstances under which this remarkable incidence holds. My hairsplitting question is: When do Soudominante and  Sous-dominante refer to the same scale degree?

For example in the chromatic mode
C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B-C' the divider of the mode is G and its predecessor (the pseudo-"soudominante") is Gb. The "sous-dominante", however, the scale degree a fifth below the finalis C) is still the F.
[NB: David Clampitt and I have shown in our paper on modes that for standard modes (generalized Ionian) the origin of the folding coincides with the divider predecessor.  
http://www.mtosmt.org/issues/mto.11.17.1/mto.11.17.1.clampitt_and_noll.html  
In this example Gb is origin and divider predecessor]

What distinguishes the diatonic case from the chromatic one is the fact, that in the diatonic case the diazeuxis, i.e. the difference intervall between fifth and fourth (generator and co-generator) coincides with the major second (the larger step interval of the mode). This is (technicalities aside) equivalent to Eytan Agmon's (1989) definition of "diatonic" in his JMT paper:  "A Mathematical Model of the Diatonic System".

In other words, Agmon's generalization of the term "diatonic" circumscribes the conditions under which the terms Soudominante and Sous-dominante are competing definitions for the same scale degree.

Sincerely
Thomas Noll

On 24.02.2012, at 23:35, Nicolas Meeùs wrote:

> Le 24/02/2012 16:55, Ninov, Dimitar N a écrit :
>> 
>> [...]
>> The term "subdominant" suggests that the IV chord is located a fifth below the tonal center, and that it is a "lower dominant". However, the term "predominant" contains no reference to the tonal center but to the dominant triad itself, as if a key had two centers. The attempt to validate a harmonic function with no relation to the tonal center seems theoretically unsupported to me.
> I don't know whether this is "theoretically" supported, but it certainly is historically very much supported. It is at the core of French theory and probably at the origin of the word "subdominant" itself.
> 
> Rousseau, in the Dictionnaire, wonders about the origin of the term. He writes (s.v. "Sous-dominante"):
> Sous-dominante ou Soudominante. Nom donné par Monsieur Rameau à la quatrième note du ton, laquelle est par conséquent au même intervalle de la tonique en descendant, qu'est la dominante en montant [...]. Je puis me tromper dans l'acception des deux mots précédents, n'ayant pas sous les yeux, en écrivant cet article, les écrits de Monsieur Rameau. Peut-être entend-il simplement, par sous-dominante, la note qui est un degré au-dessous de la dominante [...].
>     [Subdominant. Name given by Rameau to the fourth note of the key, which therefore is at the same descending interval from the tonic as the dominant is ascending. I may be mistaken in the acception of these two words [the text also discusses "sous-médiante", submediant], not having under the eyes the writings of Rameau as I write this article. He may understand by subdominant simply the note that is one degree under the dominant.]
> It often is difficult to say how the term was understood. In French today, and probably from Rameau and before, it means the degree under the dominant, not the dominant under the tonic. The earliest instance of the term that I know of is in Dandrieu's Principes de l'accompagnement, c1719. That Dandrieu understood "soudominante" as the degree under the dominant can be deduced from his usage of "sudominante", the degree above it. This has been the almost constant usage in French (and a few other languages) since then. Note that the word does not occur in Rameau's Treatise of 1722, nor, for that matter, in Fétis' Treatise: they both merely write "the 4th degree".
> 
> I won't take position about this now. But I wanted to stress that this kind of discussion has been going since centuries.
> 
> 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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