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

[Thomas Noll]

Dear Nicolas, 
The role of hierarchy is indeed a crucial subject on several levels of description. I think your concept of an overarching hierarchy may productively contribute to arguments in Handschin and Dahlhaus.   

> 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. 

This reading is correct, I think. Handschin uses the term "Beicharakter" in order to describe something which is added to the tone character in order to yield a specific modal character. He also uses the term "Beicharakter" in order to describe something which is added to the tone character in harmonic tonality to grasp the influence of the triad.

About the modal Beicharakter Hanschin writes (on page 254/255):
'Es ist klar, dass durch die verschiedenen "Setzungen", wie sie zum Beispiel die Kirchentöne repräsentieren, jeweilen ein neuer Zusammenhang über den ersten gelegt ist, ein Zusammenhang, der in höherem Masse von unserem Willen oder unserer Neigung abhängt. Wir können sagen, dass der Ton mit d-Charakter im d-Modus "Grundton" und im c-Modus zweite Stufe ist, oder wir können sagen, dass der Ton d einen verschiedenen "Beicharakter" erhält, je nachdem ob er 2. Stufe im c-Modus oder 4. im a-Modus ist. "Natürlicherweise" behauptet im Aufbau der Modi neben dem Grundton im allgemeinen die Quint oder auch die Quarte die Hauptrolle.'

In this sentence he uses in the beginning "Setzung" (agreement), but at the end he also uses "natürlicherweise" (naturally). 
It is therefore interesting that Dahlhaus draws a connection between Fetis concept of tonality and Hanschin's Theory (of the Tongesellschaft). On page 15  Dahlhaus writes:

' Als Versuche, außer den "faits isolés" auch deren "liaison nécessaire" aus der Natur der Sache oder der des Menschen zu erklären, sind das "Lipps-Meyer-Gesetz" und Jacques Handschins Theorie der "Tongesellschaft" zu verstehen.' 

In Gjerdingen's translation the passage (also page 15) reads as: 
'The "Lipps-Meyer law" and Jacques Handschin's theory of "tone association" [Tongesellschaft] chould be understood as attempts, based on the nature of acoustics or cognition, to account not just for the "isolated facts" but also for their "requisite connections".'  
  
In the concrete argument about modes Dahlhaus seems – at first sight – to reject Handschin's view that abstract diatonic line of fifths is foundational and that the modal Beicharakter is merely added. In his counterargument he refers to the historically better justified construction of the diatonic modes from tetrachords. But eventually he presents his criticism more as an anti-thesis and proposes a reconciliation of both perspectives. At the end he writes:

'The modal and abstract representations of the diatonic system are two sides of the same coin, and it would be prejudiced to think that one has to decide which takes precedence over the other. Hidden behind the notion that there must always be a first and founding phenomenon from which a second phenomenon is then derived lies a false expectation of reality.'         

> 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.

I see your concrete criticism with the mere widening of the fifth range as a historical parameter. This is quite naive. Another example like this is the speculation on the evolution of tone repertoire along the overtone series. My point about the markedness of height- and width directions is a reference to modal vs. counterpuntal aspects of tone relations. This has historical implications, but my argument is primarily a systematic one. Concerning the music-theoretical discourse I see a potential in the exchange of ideas between historical and systematic interest. We can study the historic discourse about music, come to know or make ourselves diagnoses about theoretical problems and we can introduce recent or old mathematical ideas to solve them. 

The definition, for example, of the 2 "species of the major step" and the 5 "species of minor step",  yield (in the proper combinations) the inverses of the modal combinations of the 4 species of the fifth and the 3 species of the fourth. This leads to circular definitions of the modes (as automorphisms of the free group F_2) where neither the steps nor the consonances are foundational. Dahlhaus speaks of two sides of the coin, but he does not tell us how to get such a coin. I (tend to) think, that the mathematical concept of a group automorphism fills a serious gap in Dahlhaus' attempt to close his Handschin-discussion with a dialectical argument.
Sincerely
Thomas Noll  



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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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