1. II CONCEPT SPACES
II.1 (Thu Feb 01) Case study I (rhythm): Riemann, Jackendoff-Lerdahl

2. For performance we need quantification of analytical facts = weights!
w(E,H,…)
w(E,H,…) H E

3. Meter and Rhythm (I)
Edward Aldwell & Carl Schachter (Harmony & Voice Leading. Thomson & Schirmer, London et al. 2003): Musical rhythm organizes the flow of time. (By duration, accent, and grouping)
Gustav Becking (Riemann Musiklexikon. Schott, Mainz 1967, Frieder Zaminer citing G.B.): The question concerning the basics and principles of metrical order pertains to the most controversial of theory and are a core problem of analysis.
Hugo Riemann (System der musikalischen Rhythmik und Metrik. Breitkopf & Härtel, Leipzig 1903): I have called rhythmical quality the valuation of the average times, which govern the rhythm of a piece, when compared to the normal mean value of the healthy pulse.
Jürgen Trier (Studium Generale II. 1947): Rhythm is the order in the course of articulated shapes, which is made to evoke and to satisfy a tendency of tuning in by regular repetition.
Riemann (System der musikalischen Rhythmik und Metrik. Breitkopf & Härtel, Leipzig 1903): We operate with a second basic concept (apart from the rhyth. qual.), namely that of different weights of times, the metrical quality.

4. ...experienced listener, who knows what the composer intended!
Meter and Rhythm (II)
Riemann (System der musikalischen Rhythmik und Metrik. Breitkopf & Härtel, Leipzig 1903): The counting times (beats, basic rhythmical times) gain under any circumstances only real existence by their contents.
Fred Lerdahlc & Ray Jackendoffl (A Generative Theory of Tonal Music. MIT Press, Cambridge MA 1983): In our view, and adequate account of rhythm first of all requires the accurate identification of individual rhythmic dimensions. The richness of rhythm can then be seen as the product of their interaction. The first rhythmic distinction that must be made is between grouping and meter. When hearing a piece, the listener naturally organizes the sound signals into units such as motives, themes, phrases, periods, theme-groups, sections, and the piece itself. Performers try to breathe (or phrase) between rather than within units. Our generic term of these units is group. At the same time, the listener instinctively infers a regular pattern of strong and weak beats to which he relates the actual musical sounds. The conductor waves his baton and the listener taps his foot at a particular level of beats. Generalizing conventional usage, our term for these patterns of beats is meter. -
In dealing with especially complex artistic issues, we will sometimes elevate the experienced listener to the status of a perfect listener – that privileged being whom the great composers and theorists presumably aspire to address.
...experienced listener, who knows what the composer intended!

5. Fred Lerdahl & Ray Jackendoff, Example = Mozart‘s Jupiter Symphony
Meter and Rhythm (III)
Fred Lerdahl & Ray Jackendoff, Example = Mozart‘s Jupiter Symphony
MWFR 1: Every attack point must be associated with a beat at the smallest metrical level present at that point in the piece.
MWFR 2: Every beat at a given level must also be a beat at all smaller levels present a that point in the piece.
MWFR 3: At each metrical level, strong beats are spaced either two or three beats apart.
MWFR 4: The tactus and immediaely larger metrical levels must consist of beats equally spaced throughout the piece. At subtactus metrical levels, weak beats must be equally spaced between the surrounding beats.
Steve Coleman; #1 on CD The Sonic Language of Myth

6. Critical discussion of the concepts!
Meter and Rhythm (IV)
Fred Lerdahl & Ray Jackendoff grouping rules:
GWFR 1: Any contiguous sequence of pitch-events, drum beats, or the like can constitute a group, and only contiguous sequences can constitute a group.
GWFR 2: A piece constitutes a group.
GWFR 3: A group may contain smaller groups.
GWFR 4: If a group G1 contains part of a group G2, it must contain all of G2.
GWFR 5: If a group G1 contains a smaller group G2, the G1 must be exhaustively partitioned into smaller groups.
Critical discussion of the concepts!
Compare to Krenek‘s idea of musical axioms!

7. hounted

8. Meter and Rhythm (VI) The GTTM ideology
It is a theory, which wants to yield an interface between cognitive psychology and music theory. „Muscic Theory as Psychology“
It refers to a number of important music-theoretical concepts, such as parallel, stable, harmonical identity, metrical stability, cadence, suspension, resolution, conflict, mutually consonant combination, local tonic.
They are all external to the theory. Therefore, the crucial decisions are outsourced, which makes the theory unreliable.
(Jean-Jacques Nattiez): When two passages are identical they certainly count as parallel, but how different can they be before they are judged as no longer parallel? (...) It appears that a set of preference rules for parallelism must be developed, the most highly reinforced case of which is identity. But we are not prepared to go beyond this, and we feel that our failure to flesh out the notion of parallelism is a serious gap in our attempt to formulate a fully explicit theory of musical understanding. For the present we must rely on intuitive judgments to deal with this area of analysis in which the theory cannot make predictions.

9. CT the concept of a local meter pitch X onset ? l = 3 period
no local meter!!
period
l = 2
period
l = 2 CT

10. maximal local meters of the collection X of onsets
pitch
onset
maximal local meters of the collection X of onsets
l=8
l=4
l=3
l=2

11. ? GTTM maximal local meter M local meter M length l(M) = 4 no
onset axis
maximal
local meter M no
local meter
local meter M length l(M) = 4

12. n/16 2 3 4 6 8 10 12 a b c d e

13. basis of topology: open balls U closure interior
Classical topology (open sets) of the usual plane X (= —2 )
basis of topology:
open balls V U W
B(x) x
VW
closure
interior

14. x = onset, contained in U(x) = L1 L2  L3...  Lk
Definition of topology Met(X) for set X of onsets
Met(X) is defined by the basis B(X) = set of all (non-empty) intersections U = L1 L2  L3...  Lk of maximal local meters Li in X
These intersections are also local meters und play the role of open balls B(x) of the usual plane.
x = onset, contained in U(x) = L1 L2  L3...  Lk

15. What is the „logic“ of the topology Met(X) for onset x of X?
Logic of Met(X)
What is the „logic“ of the topology Met(X) for onset x of X?
„U(x) is open ball around x.“ interpreted as „Truth value U(x) valid for x.“
„x dominates y“: Every open ball U(y) around y contains x, in symbols „x > y“.
Logical interpretation thereof:
„All truth values U valid for y, also valid for x.“
„x is at least as true as y.“
„Every maximal meter in y is also one in x.“
‚Unclean‘...!

16. n/16 a b c d e 2 3 4 6 8 10 12
6 > 0

17. ? x > y, all x,y GTTM maximal local meter M
onset axis
maximal
local meter M no
local meter
local meter M length l(M) = 4
GTTM
x > y, all x,y

18. Nerve of covering of X by maximal local meters:
Nmax(X) = { ={M0,...M} such that M0...M  Ø}  = simplex,  = dimension von  Replace onsets by simplexes = new kind of „points“! M M0 X
Have „clean“ topology Nerve(X) on Nmax(X)!

19. Nerve Nmax(X) of covering {a, b, c, d, e}

20. Have continuous „cleaning“ applications
Sp: Met(X)  Nerve(X)
x ~> Sp(x) = {M max. loc. meter with x M}
Logical interpretation: Sp(x) = collection of all truth values of x

21. 6 Met(X) Nerve(X) c Sp a 2 b e d x > y if and only if Sp(y) Í Sp(x)2 3 4 6 10 12 Sp b e c a d
Nerve(X) 3 12 6 4 2 10 a b c d e
x > y if and only if Sp(y) Í Sp(x)

22. Especially: metrical weights
Need e.g. for PerformanceRubette quantification of topological facts = analytical weights!
Especially: metrical weights
w(E,H,…)
There are three main rationales for performance: emotion, gesture, and ratio.
Emotion has strongly been preconized by Alf Gabrielsson. He maintains that "we may consider emotion, motion and music as being isomorphic" [Gabrielsson1995]. While this conjecture may please psychologists, it is completely useless to scientific investigation. In fact, such an isomorphism is a piece of poetic literature as long as the components: emotions, gestures, and music, are not described in a way to make this claim verifiable. Presently, there is no hope for a realistic and exhaustive description of emotions. Same for gestures, and as to music, the mathematical categories of local and global musical objects are so incredibly complicated that the mere claim sounds like a cynical joke. For example, the number of isomorphism classes of 72-element motives in pitch and onset (modulo octave and onset period) is ^36 [Fripertinger1993]. How could the claimed isomorphism fit in this virtually infinite arsenal?
Gestural categories as a rationale for performance have been advanced in approaches [Kronmann1987] which maintain that musical retards, for example, share a structure of Newtonian mechanics. Such approaches cannot, however, explain the agogic phenomena within a motivic movement, or the dynamical differentiation within a chord, for example. Moreover, the gestural motivation for a determined instance of performance is extremely complex: How could one deduce Glenn Gould's performane when knowing his beautiful dance of fingers, arms, and body?
This is why we shall stick to rational semantics in performance, it is the easiest and most explicit rationale. This means that we have to investigate the score text by means of metrical, rhythmical, motivic, harmonic, contrapuntal etc. analyses and to correlate these findings to the expressive shaping of performance. This is also a traditional and important requirement of rhetorics: to convey the text's meaning, and not personal emotions or gestures. Theodor W. Adorno has strongly recommended such an analytical performance approach [Adorno1963]. It is an interesting question, whether traditional performances have much to do with analytical performance, and if not, how such a performance would sound like! We shall give an example of such a performance in this talk. H E

23. metrical weight ~ rhythm
w(x) = W(Sp(x)) = S M ∈ Sp(x), m ≦ l(M) l(M)p 4 6 8 10 12 14 16 18 20 22
m = p = 2 a b c d e