1. II CONCEPT SPACES
II.2 (Thu Feb 01) Concepts and software for a theory of rhythm: The MetroRubette

2. Definition:
Global metric is the structure induced by the covering of the onsets by (maximal) local meters.
What is rhythm?

3. 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.
w(E,H,…) H E

4. Definition: The (metrical) rhythm is the weight function on the onsets, which is deduced from the global metric, as typically described by the formula
w(x) = S x ∈ M, m ≤ l(M) l(M)p
l(M) = length of local meter M m = minimal admitted length of local meters
p = metrical profile
Discuss variants!!

5. Quantification of Metrical Semantic
Profile = growth number for length contributions
Minimum = minimal admitted lengths
Elimination of too short local meters lengths < Minimum
Onset
MetroWeight(E) =

6. omitted eliminate! MINIMUM = 3 PROFILE = 2 MetroWeight(E) = 32 = 9
pitch
onset
omitted
eliminate!
l = 2
l = 3
MINIMUM = 3 PROFILE = 2
MetroWeight(E) = 32 = 9

7. w(x) = S x ∈ M, m ≦ l(M) l(M)pa b c d e
w(x) = S x ∈ M, m ≦ l(M) l(M)p 4 6 8 10 12 14 16 18 20 22
m = p = 2

8. Have the problem of different types of score objects contributing to the weight functions.
How can we take care of this distribution in a precise way?
Ideas?

9. Distributed Metrical Logic
Covering I = {X1, X2,...Xn} of X by n sets X1, X2,...Xn of onsets of instruments, barlines, pauses, l.H, r.H., different note types, etc. ( GTTM)
For each Xi have logic Met(Xi) and a corresponding weight wi with the minimal lengths limits mi und the metrical profile pi.
Each weight contribution wi is given a distributor-factor 0 ≤ i and we then define the distributed weight w by
w(x) = 1 w1(x) + 2 w2(x)+... n wn(x) = i i wi(x)

10. four predicates
left hand right hand barlines pauses

11. L S

12. Os X
MetroRubette

13. Sonification of metrical weights
J. S. Bach: Kunst der Fuge, Contrapunctus III
Joachim Stange-Elbe s a t b
Sonification of metrical weights

14. „Träumerei“ right hand, from longest to shortest minima

16. Mathias Rissi: Jazz Composition Example
URTEXT PAPAGO

17. Mathias Rissi: Jazz Composition Example
PAPAGO (Mathias Rissi/SUISA)
Mathias Rissi: Jazz Composition Example

18. Anja Volk-Fleischer‘s work (PhD 2002 thesis Die analytische Interpretation etc.)
„...lässt sich nun die metrische Kohärenz als Korrespondenz zwischen innerem und äusserem metrischen Gewicht beschreiben.“

19. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 2 The new MetroRubette on Rubato Composer software displays metrical weights for different voices
(represented in different colors) and enables the display of single maximal local meters (selected and highlighted,
shown on top) passing through selected onsets (selection top left).

20. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 3 Brahms Piano Sonata op.1 mm The piano roll representation shows the pitches of the sonata.
Here, l = 22. The long weight lines here (coinciding with the grid) show the piece as if it were in a quintuple meter.
Note the 26 consecutive pulses.

21. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 4 Brahms Piano Sonata op.1 mm , the second and third subject groups. Here, l = 22.
The anticipated quadruple meter is better reflected as a triple meter, as shown by the high weights on the ¾ grid.

22. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 5 Here, the weight of the right hand is dropped entirely, and l = 12.
The triple meter found in the above figure vanishes.

23. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 6 Metric analysis reveals meters on local and global scales. We show mm , the second group.
Here, l = 10. By our analysis, this group without the third subject group lacks a clearly defined meter

24. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 7 Piano Sonata Op.1 mm Here, l=12. Its exposition is parsed out into 2 bar quintuple hyper-metrical phrases.
The rhythmically dense second thematic group serves as a clear divider between the first and third subject areas.

25. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 8 Brahms Piano Sonata Op.1 mm Here, l = 10. The development has a very clear metrical 4/4 pulse,
as shown by the grid lines.

26. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 9 Brahms Piano Sonata Op.1 mm Here, l = 10. The right hand establishes a competing down beat.

27. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 10 mm The first group of the recapitulation. Here, l = 22.
The beat emphasis is entirely regular.

28. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 11 mm The second group. Here, l = 24. Note the lack of clear metrical structure at the beginning of the figure.

29. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 12 mm The closing theme group of the recapitulation. Here, l = 24.

30. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
fig 13 A full selection of the recapitulation. Here, l = 24.

31. Analysis of Brahms’s Sonata op. , with Bill Heinze, to be published
Summary
Metric Shift
Metric shift involves establishing a meter that is displaced from the down beat.
Metric Shift is best seen in the first bars of the sonata. Not only does the theme enforce a quintuple pattern,
but that pattern begins 2 beats off of the notated downbeat. Likewise, the right hand in the second thematic
group places more emphasis on the second eighth-note in the bar.
Metric Competition
Metric competition, which is found more frequently in the symphonies, is where two or more local meters exist
in different voices. For example, the first piano sonata's second thematic group shows a strong meter that occurs
every six quarter-notes in the right hand only. The left hand alone presents a simple quadruple meter. These two
competing structures combine to create a third.
Metric Modulation
Metric Modulation is likely the most common form of incoherence found in Brahms's first sonata. This is where
the local meter is in direct competition with the global bar-lines. The quintuple meter in the beginning of the first
sonata, or the strong triple meter in the in the second subject group are both cases where the local meter is not
represented in the bar-lines.
Dissonant Hypermeters
Finally, even when there is a clear congruence between the notated bar lines and the metric analysis, it is important
that we look at the weight across multiple bars. This metric phrase length provides insights to how to phrase the
music on a large scale. Hidden three, two and a half, or five bar metric structures could imply new interpretations
of this oft-performed piece.