II STRUCTURE THEORY

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II STRUCTURE THEORY II.1 (Tu Sept 22) General
context and tempo curves
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Structure Theory is the first big topic, dealing
with What performance is
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The oniontological topography specifies a
communication from the poietic to the neutral
position, yielding the performed music as a
sounding expression. The delicate point is the
technical way of connecting the score information
and the surrounding CSI to the sounding output.
The solution we have chosen is historically
justified in that the score has evolved from the
neumatic notation, meaning that the score is a
dance floor of gestures and therefore relates
to real physical events in an abstracted way.
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This implies that the score coordinates of onset
and pitch are an abstraction of real physical
coordinates. The nature is this abstraction is a
standardization of time and pitch, subdividing
these parameters into quanta time is a mulitple
of standard durations, usually 1/2n (1/2, 1/4,
1/8 etc.) of a measure's duration, and pitch is a
mulitple of semitones. These standard units are
not the physically meant quantities, but more or
less so, once we have fixed the gauging of time
and pitch. So the score space is a symbolic space
bearing the potential, when additional
information is provided, to generate physical
space coordinates. Such information suggests that
the transformation from score to sound is a map
on the score space, mapping symbolic events
(notes, pauses, bar lines) to physical ones. We
call it the performance transformation ?
Symbolic Space gt Physical Space The precise
definition of these spaces is of course required,
and it is by no means simple! This approach is
the most precise and efficient one to date. But
it has a number of consequences that are as
remarkable as difficult for mathematicians, music
theorists, and performers. The most difficult one
is the fact that such a map is mathematically
complex, even in the most common situations. A
number of delicate questions arise How can we
classify such maps? How can we then motivate such
a mathematical choice of spaces and map
properties in terms of common musical situations?
Which are the most elementary such maps in music?
Is it really necessary to embark in such general
math? Isn't music dealing with just the most
elementary mathematical situations? Which parts
of these maps are connected to reasonable
performance parameters, if there are such
parameters at all? Etc. Etc.
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• But pure math is not the only problem. The
  performance transformation takes place in a
  number of local and global contexts, namely
• It is a performance with different instruments,
• it is a performance of a patchwork of parts of a
  larger compositions,
• it is a performance in different dimensions
  (onset, pitch, duration, etc.) which play very
  different roles, some are dominating, others are
  more subsidiary.
• And performance is also an evolutionary process
  that evolves from sight reading to a refined
  performance.
• This means that the mathematical description of
  performance by the mapping ? must also include
  these local/global contexts.
• It will be quite challenging to realize this
  complex structural description language, but any
  simplification is impossible, as will be shown
  later. This is in fact not an artificial blow-up
  of musical situations, it is the mucial
  complexity, which enforces the setup. Etc. Etc.

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instruments
atlas
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hierarchies of parameters
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genealogy
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Tempo curves Tempo deals with the performance of
time. There is a symbolic time and a physical
time, into which the symbolic time is
transformed. E.g, quarter note time units are
transformed into seconds, minutes etc.
?E
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• What is the space of time?
• Hanspeter Danuser/Carl Dahlhaus (loc. cit. p.
  53) Potential time vs. real time. Performances
  are contained in real time, whereas the score
  contains potential time. Comments??
• Peter Desain and Henkjan Honing (CMJ 1989,
  13/3,p. 56-66 The Quantization of Musical
  Time.) Symbolic time is discrete, whereas
  physical time, tempo, and expressive timing are
  continuous. Comments??
• The space of symbolic and the space of physical
  time is the real line? of all numbers, not just
  a discrete subset, it is the line of all real
  values, without wholes, like in geometry. So
  performance of onset time is a map ?E from the
  time line of symbolic time to that of physical
  time

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What is tempo?
time e as a function of space S e e(S)
speed(S) 1/slope(S) 1/(de/dS)
meter/seconds
e e(S)
S
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What is tempo?
time e as a function of score onset time E e
e(E)
Tempo(E) 1/slope(E) 1/(de/dE) q
/seconds notated by T(E) Maelzel Metronome
M.M. q /minutes
e e(E)
E
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Carl Czernys example from his piano school
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Two pianists performance of the theme in
Mozarts Piano Sonata A major K 331. Solid line
first performance, dotted line repetition
(Gabrielsson 1987)
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• For the book project
• Look up whether there is a history of tempo.
• The Italian tradition (17th C.) and the
  verbal/symbolic descriptions.
• Maelzel's machine, Beethoven.
• Absolute and relative tempo attributes.
• The ambiguity of tempo indications, concatenation
  of relative tempi etc.
• Here are the types of tempo signs for our
  discussion in terms of temo curves
• Absolute tempo signs, such as Maelzel Metronome
  or Italian verbal descriptions such as Andante,
  Adagio, etc.
• Relative punctual tempo signs, such as the
  fermata, general pause G.P., caesura, breath mark
• Relative local tempo signs such as
• coarse indications like ritardando, rallentando,
  accelerando, stringendo, etc.
• notation of correspondence between adjacent
  tempos, such as 1/2 3/8(MM 1/4 93 MM 93
  x (1/2)/(3/8) 93 x 4/3 124)
• signs of type a tempo or istesso tempo

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• Attention, this is not the last word about
  tempi!!
• Does every piece have a tempo?
• Several tempi (Chopin's rubato -gt bound rubato
  vs. free rubato)?
• Tempo hierarchies the example of the
  conductor's tempo vs. the musician's one? Are
  tempi independent? Are there tempi relative to
  other tempi?
• Discuss these issues!

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How does time relate to tempo? Intuitively If I
drive my car, starting at position S0 on the
street, and if I observe the speed on my
tachometer until I arrive at position S1, how
much time has then elapsed? Musically If I play
a piece, starting at onset time E0 on the score,
and if I observe the musics tempo on my tempo
curve until I arrive at position E1, how much
time (in seconds, say) has then elapsed?
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T(E) 1/slope(E) 1/(de/dE) q /seconds
e1 - e0 E0? E1 slope(E) dE E0? E1 1/T(E) dE
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Two Examples
e1 - e0 E0? E1 1/T(E) dE E0? E1 1/T dE
e1 - e0 (E1-E0 )/T
e1 - e0 E0? E1 1/T(E) dE E0? E1 1/(T0
S.(E- E0) ) dE e1 - e0 1/S.ln(T1/ T0 )
Attention Here S ? 0
T(E) constant T
linear tempoT(E) T0 S.(E- E0) S
(T1-T0)/(E1-E0)
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This covers the general tempo curve by
sufficiently fine approximation!