Formal aspects of musical structures - André Riotte
Chapter 1 - The musical space
A few reminders of elementary physics
We shall not go into a detailed discussion here of the physics of musical phenomena. The basics are
described in the appropriate works [3.5]
We shall simply remind you that “natural” sources of sound are mainly solid or gaseous bodies
(metals, wood, membranes, air column), or occasionally liquid bodies (remember the “bottlephone”
and the Lasry and Baschet’s research) whose geometric proportions favour certain normal modes
(fundamental natural modes of vibration).
These modes are solicited by an input of energy, intermittent or continuous (percussion, friction,
pressure), making use of resonance phenomena. However, any source of noise can be considered a
source of music.
As far as “artificial” sources are concerned, there are two types:
Electronic generators, which devise and produce electric waves; based on intellectual
knowledge of resonance phenomena, they generally constitute “models” of natural sources. The
information produced is the continuous variation of the electrical wave, transformed into a sound
wave by electro-mechanical transducers.
Programming systems and computer programmes are abstract models in themselves, and
devise not a continuous wave but point by point information, which must then be transformed
into a continuous sound wave by means of digital-analogue converters. It is important to note
that in this case, the “virtual” duration of the information is itself parameterised, and there is no
synchronism between devising and emission; it is retranscribed in real time by a “clock”.
The “musical information” is transmitted through variations in the pressure of the surrounding air. The
quantities of energy involved in this transfer are generally very low.
Here again, electronics intervenes to transmit messages outside the acoustic range, mechanicalelectric transducer (microphone) and electronic transmission may be followed by wireless transmission
and re-translation into sound, constituting the intermediate current chain, which comprises its own
deformations (fig 1.1).
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It is important to note that the transmission chain may defer the emission of the message, in which
case various physical media (disks, cassettes, etc) constitute a fixed memory for storing the potential
sound wave.
As far as the receiver is concerned, it is a pair of ears, in itself a complex biological mechanism [3.2 ],
which transmits and interprets the information received by the brain and from the brain to the psychomotor system. That an association of human receivers bathed in a same sound message may modify
their receptivity is likely but not proven.
Lastly, as the sound message is spatio-temporal in nature, the localisation and possible mobility of the
sources and of the receivers have their importance, given the physical properties of the transmission
milieu (speed of sound, distances, etc.), as do the particularities of the space the exchange takes
place in (acoustics, architecture).
All these stages in the production and transmission of a sound, in particular those making use of
recent technologies, have given rise to the perfecting of descriptive and analytical methods that call
upon applied mathematics.
Most are related to differential analysis, since they are phenomena concerning intermittent or
sustained variation in the state of the physical milieu.
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As the approach is analytical, it has tended to isolate certain properties or parameters of the sound
produced, which we will come back to.
However, upstream processing, that is to say the medium through which the imagination organises
sounds, as well as its effects downstream on the receiver, have remained largely in the domain of
subjective description. These are the levels on which efforts are still required, and we will turn our
attention first of all to the first one.
In this chapter, we are now going to apply a similar approach to attempt a description of the abstract
musical space, that is to say the tools that exist for formalising sound parameters, with their
advantages and disadvantages.
The parameters of sound
Identifying constants in the domain of variation and signs of repetitivity in the kingdom of the
irreproducible is the usual discourse of the physicist. Whence the notion of the fundamental frequency
of a sustained sound, or note, or pitch, measured in cycles per second. A sound is conventionally
referred to as pure when the pressure wave that characterises it as a point in space is represented by
a sine curve. Pure sounds do not exist in nature, but electronic generators give a good approximation
of them. We will not recall here the theory of vibrating strings or of vibrating membranes or air
columns; we shall just give a brief reminder that the physical properties and geometry of the vibrating
milieu, in addition to the fundamental frequency, favour frequencies that are multiples of this, or
harmonics; the rate of these harmonics (ratio of amplitude to the fundamental) determines the timbre
of the composed sustained sound.
The mathematical models are taken from differential analysis: differential equations and systems of
equations, linear or non-linear, and equations with partial derivatives allow representations of the
waves produced.
When the phenomenon is established, it gives rise to a transient of varying amplitude, which
corresponds to the attack of the sound; when the energy is provided by percussion, the transient is
particularly strong, and the body (or milieu) then vibrates at its own frequencies, with a more or less
long period of decay down to total extinction.
When the properties and proportions of the vibrating milieu have been selected, (for example, a
physical system with identified constants), the frequency spectrum is discrete, and a Fourier serial
analysis allows a breakdown of the representation of the wave into a sum of sine curves. In the
general case of a noise, the spectrum may be continuous in a certain frequency band and only a
Fourier integral analysis is possible.
If we come back to a sound sustained by a source of energy and produced with musical intent, its
exhaustive description at a given point in space, from its initial appearance to its extinction, is the
representation in the form of a curve with Cartesian coordinates of the variation in air pressure over
time at the given point.
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The observation of this curve, recorded on paper (like an electrocardiogram, for example), shows
three distinct areas:
- the area in which the sound is established, during which the transient dominates
- the area in which the sound is sustained, where a periodicity establishes itself, the amplitude
of which is modulated by the slow variations of the energy supply, which correspond to the
dynamic of this sound, that is to say to its variations in intensity while it is sustained, and
modulated, where relevant, in frequency around the nominal value of the fundamental (vibrato
of string instruments, voice, etc.)
- the area of decay, where the amplitude decreases rapidly.
It is in the area where the sound is sustained that a serial Fourier analysis will show the breakdown of
the wave into a sum of sine curves, whose frequency and amplitude ratios to the fundamental are
characteristic of the timbre.
7
To say that a note from a clarinet is an A accented, louré and played mf is therefore to imply a series
of drastic simplifications with regard to the physical reality, as the timbre itself can change from one
tessitura to another (for example, bagpipes to clarinet).
It is important to keep these simplifications in mind when we use formalisms on the parameters of
sound.
In particular, the formalisms on pitch scales assume indefinitely sustained pure sounds, and only
constitute a first, restrictive approximation.
Formalisation of conventional scales. The group structure
Conventional tonal language is based on the temperament of intervals known as “natural” which are
detected as harmonics of a fundamental. Xenakis interprets this [5.8] as a compromise between two
approaches: one geometric and Pythagorean, based on equal subdivisions of a single string length,
and an arithmetic or Aristoxenian approach, which adds up the intervals between sounds.
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In any case, any sound scale built on a unitary interval between two frequencies
gives precedence to the interval
that is to say that
and which
can be formalised in an efficient
manner, according to Barbaud, from the set
the integers
, through a bijective application on
a = 0,1,...n-1.
On condition that the resulting sound scale is considered to give precedence to the octave, the use of
the structure of whole numbers modulo-n will allow us to formalise a whole series of musical properties
in this scale.
n = 12 equal tempered scale
n = 18 scale of thirds of tones,
n = 24 scale of quarter tones, etc.
If we decide to use a value for n that differs from the conventional one, it will be necessary to estimate
the distances of the resulting intervals in relation to the distances formed by the succession of the
natural harmonics of a sound.
nd
We can, of course, favour another interval, for example of the 2 harmonic, that is to say
, if
we make sure there is no “kth” degree
for, as the auditory pregnance of the octave is
stronger than that of the fifth, the modulo-n study would lead to properties that are not verified by the
ear. To come back to the set
transposition of musicians) that is to say that
we use its group structure for the law of addition (the
a) this law is associative:
b) there is a neutral element:
c) each element allows symmetry:
d) this law is always defined as
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This formalisation is well suited to classical tonal language, of which Barbaud has studied certain
harmonic properties.
For example, although the major scale
considered as the union of three subsets
is not a sub-group of
, it can be
major common chord built on the tonic
major common chord built on the dominant
major common chord built on the dominant
Similarly
represents a minor common chord
and the three forms of the “relative” minor scale of
modal form (note
are given by the unions
)
harmonic form see fig 1.2
melodic form
or, again
describes all the major and minor common chords that exist in tone “ i “
remember that
is a sub-group of
now, for
we note
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The modulo-12 congruent scales
If we build all the sub-groups of
Sub-groups
using the properties of modulo-12 congruence
No. of elements
itself
No. of sub-groups
Values of k
1
12
2
6
3
4
4
3
6
2
12
1
We obtain 28 sub-groups which are scales familiar to musicians;
- the totality of octaves built on a note
- the augmented fourths (or diminished fifths)
- the 4 augmented fifth chords
- the 3 diminished seventh chords
- the 2 whole-tone scales Debussy used
- the twelve-tone set.
As a passing note, historically, the members of the musical phrase using one of these sub-groups as a
scale used the subsets (chords) of the sub-group without harmonic restriction, implicitly drawing the
consequence of “compatibility between aggregate sets” from the properties of equidistance between
successive degrees. It is easy to verify this in all the diminished seventh passages of the Romantic
period, but also in Debussy’s whole tone scale sections (cf The prelude of the first book Voiles).
The usual logical operations from these 28 sub-groups allow us to formalise all the modulo-12 scales.
However, Xenakis [5.2] preferred to express this formalism using modulo-n residue classes, which
gives:
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for
(classe = class)
for
(classe = class, notée = noted)
for
for
for
The corresponding lattice is symbolised fig 1.4.
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We will see that this new notation has many consequences, for it gives a new degree of liberty (the
modulo-n) which means that the formalism is not restricted to modulo-12 scales, that is to say to
scales whose internal structure is repetitive from one octave to another.
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Xenakis performed the exercise of formalising the major scale using logical operations on sieves,
which gives (fig 1.3)
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as well as a whole series of scales with micro-intervals in Byzantine music. [5.7].
Although it does not correspond to the traditional language, from now on we will establish a distinction
between
a mode: scale built exclusively from the union of sub-groups of
properties
thereby combining their symmetry
and a scale, the internal dissymetries of which require the use of other logical operations for its
description.
An example of an application: Messiaen’s modes of limited
transposition
Messiaen presented [5.2], without formalising it, the musical structure of the modes he makes constant
use of in his compositions, and which he called modes of limited transposition because, due to internal
symmetries, they cannot be transposed 12 times without falling back into the same subsets.
The use of the operation union of modulo-n residue classes, (n = 2, 3, 4 or 6) makes it easy to
formalise them.
The first mode is in fact Debussy’s whole tone scale. There are only 2 transpositions,
this is the only mode which is a sub-group of
“create an indistinct tone” or as a mode in itself.
and
and
(see above). Debussy used it extensively, either to
In the latter form is the famous Prelude “Voiles”, the second of 24 Preludes for piano, which uses
mode
only, with the exception of 5 central climax bars, based on the pentatonic scale. (This scale,
according to the previous definition, is one of the most extensively present historically, appearing in a
wide range of types of music, from Chinese to African folklore. In “Voiles”, Debussy used it in the
transposition which makes use of only the black keys on the keyboard.)
The second mode is formed by the union of 2 of the three 3 residue classes from base 3.
According to the formula of combinations
, there are only 3 separate transpositions:
st
(1 Messiaen transposition)
nd
(2 Messiaen transposition) fig 1.5
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Formal aspects of musical structures - André Riotte
rd
(3 Messiaen transposition)
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Formal aspects of musical structures - André Riotte
th
Musically, its structure based on the combination of 2 diminished 7 chords and the fact that
make it a particularly ambiguous mode, in fact
and Messiaen played on this quadruple tonal ambiguity.
nd
There are abundant examples of the use of the 2 mode in his work; a random selection would
include the principal sections of one of the preludes for piano composed in 1930, Le Nombre Léger.
But Debussy had already employed it intuitively; see, for example, the second last section (bars 31 to
th
37) of the 13 prelude for piano, Brouillards, constructed on
then
(fig 1.7).
We even find two forms superimposed
and
(harmonics pedal), based on a technique familiar
th
to Messiaen, in the 14 prelude, Feuilles mortes (bars 25 to 30). (fig 1.8)
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Formal aspects of musical structures - André Riotte
th
In fact, this mode and its internal repetitions were already latent in the style of the 19 century. It is
found, for example, in the form
st
in a transition of Chopin’s 1 Ballad (bars 130 to 133. fig 1.6)
The third mode is formed by the union of 3 of the 4 residue classes of base 4, or 4 transpositions; in
fact
rd
(3 Messiaen transposition)
nd
(2 Messiaen transposition)
st
(1 Messiaen transposition)
th
(4 Messiaen transposition)
Before we continue, we can use a simple relation:
There are 4 types of basic modulo 12 sieves
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Formal aspects of musical structures - André Riotte
where
and
We have
(k+j) modulo C
whence:
These relations show the internal symmetries on which it is possible to play, that is to say the parts
common to the 2 modes, for example:
fig 1.9 gives the representation of
th
4 mode (8 notes)
th
5 mode (6 notes)
th
6 mode (8 notes)
6 transpositions
6 transpositions
6 transpositions
th
7 mode (10 notes)
6 transpositions
th
We can see that the direct interferences between the bases 3 and 4 are not used (only the 6 mode
establishes communication through the intermediary of
.
In fact:
This is the mode I myself used in Dualités for violin and piano (1964)
The common principle of these modes is to introduce partial dissymmetries into a symmetrical base 12
organisation .
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Congruent scales different to 12 (curvature modes)
We saw that the previous formalisation used only sieves sub-groups of
sub-multiples of the octave).
(that is to say, musically,
Now imagine that we use sieves with a base that is not a sub multiple of 12 (modulo 5, 7, 8, etc.)
This can be done in different ways.
1- “Curvature” modes
The sieves must be “initialised”, that is to say a reference pitch set relative to each sieve used. If we
use, for example, the combination of sieves with base 13 in the 12 note system (octave + ½ tone), we
obtain a “mode” whose fixed interval ratios increase by a half-tone for each new cycle (the system
becomes in fact coherent modulo-13). The resulting scale is therefore functionally modulating. (see fig
1.10 the mode used in “Jubilation heuristique” Riotte 1968).
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2- Non-congruent scales
First note that we can distinguish 2 types of sieves that cannot be reduced to the octave:
Those whose modulo is a prime number (5, 7, 11, 13, etc.)
Those whose modulo is a multiple of the bases 2, 3, 4, 6 which can be reduced to the octave
Whereas the first express the totality of the modulo-12 degrees if we consider a theoretical area that is
not limited to the audible range:
th
5 representing the cycle of 4 s
th
7 representing the cycle of 5 s, etc.
The second only run through a subset of the sub-group of which their modulo is a multiple;
th
minor 6
th
major 6
... etc.
This particularity, which is a result of the properties of the integers themselves, allows us to predict
“auditively” their contribution in a non-repetitive scale.
In any case it is easy to conceive, given their periodicity extended over a large number of octaves, that
logical operations on several of them constitute coherent scales that are nonetheless non-repetitive
across the audible spectrum. This property, rich in possibilities, has been used by many composers,
including Xenakis and myself.
Comment:
The formalisation of scales by sieves implies a temperament. If this temperament is built on the
octave, and comprises n degrees reduced in octave ambitus by injection (modulo-n):
- If n is a prime number, all the sieves run through the entirety of the reduced in octave ambitus
by modulo-n injection.
- If n is not a prime number, then
k, l, m prime numbers α,
β, γ positive integers.
And the sieves built on the degrees of the scale fall into 2 sub classes
The sub-class of sieves whose modulo is prime with n; these cover the entirety of the degrees
reduced in octave ambitus by injection
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Sub-class 1: 5,7,11,13, ...
The sub-class of sieves built on an incomplete combination of prime components of n and their
multiples; these only cover a subset of the degrees injected in the octave.
Sub-class 2: 2,3,4,6,8,9,10,14, ...
It is important to note that while the sub-classes are the same irrespective of the unit chosen
(semitone, quarter tone, etc.), they do not correspond to the same degrees of the scale.
Partially congruent scales. An example taken from Nomos Alpha
(Xenakis)
In his article “Vers une métamusique” (Towards a metamusic ) [5.7], Xenakis makes allusion to a
structure “outside time” (hors-temps) founded on the particular properties of selected residue classes.
That is, the set P of residue classes
we have
modulo-r based on the natural integers
prime with r
(relative integers)
whence
In relation to the multiplication operation with modulo-r reduction, the
group closed on itself, that is to say their products remain in the set.
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form a finite commutative
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Formal aspects of musical structures - André Riotte
If we take for example
, it gives P = {1,5,7,11,17,...}
In fact, we will have:
etc.
Xenakis used this property in “Nomos Alpha” for cello only, the detailed formalism of which he explains
in his article “Vers une philosophie de la musique” (Towards a philosophy of music) [5.8]
He builds a family of sieves composed from an expression function
based on logical
operations between two sieves that form a pair
Which is, for the starting function:
The residue classes are built on the quarter tone unit. In this way, he obtains a non-repetitive scale in
quarter tones (fig 1.11).
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Without going into a detailed musical analysis, we can comment on the approach used in its
conception:
Let us write to simplify
the function L (ll, 13) is written
=
Of the 3 terms it comprises, it is term 3 that provides the main fabric of the modulo-13 scale (a fifth
minus a ¼ tone); this term represents a “convex” curved mode (see definition fig 1.11 of the same
principle as the mode of “Jubilation heuristique” but in the sphere of quarter tones. In the modulo-26
area (octave + semitone), it is constituted by 2 subsets:
Perfect minor chord
th
7 chord diminished to 3 notes (offset a quarter tone higher in relation to the origin).
Term 2 adds a modulo-11 sieve obliterated by a modulo-13 mask
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Formal aspects of musical structures - André Riotte
It is therefore a convex curved mode complexified by additions of modulo-13 and modulo-11,
intermediate between the curved mode and the totally non-repetitive scale.
Xenakis builds, from a more general function
a graph that runs through the permutations of the
in function L according to an obligatory path that
constitutes what he calls a metabole, a combination of cyclic shifts of the indices of the sieves
(transpositions) and of modifications of modulos (modulation).
Totally non-congruent scales. An example taken from Anamorphoses
(Riotte 77)
Taking this to the limit, as the curved modes still have a faint hint of tonal function, it is possible to
define a coherent domain “outside time” with no trace of repetitivity.
From a set of sieves with bases greater than 6, and assuming an origin provided (see fig 1.12) , it is
possible to define, from
“perceptible”.
, subsets with logical relations between them that are auditively
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For the conception of the work, I needed to define 2 complementary subsets interpreting the semantic
aura of the remember-forget pair [5.14].
All the subsets of the work are built from sieves from
to
(the sphere of the semitone).
Here
If we define the operation
, it has elements in common with M, or
To obtain a subset J such that
, it suffices to apply the logical operation
.
Formalisation of durations
In Western music, apart from some isolated phenomena (isorythmy), as far as formalism is concerned,
th
durations remained restricted to an elementary world up until the 20 century. A world essentially
articulated on the basis of a regular subdivision of binary-tertiary based objective time, with a hierarchy
(down beats/up beats) that can be described from the residue classes (fig 1.13).
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On the other hand, the same can not be said of oriental music. We only have to think of the network of
symbols attached to the rhythmic figures in Hindu music, figures dear to Olivier Messiaen, for
example.
However, we should recognise that “unmeasured” time (that is to say suspension of metre) was
always present, albeit very sporadically, in the musical history of the classical period.
See for example.
Rameau: Prelude to the 1er recueil de précis for harpsichord. fig 1.15 free fluctuations prior to a
fixed tempo.
Mozart: Capriccio KV 395 – fig 1.16. Improvised style.
th
Beethoven: Adagio grazioso of the 16 piano sonata (op31 no.1) fig 1.17 suspension of time
(point d'orgue) efflorescence within a scale.
Furthermore, the romantic period introduced the idea of rubato, that is to say of a subjective
modulation of the beat (contraction-dilatation) and not of a veritable exploration of real numbers.
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As for the possibility of the parallel play of prime residue classes among each other, we also find
th
distant traces of this in “3/2” (e.g.: 10 piece in Davidsbundlertanze) (fig 1.14) R. Schumann p 25 of
the score.
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The real language of durations is situated at the level immediately above the concatenation of whole
values, that is to say one of the compositional materials, as opposed to what I have called the prior
“musical space” (scales for the pitches, metre for the durations).
In the classical period, during performance, the materials, founded on proportions that are generally
arithmetical, had to follow a dialectic of tension-relaxation, schematised by the alternation of
“masculine rhythms” (tonic accent on the last down beat) and “feminine rhythms” (tonic accent
followed by an inflexion on the following down beat).
They entered into the definition of elements of a morphology and a syntax with implicit rules, to which
the dynamic articulation referred, divided into phrases, sections, etc.
However, note that the existence of formally drawn up rules, such as those of strict imitation (see for
nd
example the 2 Invention for 2 voices in C minor by J.S. Bach) and of the fugue, already introduced
an abstract notion of the interaction of structures, not only in succession, but also in simultaneity
(subject-response) implying the possibility of an acoustic dissociation of the instantaneous
components of the message much finer than even an in-depth physical analysis of the sound
phenomenon would allow. On the one hand, the very idea of “voice”, which clearly shows a human
source, presupposed this distinction originally founded on the perception of simultaneous sources
characterised by timbres, this characterisation becoming gradually reduced to the much less clear
abstract notion of the layering of frequency bands (tessitura) within a family of uniform timbres
(harpsichord).
Furthermore, this simultaneity was the implicit recognition of distinct categories of time depending on
the sound source, with the possibility of a single “subject” (in the sense of the fugue) being found in
more than one voice at different stages of evolution.
It is fair to remark that this characteristic of the contrapuntal language, highly sophisticated in the
language of J.S Bach, and specific to Western music, did not continue to evolve in this direction.
On the other hand, the phenomenon of repetition is one of the universal structural bases of music, and
in this respect deserves a few elementary remarks.
Repetition
Repetition, in the widest musical sense, is the re-emission of a single sound phenomenon.
We can provisionally distinguish several levels of repetition, related to the complexity of the
phenomenon repeated, and to the circumstances of the repetition. These are the problems that are
the subject of the present chapter.
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Global level:
repetition of a musical “work”: poses problems of situation in space and time, global
memorisation at the level of the receptor, and therefore of the very signification of the musical
event: the score as symbolic message, the myth of identity.
Internal structural levels:
The section: illustrated by the reprise (the minuet or rondo of a sonata are the most significant
examples)
The phrase or member of a phrase: illustrated in the recent past by a characteristic of
Debussy’s language, analysed by Nicolas Ruwet (Note on duplications in the work of Claude
Debussy in [5.9]).
The motif: illustrated today by the American repetitive group (see feature in the review Musique
en jeu No. 26, particularly the article “Musique repetitive” by IVANKA Stoianova (1.15).
Elementary level:
Repetition of a sound “molecule” (an element which has physical reality, and therefore has all its
parameters: a note in the extended sense)
Poses the problem of periodicity, that is to say of a metric of durations, with the corollary of the
metrics of the entire musical space, the axiomatics of which were posed by Xenakis, inspired by
the axiomatic of Peano’s numbers [5.7].
These 3 levels pose fundamental structural problems that we will broach later on.
Here, we will limit ourselves to a few reflexions on these last levels. First let us observe that to the
extent that art seeks to show relations between conceptual categories hitherto considered distinct,
repetition is the first functional operation, historically speaking.
Now, any formalism is built on the same concepts: definition of categories of objects linked by identity
or by relations (operators)
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Note that all the mathematical formalisms have considered time as a separate category (the
“independent” variable). It was not until relativist theories came along and research on physical “units”
that “space-time” was considered a homogenous milieu.
As a musical event must transmit its categories at the same time as its “message”, it is natural that
repetition be a basic operation.
A repetition of objects (sounds) is the de facto definition of a set (closed or open)
Scales
Subsets
Sub-groups
Even the series, presented as a concern of non-repetition, is based on the repetition of (abstract)
relations. It can be taken further to a concept of order both on objects and their relations (balanced
cycles, see chapter 2).
It could be interesting, in formalised analysis, to find a way of measuring the categories, concepts and
operators a “musical work” uses, independently of the implementation of the actual operations.
This could be based on:
Total identity (mental view: “other conditions being equal” the human receptor will have varied)
The identity of certain components
Analogy (global impression that two phenomena are close. E.g.: scatter plot with the same
statistical distribution within a given duration.)
After having insisted on repetition or repetitivity, invention sought to bring to light its complement,
irreproducibility:
At the level of the sounds (new uses of traditional instruments, stochastics and scatter plots,
synthesis of new sounds).
The space of the sounds (the approach of the continuous through glissandi)
The structures
Open works
Cybernetics of forms
Musical beings
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The two approaches can be schematised thus:
Invariables
diversity
We can then contest the necessity for formalisation; nonetheless, experience shows that the
deliberate search for maximum diversity led to the de facto perception of elementary categories of
invariables (e.g. impression of the equal statistical distribution of sounds within certain works of the
serial school).
The concepts of repetitivity and regular repetitivity (frequency) are formalisations, metrics centred on
the highlighting of the time variable based on the bar (objectivation).
Return to formalisation
Boulez [5.5] distinguishes pulsated time and amorphous time, which could be described as an initial
approximation from proportions of natural integers or rational numbers (real...)
However, this distinction does not take account of the disturbance added to the language of durations
by Stravinsky, of which Boulez nonetheless developed an in-depth analysis of the Sacre du Printemps
which was a milestone [5.6].
Now, if the two “awarenesses” of time pre-existed in the classical period, it was more in the form of a
measured time and non-measured time, the second analysis being achieved through the awareness
of an individual time (that of the performer) functionally playing generally on the dilation of a
periodicity, possibly up to suspension (point d'orgue).
It is therefore a question of a modulation of periodicity, of which the rubato is a limited variant.
On the other hand, even in traditional folklore languages introducing prime subdivisions with the binary
(5,7,11, analysed in general by additions of 2 and 3), the function of the down beat was only rarely
contradicted; the preferential residue class kept its function throughout process.
It is this permanent contradiction that Stravinsky instituted, without, however, escaping periodicity, that
is to say the unit of measurement.
An allusion to prime periods among each other, which we will come back to in detail, prepared the way
for the awareness of a new possible formalisation, introducing to durations the equivalent of the logical
operations on the residue classes, the development of which we saw regarding the formalisation of the
pitch space.
However, the first stage necessary was the final liberation of durations, and their simultaneous play.
Olivier Messiaen was the first to develop a coherent language of “counterpoint of rhythms”, which he
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explained himself [5.2], based on the simultaneous play of prime durations among each other (see for
example the piece for organ Le Verbe, a fragment quoted by Messiaen himself [5.2] fig 1.18).
But above all it gave autonomy to rhythmic figures, by subjecting them to regular or irregular
augmentation or diminishing (multiplication and division by 2, addition and subtraction of a fraction of
th
each value), and juxtaposing them (see for example. The 9 movement of Turangalila Symphonie,
Turangalila III).
Boulez refined the possible transformations of rhythmic cells (Eventuellement [5.6] p. 160), by
conversion (monnayage), negative, engendering a rhythm inside itself, etc.
“Arithmetic” and “geometric” subdivisions
One variation to this approach, also latent in the traditional language through what musicians
improperly called “irrational” values (for in general they relate to what mathematicians call rational
numbers, in extension to natural and relative integers) is the subdivision of a given duration into prime
fractions.
The “3 against 2” already mentioned, when applied to short values (triplet) is the most familiar
example.
We could define this approach, in opposition to the previous one, as “geometric” subdivision of time
(according to Xenakis’ distinction), the other one being considered “arithmetic”.
We find it developed, for example, in the petite musique de nuit (1958) by Roman HaubenstockRamati, “mobile” for orchestra ( fig 1.19). Such as it is used in this example, it only has an indicative
value. I used it myself in a movement of the String Quartet Multiple as a controlled means of
distributing the attacks between the 4 instruments (fig 1.20 – see also abacus fig 1.21).
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It should be noted, however, that the practical realisation of this subdivision by musicians remains at
the limit of their possibilities, and can only be achieved with a certain precision in some very special
cases. For example, my main point in the previous application was a control of the succession of noncoincidental attacks contributing to the synthesis of a unique complex proportion. (Note that I have
since re-written this movement using KANT software with the help of Gérard Assayag, project leader
at IRCAM)
On the other hand, there is nothing to stop a rigorous application of such principles of “rational”
subdivision of a given unit in the case of computer calculations and synthesis.
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The two approaches, arithmetic and geometric, could also be combined, as the arithmetic approach,
we shall see later, lends itself to formalised developments on the scale of “phrases” using logical
operators, whereas the geometric approach is better suited to a finer control, for example to draw a
single complex event distributed between several voices-instruments-sources.
Subdivisions of a macro-unit of time
Since, for the moment, we have set aside the global proportions of a musical work, which we will look
at later, we will restrict ourselves to a few examples of subdivision on the scale of the section.
The basic material (rhythmic cell) is also a question of proportions. In the world of traditional binary
based durations, the application on the integers remains the rule, with the unit as the smallest binary
duration used. The rhythmic cell can then be represented by a sequence of integers:
4,5,3,1 means
If we take x for the unit.
(It is useful to remember that in such a formalism, each integer n represents two pieces of information:
the actual duration of the event in n units, and the moment this event occurs, located in relation to an
origin of time that coincides with the start of the cell at a moment equal to the sum of the previous
integers.)
It then becomes easy to imagine the basic operations possible such as the augmentation already
practiced by J.S. Bach (product of each element by a given integer), diminishing (division by a given
integer, on the restrictive condition that it is, for the n terms, a common denominator).
Messiaen had already added to this an addition (or subtraction) operation which can be generalised by
adding a given integer to each integer:
example: +2 to the above cell gives:
6,7,5,3 that is to say
Boulez, in his article “Eventuellement” [5.6] provides other, more complex cellular transformations,
which can be characterised by the successive application of elementary operators to the same cell.
A cell is characterised by its proportions. We could therefore define a family of cells or profile by an
unequalness.
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Formal aspects of musical structures - André Riotte
Take the example of 4 terms of a duration cell:
The two previous examples are particular cases of a single profile defined thus:
for which the number of solutions is finite, countable or infinite depending on whether the absolute
upper and lower limits of
are fixed or not, and where
natural integers
or
rational numbers
or
real numbers.
Here we use unequalness, or a wide order of relation. To limit the solutions, to avoid large numbers,
such a profile can be applied to a grid of pre-established durations.
A duration grid, equivalent in time to a scale in space, is a succession of k durations matching a given
structure.
It is therefore easy to conceive of grids that are congruent or non-congruent modulo-n, by analogy with
the scales.
For a concrete example used in my string quartet Multiple , refer to fig 1.22. It provides solutions for
the application of the profile
operators, chapter 2) in a grid
and its “inversion” (see the generalisation of serial
G = <8, 6, 5, 1, 2, 4, 7, 3, 8> (unit = x
)
(G is in fact a balanced cycle of modulo-8 durations, see chapter 2.)
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Formal aspects of musical structures - André Riotte
fig 1.23 gives all the solutions for the inversion of the above profile with the given global duration (sum
of the durations of the profile).
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Formalisation of intensities
As we have seen up till now, the commonly available formalisms have the disadvantage of uncoupling
the sound parameters, and considering them separately.
Now, if we were able to formalise pitches and durations, this is because there is a precise notion of
measurement available. The same can not be said for intensities.
For in fact, although in theory the intensity of a sound is a linear function of the amplitude of the sound
wave, in practice, measurement is not classical, and the only concept familiar to traditional musicians
is that characterised by classes of equivalence, that is to say the fact that one sound is more or less
intense than another.
There is a scale of values, but it is subjective and subject to interpretation.
In addition, musical reality evolves on several levels, and so we must distinguish between
- the individual intensity of sounds that participate in a complex at a given moment
- the global intensity resulting for this complex
- the dynamic evolution of these intensities.
This latter particularity historically distinguishes, as far as a formalism is concerned, intensities from
other sound parameters. Control of the breath, for instance (voice and wind instruments) or the
pressure of the bow (strings) permits crescendos, that is to say continuous variation between two
limits.
If the most natural formalism is therefore the “wide order relation” that characterises a profile, we can
consider that it applies even if the intensity varies continuously from one limit to the next successive
one.
This is how we could characterise for example the dynamic of an idea of a cell from a randomly
quantified scale
ppp
pp
p
mp
mf
f
ff
fff
1
2
3
4
5
6
7
8
In a parametric form.
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Example taken from “Transe Calme” for piano (Riotte, 1973)
(Head of main voice)
Pitch profile:
Duration profile
Dynamic:
Articulation:
The arrow is a symbol of continuous linear variation between the events: each actualisation of the idea
of the cell should be initialised (setting of the value of i in the chosen scale)
- either depending on a global average intensity in the passage under consideration
- or in relation to the intensities of the preceding cell, for example a principle of “concatenation
with covering” (CAR) (which we will mention again later), the initial intensity of the cell should
coincide with the final intensity of the preceding cell.
It is clear that in a polyphonic model, the initialisations of superimposed events should be established
from the same perspective, that is to say in relation to each other.
Lastly, as the narrowness of the scale adopted (8 to 10 terms) excludes the modulo, the limit
phenomena should commonly be taken into account, that is to say that depending on a given
initialisation, we should allow for “saturation” conditions, for example:
In the above cell where i = 7, we have:
7
7
ff
ff
that is to say
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Formal aspects of musical structures - André Riotte
The problems of timbres and attacks
In the traditional musical space of instruments, even with the extensions that are known to us today,
the notion of timbre is too closely linked to the actual quality of the sound, that is to say to a complex of
closely related physical characteristics, for a developed formalism to describe it other than as a rough
approximation. The most precise distinction regards the relative importance of the attack of the sound
– or transient - and of the sustained sound imposing a predominant frequency that the ear can
recognise.
From the organ, the instrument par excellence of the “eternal” sound – hence its para-liturgical
function at a time where the notion of absolute was synonymous with atemporal fixedness, to the
cymbal whose peal, complex spectrum and evanescence translate the elusiveness of fleeting reality,
all the transitions are latent. However, due to the way orchestral instruments are made, including the
recent extensions in percussions, the families of timbres remain at the crossroads between formal
models built on entities that can be classified with all their parameters, and the computer simulation of
instruments of which the quality of sound is only perceptible, apart from some exceptions (see the
work on sound synthesis by Risset) globally, which makes it impossible to subdivide them into wellordered classes.
In these conditions, only the properties of sets can easily be applied.
Organising a language means defining the field in which we will operate, a field in which combination
will allow us to explore all the possibilities, and the subsets that we want to consider as significant.
To set these ideas on a simple example, let us restrict ourselves to the attack of strings, considered as
a family of homogenous timbres (see fig 1.24)
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We can see that
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[ A(1), L(1), I(3), N(2), H(0), S(0) ]
characterises two notes pressed p on the fingerboard.
The quantitative space we are working in therefore generally has 6 dimensions; we still have to detect
the prohibited zones, such as A(2), L(1) or N(4), H(2).
Certain studies [3.11] seem to confirm the close relations that exist for the ear between timbre and
attack, and should enable a less subjective handling of these parameters and their correlations in the
future.
Interaction of the parameters
The previous developments situated the possibility of defining “virtual spaces” constituted by the
product of the frames of pitches and duration grids, forming a potential crossing at each node which
could situate the main events, with their characteristics of intensity, timbre and attack, corresponding,
in fact, to a 5 dimensional space. However, as we observed above, the last three parameters can only
relate to equivalence relations or even more primitive functions.
Such a description could be criticised because of the quantifications it implies, and the same goes for
the artificial dissociation of characters of sound that are intricately linked.
In particular, when we wish to work with sounds that are complex or continuously variable, research in
computer synthesis leads us to believe that the continuous should be taken into account.
Currently, when the variation is linear and only concerns one of the parameters (traditionally intensity),
a bi-dimensional representation remains possible (see for example Xenakis’ string glissandi layers).
As a provisional conclusion, remember the particular case of the approach to parameter interaction in
“mode de valeurs et d'intensities” for piano (1949) by Olivier Messiaen.
For each of the three voices (fig 1.25) it defines a pitch mode with twelve sounds distributed across
several octaves, but each pitch is in fact a 4-uple (pitch, duration, intensity, attack).
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This is therefore an intermediate model between the definition of a virtual space such as described in
this chapter and that of the local mechanisms as we shall see in the following chapter.
The principal constraints:
- each event of a voice must be immediately followed in sequence by another event of the
“mode” (without silence);
- the emission of a sound excludes, for its duration, the occurrence in the two other voices of a
sound of the same name;
in fact define a triple automaton, where each new event of a voice can be chosen from among 9
possibilities (12 minus the previous event – non repetition – and the two events in progress in the
other voices).
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