Bausch !1

Partiels as a Vessel

The spectral aesthetic movement brought many innovations to the way composers

developed musical materials. Partiels, the third installment of a suite of pieces under the name

Les Espaces Acoustique, was written by Gérard Grisey for 18 musicians. Starting with a

spectrum based on a trombone pedal of E1 (middle C as C4) and ending with crumpling papers

and whispers, in doing so, Grisey creates a trajectory over the course of the entire work: a

transition from harmonic sounds to inharmonic sounds, a concept that will be discussed in detail

later on in this paper. This occurs both on a macro level and a micro level within sections of the

piece. To facilitate these musical endeavors, abstract processes needed to be interpreted in a

musical idiom. These ideas consist of mathematical formulas and concepts that one would not

normally associate with a musical composition. These abstractions consist of the harmonic

series, harmonic versus inharmonic sounds, periodic versus statistical rhythms, additive

synthesis, filtering, distortion, and ring modulation. Through a brief analysis focusing on the use

of these concepts, Grisey utilizes Partiels as a vessel to carry out these mathematical applications

within a musical setting.

In order to understand the various abstract processes employed by Grisey, as well as

others, in this musical aesthetic, I would like to describe each approach in detail prior to

discussing their musical applications. The harmonic series, arguably the main focal point of the

spectral aesthetic, can be expressed by a simple formula: ƒx=(ƒ1•x) where ƒ1 represents the

fundamental frequency, or the lowest pitch, and x refers to the partial number. When multiplied

together, the frequency of the partial is revealed in terms of ƒx. Spectralists refer to groups of

pitches as either harmonic or inharmonic. Quite simply, a group of pitches whose frequencies are

Bausch !2

integer multiples of a fundamental frequency would be considered harmonic, whereas any

frequency in a group of pitches that is not an integer multiple of a fundamental frequency would

be deemed inharmonic; in extreme cases this results in noise, or a group of frequencies absent

from pattern. 1

This concept of harmonicity versus inharmonicity closely resembles Grisey’s reflections

of musical time in his Tempus ex Machina. In this treatise he explains his range of periodic to

statistical in the form of a chart (see Figure 1). For ease of comparison, Grisey’s term statistical 2

is synonymous with aperiodic. In this chart, he qualifies a periodic sequence of events ranging

from maximally predictable in which there is clear order, to aperiodic in which there is zero

predictability. Periodicity accounts for relationships outside of the temporal domain as well. The

harmonic series, as mentioned above, is considered a periodic sequence of events in the sense

that the next partial can be accurately foreseen.

Grisey also turns to computer music idioms to manipulate his orchestration as well as

pitch material. One of these ideas closely relates to Jean-Baptiste Joseph Fourier’s concept of

additive synthesis derived from a Fast Fourier Transform (FFT analysis). Instead of adding sine 3

1. François Rose, “Introduction to the Pitch Organization of French Spectral Music,” Perspectives of New Music 34, no. 2 (1969): 7, accessed November 29, 2015, http://www.jstor.org/stable/833469

2. Gérard Grisey, “Tempus ex Machina: A composer’s reflections on musical time,” Contemporary Music Review 2 (1987): 244

3. “Any periodic variation in a quantity, no matter how complicated, may always be represented as the sum of a number of simple sine waves with frequencies that are multiples of the basic repeat frequency of the wave (the fundamental).” Charles Taylor and Murray Campbell, “Sound,” Grove Music Online, accessed September, 2015, http://www.oxfordmusiconline.com/subscriber/article/grove/music/26289.

Bausch !3

waves, as in an FFT analysis, Grisey would add instruments together based on the spectral

information of the sound in question (i.e., one instrument per sine wave) to create a concept

labeled instrumental additive synthesis. Much like an equalizer module in an electronic music 4

studio, the concept of filtering can be applied to composition as well. An equalizer is designed to

raise/reduce the amplitude of certain frequencies within a sound. By interpreting frequencies as

pitches, we can apply filtering to composing in the same vein as a sound engineer would tune a

sound. With a simple modification to the already-mentioned harmonic series formula, ƒx(ƒ1•x)d,

the harmonic series can be stretched or compressed. This distortion – expansion or compression

– is controlled by the distortion coefficient attached at the end of the formula. If this coefficient

is greater than one, the harmonic series will be stretched, whereas, if the coefficient is less than

one, the series will be compressed. Both outcomes result in an inharmonic spectrum. Ring

modulation is a concept that when, at least, two sine tones (or pitches) are present, the addition

4. François Rose, “Introduction to the Pitch Organization of French Spectral Music,” 7.

Figure 1: Chart of periodicity from Gérard Grisey's Tempus ex Machina

Bausch !4

and subtraction of their frequencies create two new sidebands with amplitudes equivalent to the

reciprocal of the number of pitches being modulated. Ring modulation can also be expressed as a

formula: RM= (C±M), where C represents the carrier frequency and M represents the modulator

frequency.

Building on these ideas, I will now focus on how Grisey uses these concepts in context.

In Section One of Partiels (rehearsal numbers 1–11), Grisey implements several abstract

concepts to generate musical material. These concepts consist of the harmonic series,

instrumental additive synthesis, periodicity, filtering, and distortion. Although Grisey utilized a

sonograph of a trombone playing a pedal E1, the same group of frequencies could have been

derived from using the formula ƒx=(41.2 • x), substituting x for partials 1-51 (note this does not

give us amplitude information). The combination of the frequencies, coupled with their

amplitude, creates an accurate framework for Grisey to superimpose, through orchestration, the

ensemble of 18 musicians. This technique, as mentioned above, is considered instrumental

additive synthesis. The synthesis of this spectrum occurs eleven times. He uses these occurrences

to gradually shift from a harmonic spectrum to a distorted, inharmonic spectrum. Figure 2

represents the pitch material as it appears over time. The numbers above the measures refer to

rehearsal marks within the score, the open note-heads refer to harmonic pitches, and the closed

note-heads refer to inharmonic pitches. The first appearance of an inharmonic tone occurs in

rehearsal 3; a D6 is played in the vibraphone. However, if we refer back to the concept that

periodicity is directly correlated to harmonicity, this is not the first instance of inharmonicity in

this section. The contra bass introduces the fundamental, E1, with three distinctly even sforzando

quarter notes prior to each synthesis of the spectrum; this occurs in both the introduction and in

Bausch !5

Section One. These three even quarter notes are altered by one sixteenth note before rehearsal 2.

The significance in this alteration lies within our ability to predict the next musical event

resulting in an increased level of aperiodicity. From this point forward, the contra bass’s

interjections of E1 are in constant flux until the end of this section. In rehearsal 4, instead of

adding a second inharmonic tone, he introduces noise into the spectrum with the viola. Although

the viola is playing a ↓D5 (a D5 played a 1/6 tone flat), a harmonic pitch, Grisey instructs the

performer to use wide and uneven vibrato. This adds a level of both unpredictability and nuance

when compared to all previous pitches which, up to this point, have been presented without

Figure 2: pitch transformation of Section One Image by François Rose, “Introduction to the Pitch Organization of French Spectral Music”

Bausch !6

vibrato to mimic the quality of a sine-tone; the exaggerated vibrato in the viola adds, by

association with its surrounding pitches, a sense of noise into the instrumental spectrum.

Throughout this section, scratch-tones, the extended technique of applying an overabundance of

pressure with the bow to the strings, are added as an element of the noise. In conjunction with

increasing amounts of inharmonic pitches, elements of noise, and temporal variations in the

presentation of the fundamental, he incorporates a low pass filter into the spectrum. A low pass

filter reduces the high frequencies of an incoming signal; the overall trajectory of the pitches in

Figure 2 shows a gradual shift from high pitches in rehearsal 1, to low pitches in rehearsal 11.

Section Two lies between rehearsal numbers 12-22. Unlike Section One, this section

starts inharmonic and shifts to a harmonic sonority. Since this section starts inharmonic, it

warrants the use for operations that produce results outside of the harmonic spectrum. The main

process utilized here is ring modulation. The orchestrated results of this process are applied both

rhythmically and in terms of pitch. The former is a result of the difference frequency (subtraction

of two simultaneous frequencies) falling beneath the threshold of hearing. For example, Figure 3

shows an excerpt of the score around rehearsal 15. Further analysis of this excerpt shows

Figure 3: score excerpt showing ring modulation around rehearsal 15

Bausch !7

Grisey’s use of ring modulation to generate a rhythm; clarinets 1 and 2 play a dyad of D3 and C3

(1/4 sharp), respectively; these two pitches have frequencies of 146.83 Hz. and 134.65 Hz. When

plugged into our formula, RM=(|146.83| ± |134.65|), we receive two results: RM+ = |281.48|, and

RM- = |12.18|. 281.48 Hz. represents a slightly-flat #C4, this pitch does not appear in this section

and, therefore the summation tone information will be discarded. The other result, 12.18 Hz., is

well below the threshold of human hearing which implies searching for a rhythmic proportion

that closely matches 12.18 instances per second. In order to have sixteenth notes moving at

approximately 12.18 beats per minute, the most logical choice is to reduce the group of 17

sixteenth notes to reveal if it matches this frequency rate. To do this, first we need to multiply the

beat grouping of 17 with the tempo of the music. In this case, the tempo is 88 beats per minute in

2/4 meter. The group of 17 sixteenth notes is grouped into two beats, this modifies our tempo for

mathematical purposes by cutting it in half. If we multiply (17 • 44) the result is 748, meaning

each sixteenth note has a tempo of 748 beats per minute. To project this number into the tempo

of 60 beats per minute, we must divide (748/60). This results in an answer of 12.46; each 16th

note has an individual tempo of 12.46 beats per minute in the macro tempo of 60 beats per

minute. 12.46 is quite close to our original goal of 12.18. This is an explanation of one example

of the multiple instances of ring modulation throughout this section. Figure 4 represents a

condensed graph created by Chris Arrell to chart the remaining ring modulation relationships. 5

5. Chris Arrell, The Music of Sound: An Analysis of Partiels by Gérard Grisey (revision of DMA dissertation, Cornell University, 2002), 325.

a+

Bausch !8

Another instance of the ring modulation formula occurs in Section Three (rehearsals

23-27). Unlike in Section Two, the application of ring modulation to a dyad, the frequency seeds

for the ring modulation formula stem from linear tone rows. Furthermore, the additive and

subtractive results form their own rows. As seen in the serial music of the Second Viennese

school, the hauptstimme and nebenstimme symbols point out which row is the most important;

haptstimme representing the foreground, and nebenstimme representing the background. Figure 5

Figure 4: graphic, by Chris Arrell, expressing the ring modulation relationships in Section Two

Figure 5: first hauptstimme tone row with frequencies

Bausch !9

shows the initial row with frequency data located above the note heads as it appears in violin 2.

The linear quality of the musical material demands a different mathematical application; instead

of adding/subtracting dyads of frequencies, as in Section Two, the process needs to be adapted to

adjacent notes. Referring back to the frequency data in Figure 5, Table 1 includes the

mathematical outcome of the row.

In the score, at rehearsal 23, the frequencies listed in the addition row of Table 1 correspond

exactly to the pitches in violin 1 and the frequencies in the subtraction row are consistent with

the pitches in viola 1 (see Figure 6). This concept grows in complexity throughout this section.

For example, the next instance of a row adds a second hauptstimme row. Applying the same

concept to this second occurrence, the secondary rows nearly double in pitch content because of

the additional hauptstimme row. However, instead of applying to formula to subsequent pitches

Table 1: Rehearsal 23, first four pitches of hauptstimme rowOriginal Frequency, violin 2 (Hz.) 1318.51 – 987.7 – 698.46 – 1318.51

Addition, violin 1 (Hz.) – 2306.21 – 1686.16 – 2016.97 –

Subtraction viola 1 (Hz.) – 330.81 – 289.24 – 620.05 –

Table 1: Rehearsal 23, first four pitches of hauptstimme row

Figure 6: excerpt of score, first hauptstimme tone row (violin 1, violin 2, viola 1)

Bausch !10

in the row, both rows are utilized in conjunction with one another. Temporal location of the two

rows depicts which frequencies will be added/subtracted. Figure 7 provides a visual for this

example, coupled with Figure 8 to extrapolate the temporal decisions. Figure 8 presents three

important pieces of information: spatiality, pairings, and frequency. The dotted lines in Figure 8

express the temporal relationship between the two primary tone-rows. To further understand

Grisey’s compositional decisions, each pitch in the row is given a distinct label in order to

decipher the sequence in which the pairs appear. The pairs between the two series are connected

with a dashed-line warranting the following order of pairs: [1b 1a], [1a 2b], [2b 2a], [2a 3b], and

[3b 3a]. This does not mark the end of the process, but the end of the pattern (the last pitch of the

pair becomes first pitch in the next pair).

Figure 7: score excerpt of second row of rehearsal 23

Figure 8: analysis of second row showing pitch relationships

Bausch !11

Section Three, like Section One, starts harmonic and gradually shifts to inharmonic.

Since this excerpt is towards the beginning of the section it consists primarily of harmonic

pitches, with the one exception being ↑B6 (1/6 tone sharp). Each frequency is labeled in Figure 8.

Note, there are instances when there are two slightly different frequencies attached to the same

notated pitch, this is a result of rounding to the nearest harmonic pitch to maintain harmonicity.

This generative process increases in orchestration while fluctuating in speed until rehearsal 28.

This analysis provides a deeper understanding of how composers, such as Gérard Grisey,

utilize mathematics as a generator of musical ideas. An interesting observation can be made in

regards to the material that is applied to these abstract approaches: minimal amounts of music

can account for a maximal amount of musical possibilities when applied to mathematical

formulae. I would hesitate to classify Partiels as a product of minimalism but, there are many

similarities that fall under the process of composition, not necessarily the outcome, but in the

creative act of composing music in these two styles. The use of the harmonic series, harmonicity

and inharmonicity, periodicity and aperiodicity, additive synthesis, filtering, distortion, and ring

modulation does not reach the same level of success to listeners without vessels like Partiels to

deliver them in a coherent manner. Without a vessel, the listener is presented a concept, not an

artistically driven organization of sounds.

Bausch !12

Bibliography

Arrell, Chris. The Music of Sound: An Analysis of Partiels by Gérard Grisey. (revision of DMA dissertation, Cornell University, 2002), 325.

Grisey Gérard. “Tempus ex Machina: A composer’s reflections on musical time.” Contemporary Music Review 2 (2987): 239-275.

Rose François. “Introduction to the Pitch Organization of French Spectral Music.” Perspectives of New Music 34, no. 2 (1969): 6-39. Accessed November 29, 2015. http://www.jstor.org/stable/833469.

Charles Taylor and Murray Campbell, “Sound,” Grove Music Online, accessed September, 2015, http://www.oxfordmusiconline.com/subscriber/article/grove/music/26289.