Achorripsis

Achorripsis (Greek for Αχος "sound" and ρίψη "throw"; thus about: sound of movement / currents) is an orchestral work for 21 instruments composed by Iannis Xenakis (1922–2001) ; it was premiered in 1956 under the direction of Hermann Scherchen .

Stochastic music

Iannis Xenakis is a new music composer . In his work Formalized Music he describes the stochastic music with which he established an alternative way of composing achorripsis. His music corresponds to scientific considerations. Some of his compositions are based on probabilistic ideas. Xenakis used different probability distributions when composing Achorripsis, including the exponential , equal , Poisson and normal distributions . According to Xenakis, the composers of serial music who tried to embed all the characteristics of music in a principle of order with their compositions lacked scientific structures. Due to the increasing complexity of the serial compositions, according to Xenakis a scientific formalization of the music is necessary, which he published in his book Formalized Music in 1962.

construction

Achorripsis was composed for 21 instruments in 1956–1957. The piece, which lasts about seven minutes, was premiered on July 20, 1958 in Brussels under the direction of Hermann Scherchen . It was described as scandalous in the reviews, not least because of the stochastic structures. Xenakis himself describes in great detail in his font collection Formalized Music which stochastic elements his piece contains and how it was composed. He presented his composition not only in the form of notes, but also in a diagram that he himself called the Matrix of Achorripsis :

	α	β	γ	δ	ε	ζ	Ζ	η	θ	ι	ια	ιβ	ιγ	ιδ	ιε	ιζ	ιΖ	ιη	ιθ	κ	κα	κβ	κγ	κδ	κε	κζ	κΖ	κη
I.		4.5	6th	9	10				5.5							9.5	5	4th	5.5	2.5		5	6.5		4.5		5.5	10.5
II						5.5	4th		5	6th			4.5		5			3.5	4.5			5			20th			6.5
III			5			5			4th	10			14th		3.5	6.5	4.5			11.5		6th				6th	4th
IV			9		9.5		8.5			4th	5	6.5					10	6th	4th	3.5					11.5	5
V	3.5	6th	4.5		4th	5	5.5		4.5				5			4th	5.5	3.5			17th	10.5	10		4th		6.5	5
VI							10		5.5		10			4.5				5	6.5	5					10.5		6th
VII				6.5	15th		3.5			11	4.5		10				5	4.5	4th	6th					9	6th	16

No event

Single event

Double event

Triple event

Quadruple event

The matrix provides an overview of the time course of the piece, the instrumentation and the intensity of the sound events. You can see three levels in the matrix: The line level shows the time structure of the piece. At the column level a distribution of musical events (is events ) made. The smallest level, the cell level, deals, among other things, with the notated note values ​​for the respective instrument groups, the dynamic design of the individual time periods and the calculation of the glissandi in the stringed instruments glissando . The occupation of the cells with sound events is determined by the application of probability distributions . The termination of the individual parameters of the piece was calculated and noted down by Xenakis independently.

Row level

The matrix is ​​divided into seven instrument groups (rows) and 28 time units (columns). The instrument groups are composed as follows:

1. Woodwind instruments I (piccolo, Eb clarinet, bass clarinet)
2. Woodwind instruments II (oboe, bassoon, contrabassoon)
3. String instruments - Glissando (violin, cello, bass)
4. Percussion instruments (xylophone, woodblock, bass drum)
5. String instruments - pizzicato (violin, cello, bass)
6. Brass instruments (two trumpets, trombone)
7. String instruments - Arco (violin, cello, bass)

In this way, 196 fields are created in the matrix, which can be assigned musical events. Xenakis describes five different musical events, which essentially differ in terms of their dynamic intensity. He speaks of different sound densities. The sound density describes the number of sound events (tones) per measure. A double event has twice as many notes per measure as a single event . An empty field in the matrix means no event , i.e. silence. How exactly the different events are created can be read in the section on cell level. In order to distribute the various sound events on the matrix, he first had to determine how often a certain event should occur. He determined this with the help of the Poisson distribution . This represents an alternative to binomial distribution .

${\displaystyle P_{\lambda }(k)={\frac {\lambda ^{k}}{k!}}\,\mathrm {e} ^{-\lambda }}$

Here at is and generally applies . Xenakis chose the value 0.6 for the parameter and thus calculates the probability with which a cell will be occupied with a No event, Single event, Double event, Triple event, or Quadruple event . The probability with which a cell is occupied with a No event (i.e. k = 0) can be calculated , for example, by . The multiplication by 196 (number of all cells in the matrix) then results in an absolute distribution of the No event on the matrix. ${\displaystyle k\in \{0,1,2,3,4,5\}}$ ${\displaystyle \lambda \in \mathbb {R} _{>0}}$ ${\displaystyle \lambda }$ ${\displaystyle P_{0,6}(0)={\tfrac {0,6^{0}}{0!}}\,\mathrm {e} ^{-0,6}={\tfrac {1}{1}}\cdot \mathrm {e} ^{-0,6}=0{,}5488}$

Calculation of the probabilities for the 196 cells of the matrix

Event	Calculation of the distribution	Used value for the matrix
No event	${\displaystyle 0,5488\cdot 196=107,5640}$	${\displaystyle 107}$
Single event	${\displaystyle 0,3293\cdot 196=64,5428}$	${\displaystyle 65}$
Double event	${\displaystyle 0,0988\cdot 196=19,3648}$	${\displaystyle 19}$
Triple event	${\displaystyle 0,0198\cdot 196=3,8808}$	${\displaystyle 4}$
Quadruple event	${\displaystyle 0,0030\cdot 196=0,588}$	${\displaystyle 1}$
Quintuple event (not applicable)	${\displaystyle 0,0004\cdot 196=0,0784}$	${\displaystyle 0}$

So only the calculations of the proportions of the individual events were made on this level. But it has not yet been determined which instrument group will play with a certain intensity at any given time. The notes to be played have not yet been determined either. The table shows, for example, that a no event takes place in 107 of 196 cells in the matrix . This means that in a cell that has been assigned such an event , the instrument group of the respective row is not playing at the time of the specified column. In contrast, 65 cells of the matrix are played with the intensity of the single event . This also applies to the other events .

Column level

At the column level, Xenakis calculated how the events , whose absolute frequencies were previously calculated, are to be distributed over the columns and rows. For this he again used the Poisson distribution.

Distribution of the single events to the columns

Single event in one column	Calculation of the distribution	Rounded value
Number of columns with 0 single events	${\displaystyle 0,09827\cdot 28=2,7521}$	${\displaystyle 3}$
Number of columns with 1 single event	${\displaystyle 0,22799\cdot 28=6,3839}$	${\displaystyle 6}$
Number of columns with 2 single events	${\displaystyle 0,26447\cdot 28=7,4052}$	${\displaystyle 8}$
Number of columns with 3 single events	${\displaystyle 0,20453\cdot 28=5,7268}$	${\displaystyle 5}$
Number of columns with 4 single events	${\displaystyle 0,11862\cdot 28=3,3213}$	${\displaystyle 3}$
Number of columns with 5 single events	${\displaystyle 0,05504\cdot 28=1,5411}$	${\displaystyle 2}$
Number of columns with 6 single events	${\displaystyle 0,02128\cdot 28=0,5958}$	${\displaystyle 1}$
Number of columns with 7 single events	${\displaystyle 0,00705\cdot 28=0,1974}$	${\displaystyle 0}$

In the table, the distribution of the single events to the columns is calculated as an example. In the second column of the table, the rounded value 6 means that there are six columns in the matrix in which a single event occurs, which is easy to check. A general distribution of the events makes it clearer where a certain event occurs. Translated into music, it means that it is determined at what point in time and with what intensity. So it has not yet been determined which instrument group (i.e. which column) will play a certain event .

Cell level

At the cell level, the time intervals between the individual notes, the pitch intervals and the speed of the glissandi for the string instruments are calculated. Different distributions are used here:

1. The exponential distribution
2. The equal distribution
3. and the normal distribution

The sound densities are noted in the matrix in the individual cells. For example, the instrument group of flutes (first row in the matrix) in the second column plays with a sound density of 4.5 notes per second. The duration of achorripsis was set at around seven minutes, which results in an average duration of 15 seconds per column with 28 columns. At the fixed tempo of MM = 26 , one column corresponds to 6.5 bars.

In Achorripsis Xenakis stipulates that the maximum sound density should be 10 sounds per second. This value describes the maximum number of sounds per second that an orchestra is able to play. Because this value is set as the maximum, Xenakis assigns it to the quadruple event , i.e. the event with the highest sound density. From this arithmetically the sound densities of the single, double and triple events result . The density of a single event is calculated by , for example , because a quadruple event should have 4 times as many tones per second as a single event . As can be seen in the table, the values ​​of an event fluctuate . The sound densities of the double events differ from 8.5 to 11.5. Xenakis summarizes the different sound densities of events so together in classes. There are five classes of sound densities, with each class showing a certain spread. These five classes make up the various events. ${\displaystyle {\tfrac {10}{4}}}$

Range of events

Event	Δ sound density	span
No event	${\displaystyle 0-0}$	${\displaystyle 0}$
Single event	${\displaystyle 6,5-2,5}$	${\displaystyle 4}$
Double event	${\displaystyle 11,5-8,5}$	${\displaystyle 3}$
Triple event	${\displaystyle 17-14}$	${\displaystyle 3}$
Quadruple event	${\displaystyle 20-20}$	${\displaystyle 0}$

With his formalization of music, Xenakis constructed a system based on mathematical principles. He tried to understand composition as one of the basic sciences and to close a scientific gap that, according to Xenakis, had opened up in music since the beginning of the 20th century.

literature

• Xenakis, Iannis: Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, ISBN 1-57647-079-2 .
• Xenakis, Iannis: Stochastic Music. In: Gravesaner Blätter No. XXIII / XXIV, Ed .: Herman Scherchen pp. 156–168. 1962.
• Xenakis, Iannis: Basics of a Stochastic Music (II) In: Gravesaner Blätter No. XIX / XX, Ed .: Herman Scherchen pp. 128–150, 1960.
• Xenakis, Iannis: Achorripsis for orchestra. Score. Ed .: Boosey Hawkes, Berlin 1957.
• Arsenault, Linda M .: Iannis Xenakis' Achorripsis: The Matrix Game. Postdoctoral Fellow, University of Toronto, 2002.
• Childs, Edward: Achorripsis: A sonification of Probability Distributions. Proceedings of the 2002 International Cenferences on Auditory Display Kyoto, Japan 2002.

Web links

• YouTube - Iannis Xenakis - Achorripsis - Time course of the matrix
• Autograph of the Matrix from Formalized Music ( Memento from January 2, 2012 in the Internet Archive )

Individual evidence

1. ↑ Xenakis, Iannis: Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, ISBN 1-57647-079-2 .
2. ↑ Xenakis, Iannis: Stochastic Music. In: Gravesaner Blätter No. XXIII / XXIV, Ed .: Herman Scherchen pp. 156–168. 1962.
3. ↑ Xenakis, Iannis: Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, ISBN 1-57647-079-2 .
4. ↑ Xenakis, Iannis: Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, p. 28
5. ^ Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, p. 29
6. ↑ Xenakis, Iannis: Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, p. 29
7. ^ Formalized Music. Thought and Mathematics in Music. Stuycesant, New York 1962, p. 29