Sheridan College SOURCE: Sheridan Institutional Repository

Sheridan College Sheridan College

SOURCE: Sheridan Institutional Repository SOURCE: Sheridan Institutional Repository

Publications and Scholarship Faculty of Animation, Arts & Design (FAAD)

1993

New Software Composition Tools New Software Composition Tools

Bruno Degazio

Sheridan College, [email protected]

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Degazio, Bruno, "New Software Composition Tools" (1993). Publications and Scholarship. 5.

https://source.sheridancollege.ca/faad_publications/5

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NEW SOFTWAIRE COMPO.SITIONI TOOLS

ABSTRACT

Bruno Dega�io Degazio Sound Design

192 Spadina Ave.,#512 Toronto, Conado

This pap r briefly discu ·s s a number of. oft.ware tools dev,eloped at the author's studio through the courne of r� earch worl!c into algorithmic composition. Mosl of the tools dev loped are dircc:dy related to recu1. i c techniques· some, how ,,v r, a

is · from more i,,ieaera] techniques of algorithmic composition first described by Joseph Schillinger. , xamples of recursive techniques mdude:

• META-FRACT. S- s panning mwical content from recursive tru.cture

• the Lorenz. attraclor and Koch snowflake as musical g,encrators

• Iterated Function Systems as musical generator

• dynamic value :in the logistic &]nation and th Mande'lbrot set Non-recursive lOOLlll include:

• the Intelligent Interval Tool- a fonn of limited contrapuntal i

telligence

• the Harmonic Activator - Schillinger arpeggia:tion tool

"' ,1.he Arbi'rrary Pauem Gen 1-ator

• the smart d1i1ra1tion operator

• the Granulator - applying a grru. ula

yuthesis pr cess a

t the note-level

His hoped that the brief de ·criptions of these functions will stimulate the imagination of other composers,.

RECURSIVE TECHNIQUIE.S

Most of the soflware loots de eloped during the past year l1av;e 1esulted from tbe author's

continuing research into method. of algorithmic composition u ing recursive techniqu-s; ger ernlly

known as fractals. What follows are brief descriptiot1s of several of these tools.

META-FRACTALS

In past experimenL� with classic f

actals such as the Mandelbrot set and its real number counterpart the logistic equation, I have used the output of the equation, su'tably scaled� as a direct index to MIDI note number (i.e. pitch), MIDI note vel city, Of" ome oth �r MIDI parameter uch as continuous controller values. The notion of .meta-fractals is simply this: to replace tbi· simple

1:1 relationship with a more musically meaningful one. Thu , the output value becomes an inde into a set of musical component.i;, which. are ananged by the recursive process into a self-similar structure. This separa

tion of the algorit

m from the musical content allows scope £or several Lypes of musical activity not previously possible with fr.actal'i.

Such a strategy also has a relation to die concept of moment form as defined by composers such a Stockhausen and Messiaen: "a success·on of self-contained sections which do not relate to each other in any functionally implicative manner.n (Kramer, Moment Form in Twentieth Century Mu .ic). Kramer continues:

The crisis fo,r the· listener is

eme; it is no surprise that discon1inu:ous conlemporary music is, often not understood: by its. audience

To remove continuity is to -question the very

meaning of time in our culture and henc-e of human existence. This questioning is-going aH

around us, and its strongest stat,ement is found in contempor

ry art. By deaUng wilth the

grow. (Kramer, p.55.,. ^italics mine)

IL is interesting Lo not that dynamical systems. which arise from the need to rationally comprehend change in 1time,. can in this way themselves becom -- t

e mean by which the perception of tim is destroyed.

Several variations of implementation of meta fractal are possible. Som of those that r ^e

cataloged are:

i) The s· mpl - st type is similar to a 1: I mapping, but with a single layer of indirection through a lookup table. For example. an output value of, say, 48 (corresponding to MIDI note number 48 - cello ·c) could result instead in 99. or any other MIDI note number� T

e pri_nciple use of this method .u to restrict the continuous real number outpUL of the recursive process to a more musically meaningful set, for example t

e notes of a particular pitch mode or instrumental range.

U) The next le el of complexity is a. big step forward. A single output value can be ma.de to produce a mor

ve.: a small (typically two to our note) musical gesture. Motivic

composition is a medlod typical of the European masters of the 19th century

Prla:ie moc.u

1opruo (bar I)

JJ •...

aoprano (b.ir t)

ban (bar 1/

lnvettie>c

53): h• • w :

bau (bar IJ

Cot1clucU'li motil �

=· . ,.

-

toprui;, tbu l) J'h1Jobl11& moll!

?:J..

�

.. .

l

_{• •}

( fig. 1 - Rudolf Heti, anafysis of motives from Beethoven, Pathetique Sonata, from Cook, p.99)

iii) Complete musical events can also be ·pecified, for example, an arpeggiated chord

across evcraJ instrumenLS, witJ1 independent control of pitch bend and velocity for each nole. This amounts Lo a method of aJgorithmicaIJy d lermining the orchestration and the d namic two musical parameters that in tJ1 • past ha e eluded meaningful attempts at fracLal control. In effect, the recursive process can be used to create a m.oment form:

a set of phrases pieces, or musical gestures umclated by functional implications.

iv) The musical elemen s can consi ·t of components of an existin composition, e.g.

phrases from a Moza

t symphony, or pianistic gestures from a Chopin sonata.

,.

DIRECT PAOGl=lAM OUTPUT --""'"'I,

...

� ··-�

·. . ....

+^t i •• .... • • •••

. IT •• 'f1110srw10 . .:. • r n.sli1ng,

METAOUTPU ., I ·, �•,__..s;;----l• �.�

1�1 - 1

p�r,2

-�-

J!i

., 1 _I �onctuding .. � ...

• . \

t=I��

p(imt

(fig. 2 - musical example showing indirect output generated from Reti's motivic analysis, above)

v) The musical elements can consisL of short musical phrases themselves produced by a

fraclaJ procc s thus extending lhe self-. imilarity of strucLur to another level.

A

r,rlme cell

-.___.;--· - ^.,.

pr.1111. Ila.

•«� .. l)<I. cell mOILf mc,!,U ,----,

p:rlmt

cell IICU /1,i. a,otJfo re�t. (ll>v.J

' •

prtme c1:.U fbl.

pnmccell u,,. ^moilt

,---,,--, ,...----,

r �

"''"'·

=

-

!la.

,---; ...---

.___... ... ..

JU,.

D"'° upel. a,oU!

,_,....,,,....­

�- ^-- -·. ^{- . ..} .

l'rl"'c m� CU>.

(LD••.ulOG) mow

11-

( fig. 3 - details of Reti's analysis of Beethoven Pathetique)

..

l

Mcta-Fraci.ats directly address 1.he discontinuity of p rccptual levels in applying recursive processes 1.0 music Lhat I have noted (Dcgazio, 1986). They do this by allowing different processes to

dominate at different levels.

THE LORENZ ATTRACTOR

Historically, the Lorenz attractor was one of the (ir, t dynamical systems shown lo exhibil complex and unpredictable (chaotic) behavior. It arose in connecLion with weath r prediction in a paper by Edward N. Loitnz in 1963. The three inter-related equations are:

x'= a*(y-x) y'=-o*x-y- *7, z'= x*y-c*z

Whether by suggestion or purpose, this model proves, when mapped to a continuously variable para1 ieter such as pitch, volume, or timbre (e.g. FM modulator index) to b extremely suggestive of the ebb and flow of such natural phenomena as wind and waves. The qllalhy that seems to account for lhis is its near-predictability combined with its chaotic behavior. The equations in facl trace out a simple .eries of quasi sine waves - the element of predictability. The waves are, however, broken at unpredictable intervals by sudden changes of direction and amplitude -the element of chaos.

When used to directly gen rate sample data f r reproduction as audio through a commercial sampling device, the Lorenz auractor creates an unusual low frequency oscillation. Perhaps becau_,;;c of its historicaJ connection with weather predi(.,"tion, this result is again extremely

suggestive of natural phenomena, this time of an earthquake mmble. I used this sound for the San Francisco earthquake sequence in the TMAX film Blue Planet.

Mapped to MIDI dal.a, anol:her application wa to h raid the approaching storm in the media opera Te la: The Man Who lm

en.ted the Twentieth Centu,)1. In this instance the attractor's outputs were applied to pitch bend ( detunJng) across three channels of synthesizers producing the aural equivalent of Tesla's inlerfercncc waves within th Earth. A third application was to simulate a natural choral vibrato wheo applied as pitch bend to six channcL'i of sampled vocal sounds. The resull wa� for more a.Uractivc to the ear than a regular LFO induced vibral:O. The multiple outputs are correlated in such a way that th<..:y never me ·tat a common poinL

RECURSIVE PATTERN GENERATOR

MIDIFORTH's

allows the creation of musical analogs of the Koch snowflake, a classic fractal de crihed by Mand lbrot in

lt works by recursively layering a shorl musical motif (up lo 64 notes long, but typically only three or four)

pecifi d by the compo, er in th form or semitone transpositions from a root

Hee11rsl11e Pattern 6enen,ior

N1nn1!01r of He111enh, j5

l=--===ai Diui,ion ractor: 1�2-�

D net1-t1ue mode

( 1>11ocus, PHilHSC RE cues, uni'] �

Stnrt. 'It!

n1

IEnd at: l�99_!il_�

( 1t11ncel

J

(fig. 4 - Rccu ivc Pa1tcm Goncrator (von Koch curve) dialog box)

DYNAMIC VALUES IN THIE LOGISTIC EQUATION AND MANDELBROT SET

Choosing a ingl · value for lambda or C i lhe simpl

way of composing with these equations. It has the disadvantage, however, that for many scltin� the. oulput i simple or periodic., i.e. it

consisLS of some small numb-r of recurring values. ln order to maintain the unity provided by such periodicity bul also gain the variety of continual variation il is possible to vary

or C across a small range through the course of lhe composition. This generates a more interesting

airiety of musicaJ matcriaJ, th · behaviour of which depends on the starling point and range across which it is varied. Because the variation is continous and in one direction Lhoughout an emfae work, the changes produced arc of

en of a fundamemal or . rructural nature. For example, the first movement of

is clearly outlined by the graduaUy increasing pitch range as

aries from .895 to .92698. This is mosl evident in the bass line which moves gradua1ly from

pitch areas centred on B-flal-2 down to C-2.

121

"

. ... ,j�-,!,- ,, .• ;�

.,_ ,- .all!• .... _ .... •• __ .-..:.•.r_� ...

...,...___ -· ^...

^.. "

n

. .

1, UZ 2U :SH

u,

,u

**!I lr1:11h1 NUIIU f!eU:111tch 1''1 11otctsJ tot,1,

Clll� lf;VS -flOve Mitllil 11llr1H, <fSC> tll H.1 't .•

n, 1 r:2: -

n, u -

n. Fl'1· c.btnae Held., FI - 11r:hrt sc.r,un. Fil - redr• sci:-Hn.

(lig.5 - pitch outf ,c o Digilaf Rilual.s 1 • Proce ion)

Th_ orbiL..; produced by 1.he equation. the cause of th" mu ical pat1teming that i the main interest of

&he process, can be made, by careful selection of tJ1e start and end poinL�. Lo cros several cyclica]

and chaotic boundaries through il.s path. This produces a ar'ely of "textures" tha can be exploited lo define the structur of th composition. Forexarnp1e, in HETEROPHONY 4875, alternating pcnodic and chaotic regimes produce the algorithmic equivalent of a rondo.

Ap·ply Loyl•lic Eqm,li�n �Slro119 Artlr ,,gr)

j

I

I

'Sli.rt 11t:

!

_j

He ght:

!

I[

6P.rter11hi Olr'llil lllilMfl lhll �1nilelb or -1 STIIRl RUIL: -!l.lf15DIJOOOOIIIIOOOtllilE-Ot

$TART IM!!GINflAY: 4.0000()!)!)!)000D!!OOOE-01 E D RERL: -l.8·1SOOOOODOOOOOtlilt-(U E Ill :IMR6 t-llIBY: 4. IIDODOODOOODODOOOE-01

MAG lrlCAnn...i

1Co11rrn: !,21.00H

I

fl119: �

IOfhel:

E==:)

' ENEIHIT^IE ORB ns

D In -tia!tn iioth rhn11•e• first

storl a,: j1

I

End 111: :-19=19""g,

===,;1

[ cen el

J

(fig. 6 - Logi tic r�uation d�alog box, wilh time. varying Ia1uhda, and Mandelbrot 'et dialog box., with time varying 'X and Y')

A similar logic prevails within the Mandelbrot set, with the additional complication that tJ1e patlls are two dnnensional .. The interior of !he seL hru fortunately been well mapped in terms of its orbital structure (fig. 7). Note the unusual additive relationship, as in a fjbonacci series between adjacent cyclical regions in the interior of the seL

(fig. 7 -map of orbital bchavior of interior of Mandelbrot set, from Pciticn)

QUATERNIONS AS MUSICAL GENERATORS

Quaternions are a four dimensional extension of previous work with the logistic equation (one dimensional, reaJ numbers) and the Mandelbrot set (two dimensional, complex numbers). The extension of these recursive processes into a four dimensional space allows simuJtaneous coordinated control of up to four parameters, or of four simultaneously occurring musical

elements.. For example, pitch, duration, dynamic· and timbre are arguably the four most important musical parameters. A recursive process employing

quatemions

an

instruments of a string quartet or similar ensemble. Such a riecursive process is specified by the equation:

Preliminary exp^eriments with this equaci.011 have been promising. The principal problem of exploring a four dimensional par^ameter space is the lack of any sort of built-in sense of 'direction'.

Additionally, the quaternion parameter space has been much less explored and the system itself

much l ss . 'ludied than the logistic equation or the 1\1.andclbrol set, for both of ,vhi.ch exist detailed param rer map ·

IIJERAJED 'FUN, ⁱ CJ'II0 ¹ N SYSTE1MS A. S !MUSICAL GENERA TO·RS

A n1ethod of computing many of the clas

ic fra ·tails has been de\'eloped by Michael Bamsley of ilic Gcorhia Institute of · echnolo,gy. Known as iteratedftmction lystems. this method bears pr-omi� e as a. general syste.m ibr computing any desired fractal pal1tcrn

as opposed to the ad lw

e system of many different techniques in use at present. One J;eature of Barnsley's meiliod is that a continuous gradation of types is possible This leads to the concept of fractal interpalatio:n - the ability to generate fract:aUy consii.;tent daita ,(i.e. of equal fractal dimension) from a small set of given daur+ Thus a structural outline can he specified by a mall number of pitches, the d.etaiJs of which are filled im by the process in a fr&ctaJJy consistent manner.

NON-,RECURS.IVE TECHNIQU E:S

Certain oth .r .- ftware, tools have arisen ou· of mu ·ical needs in the p.ast y,. air;, which d., nol

how ver pertain t arny specific recursive techruq1rae. Two .are:

HAR'MONIIC ACTIVAJOR (:SCiHILllNGl:R' A I RPEGGIA110Nl ,o,oL:)

This tool implements the Schillingerian notion of ha_llllonic activation. By specifying a ·pre-defined

,chord erie�. and a rhyllnnic skeleton� the hannonic structure is 'acti a

ed · through airpeggiation, It

tttke. as its paramet

li :· a, _t ofchords, an arpeggiation pauem, a d:1yillnnic value and a total duration. Continual varialion can b · adde,d through Schillinger''s '<:ydica1 permutations'. For

,�,mmple,. given a ries of 5-voicc chords, an arpcgg:iation paUem ,spelled Irom the lowest voice)

of a b

d ,e, and a total duration of 15 notes,. the arpeggiation

would generate thls accompaniment:

ab ·d bed ea cdeab

The pitch pattern is specified a chord elements. numb :ired from the i

voic · .. An independent

velocity pattern can als

be applied.

H11rmoni At Uu11 tor

PiUHELEME� Rererence p!lr11se: M L00Y2 Number of Chol'd Noles:

EJ

umller or l'ultam ate.:

EJ

D Sbiilinge.-1termutetlon1

,-i.,me orpht'<IW to be ptocuud:

jriewphrue

IIRCel

( fig. 8 - Harmonic Activator dialog box)

ARBITRARY PATTERN GENERATOR

The a

hiLrary Pattern Generator is a simple and useful lool. It is used whenever a simple repeating pattern is required, such as a sequence of repeating melodic shapes or dynamic accents. These are

·Often useful starting points for other algorithmic process. The simple nature of the repetition can be made somewhat more complex by Lhe application of SchilUnger' s cyclical permutations, as in the Ha1rmonic Activator.

llrbltHry P111tam GenMo1ur

(fig. '9 - Arbitr.try Paueru Generator dialog box)

SMART DURATION OPERATOR

When any of the above processes is applied to rhythm or duration the results can be less than

satisfactory because of Ote way a continuous value (e.g. the output of a recursive function) maps

into a discontinuous par.ameter space (musical durations). For trus reason a means of restricting

operations involving rhythm'c aspects of music (durations, positions within the measure) to a

mu icalJy relevant set of values was deve1oped. Three such ets of values, as defined by

Schillinger. are:

the 2 power series - 1/4 1/2

1

the 3 power series - 1/9 1/3

1

Lhe 5 power series - l/25 1/5

1

2

4

::;

5

8 etc

27 etc

The smart duration operator automatically maps operations applied to durations and rhythms to memher. of one of these series, or of some oLher limited set of rhythmically meaningfuJ values.

THE INTELLIGENT INTERVAlTOOL

The Int.elligent Interval Tool is a simple mcth d of providing any

cnnlrapunta!/haimonic intelligence

It can modify an existing musical line by reference to a econd line (the canLus firmus) and a table of

intervals .. The 'allowed' and 'disallowed' pitch intervals can vary through the composition and are specified on a pitch-class-against-piLch-class basis in the form of a t.able:

P.CI.,•• 1 M

(: C D [l# E 6 ll# R fl# (I

£#" C# bit E F P' G# ⁸ R• ^{II C}

D D G# ( F f.lt 6 G# A ii# ii t Cill

0# D" f F f# 6 611 R A#' B r. (# D

[ [ f F# G GIi! R

""'

f f f# 6 G# Ft H# e C (:# II 0#

r# f# 6 Git A Ftf:I B C C# D II# E f

6 6 6 ... R A# B C c• ^D D* ^[ ,F ftt

(i# (!;it II ft# e C c• 0 (!It [ r f# 6 H�m,on1ni· Elnmla.

ll fl II"' B C C# D 011 E r F# G GN �lortot;� Dldot: �

nit ^fl# R C C# D 0# [ F r• G f)# n

B 8 C C# ll Oit E f f# 6 G# A Ii#

Referente Phro,e: Mltoiii''2

( OP!I dn1en111tr

I ^I

I

I

l

[ Si,11u l11ti,ro111 Table

l !

I I

- --

(fig. 10 - Iuterval Co:rrectim l'ablc and I Iarmoniz.atiou dial.O" box.)

Modification of thi inlierval list aUow the imulation of styles a.s varied as organum (only fourths, fifths and octaves allowed) to dodecophony (fourths, fifths and octaves disallowed). Simple extensions to this Looi allow an alternate. set of jnt.ervals to be sp cified for metrical context sensitivity. Thus, there can exist allowed intervals for both 'strong' and 'weak' beats allowing the speciHcation of traditional forms of consonance-dissonance relationships such as passin•• tones and appogiaturae. Editing functions lllsed witJh the interval correction table allow the copying of interval structures (i.e. chords) from one root to another (transposition), and copying pitch classes from one root to another (inversion).

GRAN UILATED MUSIC

This tool consists of the application Lo MIDI data of an audio synthesis technique known as

granular synthesis. The basic procedure is Lo take a MIDI fi1e consisting possibly of some well known piece of mu ic, and stretching it out through Lhe r

p tition of small blocks of musical material. A ''window size" of, say, two bars, is decided upon. The computer then progresses through the source MIDI file per

orming the fir t two bars, then repeats starting, say an ,eighth note later, again playing two bars form that point, advances another eighth note and perf onns two bars from that point and so on, until the end of the file is reached. With suitable selection of window size and advancement rate. sttetch factors of s eral thousand times may achieved while still retaining recognizable musical feaLures in the source material. The idea is a direct application of a technique used by others, notably Harry Truax, at the audio level to achieve vastly time-stretched sounds while keeping frequency components within the human audio range.

(fig. 11 - measures 5-6 of Joplin's Magnetic: Rag)

(fig. 12 - abo c measures trctched out to 12)

ACKNOWLEDGEMENTS

The work described in this pap �r was funded with the ass.istance of the Ontario Arts Council,

Electronic Media Program and the Canada Council, Computer futegrated Media Program. I am also grateful to Gustav Ciamaga and to John Free for discussion of detaiJs rc6rarding musicaJ uses and suflware

BIBLIOGRAPHY

Barnsley, M.

Becker, K-H Cook,N.

D ga:zio, B.

Degazio, B.

Degazio, B.

Degazio, B.

Kramer, J.

Lauw r1er, H.A.

Norton A Peitgen, H-0

Fractals Everywhere Academic Press, Bo. ton, 1988 Dynamical Systems and Fractals Camhridfl'C nivcrsity Press 1989 A

Guide to

Musical Aspects of Fractal Geometry

Procce.dings of the International Computer Music Conference The Hague 1986

The Development of Context Sensitiv;ty in the Midiforth Computer Music System

Procee-diugs of the Jntcrnational ompmer Music onfercnce Cologne, 1988

The Schillinger System of Musical Composition and Contemporary Computer Music

Proceedings of Diffosiun Conference/Festival of the Canadian Electroac.ousti.c Community, 1988

Algorithm;c Techniques in Digital Rituals

Proceeding of Per pectivcs - onforence/ cstival of the Canadian Electroacoustic Community J 991

Moment Form in Twentieth Century Music

The Musica.J Quartcdy Apri1 1978

Two Dimensional Iterative Maps

in A.V.Holden Chaos, Princeton University Press, 1986 Generation and Display of Geometric

Fractals

Computer Graphics 16-3, 1982 The Betwty of Fractals

Springer-Vcrlag, New York, 1986

Peitgen, H-O, ed.

Pressing, J.

Reti, R.

Schillinger, J.

Schillinger, J.

Stockhausen, K.

The Science of Fractal Images Springer-Verlag, New York, 1988

Nonlinear Maps as Generators of Musical Design Computer Music Journal, 12-2, 1988

The Thematic Process in Music New York, 1951

The Schillinger System of Musical Composition Da Capo Press, New York, 1978

The Mathematical Basis of the Arts Da Capo Press, New York, 1976

"Momentform" in Texte zur elektronischen und instrumenten musik

Cologne, 1963-1971