{"title": "Connectionist Music Composition Based on Melodic and Stylistic Constraints", "book": "Advances in Neural Information Processing Systems", "page_first": 789, "page_last": 796, "abstract": null, "full_text": "Connectionist Music Composition Based on \n\nMelodic and Stylistic Constraints \n\nMichael C. Mozer \n\nDepartment of Computer Science \nand Institute of Cognitive Science \n\nUniversity of Colorado \n\nBoulder, CO  80309-0430 \n\nTodd Soukup \n\nDepartment of Electrical \nand Computer Engineering \n\nUniversity of Colorado \n\nBoulder, CO  80309-0425 \n\nAbstract \n\nWe describe  a  recurrent  connectionist  network,  called CONCERT,  that  uses  a \nset  of  melodies  written  in  a  given  style  to  compose  new  melodies  in  that \nstyle. CONCERT  is  an extension of a traditional algorithmic composition tech(cid:173)\nnique  in  which  transition  tables specify  the  probability  of the  next  note  as  a \nfunction of previous context.  A central ingredient of CONCERT  is the use of a \npsychologically-grounded representation of pitch. \n\n1  INTRODUCTION \n\nIn creating music, composers bring to bear a wealth of knowledge about musical conven(cid:173)\ntions.  If we hope to build automatic music composition systems that can mimic the  abili(cid:173)\nties  of human  composers,  it  will  be  necessary  to  incorporate  knowledge  about  musical \nconventions into the  systems.  However, this knowledge  is difficult to express:  even hu(cid:173)\nman composers are unaware of many of the constraints under which they operate. \n\nIn this paper, we describe a connectionist network that composes melodies.  The network \nis  called  CONCERT,  an  acronym  for  connectionist  composer  of  erudite  tunes.  Musical \nknowledge  is incorporated  into CONCERT  via  two  routes.  First, CONCERT-is  trained  on  a \nset  of  sample  melodies  from  which  it  extracts  rules  of note  and  phrase  progressions. \nSecond, we have built a representation of pitch into CONCERT that is based on psychologi(cid:173)\ncal  studies  of human  perception.  This  representation,  and  an  associated  theory  of gen(cid:173)\neralization  proposed  by  Shepard  (1987),  provides CONCERT  with  a basis for jUdging  the \nsimilarity among notes, for selecting a response,  and for restricting the  set of alternatives \nthat can be considered at  any  time. \n\n2  TRANSITION TABLE APPROACHES TO COMPOSITION \n\nWe  begin  by  describing  a  traditional  approach  to  algorithmic  music  composition  using \nMarkov  transition  tables.  This simple but interesting technique  involves selecting notes \nsequentially  according to a table that  specifies the probability of the next note  as a func-\n\n789 \n\n\f790  Mozer and Soukup \n\ntion of the  current note (Dodge  &  Jerse,  1985).  The tables may be hand-constructed ac(cid:173)\ncording to certain criteria or they may  be set up to embody a particular musical style.  In \nthe  latter case,  statistics  are  collected over a  set of examples (hereafter, the  training set) \nand the table entries are defined to be the transition probabilities in these examples. \n\nIn  melodies  of any  complexity,  musical  structure  cannot  be  fully  described  by  pairwise \nstatistics.  To capture additional  structure,  the  transition  table can  be generalized  from  a \ntwo-dimensional array  to n  dimensions.  In  the n -dimensional table,  often referred  to  as \na table of order n -1, the  probability of the next note is indicated as a function of the pre(cid:173)\nvious n -1 notes.  Unfortunately,  extending the transition  table  in  this  manner  gives rise \nto  two  problems.  First,  the  size of the  table explodes  exponentially  with  the  amount of \ncontext and  rapidly becomes unmanageable.  Second, a table representing the high-order \nstructure masks whatever low-order structure is present. \n\na context-sensitive grammar -\n\nKohonen  (1989)  has proposed  a scheme by  which only the relevant high-order  structure \nis  represented.  The scheme  is  symbolic algorithm  that,  given a training set of examples, \nproduces a  collection of rules -\nsufficient for  reproduc(cid:173)\ning  most  or  all  of the  structure  inherent  in  the  set.  However,  because  the  algorithm  at(cid:173)\ntempts to produce deterministic rules -\nthe \nalgorithm will not discover regularities unless they are absolute;  it is not equipped to deal \nwith  statistical  properties of the  data.  Both Kohonen's musical  grammar and  the  transi(cid:173)\ntion  table  approach  suffer  from  the  further  drawback  that  a  symbolic  representation  of \nnotes  does  not  facilitate  generalization.  For  instance,  invariance  under  transposition  is \nnot  directly  representable.  In  addition,  other similarities  are  not  encoded,  for  example, \nthe congruity of octaves. \n\nrules that always apply  in a given context -\n\nConnectionist  learning  algorithms offer  the  potential  of overcoming  the  various  limita(cid:173)\ntions of transition table approaches and Kohonen musical grammars.  Connectionist algo(cid:173)\nrithms  are  able  to  discover  relevant  structure  and  statistical  regularities  in  sequences \n(e.g.,  Elman,  1990;  Mozer,  1989),  and  to  consider varying amounts of context,  noncon(cid:173)\ntiguous context, and combinations of low-order and high-order regularities.  Connection(cid:173)\nist  approaches  also  promise better generalization through the use of distributed  represen(cid:173)\ntations.  In  a  local  representation,  where  each  note  is  represented  by  a  discrete  symbol, \nthe sort of statistical contingencies that can be  discovered are  among notes.  However,  in \na distributed representation, where each note is represented by a set of continuous feature \nvalues,  the  sort  of contingencies that  can be discovered  are  among features.  To the  ex(cid:173)\ntent  that  two  notes  share  features,  featural  regularities  discovered  for  one  note  may \ntransfer to the other note. \n\n3  THE CONCERT ARCHITECTURE \n\nCONCERT  is  a  recurrent  network  architecture  of the  sort  studied  by  Elman  (1990).  A \nmelody is presented to it, one note at  a time, and its task at each point in time is to  predict \nthe  next  note  in  the  melody.  Using  a  training  procedure  described  below,  CONCERT's \nconnection  strengths  are  adjusted  so  that  it  can  perform  this  task  correctly  for  a  set  of \ntraining  examples.  Each  example  consists  of  a  sequence  of  notes,  each  note  being \ncharacterized by  a pitch and  a duration.  The  current  note  in  the  sequence is represented \nin  the  input  layer  of CONCERT,  and  the  prediction  of the  next  note  is  represented  in  the \noutput layer.  As Figure 1 indicates, the next note is encoded  in  two different ways:  The \nnext-note-distributed  (or NND)  layer  contains CONCERT'S  internal  representation  of the \n\n\fConnectionist Music Composition Based on Melodic and Stylistic Constraints \n\n791 \n\nnote, while the next-note-Iocal (or NNL) layer contains one unit for each alternative.  For \nnow, it should suffice to say that the representation of a note in the NND layer,  as well as \nin  the input layer, is distributed, i.e., a note  is indicated by  a pattern of activity across the \nunits.  Because such patterns of activity can be quite difficult to  interpret, the NNL layer \nprovides an alternative, explicit representation of the possibilities. \n\nNnlNotc \n\nLocal \n(NNW \n\n:~ \n\ncOllin,. \n'-------,.--,r---' \n\n'I \n\nI \n\n\\ \n\nFigure 1:  The CONCERT Architecture \n\nThe  context  layer  represents  the  the  temporal  context  in  which  a  prediction  is  made. \nWhen a new note is presented in  the input layer,  the current context activity pattern is in(cid:173)\ntegrated  with  the  new  note  to  form  a  new  context  representation.  Although  CONCERT \ncould readily be wired up  to behave as a k -th order transition table,  the architecture is far \nmore  general.  The  training  procedure  attempts  to  determine  which  aspects  of the  input \nsequence are  relevant for  making future  predictions and  retain only this task-relevant  in(cid:173)\nformation  in the  context layer.  This contrasts with Todd's (1989) seminal work on con(cid:173)\nnectionist composition in which the recurrent context connections are prewired and fixed, \nwhich  makes the  nature  of the  information Todd's model  retains  independent  of the  ex(cid:173)\namples on which it is trained. \n\nOnce CONCERT has been  trained, it can be run in composition mode to create new pieces. \nThis  involves  first  seeding CONCERT  with  a  short  sequence  of notes,  perhaps  the  initial \nnotes of one of the  training examples.  From this point on, the output of CONCERT can be \nfed  back  to  the  input,  allowing  CONCERT  to  continue  generating  notes  without  further \nexternal  input.  Generally,  the output of CONCERT does not specify  a single  note with  ab(cid:173)\nsolute certainty; instead, the output is a probability distribution over the set of candidates. \nIt is thus necessary  to select a particular note in accordance with this distribution.  This is \nthe role of the selection process depicted in Figure 1. \n\n3.1  ACTIVATION RULES AND TRAINING PROCEDURE \n\nThe activation rule for  the context units is \n\nwhere c,. (n) is  the  activity of context unit i  following processing of input note Il  (which \n\nj \n\nj \n\n(1) \n\n\f792  Mozer and Soukup \n\nwe refer to as  step n), Xj (Il) is the activity of input unit j  at  step n, Wij  is the connection \nstrength from  unit  j  of the  input to  unit  i  of the  context  layer,  and  Vij  is  the  connection \nstrength from  unit j  to unit i  within the context layer, and s  is a sigmoid activation func(cid:173)\ntion rescaled to the range (-1,1).  Units in the NND layer follow a similar rule: \n\nnlldj (n ) .. s [LUjj Cj (n)] , \n\nj \n\nwhere mzdj (n ) is the activity of NND unit i  at step nand Uij  is the strength of connection \nfrom context unit j  to NND unit i . \n\nThe transformation from  the NND layer to  the NNL layer is achieved by  first  computing \nthe  distance  between  the  NND  representation,  nnd(n),  and  the  target  (distributed) \nrepresentation of each pitch i, Pi: \nd j  = I nnd(n) - pj I , \n\nwhere  1\u00b71  denotes  the  L2  vector  norm.  This distance  is  an  indication  of how  well  the \nNND  representation  matches  a  particular  pitch.  The  activation  of  the  NNL  unit \ncorresponding to pitch i, nlllj, increases inversely with the distance: \n\n1l111j(n) = e-d'/\"Le-d}  . \n\nj \n\nThis  normalized  exponential  transform  (proposed  by  Bridle,  1990,  and  Rumelhart,  in \npress) produces an  activity  pattern over the NNL units in which each  unit  has  activity in \nthe  range  (0,1)  and  the  activity  of all  units sums  to  1.  Consequently,  the  NNL  activity \npattern can be interpreted  as  a probability distribution -\nin this case,  the probability that \nthe next note has a particular pitch. \n\nCONCERT  is  trained  using the back propagation unfolding-in-time procedure  (Rumelhart, \nHinton, &  Williams, 1986) using the log likelihood error measure \n\nE  = - L log nnltgt (l1,p) , \n\np,n \n\nwhere p  is  an  index  over  pieces  in  the  training  set  and  11  an  index  over  notes  within  a \npiece; tgt  is the target pitch for note 11  of piece p . \n\n3.2 \n\nPITCH REPRESENTATION \n\nHaving  described  CONCERT's  architecture  and  training  procedure,  we  turn  to  the \nrepresentation  of pitch.  To  accommodate  a  variety  of music, CONCERT  needs  the  ability \nto represent a range of about four octaves.  Using standard musical notation, these pitches \nare labeled as follows:  C1,  D1,  ... ,  B1,  C2,  D2,  ...  B2,  C3, ...  C5, where  C1  is the \nlowest  pitch  and  C 5  the  highest.  Sharps  are  denoted  by  a  #,  e.g.,  F#3.  The  range \nC1-C5 spans 49 pitches. \n\nOne might argue that the choice of a pitch representation is not critical because back pro(cid:173)\npagation  can,  in  principle,  discover  an  alternative  representation well  suited  to  the  task. \nIn  practice,  however,  researchers have found that the choice of external  representation  is \na  critical  determinant  of the  network's  ultimate  performance  (e.g.,  Denker  et  aI.,  1987; \nMozer,  1987).  Quite simply,  the  more  task-appropriate  information  that  is  built into the \nnetwork,  the  easier  the job the  learning  algorithm  has.  Because  we  are  asking the  net-\n\n\fConnectionist Music Composition Based on Melodic and Stylistic Constraints \n\n793 \n\nwork  to  make  predictions  about  melodies  that  people  have  composed  or  to  generate \nmelodies  that  people  perceive  as  pleasant,  we  have  furnished  CONCERT  with  a \npsychologically-motivated representation of pitch.  By this, we  mean  that notes that peo(cid:173)\nple  judge  to  be  similar  have  similar  representations  in  the  network,  indicating  that  the \nrepresentation in the head matches the representation in the network. \nShepard  (1982)  has  studied  the  similarity of pitches  by  asking  people  to judge the  per(cid:173)\nceived  similarity  of  pairs  of  pitches.  He  has  proposed  a  theory  of  generalization \n(Shepard,  1987) in which  the similarity of two items is  exponentially related to their dis(cid:173)\ntance  in  an  internal  or  \"psychological\"  representational  space.  (This  is  one justification \nfor the NNL layer computing an exponential function of distance.)  Based on psychophy(cid:173)\nsical  experiments,  he  has  proposed  a  five-dimensional  space  for  the  representation  of \npitch, depicted in  Figure 2. \n\nOIl\":\" \nD1  .... \nC'I4-\nCI-\n\nPilch Height \n\nChromalic Circle \n\nCircle of Fiflhs \n\nFigure 2:  Pitch Representation Proposed by Shepard (1982) \n\nIn  this  space,  each  pitch  specifies  a  point  along  the pitch  height (or PH)  dimension,  an \n(x ,y) coordinate on the chromatic circle (or CC), and  an  (x ,y) coordinate on the circle of \nfifths  (or  CF).  we  will  refer  to  this  representation  as  PHCCCF,  after  its  three  com(cid:173)\nponents.  The pitch  height component specifies the logarithm of the frequency  of a pitch; \nthis logarithmic transform places tonal half-steps at equal spacing from one another along \nthe  pitch  height  axis.  In  the  chromatic  circle,  neighboring  pitches  are  a  tonal  half-step \napart.  In  the  circle  of fifths,  the  perfect  fifth  of a  pitch  is  the  next  pitch  immediately \ncounterclockwise.  Figure  2  shows  the  relative  magnitude  of the  various  components  to \nscale.  The proximity of two pitches in the five-dimensional PHCCCF space can be deter(cid:173)\nmined simply by computing the Euclidean distance between their representations. \n\nA straightforward scheme for  translating the  PHCCCF representation into an activity pat(cid:173)\ntern  over  a  set  of connectionist  units  is  to  use  five  units,  one  for  pitch  height  and  two \npairs to encode the (x ,y ) coordinates of the pitch on the two circles.  Due to several prob(cid:173)\nlems,  we  have  represented  each  circle  over  a set  of 6 binary-valued  units  that  preserves \nthe  essential  distance  relationships  among  tones  on  the  circles  (Mozer,  1990).  The \nPHCCCF representation  thus consists of 13 units altogether.  Rests (silence)  are assigned \na  code  that  distinguish  them  from  all  pitches.  The  end  of a  piece  is  coded  by  several \nrests. \n\n\f794  Mozer and Soukup \n\n4  SIMULATION EXPERIMENTS \n\n4.1  LEARNING THE STRUCTURE OF DIATONIC SCALES \n\nroughly  75%  of the  corpus  -\n\nIn  this simulation, we  trained CONCERT on a set of diatonic scales in  various keys over a \none octave range,  e.g.,  01  El  F#1  Gl  Al  Bl  C#2  02.  Thirty-seven such scales \ncan be  made  using pitches  in  the  C l-C 5  range.  The training set consisted  of 28  scales \nselected  at  random,  and  the  test  set  consisted  of the \n-\nremaining  9.  In  10  replications  of  the  simulation  using  20  context  units,  CONCERT \nmastered  the  training  set  in  approximately  55  passes.  Generalization  performance  was \ntested by presenting the scales in the test set one note at a time and examining CONCERT's \nprediction.  Of the 63  notes to be predicted in  the  test set, CONCERT achieved remarkable \nperformance:  98.4%  correct.  The few  errors were  caused by  transposing notes one  full \noctave or one tonal half step. \n\nTo compare CONCERT with a transition table approach, we built a second-order transition \ntable from  the training set data and measured  its performance on the test set.  The transi(cid:173)\ntion table prediction (i.e., the note with highest probability) was correct only 26.6% of the \ntime.  The  transition table  is  somewhat of a straw man  in  this environment:  A  transition \ntable  that  is  based  on  absolute  pitches  is  simply  unable  to  generalize  correctly.  Even  if \nthe transition table encoded relative pitches, a third-order table would be required to  mas(cid:173)\nter the environment.  Kohonen's musical grammar faces the same difficulties as a transi(cid:173)\ntion table. \n\n4.2  LEARNING INTERSPERSED RANDOM WALK SEQUENCES \n\nThe  sequences  in  this  simulation  were  generated  by  interspersing  the  elements  of two \nsimple random walk sequences.  Each interspersed sequence had the following form:  a l' \nbi>  a2,  b 2,  ... , as,  bs, where al and  b 1 are  randomly  selected pitches,  ai+l  is  one step \nup or down  from  aj  on the  C major scale,  and likewise for  bi +1 and  bj \u2022  Each  sequence \nconsisted of ten notes.  CONCERT, with 25  context units, was trained on 50 passes through \na set of 200 examples and was then tested on an  additional 100.  Because it is impossible \nto  predict  the  second  note  in  the  interspersed  sequences  (b 1)  from  the  first  (a 1),  this \nprediction was  ignored  for  the purpose of evaluating CONCERT's performance.  CONCERT \nachieved a performance of 91. 7% correct.  About half the errors were ones in which CON(cid:173)\nCERT  transposed  a correct  prediction by  an  octave.  Excluding these errors,  performance \nimproved to 95.8% correct. \n\nTo  capture  the  structure  in  this  environment,  a  transition  table  approach  would  need  to \nconsider at least the previous two notes.  However,  such a transition  table is  not likely to \ngeneralize  well because,  if it  is  to  be  assured  of predicting a  note  at  step  11  correctly,  it \nmust observe the note at  step 11 -2 in  the context of every possible note  at  step 11  -1.  We \nconstructed a second-order  transition table from CONCERT'S  training set.  Using a testing \ncriterion analogous to that  used to evaluate CONCERT,  the transition  table achieved a per(cid:173)\nformance  level on the test set of only 67.1 % correct.  Kohonen's musical grammar would \nface the same difficulty as the transition table in this environment. \n\n\fConnectionist Music Composition Based on Melodic and Stylistic Constraints \n\n795 \n\n4.3  GENERATING NEW MELODIES IN THE STYLE OF BACH \n\nIn  a  final  experiment,  we  trained  CONCERT  on  the  melody  line  of a  set  of ten  simple \nminuets and  marches by  J.  S.  Bach.  The pieces had several voices,  but the melody gen(cid:173)\nerally appeared  in the treble voice.  Importantly, to naive listeners the extracted  melodies \nsounded pleasant and coherent without the accompaniment. \n\nIn  the  training data,  each  piece  was  terminated  with  a rest  marker  (the  only  rests  in  the \npieces).  This allowed CONCERT  to  learn  not only  the notes within a piece but also when \nthe end of the piece was reached.  Further, each major piece was transposed to the key of \nC major and each minor piece to the key of A minor.  This was done to facilitate learning \nbecause  the pitch  representation  does  not  take  into account  the  notion  of musical  key;  a \nmore sophisticated pitch representation might avoid the necessity of this step. \n\nIn  this simulation, each note was represented by a duration as  well as a pitch.  The dura(cid:173)\ntion  representation  consisted  of five  units  and  was  somewhat  analogous  the  PHCCCF \nrepresentation  for  pitch.  It allowed  for  the  representation  of sixteenth,  eighth,  quarter, \nand  half notes,  as  well as  triplets.  Also  included  in  this  simulation were  two  additional \ninput ones.  One indicated whether the piece was  in  a major versus minor key,  the  other \nindicated whether the  piece was  in 3/4 meter versus  2/4 or 4/4.  These inputs were  fixed \nfor a given piece. \n\nLearning the examples involves predicting a total of 1,260 notes altogether, no small feat. \nCONCERT was trained with 40 hidden units for  3000 passes through the  training set.  The \nlearning rate was gradually  lowered from  .0004 to  .0002.  By  the completion of training, \nCONCERT  could  correctly  predict  about  95%  of  the  pitches  and  95%  of the  durations \ncorrectly.  New pieces can be created by  presenting a few  notes to start and then running \nCONCERT in  composition mode.  One example of a composition produced by CONCERT  is \nshown  in  Figure 3.  The  primary  deficiency  of CONCERT's  compositions  is  that  they  are \nlacking in global coherence. \n\nIF  J \n\ntiS  I \nF  18 ( r E  r \n\nJ  r \n\nII \n\nFigure 3:  A Sample Composition Produced by CONCERT \n\n\f796  Mozer and Soukup \n\n5  DISCUSSION \n\nInitial results from  CONCERT are encouraging.  CONCERT is able to  learn musical structure \nof varying  complexity,  from  random  walk  sequences  to  Bach  pieces  containing  nearly \n200 notes.  We presented two examples of structure that CONCERT can learn but that can(cid:173)\nnot be captured by a simple transition table or by Kohonen's musical grammar. \n\nBeyond a  more  systematic examination of alternative  architectures, work on CONCERT  is \nheading in two directions.  First, the pitch representation is being expanded to account for \nthe perceptual effects of musical context and musical key.  Second, CONCERT is being ex(cid:173)\ntended to better handle the processing of global structure in music.  It is unrealistic to ex(cid:173)\npect that CONCERT, presented with a linear string of notes, could induce not only local re(cid:173)\nlationships among the notes, but also more global phrase structure, e.g., an AABA phrase \npattern.  To  address  the  issue  of  global  structure,  we  have  designed  a  network  that \noperates at several different temporal resolutions simultaneously (Mozer, 1990). \n\nAcknowledgements \n\nThis  research  was supported by  NSF grant  IRI-9058450,  grant 90-21  from  the  James  S. \nMcDonnell  Foundation.  Our  thanks  to  Paul  Smolensky, Yoshiro Miyata,  Debbie Breen, \nand  Geoffrey  Hinton  for  helpful  comments  regarding  this  work,  and  to  Hal  Eden  and \nDarren Hardy for technical assistance. \n\nReferences \n\nBridle,  J.  (1990).  Training stochastic model  recognition algorithms  as  networks can lead  to  maximum mutual \n\ninformation  estimation  of parameters.  In  D.  S.  Touretzky  (Ed.), Advances in  neural information pro(cid:173)\ncessing systems 2 (pp. 211-217).  San Mateo, CA:  Morgan Kaufmann. \n\nDodge, c.,  &  Jerse,  T.  A.  (1985).  Computer  music:  Synthesis,  composition,  and performance.  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(1989).  A  ~onnectionist  approach  to  algorithmic  composition.  Computer  Music  Journal,  13, \n\n27-43. \n\n\f", "award": [], "sourceid": 429, "authors": [{"given_name": "Michael", "family_name": "Mozer", "institution": null}, {"given_name": "Todd", "family_name": "Soukup", "institution": null}]}