Tillmann, B., Bharucha, J. J., & Bigand, E. (2000). Implicit Learning of Tonality- A Self-Organizing Approach. PSYCHOLOGICAL REVIEW-NEW YORK-, 107(4), 885-913

7/29/2019 Tillmann, B., Bharucha, J. J., & Bigand, E. (2000). Implicit Learning of Tonality- A Self-Organizing Approach. PSYCH

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Psychological Rev iew Copyright 2000 by the American Psychological Association, Inc.2000, Vol. 107, No. 4, 885-913 0033-295X/00/$5.00 DO I: 10.1037//0033-295X.107.4.885

Im plicit Learning of Tonality: A Self-Organizing Ap proachBarbara TillmannU n i v e r s i t6 d e B o u r g o g n e a n d D a r t m o u t h C o l l e g e

Emmanuel BigandU n i v e r s it 6 d e B o u r g o g n e

J a m s h e d J . B h a r u c h aD a r t m o u t h C o l l e g e

Ton al music is a highly structured system that is ubiquitous in o ur cultural environment. W e demonstratethe acquisition o f implicit knowledge of tonal structure through n eural self-organization resulting frommere exposure to simultaneous and sequential combinations of tones. In the process of learning, anetwork w ith fundamental neural constraints com es to internalize th e essential correlational structure oftonal music. Af ter learning, the network w as run through a range o f experiments fro m the literature. Themo del provides a parsimonious account of a variety o f empirical findings dealing with the processing o ftone, chord, and key relationships, including relatedness judgments, m emo ry judgments, and expectan-cies. It also illustrates the plausibility of activation being a unifying mechanism underlying a range ofcognit ive tasks .

N a t u r a l e n v i r o n m e n t s c o n t a i n h i g h l y s t r u c t u r e d s y s t e m s t ow h i c h w e a r e e x p o s e d i n e v e r y d a y l i f e . T h e h u m a n b r a i n i n t e r n a l -i z e s t h e s e r e g u l a r i t i e s b y p a s s i v e e x p o s u r e , a n d t h e a c q u i r e d i m -p l i c i t k n o w l e d g e i n f l u e n c e s p e r c e p t i o n a n d p e r f o r m a n c e . A s p e c t so f l a n g u a g e a n d m u s i c p r o v i d e t w o e x a m p l e s o f h i g h l y s t ru c t u re ds y s t e m s t h a t m a y b e l e a r n e d i n a n i n c i d e n t a l m a n n e r . I n e a c h c a s e ,t h e r e i s a p a r ad o x . O n t h e o n e h a n d , a t h o r o u g h f o r m a l d e s c r i p t i o no f t h e s t r u c t u r e h a s p r o v e n t o b e e x t r e m e l y c h a l l e n g i n g . O n t h eo t h e r h a n d , n a t i v e s p e a k e r s o r n o n m u s i c i a n l i s t e n e r s i n t e r n a l iz e t h er e g u l a r i t i e s u n d e r l y i n g l i n g u i s t i c o r m u s i c a l s t r u c t u r e s w i t h a p p a r -e n t e a s e . A s u b s t a n t i al c o r p u s o f r e s e a r c h h a s b e e n d e v o t e d t o t h el e a r n i n g p r o c e s s o f l a n g u a g e , b u t l i t t l e h a s b e e n d e v o t e d t o t h el e a r n i n g o f m u s i c . T h e c e n t r a l p u r p o s e o f t h e p r e s e n t a r t i c l e i s t oi n v e s t ig a t e h o w i m p l i c i t k n o w l e d g e o f s o m e b a s i c f e a t u re s o fW e s t e r n m u s i c a l g r a m m a r m a y b e a c q u i r e d a n d m e n t a l l yr e p r e s e n t e d .

Barbara Tillmann, Department o f Psychology, U niversit6 de Bour-gogne, LEA D-C NR S, Dijon, France, and Department of Psychological andBrain Sciences, Dartmouth Colle ge; Jamshed J. Bharucha, Department ofPsychological and Brain Sciences, Dartmouth College; Emmanuel Bigand,Department of Psychology, Universit6 de Bourgogne, LEAD-CN RS.

This research was supported in part by National Science FoundationGrant SBR-9601 287 and National Institutes of Health Grant 2P50NS1777 8-18 to Jamshed J. Bharucha and by a grant from the InternationalFoundation for Music Research to Emmanuel Bigand. Barbara Tillmannhas been supported by th e French M inistry o f Education and Research andby the Deutsche Akademische Austauschdienst DAAD. We thank Herv6Abdi, Pierre Perruchet, and Philippe Schyns for insightful comments atdifferent stages o f the present wo rk and Carol Krumhansl and Fred Lerdahlfor comments on the article.

Correspondence concerning this article should be addressed to BarbaraTillmann, Department of Psychological and Brain Sciences, DartmouthCollege, 6207 Moore Hall, Hanover, New Hampshire 03755. Electronicmall may be sent to [email protected] 5

W e p r e s e n t a c o n n e c t i o n i s t m o d e l t h a t (a ) s i m u l a t e s t h e i m p l i c i tl e a r n i n g o f p it c h s t r u ct u r e s o c c u r r i n g i n W e s t e r n h a r m o n y a n d ( b )a c c o u n t s f o r a r a n g e o f e m p i r i c a l f i n d i n g s i n m u s i c p e r c e p t io n . T h ea r t i c l e i s o r g a n i z e d i n f o u r p a r ts . F i r s t , w e s u m m a r i z e t h e r e g u l a r -i t i e s u n d e r l y i n g t h e W e s t e r n t o n a l m u s i c a l s y s t e m t h a t m a y b ei n t e r n a l i z e d b y i m p l i c i t l e a r n i n g a n d r e v i e w e x i s t i n g m o d e l s o ft o n a l k n o w l e d g e p r e s e n t a t i o n . S e c o n d , w e p r o p o s e a c o n n e c t i o n i s tm o d e l , b a s e d o n s e l f - o r g a n i z i n g m a p s ( S O M s ) , t h a t s i m u l a t e s t h el e a r n i n g o f t o n a l r e g u l a r it i e s b y m e r e e x p o s u r e . T h i r d , w e p r e s e n tt e s t s o f t h e t r a i n e d n e t w o r k w i t h e x p e r i m e n t a l t a s k s o n t h e p e r -c e p t i o n o f t o n a li t y . F o u r t h , w e d i s c u s s t h e p r o p o s e d m o d e l a n ds o m e o f i t s f u t u r e d e v e l o p m e n t s .

I M P L I C I T L E A R N I N G O F R E G U L A R I T I E S I NW E S T E R N T O N A L M U S I C

I n t h e a c q u i s i t i o n a n d r e p r e s e n t a ti o n o f k n o w l e d g e , a d i s t i n c t i o ni s m a d e b e t w e e n e x p l i c i t a n d i m p l i c i t . E x p l i c i t l e a r n i n g i s t h ea c q u i s i t i o n o f d e c l a r a t i v e k n o w l e d g e . T h i s i s t h o u g h t t o o c c u r b yh y p o t h e s i s t e s t i n g o r r u l e i n s t r u c t i o n . I m p l i c i t l e a r n i n g i s t h ea c q u i si t io n o f k n o w l e d g e i n a n i n c i d e n ta l m a n n e r w i t h o u t c o m -p l e t e v e r b a l i z a b l e k n o w l e d g e o f w h a t i s le a r n e d ( S e g e r , 1 9 9 4 ).I m p l i c i t l e a r n i n g i s s e e n a s a f u n d a m e n t a l c h a r a c t e r is t i c o f t h ec o g n i t i v e s y s t e m , e n a b l i n g t h e a c q u i s i t i o n o f h i g h l y c o m p l e x i n -f o r m a t i o n t h a t m a y n o t b e a c q u i r e d i n an e x p l i c i t w a y ( R e b e r ,1 9 8 9 ) . I m p l i c i t l e a r n i n g p r o c e s s e s h a v e b e e n s t u d i e d i n t h e l a b o -ra tory w i th a r t i f i c ia l m a te r ia l ba se d on s ta t i s t i c a l r e gula r i t i e s . Oneo f t h e m o s t f r e q u e n t l y u s e d e x p e r i m e n t a l s i t u a t io n s c o n s i s t s o fp r e s e n t i n g p a r t i ci p a n t s w i t h s e q u e n c e s o f e v e n t s g e n e r a t e d b y a na r t i f i c i a l l y d e f i n e d g r a m m a r . F o r e x a m p l e , a f i n i t e s t a t e g r a m m a rg e n e r a t e s c o m p l e x l e t t e r s t ri n g s b a s e d o n a r e s t r i c te d s e t o f l e t t e r s( R e b e r , 1 9 6 7 ) . A f t e r p a s s i v e e x p o s u r e t o g r a m m a t i c a l l e t t ers t r in g s , p a r t i c ip a n t s w e r e b e t t e r t h a n c h a n c e a t d i f f e r e n t i a t i n g n e wg r a m m a t i c a l l e t t e r s t r i n g s f r o m n e w n o n g r a m m a t i c a l o n e s . M o s tw e r e u n a b l e t o e x p l a i n t h e r u l e s u n d e r l y i n g t h e g r a m m a r i n v e r b a l


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88 6 TILLMANN, BHARUCHA, AND BIGANDf r e e r ep o r t s ( e .g . , A l t m a n n , D i e n e s , & G o o d e , 1 9 9 5 ; D i e n e s ,Broadben t , & Ber ry , 1991 ; Reber , 1967 , 1989) .

A r t i f ic i a l m a t e r i a ls a r e s i m p l e r t h a n e n v i r o n m e n t a l s e q u e n c e s o fe v e n t s . H o w e v e r , t h e s a m e b a s i c p r i n c i p l e s o f l e a r n i n g m a y s e r v ea s a m o d e l f o r u n d e r s t a n d i n g th e i m p l i c i t l e a r n i n g p r o c e s s e s i nn a t u r a l e n v i r o n m e n t s ( W i n t e r & R e b e r , 1 9 9 4 ) . F o r e x a m p l e , s e v -e r a l s t u d i e s a t t e m p t e d t o b r i d g e i m p l i c i t l e a r n i n g o f a r t i f i c i a lg r a m m a r a n d l a n g u a g e l e a r n i n g . S a f f r a n , N e w p o r t , A s l i n , T u n i c k ,a n d B a r r u e c o ( 1 9 9 7 ) s h o w e d w i t h a u d i t o r y s e q u e n c e s t h a t p a r t i c -i p a n t s w e r e a b l e t o u s e t h e s t a t is t i c a l r e g u l a r i t i e s s u c h a s t r a n s i t i o np r o b a b i l i t i e s o f s y l l a b l e s . A f t e r t h e p r e s e n t a t i o n o f a n a r t i f i c i a ll a n g u a g e - l ik e a u d i t o r y s e q u e n c e ( e .g ., b u p a d a p a t u b i t u t i b u . . . ) ,c h i l d r e n a n d a d u l t s p e r f o r m e d a b o v e c h a n c e i n d i s t i n g u i s h i n ga r t i f ic i a l w o r d s ( e . g. , b u p a d a , p a t u b i ) f r o m n o n w o r d s . E v e n a b r i e fe x p o s u r e t o a c o m p l e x n a t u r a l l a n g u a g e i n d u c e s a s e n s i t i v i t y t os t r u c t u r a l c o n s t r a i n t s : A f t e r a 1 2 - m i n p r e s e n t a t i o n o f a c a r t o o n f i l mn a r r a t e d b y a n a t i v e M a n d a r i n s p e a k e r, D u t c h a d u l t s d i s c r i m i n a t e da b o v e c h a n c e b e t w e e n r e a l M a n d a r i n w o r d s a n d p s e u d o w o r d s( Z w i s t e r l o o d , 1 9 9 0 , r e p o r t e d b y A l t m a n n e t a l. , 1 9 9 5 ) . F u r t h e rr e s u l t s o b t a i n e d f o r c h i l d r e n ' s s e n s i t i v i t y t o o r t h o g r a p h i c r e g u l a r -i t i e s ( P a c t o n , P e r r u c h e t , F a y o l , & C l e e r e m a n s , i n p r e s s ) s u p p o r tt h e e x t e n s i o n o f i m p l i c i t le a r n i n g c o n c l u s i o n s i s s u e d f r o m a r t i f ic i a ll a b o r a t o r y r e s e a r c h t o n a t u r a l m a t e r i a l .

I n s e v e r a l d o m a i n s , i m p l i c i t l e a r n i n g h a s b e e n s t u d i e d w i t h b o t ha r t i f i c i a l a n d n a t u r a l e v e n t s . I n t h e m u s i c d o m a i n , o n l y a f e ws t u d i e s i n v e s t i g a t e d i m p l i c i t l e a r n i n g d i r e c t l y w i t h a r t i f ic i a l s t r i n g so f m u s i c a l e v e n t s ( B i g a n d , P e r r u c h e t, & B o y e r , 1 9 9 8 ) . H o w e v e r ,m a n y s t u d i e s a d d r e s s e d t h i s i s s u e i n d i r e c t l y w i t h m u s i c p r o c e s s i n g( s e e D o w l i n g & H a r w o o d , 1 9 8 6 , f o r a r e v i e w ) . W e s t e r n m u s i c a lg r a m m a r i s m o r e c o m p l e x t h a n t h e f i n i te s t at e g r a m m a r s u s e d i ni m p l i c i t l e a r n i n g s tu d i e s . I t m a y b e c o n c e i v e d o f a s a t h r e e - l e v e lh i e r a r c h i c a l g r a m m a r t h a t g e n e r a t e s s t r o n g r e g u l a r i ti e s i n m u s i c a lp i e c e s . L e t u s c o n s i d e r s o m e o f t h e m o s t b a s i c r u l e s a n dregu la r i t ie s . 1I n W e s t e r n m u s i c , a r e s t r ic t e d s e t o f 1 2 p it c h c l a s s e s ( r e f e r r ed t oa s t h e t o n e s C , C ~ / D ~ , , D , D ~ / E ~ , E , F , F / G b , G , G I I / A b , A ,A ~ / B b, B ) a r e c o m b i n e d i n h i g h l y c o n s t r a in e d w a y s . T h i s s e to f 1 2 t o n e s i s o r g a n i z e d i n s u b s e t s o f s e v e n , c a l l e d d i a t o n i c s c a l e s .D e p e n d i n g o n t h e p a t t e r n o f i n t e r v a l s s e p a r a t i n g t h e s e v e n t o n e s ,d i a t o n ic s c a le s m a y b e m a j o r o r m i n o r . A p p l y i n g t h e t w o d i a to n i cs c a l e s t o e a c h o f t h e 1 2 p i t c h c l as s e s l e a d s t o t h e d i s t i n c t i o n o f 1 2m a j o r a n d 1 2 m i n o r m u s i c a l k e ys . T h e t o n e s t h a t f o r m a m u s i c a lk e y ( e . g. , B - C t I - D ~ - E - F ~ - G ~ - A ~ ) a re m o r e l i k e l y t o c o - o c c u r i nW e s t e r n m e l o d i e s t h a n t o n e s t h a t d o n o t f o r m a m u s i c a l k e y ( e . g . ,B - C - D ~ - E - F I I - G ~ - A ) . F o r e a c h s c a l e, s e v e n d i a t o n i c ch o r d s m a yb e d e f i n e d o n e a c h o f t h e s e v e n d e g r e e s o f t h e s c a l e a c c o r d i n g t os p e c i f i c h a r m o n i c r u l e s . A c h o r d i s a s i m u l t a n e i t y o f t h r e e t o n e s ,u s u a l l y c a l l e d t h e r o o t , t h i r d , a n d f i f th . I n t h e m a j o r k e y s , t h e c h o r db u i l t o n t h e f i r s t , f o u r t h , a n d f i f t h s c a l e d e g r e e ( I , I V , a n d V ) a r em a j o r , t h o s e b u i l t o n t h e s e c o n d , t h i r d , a n d s i x t h s c a l e d e g r e e a r em i n o r ( i i , i i i , v i ) , a n d t h e c h o r d b u i l t o n t h e s e v e n t h d e g r e e i sd i m i n i s h e d ( v i i ) .2 C h o r d s f o r m a s e c o n d o r d e r o f m u s i c a l u n i ts ,a n d t h e i r o c c u r r e n c e i s s t r o n g l y c o n s t r a i n e d b y W e s t e r n m u s i c a lg r a m m a r . C h o r d s b e l o n g i n g to t h e s a m e m u s i c a l k e y a r e m o r el i k e l y t o c o - o c c u r i n a g i v e n m u s i c a l p i e c e t h a n c h o r d s b e l o n g i n gt o d i f f e r e n t k e y s .

K e y s d e f i n e a t h i r d o r d e r o f m u s i c a l u n i t s . S o m e k e y s s h a r en u m e r o u s c h o r d s a n d t o n e s . F o r e x a m p l e , t h e C - m a j o r k e y s h a r e sf o u r c h o r d s a n d s i x t o n e s w i t h t h e G - m a j o r k e y , tw o c h o r d s a n d

f i v e t o ne s w i t h t h e D - m a j o r k e y , a n d o n l y o n e t o n e w i t h t h eF i t - m a j o r k e y . K e y s s h a r i n g c h o r d s o r t o n e s a r e s a i d to b e h a r -m o n i c a l l y r e la t e d . T h e s t r e n g t h o f t h e s e h a r m o n i c r e l a t i o n s h i p sd e p e n d s o n t h e n u m b e r o f s h a r e d c h o r d s o r t o n e s . I n m u s i c t h e o r y ,k e y s a r e c o n c e i v e d s p a t i a l l y a s a c i r c l e , r e f e r r ed t o a s t h e c y c l e o ff i f t h s ( F i g u r e 1 ) . T h e n u m b e r o f s t e p s s e p a r a t i n g tw o k e y s o n t h i sc i r c l e ( w h a t e v e r t h e d i r e c t i o n o f t h e r o t a t i o n ) d e f i n e s t h e i r h a r -m o n i c d i s t a n c e . T h e s e i n t e r k e y d i s t a n c e s f o r m t h e b a s i s f o r s t r o n gr e g u l a r i t i e s i n p i e c e s o f W e s t e r n m u s i c . K e y c h a n g e s a r e m o r el i k e l y t o o c c u r b e t w e e n c l o s e l y r e l a t e d k e y s ( e . g . , C a n d F o r Gm a j o r ) t h a n b e t w e e n l e s s r e l a t e d o n e s ( e . g . , C a n d F ~ m a j o r ) .I n t e r k ey d i s t a n c e s a r e a l s o d e f i n e d b e t w e e n m a j o r a n d m i n o r k e y s .A m a j o r k e y ( e . g . , C m a j o r ) i s h a r m o n i c a l l y r e l a t e d t o b o t h i t sr e l a t i v e m i n o r k e y ( a m i n o r ) a n d i t s p a r a l le l m i n o r k e y ( c m i n o r ) .T h e s e m u l t i l e v e l r e l a t i o n s a m o n g t o n e s a n d c h o r d s , c h o r ds a n dk e y s , a n d m a j o r a n d m i n o r k e y s d e f i n e a c o m p l e x s e t o f p o s si b l er e l a t i o n s b e t w e e n m u s i c a l e v e n t s ( se e K r u m h a n s l , 1 9 9 0 ; L e r d a h l ,1 9 8 8 ) , a n d t h e y s t r o n g l y c o n s t r a i n t h e t r a n s i t i o n p r o b a b i l i t i e s b e -t w e e n m u s i c a l e v e n t s a s a p i e c e u n f o l d s ( F r a n c ~ s , 1 9 5 8 ; P i s t o n ,1978) .A n o t h e r i m p o r t a n t fe a t u re o f W e s t e r n m u s i c a l g r a m m a r i s t h att o n e s a n d c h o r d s h a v e d i f f e r e n t s t r u c t u r a l f u n c t i o n s w i t h i n a k e y .A c c o r d i n g t o M e y e r ( 1 9 5 6 ) , " I n t he m a j o r m o d e i n W e s t e r n m u s i ct h e t o n i c t o n e 3 i s t h e t o n e o f u l t i m a t e r e s t t o w a r d w h i c h a l l o t h e rt o n e s t e n d t o m o v e . O n t h e n e x t h i g h e r l e v e l t h e t h i r d a n d f if t h o ft h e s c a l e , t h o u g h a c t i v e m e l o d i c t o n e s r e l a t i v e t o t h e t o n i c , j o i n t h et o n i c a s s t r u c t u r a l t o n e s ; a n d a l l t h e o t h e r t o n e s , w h e t h e r d i a t o n i co r c h r o m a t i c , t e n d t o w a r d o n e o f t h e s e " ( p p . 2 1 4 - 2 1 5 ) . T h e s ed i f f e r e n c e s i n m u s i c a l f u n c t i o n s c r e a t e w i t h i n - k e y h i e r a r c h i e s .I n t e r e s t i n g l y , w i t h i n - k e y h i e r a r c h i e s a r e s t r o n g l y c o r r e l a t e d w i t ht h e f r e q u e n c y o f o c c u r r e n c e o f t o n e s i n W e s t e r n m u s i c a l p i e c e s .T o n e s t h a t o c c u r w i t h g r e a t e r f r e q u e n c y ( t h e t o n i c , t h e f i f th , a n d ,t o a l e s s e r e x t e n t , th e t h i r d ) a r e t h o s e d e s c r i b e d b y m u s i c t h e o r y a sb e i n g t h e m o s t i m p o r t a n t i n a g i v e n k e y . F r o m a p s y c h o l o g i c a lp o i n t o f v i e w , t h e h i e r a r c h i c a l l y i m p o r t a n t t o n e s o f a k e y a c t a ss t a b l e c o g n i t i v e r e f e re n c e p o i n t s ( K r u m h a n s l , 1 9 7 9 , 1 9 9 0 ) tow h i c h t h e o t h e r t o n e s a r e a n c h o r e d ( B h a r u c h a , 1 9 8 4 ) .

A w i t h i n - k e y h i e r a r c h y i s a l so f o u n d a m o n g t h e s e v e n c h o r d s o fa k e y ( B h a r u c h a & K r u m h a n s l , 1 9 8 3 ; K r u m h a n s l , B h a r u c h a , &C a s t e l l a n o , 1 9 8 2 ). C h o r d s b u i l t o n t h e f i rs t , f i ft h , a n d f o u r t h s c a l ed e g r e e s ( r e f e r r e d t o a s , r e s p e c t i v e l y , t h e t o n i c , d o m i n a n t , a n ds u b d o m i n a n t c h o r d s ) u s u a l l y h a v e a m o r e c e n t r a l s y n t a c t ic f u n c -t i o n t h a n t h o s e b u i l t o n t h e o t h e r s c a l e d e g r e es . F o r e x a m p l e , ad o m i n a n t c h o r d f o l l o w e d b y a t o n i c c h o r d i s a n a u t h e n t ic c a d e n c e ,w h i c h m a r k s f i n a l i t y . I n c o n t r a s t , a s u b d o m i n a n t c h o rd f o l l o w e db y a d o m i n a n t c h o r d i s a h a l f c a d e n c e , w h i c h m a r k s a t e m p o r a r y

i A complete description of the Western musical system is beyo nd thescope of this article. The main purpose of the following section is tohighlight the organizational features that give rise to the most salientregularities in Western musical pieces.2 Major chords consist of intervals of a m ajor third (four semitones) anda perfect fifth (seven sem itones with respect to a reference pitch [the root]).In minor chords, the third is minor (three semitones), and in diminishedchords the third is minor and the fifth is diminished (six semitones).3 The tonic is the tone on the first degree of the scale. It also givesits name to the key. For example, the tonic of the key of C major is thetone C.


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IMPLICIT LEARNING OF TONALITY 887

C

A b ' o " \Ob /F#

Figure 1. Cycle o f fifths representing the distances between major keys.

end ing . Bo th mus ic theor i s t s (De l i~ge , 1984 ; Schenk er , 1979) andmus ic ps ycho log is t s (S loboda , 1985) cons ide r tha t the au then t iccadence ac t s a s a bas ic syn tac t i c s t ruc tu re in Wes te rn mus ic ( seeDel i~ge , 1984 , and S loboda , 1985 , fo r a fu r the r deve lopment ) .

A c r i t i ca l f ea tu re o f We s te rn mu s ic i s tha t the func t ions o fm u s i c a l e v e n t s c h a n g e w i t h t h e k e y c o n t e x t . A C - m a j o r c h o r dfunc t ions as a s t ab le ton ic chord in a C-m ajor con tex t and as a l e sss t a b l e d o m i n a n t o r s u b d o m i n a n t c h o r d i n t h e F - o r G - m a j o r k e yc o n t e x ts , r e s p e c ti v e l y . S i m i l a r l y , a G - C c h o r d s e q u e n c e f o r m s a na u t h e n ti c c a d e n c e i n a C - m a j o r k e y c o n t e x t b u t n o t in a G - m a j o rk e y c o n t e x t . T h i s c o n t e x t d e p e n d e n c y i s a f u n d a m e n t a l a s p e c t o ft h e W e s t e r n t o n a l s y s t e m : U n d e r s t a n d i n g t h e f u n c t i o n o f e v e n t swi th respec t to the mu s ica l con tex t i s c ruc ia l to a ssess ing mus ica lg r a m m a t i c a l i t y .

D e s p i t e t h e c o m p l e x i t y o f t h e s y s t e m , s e n s i t i v i ty to m u s i c a ls t ruc tu re does no t r equ i re exp l ic i t lea rn ing . Because m us ica l lyna ive l i s t eners a re cons tan t ly exposed in everyday l i f e to theregu la r i t i e s under ly ing the mus ic o f the i r cu l tu re , they acqu i rei m p l i c i t k n o w l e d g e o f t h e m ( B h a r uc h a , 1 9 8 4 ; D o w l i n g & H a r -woo d , 1986 ; F ranc~s , 1958) . Th is imp l ic i t know ledge emb odiesthe func t ions o f tones and chords in a key (T i l lmann , B igand , &Madure l l , 1998) , the re la t ions be tween d i f fe ren t keys (Bar t l e t t &Dowl ing , 1980 ; Cuddy & Thompson , 1992a , 1992b ; Thompson &Cuddy , 1989) , and the change in func t ion o f even ts depend ing onthe key con tex t (Bharucha & Krumhans l , 1983 ; B igand , 1993 ;Bigand , 1997 ; B igand & P ineau , 1997 ; Krumhans l e t a l . , 1982) .T h e i n t e r n a li z e d r e p re s e n t a t io n i n f l u e n ce s m u s i c a l m e m o r y ( B i g -and & P ineau , 1996 ; Cuddy , Cohen , & Mewhor t , 1981 ; Dowl ing ,1978 ; Dowl ing , 1991) , mus ica l expec tanc ies (Bharucha &Stoeck ig , 1986, 1987; Cudd y & Lunn ey , 1995) , and the res to ra t ionof miss ing mus ica l even ts (DeW it t & Sam uel , 1990). Resu l t sgenera l ly revea l s t rong cons i s tency fo r l i s t eners wi th d i f fe ren tl e v e l s o f m u s i c a l e x p e r t i s e . A l t h o u g h m u s i c i a n s u s u a l l y e x h i b i tbe t t e r pe r fo rmance than nonmus ic ians , the i r overa l l r e sponsesshow the same pa t te rns ( see a l so Bigand , Pa rncu t t , & Lerdah l ,1996; A . Cohen , 1994; Croonen & H outsma , 1994) . Ques t ionna i red a t a e v e n s u g g e s t t h at t o n a l k n o w l e d g e i s t a c i t fo r b o t h g r o u p s o fpar t i c ipan t s (Hol le ran , Jones , & B ut le r , 1995). F ina l ly , even t -re la ted po ten t ia l s (ERP) s tud ies p rov ide fu r the r ev idence tha tm u s i c i a n s a n d n o n m u s i c i an s s h o w s i m i l a r e l e c t r o p h y s i o l o g i c a l

responses to sub t le changes in the ha rmonic func t ion o f a t a rge tchord (Regnau l t , B igand , & Besson , in p ress ) .

Our main purpose in th i s a r t i c le i s to inves t iga te how th i si m p l i c i t k n o w l e d g e o f W e s t e r n p i t c h r e g u l a r it i e s m a y b e r e p r e -s e n t e d a n d l e a r n e d t h r o u g h p a s s i v e e x p o s u r e t o m u s i c a l e x e m p l a r s .W e a r g u e t h a t a m o d e l o f d i s t r ib u t e d k n o w l e d g e o f f e rs a p o s s i b l ee x p l a n a t o r y f ra m e w o r k t h a t a c c o u n t s f o r l e a r n in g i n t h e a b s e n c e o fe x p l i c i t t u t o ri n g a n d t h a t s u g g e s ts a n u n d e r l y i n g m e c h a n i s m - -a c t i v a t i o n - - t h a t u n i f i e s a r a n g e o f p s y c h o l o g i c a l t a s k s. I t s h o u l d b eno ted tha t , beyon d the com plex regu la r i t i e s o f p i t ch s t ruc tu re , theW e s t e r n t o n a l s y s t e m a l s o c o n t a i n s c o m p l e x t e m p o r a l r e g u l a r it i e sdef ined by met r ica l and rhy thmic s t ruc tu res . In mus ic cogn i t ion ,the p rocess ing o f p i t ch and tempora l s t ruc tu res have o f ten beenc o n s i d e r e d i n d e p e n d e n tl y . E v e n t h o u g h t h i s m e t h o d o l o g i c a l i n d e -pendence rem ains a mat te r o f deba te (Bol tz , 1999 ; Jones & Bol tz ,1989; Pe re tz & Kol insky , 1993) , we focu s on p i t ch regu la r i t i e son ly and no t on t empora l r egu la r i t i e s . However , we re tu rn to th i si s s u e i n t h e G e n e r a l D i s c u s s i o n b y c o n s i d e r in g p o s s i b l e e x t e n s i o n sof the connec t ion i s t mode l to in tegra te some tempora l r egu la r i t i e so f W e s t e r n m u s i c .

M O D E L S O F D IS T R I B U T E D K N O W L E D G ER E P R E S E N T A T I O N

C o n n e c t i o n i s t m o d e l s h a v e t w o p r i n c i p a l a d v a n ta g e s o v e r t r a -d i t i o n a l r u l e - b a s e d m o d e l s : ( a ) T h e r u l e s g o v e r n i n g t h e d o m a i n a r en o t e x p l i c i t b u t r a t h e r e m e r g e f r o m t h e s i m u l t a n e o u s s a t i s f a ct i o n o fmul t ip le cons t ra in t s r ep resen ted by ind iv idua l connec t ions , and (b )these cons t ra in t s themse lves can be l ea rned th rough pass ive expo-sure . In the f i e ld o f a r t if i c ia l g ram mar l ea rn ing , a kno wledg er e p r e s e n t a t i o n o f t r a i n i n g s t i m u l i m a y e m e r g e f r o m a s s o c i a t i v el e a r n in g m e c h a n i s m s o f c o n n e c t i o n i s t m o d e l s . A u t o a s s o c i a t o r n e t -works m emo r ize s t imul i genera ted by a r t i f i c ia l g ramm ars , c lass i fynew s t imul i , and s imula te exper imenta l r e su l t s even be t te r thanexem pla r -base d mo de ls (Dienes , 1992). Th is approach a l lows us toin te rp re t the no t ion o f "abs t rac t knowledge" d i f fe ren t ly than on lyi n t h e s t r o n g s e n s e o f r u l e - l i k e k n o w l e d g e o r o f s i m p l e s e n s i t i v i tyto s to red examples (C lee remans , 1994) . In the l anguage domain ,M c C l e l l a n d e t a l. d e v e l o p e d o n e c l a s s o f n e u r a l n e t m o d e l s o fknow ledge represen ta t ion . These in te rac t ive -ac t iva t ion mod e ls o fword recogn i t ion (McC le l land & Rum elhar t , 1981 ; Rum elhar t &M c C l e l l a n d , 1 9 8 2) a n d s p e e c h r e c o g n i t io n ( E l m a n & M c C l e l l a n d ,1984; McC le l land & Elm an , 1986) s im ula te the in te rac t ion be -tween kn owled ge and pe rcep t ion wi thou t s to r ing l ingu is t i c ru lesexp l ic i t ly . Three l eve l s o f un i t s r ep resen t f ea tu res , l e t t e r s (o rp h o n e m e s ) , a n d w o r d s . R u l e - l i k e b e h a v i o r e m e r g e s f r o m t h e i n -te rac t ions o f a se t o f word un i t s and le t t e r (o r phoneme) un i t s .

I n t h e m u s i c d o m a i n , a g r o w i n g n u m b e r o f n e u r a l n e t w o r km o d e l s h a v e b e e n d e v e l o p e d d u r i n g t h e l a s t d e c ad e . M o d e l s o f t h i sna tu re have been o f fe red fo r p i t ch pe rcep t ion (Sano & Jenk ins ,1991; Tay lo r & Greenhou gh , 1994) , oc tave equ iva le nce (Bharucha& Me nc l , 1996) , chord c lass i f i ca t ion (Laden & Keefe , 1991) , andmelo d ic sequence l ea rn ing (Bharucha & Todd , 1989 ; Krumhans l ,Louh ivuor i , To iv ia inen , J~ 'v inen , & Eero la , 1999 ; Page , 1994) .S o m e m o d e l s s i m u l a t e m o r e c o m p l e x a s p e c t s o f m u s i c l e a r n i n ga n d p e r c e p t i o n , s u ch a s c a t e g o r i z a t io n a n d m e m o r y o f f e a t u r epa t te rns . G je rd ingen (1990) expo sed a four - leve l ne two rk based onG r o s s b e r g ' s ( 1 9 87 ) a d a p t i v e r e s o n a n c e t h e o r y t o e a r l y w o r k s o fMozar t . The inpu t l aye r codes mus ic theore t i c concep ts ( i .e . ,


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888 TILLMANN, BHARUCHA, AND BIGANDharmonic tritone, contrapuntal dissonance) and low-level musicfeatures (i.e., melodic contour, p itch of the major diatonic scaleplus a unit for the alterations of flat and sharp). The four levelsinclude dynamic short-term memory (Level 1) that leads to theformat ion of stable categories (Level 2), and a second temporarystore (Level 3) that categorizes on a higher level (Level 4). Themodel develops memories of critical feature patterns and derivescategorizations comparable to complex music theoretic concepts(i.e., voice-leading combinations).

Few models attempt to formalize how the multiple relationshipsamong tones, chords, and keys may be represented in a singleframework. Griff ith' s (1994) model simulates how keys are in-duced from patterns of pitch use and how abstract p itch identi tiesare established from interval use. Supervised and unsupervisedparadigms are used in a modular combination, allowing the modelto use its own derived information to guide subsequent processes.The model formalizes an inductive mechanism for learning keyand scale degrees from melodic sequences. It mostly concentrateson the links between pitch and key and is not designed to addressthe relationships between tones and chords and chords and keys.The output of the model is compared with music theory but notwith empirical data.

Leman (1995; Leman & Carreras, 1997) simulated the percep-tual learning of tonal centers by presenting chords and real musicalpieces to a framework of two modules. I n these simulations, themusical acoustical signal is first processed by an auditory model,and the transformed information defines the input for a self-organizing map. In Leman (1995), three auditory models definethree types of input vectors for a variety of chords, and a self-organizing map is trained with these different inputs. Whatever theauditory model used, map units specialize in the detection ofchords. After training, the activations on the map in response to agiven stimulus reflect the harmonic relatedness of the representedchords to the stimulus. On the basis of activation regions found onthe map, the tonal centers are inferred. The trained network isexposed to musical pieces, and the detected changes in tonalcenters are compared with music theoretical analyses. In Lemanand Carreras (1997), the input signal is derived from neural firingpatterns i n response to real sound recordings of Bach pieces. A nSOM is trained to extract the regularities in these input patterns.After training, tonal centers activated by a given musical stimulusare inferred on the basis of activation regions on the map. Theoutput of the model conforms generally to music theory, and someempirical data are simulated by the model.

Leman's (1995) and Leman and Carreras's (1997) models focuson chords and on tonal centers but do not account for the relation-ships between tones, chords, and keys. The main interest is to showthat higher order units of Western music (i.e., chords or tonalcenters) may be learned by passive exposure to a rich acousticinput. The SOMs extract the invariant features of musical soundsthat lead to the formation of abstract units. In other words, thesemodels formalize how learning processes may be driven by psy-choacoustic features in a bottom-up manner. However, they do notinvestigate how the acquired knowledge may, in turn, influence theprocessing of musical events, leading to predictions that can betested experimentally. The crucial benefit of learning is the use ofknowledge to react to environmenta l stimuli better. Top-downinfluences facilitate the processing of environmentalevents, as hasbeen shown in different domains of cognition, including music.

For example, once the key of a musical context is recognized, thetones belonging to that key are perceived as more stable than othertones, even if they were not present in the stimulus context(Franc~s, 1958; Krumhansl, 1990). A model of knowledge repre-sentation should be able to account for these top-down effects andfor the way they combine with bottom-up influences. In wordrecognition, connectionistmodels simulate the influence of knowl-edge by interactive activation between higher level units (words)and lower level units (letters; McClelland & Rumelhart, 1981). Inmusic, Bharucha's (1987a, 1987b) model of spreading activation(referred to as MUSACT) relies on a comparable architecture.

In this model, a pattern of connect ions constitutes a knowledgerepresentation of Western harmony (Bharucha, 1987b, 1994). Theunits of the network are organized in three layers corresponding totones, chords, and keys (Figure 2, top). Each o f the 12 tone unitsis connected to three major and three minor chords, of which thattone is a component. Analogously, each chord unit is connected tothree major key units representing keys of which it is a member.Western musical rules are not stored explicitly but emerge fromthe activation that reverberates by connected links between tone,chord, and key units.When a chord consisting of three tones (e.g., C-E-G) is played,the units representing these tones are activated, and phasic activa-tion is sent toward the chord units (see Figure 2, top). The chordunit connected to all three tones receives the strongest activation(the C-major chord in this example). During a second cycle (Fig-ure 2, middle), phasic activation from the active chord unitsspreads toward the key uni ts (bottom-up activation) and starts toreverberate toward tone unit s (top-down activation). During thenext cycle (see Figure 2, bottom), activated key units send top-down activation to chord units that simul taneously receive activa-tion from the tone units. Phasic activation continues to spreadthrough the network between all layers until an equil ibrium isreached. Early activation cycles reflect bottom-up influences: Ac-tivated chord units contain at least one of the component tones ofthe stimulus chord. For example, after a C-major chord, the chordunit E will be more activated than the chord unit D because itshares one tone with the stimulus chord (i.e., the tone E). It isnoteworthy that the E-major chord unit is more activated than theD-major chord unit, even though the C-major chord and theE-major chord are harmonical ly less related in theory (they haveno parent key) than are the C-major chord and the D-major chord(which both belong to the G-major key). Similarly, key units areactivated if they have an activated chord as a member (Figure 3,left). During reverberatory cycles from ini tial activation to equi-librium, activation patterns change qualitatively. At equilibrium,the state of the network incorporates the influences of top-downprocesses and mirrors theoretical Western hierarchies. Activat iontends to decrease with increasing harmonic distance betweenchords around the cycle of fifths. These top-down influences areclearly illustrated by the D- and E-major chord units. At equilib-rium, the chord uni t D receives stronger activation than the chordunit E. Similarly, the activation of key units decreases monotoni-cally with increasing harmonic distance between major keysaround the cycle of fifths (see Figure 3, right).

For chord sequences, activat ion result ing from each chord isaccumulated. After the offset of an event , the activat ion begins todecay exponentially over time. If another event occurs beforeactivat ion has decayed appreciably, the phasic activation resulting


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IMPLICIT LEARNING OF TONALITY 88 9

Figure 2. Bottom-up and top-down activation spreading in the MUSACT model after the presentation of aC-major chord. Top: Activated tone units send bottom-up activation to connected chord units (first cycle).Middle: Phasic activation spreads to key units and reverberates to tone units (second cycle). Bottom: Chord unitsreceive activation from both key units and tone units (third cycle). From "MUSACT: A Connectionist Model ofMusical Harmony," by J. J. Bharucha, 1987a, in Program of the Ninth Annual Conference of the CognitiveScience Society, pp. 508-517 , Figure 7, Hillsdale, NJ: Erlbaum. Copyright 1987 by the Cognitive ScienceSociety, Incorporated. Used by permission.

from that next event is added to the residual activation from theprevious event, thereby creating a pattern o f activation that can beinfluenced by an entire sequence of events, weighted according torecency. In other words, the activation of a unit i in the network isa function of not just the most recent event e but also of the

previo us event, e - 1, the activation of e - 1 being itself afunction of eve nt e - 2 and s o on. The total activation, aj. e, of aunit i (a tone, a chord, or a key) after an event e is an additivefunction of three quantities: (a) the bottom-up activation causeddirectly by the stimulus itself (i.e., the tones), (b) the indirect


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890 TILLMANN, BHARUCHA, AND BIGAND

I . . . o U n s~-" / M \ - - - -- a j o r e y n it s , ~ / ~ ' \ - - - M a j o r e y U ni ts

- , Y , , V , V , X I !F # C # G # D # A # F C G D A E B F # C # G # D # A # F C G D A E BG b D b A b E b B b G b D b A b E b B bMajor Chord and Key Units in Network Major Chord and Key Units in Network

F i g u r e 3 . Left: The state of the network just after hearing the tones C , E, and G (a C-majo r chord) bu t beforeactivation has had a chance to reverberate back from key units to chord units. Right: The state of the networkafter activation has reverberated to a state of equilibrium. F rom "M usic Cog nition and P erceptual Facilitation:A Connectionist Framework," by J. J. Bharucha, 1987b, M u s i c P e r c e p t i o n , 5, p. 1, Figures 8 and 9. Copyright1987 by The Regents of the University of California. Adapted with permission.

a c t i v a t i o n r e c e i v e d f r o m o t h e r u n i t s i n r e s p o n s e t o e v e n t e ( i .e . , th ep h a s i c a c t i v a t i o n s p r e a d i n g i n t h e s y s t e m ) , a n d ( c ) t h e d e c a y e da c t i v a t i o n c a u s e d b y p r e v i o u s e v e n t s e - 1 (b e i n g i t s e l f a f u n c t i o nof e ve nt e - 2 a nd so on) . T he to ta l a c t iv a t ion , a i , e , o f a uni t i i sg i v e n b y t h e f o l l o w i n g e q u a ti o n :

q

a i , e = A + E A a i . e , c + a i . e - 1 ( 1 - d ) t (1 )c= 1

wh e re A re pre se nt s the s t im ulus a c t iva t ion ; Y, Aa . . . i s the sum ofp h a s i c a c t i v a t i o n o f u n i t i i n r e s p o n s e t o e v e n t e , a c c u m u l a t e d o v e rt h e q r e v e r b a t o r y c y c l e s t h a t a r e n e c e s s a r y t o r e a c h e q u i l i b r i u m ; dr e p r e s e n t s t h e r a t e ( v a r y i n g b e t w e e n 0 a n d 1 ) a t w h i c h a c t i v a t i o nd e c a y s a f t e r t h e o f f s e t o f t h e l a s t e v e n t ; a n d t i s th e t i m e e l a p s e ds inc e the l a s t o f f se t .

M U S A C T i s a n i d e a l i z e d s i m p l i f i c a t i o n o f a s p e c t s o f t h e W e s t -e r n m u s i c a l g r a m m a r b e c a u s e i t d o e s n o t i n c o r p o r a t e u n i t s f o rm i n o r k e y s a n d s e v e r a l o t h e r t y p e s o f c h o r d s t h a t m a y o c c u r i nW e s t e r n m u s i c ( i .e . , d i m i n i s h e d c h o r d s ) . A f u r t h e r s i m p l i f i c a t i o n i st h e a b s o l u t e p i t c h c l a s s c o d i n g a t t h e i n p u t l a y e r a n d t h e r e p r e s e n -t a t i o n o f c h o r d u n i t s i n d e p e n d e n t o f t h e r e l a t i v e p o s i t i o n o f t h ec o m p o n e n t t o n e s . 4 I n s p i t e o f t h e s e s i m p l i f i c a ti o n s , t h e m o d e lp r o v i d e s a r e l e v a n t f r a m e w o r k f o r u n d e r s t a n d i n g h o w m u s i c a lk n o w l e d g e m a y b e m e n t a l l y r ep r e s e n te d a n d h o w t h is k n o w l e d g e ,o n c e a c t i v a t e d b y a g i v e n m u s i c a l c o n t e x t , m a y i n f l u e n c e t h ep r o c e s s i n g o f to n a l s t r u c t u re s ( B h a r u c h a , 1 9 8 7 b ).

S u p p o r t f o r t h e M U S A C T m o d e l h a s c o m e f r o m e m p i r i c a ls t u d i es u s i n g a h a r m o n i c p r i m i n g p a r a d i g m . T h e r a t i o n a l e o f th e s es t u d i e s i s t h a t a p r e v i o u s c h o r d p r i m e s h a r m o n i c a l l y r e l a t e d c h o r d ss o t h a t t h e ir p r o c e s s i n g i s s p e e d e d u p . T h e e x t e n t t o w h i c h a c h o r di s p r i m e d b y a c o n t e x t i s a f u n c t i o n o f t h e a c t i v a t i o n o f t h e u n i tr e p r e s e n t i n g t h i s c h o r d i n t h e m o d e l . T h e m o r e a c h o r d u n i t i sa c t i v a t e d , t h e m o r e t h e c h o r d i s p r i m e d . T o t e s t t h i s h y p o t h e s i s ,B h a r u c h a a n d S t o e c k i g ( 1 9 8 6 , 1 9 8 7 ) a s k e d p a r t i c i p a n t s t o d e c i d ea s q u i c k l y a s p o s s i b l e w h e t h e r o r n o t a t a r g e t c h o r d f o l l o w i n g ap r i m e c h o r d w a s i n t u n e ( s e e a l s o T e k m a n & B h a r u c h a , 1 9 9 2 ) .P a r t i c ip a n t s h e a r d a p r i m e c h o r d f o l l o w e d b y a h a r m o n i c a l l yc l o s e l y o r d i s t a n t ly r e l a t e d t a r g e t c h o r d . F o r e x a m p l e , i f t h e p r i m ec h o r d i s C m a j o r , B b m a j o r w o u l d b e a r e l a t ed t a r g e t a n d F f f m a j o r

a n u n r e l at e d t a rg e t. I n M U S A C T , t h e B b - m a j o r c h o r d w o u l d b em o r e s t r o n g l y p r i m e d b y a C - m a j o r c h o r d t h a n w o u l d b e t h eFi t -m a jor c hord ( s e e F igure 3) . Em pi r ic a l da ta c onfLr rne d th i sp r e d i c t i o n : T h e p r i m i n g e f f e c t w a s s h o w n b y ( a) a b ia s t o j u d g et a r g et s t o b e i n t u n e w h e n r e l a t e d t o t h e p r i m e a n d o u t o f tu n e w h e nu n r e l a t ed , a n d ( b ) s h o r t e r re s p o n s e t i m e s f o r i n - t u n e t a r g et s w h e nr e l a t e d a n d f o r o u t - o f - t u n e t a r g e t s w h e n u n r e l a t e d . A p r e v i o u sm u s i c a l c o n t e x t ( a s i n g l e c h o r d i n t h e s e e x p e r i m e n t s ) t h u s g e n e r -a t e s e x p e c t a n c i e s f o r r e l a t e d c h o r d s t o f o l l o w , r e s u l t i n g i n g r e a t e rc o n s o n a n c e a n d f a s t e r p r o c e s s i n g f o r e x p e c t e d c h o r d s .

T h e p r i m i n g o f a c h o r d d e p e n d s n o t o n l y o n t h e p r e v i o u s c h o r db u t a l s o o n i t s h a r m o n i c f u n c t i o n w i t h i n a n e x t e n d e d t e m p o r a lc o n t e x t ( B i g a n d & P i n e a u , 1 9 9 7 ; B i g a n d , M a d u r e l l , T i l l m a n n , &P i n e a u , 1 9 9 9 ; T i l l m a n n , B i g a n d , & P i n e a u , 1 9 9 8 ) . B i g a n d a n dP i n e a u ( 1 9 9 7 ) p r e s e n t e d p a r t ic i p a n t s w i t h e i g h t - c h o r d s e q u e n c e s .T h e e x p e c t a t i o n s f o r t h e l a s t c h o r d ( t h e t a r g e t ) w e r e v a r i e d b yc h a n g i n g t h e h a r m o n i c c o n t e x t c r e a t e d b y t h e f i r s t s i x c h o r d s . T h et a r g e t a n d i t s f l a n k i n g c h o r d w e r e a l w a y s h e l d c o n s t a n t . I n o n ec o n t e x t , t h e l a st c h o r d a c t e d a s a n h a r m o n i c a l l y s t a b l e t o n i c c h o r d ,p a r t o f a n a u t h e n t i c c a d e n c e ( V - I ) . I n t h e o t h e r c o n t e x t , t h e l a s tc h o r d t o o k t h e f o r m o f a l e s s s t a b l e f o u r t h h a r m o n i c d e g r e ef o l l o w i n g a n a u t h e n t i c c a d e n c e ( I - IV ) . P a r t i ci p a n t s w e r e f a s t e r i nt h e i r i n t o n a t i o n j u d g m e n t o f t h e t a r g e t c h o r d w h e n i t a c t e d a s at o n ic c h o rd . S i m u l a t i o n s w e r e p e rf o r m e d w i t h M U S A C T u s i n g t h ef i r st s e v e n c h o r d s o f t h e s e q u e n c e s , a n d t h e a c t i v a t i o n o f t h e t a rg e tc h o r d u n i t w a s r e a d o f f ( B i g a n d e t a l ., 1 9 9 9 ) . T h e t a r g e t c h o r d u n i tr e c e i v e d s t r o n g e r a c t i v a t i o n i n t h e f o r m e r c o n t e x t ( i . e . , w h e n t h e

4 The m odel does not deny th e structural importance o f harmonic voi celeading (i.e. , the melodic motion between the tones of successive chords)and chordal position (open vs. close) in W estern music theory. It postulatesthat the introduction of these factors is not critical to account for howmusical knowledge may be mentally represented and how this knowledge,once activated b y a give n musical context, may influence the processing o ftonal structures. Empirical studies that have manipulated v oice leading,chordal position, or the harmonic spectnun of the sound (e.g., piano-likesound vs. pure tones) suggest a weak influence of these factors on chordprocessing that is overridden b y harmonic relationships (Bigand, Tillmann,Manderlier, & Poulain, 2000; Rosner & Narmour, 1992; Stoeckig, 1990).


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IMPLICIT LEARNING OF T ONALITY 891t a rge t ac ted as a s t ab le ton ic chord) than in the l a t t e r con tex t . Thea c t i v a ti o n p a tt e r n th u s m i m i c s h u m a n p e r f o r m a n c e a n d m a n a g e s t oaccoun t fo r g loba l con tex t p r iming e f fec t s . O ther r esea rch ind i -c a t e s t h at M U S A C T a l s o a c c o u n ts f o r p r i m i n g e f f e c ts a t s e v e r a ll eve l s o f g loba l mus ica l con tex t (B igand e t a l . , 1999) o r tha t occurw h e n b o t h g l o b a l a n d l o c a l c o n t e x t a r e f a c t o r i a l ly m a n i p u l a t e d(T i l lmann , B igand , & P inean , 1998) .

M U S A C T h i g h l i g h t s a c r u c i a l i s s u e o f W e s t e r n m u s i c : w h e t h e rt h e r e l a t i o n s b e t w e e n c h o r d s a r e d r i v e n b y s i m i l a r it i e s b a s e d o na c o u s t i c p r o p e r ti e s o f t o n e s o r b y i m p l i c i t k n o w l e d g e o f c u l t u r a lconven t ions and usage ( see Bigand e t a l . , 1996 , and Pamcut t ,1989; fo r a d i scuss ion) . MU SA CT d isen tang les these two fac to r sb y c h a r t in g t h e t i m e c o u r s e o f b o t t o m - u p a n d t o p - d o w n i n f l u e n c es .I t p red ic t s tha t the ac t iva t ion pa t t e rn re f lec t s bo t tom-up in f luen cesa t ea r ly ac t iva t ion cyc les , whereas top-down in f luences a re p re -d o m i n a n t w h e n t h e m o d e l h a s e n o u g h t i m e t o r e a c h e q u i li b r i u m .Tekm an and B harucha (1998) t e s ted th i s p red ic t ion by p i t t ingshared tones aga ins t conven t iona l r e la tedness . Two types o f t a rge tw e r e s e l e ct e d : O n e w a s p s y c h o a c o u s t i c a l l y m o r e s i m i l a r t o t h ep r i m e , t h e o t h e r m o r e c l o s e l y r e l a t e d o n t h e b a s i s o f h a r m o n i cc o n v e n t io n . F o r e x a m p l e , a C - m a j o r p r i m e s h a r e s a t o n e w i t h a nE - m a j o r t a r g e t b u t d o e s n o t s h a r e a t o n e w i t h a D - m a j o r ta r g e t; y e tD m a j o r i s m o r e c l o s e l y r e l a t e d to t h e p r i m e i n c o n v e n t i o n a l us a g e .P r i m i n g r e s u l ts r e v e a l e d f a c i l i t a t io n f o r p s y c h o a c o u s t i c a l l y s i m i la rta rge t s when they fo l lowed a f te r a shor t (50 ms) s t imulus onse ta s y n c h r o n y ( S O A ) a n d f a c i li t a t io n f o r c o n v e n t i o n a l l y r e la t e d t a r -g e t s a f t e r a lo n g e r S O A ( 5 0 0 m s o r l o n g e r) . A l t h o u g h b o t h p s y -c h o a c o u s t i c s i m i l a r i t y a n d c o n v e n t i o n a l r e l a te d n e s s d r i v e p r i m i n g ,t h e i n f l u e n c e o f t h e f o r m e r i s s h o r t l iv e d , p r e c i s e l y a s p r e d i c t e d b yt h e t e m p o r a l c o u r s e o f a c t iv a t i o n i n t h e M U S A C T m o d e l .

A s o r i g i n a l l y c o n c e i v e d , th e m o d e l w a s b a s e d o n m u s i c t h e o -re t i c cons t ra in t s ; ne i the r the conn ec t ions nor the i r w e igh ts r esu l tedf rom a l ea rn ing p rocess . In th i s r e spec t , the mode l r ep resen ted thei d e a l i z e d e n d s t a te o f a n i m p l i c i t le a r n i n g p r o c es s . T o b e c o m p e l -l i n g , a c o g n i t i v e m o d e l o f W e s t e r n h a r m o n y s h o u l d s i m u l a t e t h ei n t e r n a l i z a t i o n o f W e s t e r n p i t c h r e g u l a r i t i e s b y m e r e e x p o s u r e ,a l l o w i n g t h e c o n n e c t i o n w e i g h t s t o a d a p t t o t h e m u s i c a l e n v i ro n -m e n t . I t h a s b e e n s u g g e s t e d th a t M U S A C T i d e a l l y c a n i n t e r n a li z ethese regu la r i t i e s by pass ive se l f -o rgan iza t ion (Bharucha , 1991,1992) . In the nex t sec t ion , we de sc r ibe how th i s can happen .S p e c i f i c a l l y , w e s h o w h o w a h i e r a r c h i c al S O M m a n a g e s t o l e a r nt h e W e s t e r n p i t c h r e g u l a r i ti e s , c o m p a r a b l e t o t h o s e i n M U S A C T .T h e l e a r n e d m o d e l w i l l t he n b e t e s t e d f o r i ts c a p a c i t y t o s i m u l a t ea v a r i e t y o f e m p i r i c a l d a t a c o n c e r n in g t h e p e r c e i v e d r e l a ti o n s h i p sb e t w e e n a n d a m o n g t o n e s , c h o r d s, a n d k e y s ( c f. S I M U L A T I O N SO F E M P I R I C A L D A T A s e c t i o n ) .

A M O D E L T H A T L E A R N S W E S T E R N H A R M O N YB Y S E L F - O R G A N I Z A T I O NG e n e r a l P r i n c ip l e s o f S O M s

I n c o n n e c t i o n i st m o d e l s , u n s u p e r v i s e d l e a r n in g a l g o r i t h m s e x -t rac t s t a t i s t i ca l r egu la r i t i e s and enco de even ts tha t occur o f tentoge the r (Grossberg , 1970, 1976; Koho nen , 1995 ; Rume lhar t &Zipse r , 1985 ; Von der Malsberg , 1973). These a lgor i thm s a re we l ls u i t e d to m u s i c p e r c e p t i o n b e c a u s e t h e o r g a n i z a t i o n o f c h o r d s o rt o n a l i ti e s p r e s u m a b l y o c c u r s w i t h o u t s u p e r v i s io n . O n e u n s u p e r -v i s e d l e a r n in g a l g o r i th m i s t h e S O M p r o p o s e d b y K o h n n e n ( 1 9 9 5 ) .

I t c r e a te s t o p o g r a p h i c m a p p i n g s b e t w e e n t h e i n p u t d a t a a n d n e u r a lne t un i t s o f a map . Fo r two s im i la r inpu t pa t t e rns , the ma p un i t st h a t r e s p o n d m a x i m a l l y a r e l o c a t e d n e a r e a c h o t h er . T h i s c o n f o r m sto p r inc ip les o f co r t i ca l in fo rmat ion p rocess ing , such as the fo r -mar ion o f spa t i a l o rde r ing in sensory p rocess ing a reas ( i .e . , so -matosensory , v i s ion , and aud i t ion) . In the p r imary v i sua l co r tex ,the o r ien ta t ion o f s t imul i to which ce l l s r e spond bes t changes in anorder ly fash ion ac ross the con tex t : Nearby ce l l s r e spond bes t tos imi la r o r ien ta tions (Hube l & W iese l , 1962). T he aud i to ry cor texd i s p l a y s a t o n o t o p i c o r g a n i z a t io n i n w h i c h c e l l s r e s p o n d i n g b e s t t od i f fe ren t f requenc ies a re a r ranged in an o rder ly fash ion (Brugg e &Rea le , 1985; Wess ing er , Buono core , Kussm aul , & Mangun , 1997) .I n t h e a u d i t o r y s y s t e m , t o n o t o p i c o r g a n i z a t i o n c a n b e f o u n d a ta lmos t a l l ma jo r s t ages o f p rocess ing ( i .e . , inner ea r , aud i to rynerve , coch lea r nuc leus , aud i to ry cor tex) .

S O M i s b a s e d o n c o m p e t i t i v e l e a r n in g , a n a l g o r i t h m f o r d a t a -d r iven se l f -o rgan ized l ea rn ing . Wi th th i s a lgor i thm, the neura l ne tun i t s g radua l ly becom e sens i t ive to d i f fe ren t inpu t s t imul i o rca tegor ies (Rume lhar t & Z ipse r , 1985). T he spec ia l i za t ion occursb y c o m p e t i t i o n a m o n g t h e u n i t s. W h e n a n i n p u t a r r i v e s, t h e u n i ttha t i s bes t ab le to represen t it w ins the comp et i t ion . The winn ingun i t i s then a l lowed to l ea rn the represen ta t ion o f th i s inpu t evenbe t te r , a s i s desc r ibed la te r . The un i t ' s r e sponse wi l l be subse -quen t ly s t ronger fo r th i s same inpu t pa t t e rn and weaker fo r o the rpa t te rns . In a s im i la r way , o the r un i t s l ea rn to spec ia l i ze to respondto o the r inp u t pa t t e rns .

T h e c o m p e t i t i v e l e ar n i n g a l g o r i t h m c a n b e g e n e r a l iz e d , i f t h e r eex i s t s a spa t i a l l ayou t o f the un i t s . On an SOM, fo r example , theun i t s a re loca ted on a d i sc re te l a t ti ce . The ge nera l i za t ion impl iestha t no t on ly does the w inn ing un i t l ea rn , bu t i t s ne ighb or un i t s a rea l so a l lowe d to l ea rn . Ne ig hbor un i t s wi l l g radua l ly spec ia l i ze torepresen t s imi la r inpu ts and the represen ta t ion become s o rdered onthe map . Af te r l ea rn ing , each un i t i s spec ia l i zed to de tec t a pa r -t i cu la r inpu t pa t t e rn , and a topograph ic o rgan iza t ion o f the inpu tda ta can be d i scov ered on the map , such tha t s im i la r inpu t pa t t e rnsac t iva te nea rby map un i t s .

In such a neura l ne twork , the inpu t l aye r and the two-d i m e n s i o n a l m a p l a y e r a r e f u l l y i n te r c o n n e c t e d b y s y n a p s e s . B e -fo re l ea rn ing , the connec t ion s t reng ths ( the w e igh ts , w) a re in i t i a l -i zed to random va lues . When a s t imulus i s p resen ted , the inpu tun i t s , i , tha t a re tuned to the fea tu res o f the s t imulus , a re ac t iva ted .These ac t iva t ions o f the inpu t un i t s a ( i ) spread to the m ap un i t s , j ,by the connec ted l inks , w ( i , j ) . E a c h u n i t o f t h e m a p l a y e r , j ,accumula tes the ac t iva t ion i t r ece ives f rom the inpu t un i t s . Thea c t i v at i o n o f e a c h m a p u n i t a ( 1 ) i s g i v e n b y :

a ( j ) = Y ~ a ( i ) * w ( i , j ) . (2 )iThe u n i t j w i th the h ighes t ac t iva t ion de f ines the winn ing un i t .Dur ing the l ea rn ing phase , the assoc ia ted we igh t vec to r s o f thewinn ing un i t w and those wi th in a ne ighborho od se t N a re upda ted .T h e w e i g h t s o f u n i t s o u t s i d e th e n e i g h b o r h o o d s e t a r e k e p t c o n -s tan t . Learn ing cons i s t s o f upda t ing the we igh ts f eed ing in to thewinn ing un i t and i t s ne ighbors wi th the fo l lowing a lgor i thm (Ko-honen, 1995):

w < , ) + " ) ) ( t) * a ( t ) ( i )w(,+ 1) = (3 )w(0 + ))(,).a(,) (i)ll '


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892 TILLMANN, BHARUCHA, AND BIGANDwhere w~t+l~ epresen t s the we igh t vec to r a t t ime t + 1 , w~o a t t imet , and a~o epresen t s the l ea rn ing ra te . Th is l ea rn ing ru le re in fo rcesl i n k s c o m i n g f r o m a c t i v e i n p u t u n i t s a n d w e a k e n s l i n k s c o m i n gf rom inac t ive inpu t un i t s . In o the r words , i t moves the we igh tvec to rs c lose r to the inpu t vec to r , mak ing the winn ing un i t and i t sn e i g h b o r s m o r e l i k e l y t o w i n t h e c o m p e t i t i o n w h e n t h i s i n p u t o rone s imi la r to i t i s p resen ted aga in .T h e n e i g h b o r h o o d N i s s e t t o b e w i d e a t t h e b e g i n n i n g o fl e a r n i n g . D u r i n g l e a r n i n g , i t d e c r e a s e s m o n o t o n i c a l l y u n t i l i tc o n s i s t s o f t h e w i n n i n g u n i t a l o n e . A s l e a r n i n g b e g i n s , a l a r g en e i g h b o r h o o d a l l o w s a g l o b a l o r g a n iz a t i o n t o e m e r g e . W i t h as m a l l e r n e i g h b o r h o o d r a d i u s , th e u n i t s b e c o m e a d a p t e d t o t h ei n d i v i d u a l p a t t e r n s a n d i t s c l o s e r e l a t i v e s , a n d a l o c a l o r g a n i -z a t i o n e m e r g e s .

B e f o r e l e a r n i n g , t h e r e i s n o p a r t i c u l a r o r g a n i z a t i o n a m o n g t h em a p u n i t s . W h e n t h e n e t i s t r a i n e d b y r e p e a t e d p r e s e n t a t i o n o ft h e i n p u t d a t a , i t b e g i n s t o s e l f - o r g a n i z e . A t o p o g r a p h i c p a t t e r nb e g i n s t o a p p e a r , s u c h t h a t u n i t s t h a t a r e t o p o g r a p h i c a l l y c l o s ei n t h e a r r a y w i l l b e a c t i v a t e d b y s i m i l a r i n p u t s t i m u l i . S O M c a nb e c o n c e i v e d o f w i t h o n e m a p l a y e r o r b e a d a p t e d to m u l t i l a y e rh i e r a r c h ic a l s e lf - o r g a n i z in g m a p s ( H S O M s ; L a m p i n e n & O j a ,1992).

S i m u l a t in g I m p l i c i t L e a rn i n g o f W e s t e r nM u s i c W i t h S O M s

F o r t h e s i m u l a t io n s o f i m p l i c i t l e a r n in g o f W e s t e r n h a r m o n y ,w e d e f i n e d a h i e r a r c h i c a l S O M . I t s s t r u c t u r e i s s i m i l a r t om o d e l s o f l a n g u a g e p e r c e p t i o n ( M c C l e l l a n d & R u m e l h a r t ,1 9 8 1 ; E l m a n & M c C l e l l a n d , 1 9 8 4 ) a n d o f m u s i c p e r c e p t i o n( B h a r u c h a , 1 9 8 7 b ). T h e i n p u t l a y e r is t u n e d t o o c t a v e e q u i v a -l e n t p i t c h c l a s s e s . T h e s e c o n d a n d t h i r d l a y e r s w i l l l e a r n t os p e c i a l i z e in t h e d e t e c t i o n o f c h o r d s a n d k e y s , r e s p e c t i v e l y . T h eh i e r a r c h i c a l m a p i s i n s p i r e d b y f e a t u r e d e t e c t o r s f o u n d i n t h eb r a i n , w i t h e l e m e n t a r y f e a t u r e d e t e c t o r s a t t h e s e n s o r y p e r i p h -e r y ( e . g . , f r e q u e n c y ) , a n d m o r e a b s t r a c t f e a t u r e d e t e c t o r s i n t h ep r i m a r y a u d i t o r y c o r t e x , f o r e x a m p l e , p i t c h ( P a n t e v , H o k e ,L i i t k e n h 6 n e r , & L e h n e r t z , 1 9 8 9 ) o r c o n t o u r ( W e i n b e r g e r &M c K e n n a , 1 9 8 8 ). T h e a b s t r a c t p i t c h - c l a s s c o d i n g w a s b a s e d o nt h e d e m o n s t r a t e d a b i l i t y o f n e u r a l n e t m o d e l s t o e x t r a c t p i t c hf r o m f r e q u e n c y ( M . A . C o h e n , G r o s s b e r g & W y s e , 1 9 9 5 ; S a n o& J e n k i n s , 1 9 9 1 ) a n d t o l e a r n o c t a v e e q u i v a l e n t p i t c h c l a s s e s( B h a r u c h a & M e n c l , 1 9 9 6 ) . M . A . C o h e n e t a l . ' s ( 1 9 9 5 ) m o d e lt r a n s f o r m s a s p e c t r a l re p r e s e n t a t i o n o f a n a c o u s t i c s o u r c e i n t o as p a t i a l d i s t r i b u t i o n o f p i t c h s t r e n g t h s . I n B h a r u c h a a n d M e n c l ' s( 1 9 9 6 ) m o d e l , o c t a v e c a t e g o r i e s w i t h a b s t r a c t p i t c h c l a s s e s a r el e a r n e d b y s e l f - o r g a n i z a t i o n o p e r a t i n g o n a s p e c t r a l r e p r e s e n -t a t i o n . O u r p r e s e n t m o d e l c a n b e c o n c e i v e d o f a s s u b s e q u e n t t ot h e s e p h a s e s o f a u d i t o r y p r e p r o c e s s i n g ( c f . B h a r u c h a , 1 9 91 , f o ra n e x p a n s i o n o f B h a r u c h a , 1 9 8 7 b ) .

T h e s e l f - o r g a n i z i n g a l g o r i t h m p e r m i t s t h e f o r m a t i o n o f u n i tsr e p r e s e n t i n g e v e n t s t h a t a r e f r e q u e n t l y a s s o c i a t e d . I n m u s i c , t h i sa s s o c i a t i o n c o n s i s t s o f e it h e r t h e s i m u l t a n e o u s o c c u r r e n c e o ft o n e s ( w h i c h d e f i n e a c h o r d ) o r t h e t e m p o r a l p r o x i m i t y o fe v e n t s ( s u c h a s a r p e g g i a t e d t o n e s o f a c h o r d o r t h e c h o r d sf o r m i n g a k e y ) . I n t h e p r o p o s e d m o d e l , s e l f - o rg a n i z a t i o n l e a d st o a h i e r a r c h i c a l e n c o d i n g i n w h i c h t o n e s o c c u r r i n g t o g e t h e r a r er e p r e s e n t e d b y c h o r d u n i t s a n d , s i m i l a r l y , c h o r d s o c c u r r i n gt o g e t h e r a r e r e p r e s e n t e d b y k e y u n i t s . T o m o d e l t h e i n f l u e n c e o f

k n o w l e d g e o n p e r c e p t i o n , t h e n e u r a l n e t s t r u c t u r e r e s u l t i n gf r o m l e a r n i n g s i m u l a t i o n s i s t h e n u s e d w i t h a s p r e a d i n g a c t i v a -t i o n m e c h a n i s m . A f t e r th e p r e s e n t a t i o n o f a s t i m u l u s , a c t i v a t i o nr e v e r b e r a t e s b e t w e e n t h e t h r e e l a y e r s u n t i l a n e q u i l i b r i u m i sr e a c h e d . T h e u s e o f a n e u r a l n e t s t r u c t u r e a s a r e v e r b e r a t i o ns y s t e m i m p l i e s t w o c o n s t r a i n t s .

The f i r s t cons t ra in t dea l s wi th the s imula t ion o f top-down in -f luences and has i t s o r ig in in p rev iou s ly p ropo sed sp read ing ac t i -va t ion mode ls ( i.e . , Bharucha , 1987b ; Mc Cle l land & Rum elhar t ,1981) . Thre e - layer mode ls o f word reco gn i t ion ( i .e . , f ea tu res ,l e t t e r s , words ) and o f mus ica l knowledge represen ta t ion ( i .e . ,tones , chords , keys ) manage to s imula te top-down in f luences byfavor ing the sp read o f ac t iva t ion be tween the h igher l eve l s o frepresen ta t ion , namely the second and the th i rd l ayers . In thew o r d - r e c o g n i t io n m o d e l , t h e i n t e ra c t i v e p r o c e ss i n g c o n c e r n s o n l ythe l e t t e r and word l eve l s , w i thou t f eedb ack to the fea tu re l eve l . InM U S A C T , t h e w e i g h t s o f t h e c o n n e ct i o n s b e t w e e n c h o r d a n d k e ylayers a re s t ronger than be tween ton e and cho rd l ayers , y ie ld ing anin i t i a l in f luence o f the tone inpu t , fo l lowed by a s t rong in f luenceo f t h e t w o a b s t r ac t l a y e r s a n d o n l y a w e a k a d d i t i o n al i n fl u e n c e o fthe tone l ayer . A s im i la r cons t ra in t i s implem ented in the s imula -t ions p resen ted l a te r .

The second cons t ra in t r e su l t s f rom un i t s on the SO M tha t a re no tspec ia l i zed in the de tec t ion o f a s t imulus a f te r l ea rn ing . Becausereverbera t ion shou ld occu r on ly be tween s pec ia l i zed un it s , theconnec t ions feed in g in to unspec ia l i zed un i t s ( e .g . , un it s tha t do no twin fo r any o f the t r a in ing pa t te rns ) a re t r ea ted by a p run ingm e c h a n i s m . P r u n i n g i s a p r o c e d u r e s u p p o r t e d b y t h e d e v e l o p m e n -ta i neura l p r inc ip le tha t connec t ions w eaken f rom d i suse , and i t i sg e n e r a l l y u s e d i n s u p e r v i s e d le a r n i n g b y w e i g h t d e c a y o r w e i g h te l imina t ion (Haykin , 1994 ; LeCnn , Denker , & So l la , 1990; Se -t iono, 1997).

Four l ea rn ing s imula t ions respec t ing bo th cons t ra in t s a re p re -s e n te d . A l l s i m u l a t i o n s w e r e b a s e d o n t h e s a m e n e t w o r k a r c h i t e c-tu re tha t l ea rned wi th the se l f -o rgan iz in g a lgor i thm. The ne tw orkwas t r a ined wi th e i the r s imple ha rmonic mate r ia l ( c f . Learn ingS i m u l a t i o n s W i t h S i m p l e H a r m o n i c M a t e r i a l s e c t i o n ) o r m o r erea l i s t i c chord sequences (o f . Learn ing S imula t ions Wi th Shor tChord Sequences sec t ion) . The inpu t was de f ined by e i the r as p a r s e c o d i n g o r a p s y c h o a c o u s t i c a ll y r i c h e r c o d i n g s c h e m e(sparse inpu t cod ing in s imula t ions SIC-1 and SIC -2 vs . r ich inpu tcod ing in s imula t ions RIC-1 and RIC-2) . We s ta r t by cons ide r ingl e a r n i n g b a s e d o n s i m p l e h a r m o n i c m a t e r i a l a n d a s p a r s e i n p u tc o d i n g .

Learning Simulations With Simple Harmonic MaterialSIC-1

Network architecture. A th ree - layer h ie ra rch ica l sys tem wasdef ined as fo l lows : The inpu t l aye r cons i s ted o f 12 un i t s , thes e c o n d l a y e r w a s a m a p o f 3 6 u n i t s , a n d t h e t h i rd l a y e r w a s a m a pof 16 un i t s . The inpu t un i t s were tuned to the 12 chromat ic sca letones , r ep resen t ing oc tave equ iva len t p i t ch -c lass de tec to r s . Theun i t s o f the f ' tr st and secon d layers were fu l ly in te rconnec ted w i tha connec t ion mat r ix . The un i t s o f the second and th i rd l ayers weref u l l y i n t e rc o n n e c t e d w i t h a s e c o n d c o n n e c t i o n m a t r i x . A l l c o n n e c -


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IMPLICIT LEARNING OF TONALITY 893tions were bidirectional, and their strengths were initialized torandom values between 0 and 1 before learning.5

Input coding. During training, chords (consisting of threetones) were presented to the input layer (see later discussion). Inthe sparse input coding, the presence of tones was coded directly:A tone un it was activated if the tone to which it was tuned occurredin the stimulus, and was set to 0 otherwise. By convention, the 12tone uni ts were ordered as follows: A-A $-B-C-C ~-D-D ~-E-F-F ~-G-G~. The three component tones of a C-major chord wererepresented by the following vector: {0-0-0-1-0-0-0-1-0-0-1-0}.

Training. The network was allowed to self-organize as indi-cated previously (see Equations 2 and 3). At the outset of training,the neighborhood radius was set to 5 and 3.6 for chord and keylayers, respectively (for both maps the neighborhood was definedby euclidean distances between units). During training, the neigh-borhood radius decreased until reaching 0, at which point only thewinning uni t learned. The learning rate ~1 was decreased over thecourse of learning. Whenever the neighborhood radius decreased,the learning rate was divided by two. In the convergence phase(i.e., when only the winning unit learned), the learning rate de-creased over the num ber of training cycles (one training cycleconsisted of the presentation of the whole set of stimuli), c, asfollows:

n ' = l l ( c + l/n). (4)Training consisted of two phases. In the first tra ining phase, the

second layer was trained with sets of three tones (e.g., C-E-G)corresponding to the 12 major and 12 minor chords of Westernmusic. Each triplet of tones was presented separately to the inputlayer. In this phase, units of the second layer learned to detectchords. In the second training phase, the third layer was trainedwith 12 sets of 6 chords presented to the tone layer. One set ofchords consisted of 3 minor and 3 major chords of a given key(e.g., the 3 major chords C, F, and G and the 3 minor chords d, e,and a, all of which belong to the C-major key). The six chordswere presented individually o the tone layer. For each input chord,the activation of the winning chord unit in the second layer(referred to by the index b) was stored in memory without decayuntil the end of the presentation of the chord set. The pattern ofindices, b, defined the input for the training of the third layer. Inthis second phase, the units of the third layer learned to detectkeys. In both phases the training patterns were presented in randomorder during each training cycle. 6

Cal ibra t ion o f the maps . During training, units became spe-cialized in the detection of chords in the second layer and in thedetection of sets of chords (keys) in the third layer. For bothtraining phases, the weight changes decreased over the trainingcycles and with decreasing neighborhood. When weights con-verged to stationary values, the maps were calibrated by namingeach winning unit after the stimulus for which it won. For exam-ple, the unit that won for the three tones C-E-G was called theC-major chord unit. After training, the average quantization error(i.e., the mean of the euclidean distances between each input vectorand the weight vector of its corresponding winning unit) was lessthan .01 for each map.

Topograph ic organ iza tion and connec t ion matr ices . The cal -ibrat ion phase revealed a topographic organization of both maps.After training, chords that share component tones were representedby neighboring units in the second layer (Figure 4, left). Chords

c# E g# C eC# (3# c Gf gF d a,# a# D#a F# d#

A f# D b B

EA

DG

B F # C #

G#

D#C F A#

Figure 4. The calibration maps of the second layer (left) and the thirdlayer (right) of the network SIC-1. For the second layer, winning units arelabeled by chord names (minorchords in lowercase letters, major chords inuppercase letters). For the third layer, names of winning units indicatemajor keys.

that do not share tones were not represented by neighboring units.In the third layer (see Figure 4, right), key units were organized ina way that reproduces the topology of the cycle of fifths (also seeFigure 1). Keys sharing chords and tones were represented close toeach other on the map. The quality of the learned representationwas tested for both maps. In a coherent learning solution, thedistance between specialized units on the map should reflect therelationships between the input stimuli. To address this issue, theeuclidean distances between all specialized units E and the corre-lations between all input vectors C were calculated separately foreach map. The correlation between the two indices C and E wasr(298) = .69 for the chord layer and r(76) = .95 for the key layer.The learned representations on the two-dimensionalmaps reflectedthe regularities within the input spaces.

After training, each tone uni t was strongly linked to six winningunits in the chord layer, and each chord uni t was strongly linked tothree winning units in the key layer. The connections feeding intounspecialized units were eliminated during the pruning phase. Inthe prun ing procedure, all stimuli (chords and sets of chords) werepresented in random order to the input layer. After the presentationof each stimulus, the weights feeding into losing units were de-creased, and the weights feeding into the winning unit were rein-forced by a parameter varying as a function of the number ofwinning units in the layer. After pruning, only connections feedinginto winning units remained.

Feed- forward and reverbera t ion sys tem. After learning, themodel was used both as a feed-forward system and as a reverber-

5 For chord and key layers, the number of units on the maps were setsomewhat relative to the number of stimuli in the training set. In supple-mentary simulations with SIC-l, the second and third layers were eachdefined by a map of 36 units. The outcome suggested that the basic spatialstructure learned by the key map holds for maps of both sizes (i.e., 16and 36 units). In the input layer, the invariant pitch-class coding leads usto conceive melodies as sets of scale degrees in reference to the tonic andto consider chords as being invuriantunder inversions.6 The numberof necessary raining cycles varied from 200 to 1,000 overthe four learning simulations.At the outset of training, the learningratewas set to .5 and .1 for chord and key layers, respectively, n the reportedsimulations.Other simulationswith SIC-1 and SIC-2 showed that a learn-ing rate of .2 for both layers led to similar results.


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894 TILLMANN, BHARUCHA, AND BIGANDation system. Reverberation was defined by phasic activationspreading between the uni ts by the weights until equilibrium wasreached. Equil ibrium was defined to be reached when phasicactivat ion was less than a threshold of .005 for each unit (cf.Bharucha, 1987b). Because of the normaliza tion process (seeEquation 3), the learned weights of the two matrices generatedreverberat ing phasic activation that exceeded the bounds withinwhich equilibration occurs. 7 The phasic activation reverberatingwithin each of the matrices was thus scaled by a parameter de-signed to favor reverberat ion in the chord-key matrix (see the firstconstra int discussed earlier). This parameter was set first for thechord-key matrix by placing it within the boundary conditions forequilibration for this matrix alone. The corresponding parameterfor the tone-chord matrix was then titrated so that the system as awhole reached equilibrium within approximately 100 reverbera-tion cycles.

When used as a feed-forward system, the stimulated tone unitssend activation toward chord units, which in turn send activationtoward key units. This feed-forward activation reflects bottom-upinformation (tones present in the stimuli) and does not incorporatetop-down influences. Figure 5 (left) represents activation levels forthe 12 major chord units and the key units. When the three tonesof a C-major chord are presented, the unit in the second layer thatbecomes most active is the one specialized in the detect ion of theC-major chord, followed by the units specialized for major chordssharing one of the three tones. Units of chords that do not sharecomponent tones were not activated. A similar pattern was foundfor the key layer: The most activated unit was the C-major key,followed by the F-major key, the G-major key, and the other keyslinked to one of the activated chord units. The key of F~ was notactivated, because it does not share any tones with the C-majorchord.

When used as a reverberation system, the pattern of activationchanges qualitatively because of the top-down influence oflearned, schematic structures (see Figure 5, right). In the chordlayer, the difference between bottom-up and top-down activationwas manifest in the comparison of activations for the E- andD-major chords. When the three tones of a C-major chord werepresented, feed-forward activation caused the E-major chord uni tto be more strongly activated than the D-major chord unit. Duringreverberation, however, the D-major chord unit overtook theE-major chord unit by virtue of its closer harmonic relationship to

[ eedforwardsystem [ [reverberation ystem [majorchord units

F# C#G#D#A#F C G D A E B F# C# G# D# A# F C G D A E Bkey units

F# C#G # D#A# F C G D A E B F# C#G # D#A# F C G D A E B

Figure 5. Activationsof major chord units and key units n the networkSIC-1 after the presentationof a C-major chord for bottom-up activationonly (left) and for when the network has reached equilibrium(right).

C major as mediated by the key units. Across all major chord units,activation decreased monotonically with distance around the cycleof fifths from C, with the lowest for F~ and B. In the third layer,the most activated key unit was C major (for which the presentedchord is the tonic), and activation decreased monotonically withdistance along the cycle of fifths and, equivalently, with distancealong the topographic key map.R I C - 1

The sparse input coding considered only the presence of thecomponent tones of the chords. However, each tone contains acomplex harmonic spectntm, which may influence the perceptionof harmony (Helmholtz, 1885/1954; Parncutt, 1989; Terhardt,1974). Consequently, a richer input coding scheme was used in thefollowing learning simulation (RIC-1). This coding, based on thepsychoacoustic theory of Pamcutt (1988), was used by Leman(1995) in his simple auditory model. In Parncut t's theory, subhar-monic virtual pitches are assigned to each component of a complextone, and the frequency of the most commonly generated subhar-monic determines the perceived pitch. Harmonic relationshipsbetween two chords are estimated by two indices based on virtua lpitch cues: pitch salience (intensity of a virtual pitch cue) and pitchcommonality (number of cues shared by two pitch vectors). Fol-lowing Parncutt (1988) and Leman (1995), a chord was coded asa pitch-class vector whose values are weighted sums of the sub-harmonics corresponding to those pitch classes. For example, aC-major chord was represented by the following pattern: {.85-.2-0-1.83-.1-.45-.33-1.1-.7-.25-1-.33} (representing the tone uni tsA-AIC-B-C-C~-D-D~-E-F-F~-G-G~).A learning simulation, asdescribed for SIC-l, was run again with this richer input coding.

Topograph ic organ iza t ion and connec t ion matr ices . The richinput coding influenced only the topographic organization of thesecond layer and the matrix connecting the tone layer to the chordlayer. Twenty-four units in the second layer were specialized in thedetection of the chords. As in SIC-l , neighboring units had tonesin common. However, the richer coding yielded a more globalorganization: The map was globally divided in two (upper andlower halves), each half corresponding to chords from one side ofthe cycle of fifths (Figure 6, left). This outcome was in accord withPetroni and Tricarico (1997), who analyzed the influence of inputcoding on the location of winning units. They found that, incontrast to a simple local representation (that indicates which pitchis activated in the triad), adding the subharmonic sum of the tonesproduced an organization close to the cycle of fifths. In oursimulations, 11 of the 12 tone units were linked to a winning chordunit, even if some of the links remained small. This pattern ofconnec tions reflected the new input coding in which 1 of the 12subharmonics was not present in the coding of one chord: Forexample, in the C-major chord the tone unit B was not activated

7 A reverberationsystem with two matrices reaches equilibrium f twoconditions are satisfied. First, each matrix used separately as a reverbera-tion system reaches equilibriumon its own. This constraint s satisfied ifthe maximumeigenvalueof the matrix is less than l . Second, the mean ofweights over both matrices is smail enough to permit an equilibrium orreverberation in the total system. This constraint s satisfied if the phasicactivation sent back from a layer in a second cycle is smaller than thephasic activationsent in the first cycle.


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IMPLICIT LEARNING OF TONALITY 895

E c# f#e AC a dc F

~q# f a#B g# 2#

D bmajor chord units

~,# g F# C#G# D#A# F C G D A E B

F# d# .~

F # c # G # r ~ A # F C 6 D A E B

Figure 6. The calibrationmap of the second ayer of the network RIC-1(left). Winning units are labeled by chord names (minor chords in lower-case letters, major chords in uppercase letters). Activationsof major chordunits and key units after the presentationof a C-majorchord for bottom-upactivation only (right).

(see prior discussion). The first connection matrix thus developedquite differently in this simulation from the way it did in SIC-1.Interestingly, these differences had no inf luence on the formationof the second matrix. The second matrix and the topographicorganization of the third layer had the same characteristics as inSIC-1.Feed-forward and reverberation system. The feed-forward ac-tivation for major chord and key units revealed a slightly differentpattern from SIC-1 (see Figure 6, right). In the chord layer, thesedifferences were observed especially at the A t - and D-major units,whose activations were higher than in SIC-1 although still lowerthan the G~- or A-major units. The global shape of the curve wasthus preserved with a stronger feed-forward activation for theE-major chord than for the D-major chord. The key layer differedin two aspects from the feed-forward simulation in SIC-1. First,activation decreased with distance around the cycle of fifths,because the At key was more strongly activated than the D~ key.Second, the difference between the dominan t key and the sub-dominant key disappeared.

For both layers, subharmonic information changed bottom-upprofiles so that they were closer to the shape of top-down profilesthan with sparse coded inputs (see Figure 5, left). With reverber-ation, however, the act ivation profiles of chord and key layerswere analogous for rich and sparse coding (see Figure 5, right).The comparison between SIC-1 and RIC-1 suggests that the as-ymptotic behavior of the reverberating model is not particularlysensitive to the richness of the input, even though the bottom-upeffects are. The top-down processes driven by the more abstractlearned knowledge of Western harmony impose a pattern of acti-vation that is similar regardless of whether the structure within theharmonic and subharmonic series is used by the system.

Learning Simulations With Short Chord SequencesIn simulations SIC-1 and RIC-1, the model was exposed to sets

of six chords presented randomly. This material may be viewed assomewhat artificial for two reasons. First, it did not mirror thetransition probabilities of chords in Western music. Second, eachof the chords was presented with equal salience. The material usedin the following simulations, SIC-2 and RIC-2, was more ecolog-

ically valid for learning the connection matrix between chord andkey units. Short chord sequences were presented to the network,and the activation of chord units decayed as the chords recededinto the past, simulating a decaying memory.

Sequences of seven chords were constructed by conforming tostandard harmonic root progressions (Piston, 1978) and statisticalchord distributions (Budge, 1943). The selection of chords wasrestricted to the following set of chords in a major key: I, ii, ii i, IV,V, and vi. Tonic chords (I) occur more often than dominant chords(V), followed by subdominant chords (IV), then vi, and ii, andleast often iii. Ten sequences were constructed by selecting chordsat random using a probability distribution based on the priorconstraints. The last chord in each sequence was the tonic chord ofthe key. Consistent with conventions of harmonic progression,each tonal sequence ended with a final V-I or IV-I cadence. Thesesequences were transposed to all 12 major keys, resulting in 120train ing sequences.

The chords of a sequence were presented to the input layer oneby one. For each input chord, the activation of the winning chordunit (referred to by the index b) was stored in memory until the endof the sequence and was decreased by a decay parameter. Thedecreasing activation pattern of these indexes b defined the inputfor the third layer. Simulations were run with the sparse inputcoding (SIC-2) and the rich input coding (RIC-2).

SIC-2Topographic organization and connection matrices. The new

training material influenced only the matrix linking chord to keyunits. In the calibration phase, the winning key uni ts were deter-mined by presenting sets of six chords (together forming a key) tothe network. Presenting the training sequences after learning cre-ated the highest activation in the key unit representing the tonickey of the sequence. The representation of the key units in the thirdlayer mirrored the cycle of fifths. As in the simulations RIC- 1 andSIC-1 with artificial tonal material, keys sharing chords and toneswere still represented close to each other on the map. Each winn ingkey uni t was linked to six chord units representing chords belong-ing to that key. The strengths of these connections mirrored thestatistical chord distributions of the stimuli: Links from the tonicchord unit were the strongest (.663), followed by those from thedominant chord (.542), the subdominant (.34), then vi (.296), i i(.24), and iii (.08). The strengths of the connections depended onthe functions of the chords in the corresponding key: The samechord (e.g., C major) had a stronger link to the key for which it isthe tonic chord (the C-major key) than to a key for which it is thesubdominant chord (the F-major key).Feed-forward an

Documents

Tillmann, B., Bharucha, J. J., & Bigand, E. (2000). Implicit Learning of Tonality- A Self-Organizing Approach. PSYCHOLOGICAL REVIEW-NEW YORK-, 107(4), 885-913