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J E A N N E B A M B E R G E R

T U R N I N G M U S I C T H E O R Y O N I T S E A R

D o w e h e a r w h a t w e se e ; D o w e s e e w h a t w e s a y ?

R e v i s it in g s o m e o f t h e h u n d r e d s o f c h il d r e n 's d r a w i n g s o f s i m p l e r h y t h m s

a n d m e l o d i e s t h a t I h a v e c o l l e c te d o v e r th e y e a r s , I f o u n d m y s e l f m a r v e l i n g

a l l o v e r a g a i n a t h o w i t is w e e v e r le a r n t o t u r n t h e c o n t i n u o u s f lo w o f o u r

o w n s i n g i n g o r o u r i n n e r , b o d i l y f e e l f o r c o n t i n u o u s , r h y t h m i c a c t i o n s

c l a p p i n g , b o u n c i n g a b a l l, s w i n g i n g o n t h e p a r k s w i n g - i n to s t a t ic ,

d i s c re t e desc r i p t i ons t ha t ho ld s ti ll t o be l ooke d a t ou t t he re . I con t i nue

t o b e f a s c i n a te d b y c h i l d r e n 's s p o n t a n e o u s i n v e n ti o n o f n o t a ti o n s b e c a u s e

t h e y s h o w t h e c o m p l e x i t y o f t h i s c o n c e p t u a l w o r k a n d t h e e v o l u t i o n o f

l e a m i n g i n v o lv e d

a s i t i s h a p p e n i n g

S o m e t i m e s th is c o m p l e x i t y e m e r g e s

b y c o m p a r i n g o n e c h i l d ' s w o r k w i th t h at o f a n o th e r ' s, a n d s o m e t i m e s

i t c a n b e s e e n in w a t c h i n g o n e c h i ld a s f r o m m o m e n t - t o - m o m e n t s h e

t r a n s fo r m s f o r h e r s e l f t h e v e r y m e a n i n g o f t h e p h e n o m e n a s h e i s w o r k i n g

wi th (Ba m berg er , 1991 a ) .

B u t i n l o o k i n g b a c k a t al l t h e s e d r a w i n g s , I w a s s t r u c k o n c e a g a i n b y

h o w e a s y i t i s t o m i s s t h e s e m a r v e l o u s t r a n s f o r m a t i o n s f r o m a c t i o n i n t o

d e s c r i p t i o n i f w e l i m i t o u r l o o k i n g a n d t a k e a s g i v e n s j u s t t h o s e k i n d s o f

e n t it ie s t h a t a r e s e l e c t e d b y o u r c o n v e n t i o n a l n o t a t i o n a l s y m b o l s . I n d e e d ,

t h e c h i l d r e n ' s d r a w i n g s r e v e a l f o r s c r u t i n y t h i n g s t h a t w e h a v e f o r g o t t e n

w e e v e r d id n o t k n o w . C a n y o u i m a g i n e w h a t i t w a s l i k e b e f o r e y o u l e a r n e d

h o w t o r e a d - w o r d s , n u m b e r s , e q u a t i o n s , s t a n d a r d m u s i c n o t a t io n ? T h a t

p i e c e o f t h e p a s t f o r m o s t o f u s is s i m p l y

w i p e d o u t

A n d I a m g o i n g t o

a r g u e t h a t th i s w i p e - o u t p h e n o m e n o n p l a y s a b i g r o l e i n o u r i n s tr u c t i o n a l

d i s -ab i l i t i es .

W a t c h in g c h i l d r e n in v e n t w a y s o f r e p re s e n t i n g th e m u s i c t h e y h a v e

m a d e , a n d a l s o l is t e n i n g c l o s e l y t o th e e f f o r ts o f m y M I T s t u d e n t s a s t h e y

t r y to s a y w h a t t h e y h e a r in m o r e c o m p l e x c o m p o s i t io n s , h a s h e l p e d t o

t u r n m y b e g i n n i n g m u s i c t h e o r y c l a s s e s o n t h e i r e a r s . A n d t o s u p p o r t th i s

s m a l l r e v o l u t i o n , I h a v e d e v e l o p e d a n i n t e r a c t i v e , c o m p u t e r - b a s e d m u s i c

e n v i r o n m e n t w h e r e b e g i n n i n g m u s i c s tu d e n ts a r e c o m p o s e r s . I n t e rr o g a t in g

t h e i r in t u i ti v e a b il it ie s t o c o m p o s e c o h e r e n t m e l o d i e s a n d r h y t h m s , a s k i n g

q u e s t i o n s t h a t t h e y p u t to t h e m s e l v e s t o a c c o u n t f o r w h a t t h e y a r e a b l e t o d o ,

t h e y a r e d e v e l o p i n g t h e s e p o w e r f u l m u s i c a l i n tu i ti o ns a n d g o i n g b e y o n d

In ternat ional Journal o f Computers for Mathem at ical Learning 1: 33-55, 1996.

~) 1996

Klu we r Aca dem ic Publishers Prin ted in the Netherlands



3 4 J E A N N E B A M B E R G E R

t h e m . P e r h a p s w h a t I h a v e l e a r n e d c a n b e h e l p f u l t o t h o s e d e v e l o p i n g

e d u c a t i o n a l c o m p u t e r e n v i r o n m e n t s i n o t h e r a re a s , as w e l l.

S TA T IN G T H E P R O B L E M

T h e r e i s a q u e s t i o n b e i n g a s k e d m o r e a n d m o r e f r e q u e n t l y i n t h e m u s i c

e d u c a t i o n l i t e r a t u r e ( a n d O f t e n e c h o e d i n t h e m a t h a n d s c i e n c e e d u c a t i o n

l i t e r a t u r e ) : W h y d o f u n d a m e n t a l s c o u r s e s w h i c h a r e i n t e n d e d t o b e g i n a t

t h e b e g i n n in g , s o o ft e n b e c o m e p r o b l e m a t i c ? W h y d o t h e y s o o f t e n s e e m

i rr e le v a n t e s p e c i a l l y t o t h o s e s t u d e n t s w h o h a v e a l r e a d y b e e n i d e n t i f ie d a s

g i f te d ? W h y d o t h ey b e c o m e m o r e li ke t h e ra p y s e s s i o ~ t re at in g s t u d e n t

s t re s s in s t e a d o f a n e n v i r o n m e n t i n w h i c h s t u d e n t s a r e d e v e l o p i n g t h e

h e a lt h y , p o w e r f u l i n t u i ti o n s t h a t th e y b r i n g w i t h t h e m t o t h e s e c la s s e s ?

F r o m e v e r y t h i n g I h a v e l e a r n e d s o fa r, m y b e s t h u n c h i s t h a t t h e s e

p r o b l e m s a r i s e b e c a u s e w e h a v e b e e n m a k i n g s o m e c r i t i c a l l y m i s t a k e n

a s s u m p t i o n s a b o u t o u r s t u d e n t s ' h e a l t h y m u s i c a l i n t u it i o n s - w h a t t h e y

k n o w h o w t o d o a l re a d y. W e a r e a s k in g s t u d en t s t o b e g i n w i t h w h a t w e

b e l i e v e a r e t h e

s implest

k i n d s o f e l e m e n t s , b u t w h i c h f o r t h e m m a y b e t h e

m o s t d i f fi c u lt . In d o i n g s o , I t h i n k w e a r e c o n f u s i n g smalles t e l e m e n t s - i n

m u s i c , i s o la t ed , d e - c o n t e x t u a l iz e d p i t c h a n d d u r a ti o n v a l u e s - w i t h w h a t w e

a s s u m e a r e a l s o th e

s implest

e l e m e n t s . W e f o c u s o n t h e s e s m a l l , d i s c r e t e

e l e m e n t s p a r t l y b e c a u s e t h e y a r e t h e e a s i e s t t o define a n d t h u s a l s o t h e

e a s i e s t t o a s s e s s w i t h r e s p e c t t o w h e t h e r s t u d e n t s h a v e l e a r n e d t h e m o r n ot .

B u t p r o b a b l y m o r e i m p o r t a n t, t h e s y m b o l s t h at r e p re s e n t t h e s e e l e m e n t s

a re th e t o o l s o f t h e t ra d e f o r s e a s o n e d m u s i c i a n s - t h e y a re w h a t w e d e p e n d

o n f o r c o m m u n i c a t i n g w i t h o n e a n o th e r, f o r saying w h a t w e h e a r d a n d

f o r t e l l i n g o t h e r s w h a t t h e y s h o u l d h e a r a n d p l a y . B u t i n d o i n g s o , w e a r e

n o t d is t in g u i s h in g b e t w e e n o u r o w n m o s t f am i l i a r un i ts o f

description

t h e

n o t e s s h o w n i n a s c o r e , a n d t h e i n t u i t i v e , c o n t e x t u a l u n i t s o f percept ion

t h o s e w h i c h y o u n g c h i l d r e n a n d m o s t a d u l ts , t o o , a r e a t t e n d i n g t o in

m a k i n g s e n s e o f t h e m u s i c a ll a r o u n d t h e m .

T o b e e v e n m o r e p r o v o c a t i v e , I w i l l a rg u e t h a t t h e k i n d s o f e le m e n t s a n d

r e l at i o n s t h a t u n t r a i n e d l i s te n e r s a n d a l s o s e l f - ta u g h t p l a y e r s a r e a t t e n d i n g

t o in m a k i n g m u s i c a l s e n s e , a re c l o s e r t o t h o s e w e a s s o c i a t e w i t h t h e a r t is t

w h o p l a y s r e a l l y m u s i c a l l y - t h e a b i l it y t o s h a p e a p h r a s e , to f o l l o w t h e

m u s i c a l l in e , a n d t o e x p r e s s i v e l y p r o j e c t f e e l i n g a n d m e a n i n g . T h i s i s w h a t

I t h i n k t h e v i o l i n i s t L o u i s K r a s n e r 1 w a s g e t t i n g a t w h e n I h e a r d h i m s a y t o

i Krasner was an em inent violinist who d ied in 1994. In recent years he taught at the

New E ngland Conservatory of M usic and it was there at one of his master classes that I

overheard this comm ent.



TURNING MUSIC THEORY ON ITS E R

35

a s t u d e n t, F o r g e t a b o u t th e n o t e s a n d p l a y th e m u s i c . B u t o f c o u r s e t h e r e

i s a p a r a d o x , h e r e : t h e s t u d e n t h a d t o h a v e t h e n o t e s b e f o r e s h e c o u l d f o r g e t

abo u t t hem . I t is t h is pa rad ox t ha t is , I t h ink , a t the c r ux o f t he m at t e r .

P R O B L E M 2: T H E W I P E O U T P H E N O M E N O N

T h e p r o b l e m is t h a t o n c e w e h a v e th o r o u g h l y i n t e rn a l iz e d c o n v e n t i o n a l

s y m b o l i c e x p r e s si o n s a s s o c i a te d w i t h a p r o f e s s io n a l c o m m u n i t y o f u s e rs

t h e th i n g s w e n a m e a n d o u r c a te g o r i e s o f a n a ly s i s - t h e y b e c o m e th e

t o o l s o f o u r t r a d e , a n d w e a r e n o l o n g e r a w a r e t h a t i n i n t e r n a l i z i n g t h e m

w e h a v e a ls o m a d e a t ac i t on to log i ca l com mi tmen t : P u t m o s t s t r o n g ly , w e

c o m e t o b e l i e v e in t h e o b j e c t s a n d r e la t i o n s w e n a m e w i t h o u r d e s c r i p t iv e ,

s y m b o l i c c o n v e n t i o n s a s j u s t t h o s e w h i c h e x i s t in o u r p a r t i c u l a r d o m a i n .

T h r o u g h p r a c t i c e , t h e s e a r e t h e o b j e c t s , f e a t u r e s , a n d r e l a t i o n s t h a t t a c i tl y

s h a p e t h e t h e o r y a n d s t ru c t u re o f th e o u r d o m a i n - h o w w e t h in k , w h a t

w e k n o w a n d t e a c h to o th e r s, a n d t h us w h a t w e t a k e t o b e k n o w l e d g e .

O u r u n i ts o f descr ip t i on c o m e p e r il o u s ly c l o s e t o b e c o m i n g o u r u n it s o f

p e r c e p t i o n w e h e a r an d s e e w h a t w e c a n say .

M y r e s e a r c h a n d t h at o f o t h e rs s t r o n g ly s u g g e s ts t h a t w h a t w e c a n

s a y w i t h s t a n d a r d m u s i c n o t a t i o n - t h e e l e m e n t s a n d r e l a t io n s r e f e r r e d t o

b y i ts c o n v e n t i o n a l s y m b o l s , d i f f e rs in s p e c i f ic a n d p r o v o c a t i v e w a y s f r o m

t h e n a t u r a l k i n d s t h a t i n h a b i t t h e s p o n t a n e o u s d e s c r i p t i o n s m a d e b y m u s i -

c a l l y u n t r a i n e d c h i l d r e n a n d a d u l t s ( a n d I m e a n h e r e , s p e c i f i c a ll y t h o s e w h o

h a v e n o t b e e n t a u g h t to r e a d s t an d a r d m u s i c n o t a ti o n ( S M N ) ) . M o s t i m p o r -

t a n t , t h e e v i d e n c e s h o w s t h a t s p o n t a n e o u s l y i n v e n t e d n o t a t i o n s n o t o n l y

g ive p r i o r i t y t o sp ec i f i ca l l y d i f f e ren t bu t a l so equa l l y va l i d m u s i c a l f e a t u re s

a n d r e la t i o n s a s c o m p a r e d w i t h th o s e t h a t i n h a b i t S M N ( B a m b e r g e r , 1 9 8 6 ;

1991a) .

T h e n a t u r e o f t h e s e d i f f e re n c e s is e m b e d d e d i n o u r e v e r y d a y e x p e r i e n c e .

F o r e x a m p l e , in m o v i n g a r o u n d i n r e a l l if e , w e m u s t d o s o i t s e q u e n t i a ll y ,

o n e t h in g a f t e r t h e o t h e r - t h e n e x t s te p c a n n o t c o m e B E F O R E t h e la s t

s te p ; th e 4 t h o f J u l y is a l w a y s A F T E R t h e 3 r d a n d B E F O R E t h e f if th ; a n d

W e d n e s d a y a l w a y s c o m e s A F T E R T u e sd a y . T h e s e c a l e n d a r s e q u e n c e s a re ,

o f c o u r s e , s y m b o l i c a r ti f a c ts a n d t h e y a r e c e r t a in l y u s e f u l t o b e l i e v e i n - w e

c a n c o m p a r e o n e T u e s d a y w i t h an o t he r , c o u n t u p T u e s d a y s , u s e T u e s d a y

a s a t a g to r e m e m b e r w h a t w e h a v e to d o , b e c a u s e T u e s d a y is a l w a y s

t h e s a m e , th e s a m e p l a c e in t h e s e q u e n c e - b u t o n l y p a rt ly . C o n s i d e r t h e

f o l l o w i n g e v en t s : T h i s y e a r t h e 4 t h o f J u l y w a s a T u e s d a y , b u t i t F E L T

m o r e li k e a S u n d a y . S o th e N E X T d a y I g o t m i x e d u p; I k e p t th i n k i n g it

w a s a M o n d a y . A f t e r a ll, M o n d a y d o e s c o m e a f t e r S u n d a y . I n t h e p a s s i n g

o f t im e i n r e a l l if e w e t a k e t h i n g s a s t h e y c o m e , o n e a t a t im e a n d o n e t h i n g



3 6 JEANNE BAMBERGER

a f t e r t h e o th e r . B u t w h a t w e t a k e a s a t h i n g , t h e s y m b o l i c a r ti f a c t o r t h e

p e r c e p t i o n o f it , h o w w e u s e i t, in f l u e n c e s w h a t w e t a k e to b e c o m i n g n e x t .

1 2 c o m e s A F T E R 5 i f y o u a re c o u n t i n g , b u t th e b u n c h o f 1 2 - g r ap e s m i g h t

c o m e B E F O R E t h e b u n c h o f 5 -g r ap e s i f y o u a re h u n g ry . A l i c e a n d t he

Q u e e n s m a k e t h e p o i n t. T h e W h i t e Q u e e n s a y s:

. . . we had SUCH a thunder s torm last Tuesday - I m ean one of the last set of Tuesdays,

you know.

Alice was puzzled. In our country, she remarked, there's only one day at a time.

The Red Q ueen said, That's a poor thin way of doing things. Now here, we mo stly

have days and nights two or three at a time, and som etimes in the winter we take as many

as five nights together - for warm th, you know (Carroll, 1960).

E v e r y d a y l if e i s w h a t A l i c e is t a lk i n g a b o u t a n d t h a t is w h a t i s r e f l e c te d

i n e v e r y d a y d r a w i n g s o f c h i ld r e n a n d a d u l ts w h o a re n o t t r a in e d m u s i c ia n s .

T h e Q u e e n s a r e l i k e t h e o t h e r s w h o l i v e a n d b e l i e v e i n a s y m b o l i c w o r l d

w h e r e t h e n a m e s f o r th i n g s ( n i g h ts , d a y s , n o te s ) n a m e p r o p e r t y c l a s se s a n d

t h e s e a r e u s e f u l b e c a u s e t h e y c a n , in t a l k i n g a n d i n p a p e r - s p a c e , b e p u t i n

a n y o r d e r, i n b u n c h e s , s e v e r a l a t o n e t i m e . B u t e v e n i n t a l k i n g o r n o t a t i n g ,

t h e n a m e s h a v e t o c o m e o n e a f t e r t h e o th e r, n o t in t h e b u n c h e s t h e n a m e s

n a m e . I t a ll d e p e n d s o n w h a t y o u c a n a n d w a n t t o d o w i th t h i n g s - b u t

y o u n e e d t o b e a b l e t o t e l l t h e d i f f e r e n c e .

O u r c o n v e n t i o n a l u n i ts o f d e s c r i p t i o n f u n c t i o n l ik e l e n s e s th a t s h a p e ,

s e le c t , s o r t o u t, a n d s e g m e n t t h e w o r l d . A n d l ik e e y e - g l a s s e s , a s lo n g a s

t h e y a r e l e f t a l o n e , a s l o n g a s t h e y a r e n o t p e r t u r b e d , w e a r e h a p p y t o j u s t

l o o k

th rough

t h e m . I t is o n l y w h e n s o m e t h i n g g o e s w r o n g - t h e l e n s g e t s

c r a c k e d , s c r a t c h e d , f o g g e d u p - t h a t w e a r e f o r c e d t o look t w h a t w e

n o r m a l l y l o o k t h r o u g h . F o r i n s t a n c e , w h e n m u s i c i a n s l o o k t h r o u g h t h e i r

c o n v e n t i o n a l e y e - g l a s s e s a t c h i l d r e n ' s i n v e n t e d n o t a t i o n s , t h e s e i n v e n t i o n s

s e e m s i m p l y w r o n g . W i t h t h e s y m b o l i c c o n v e n t io n s s h a p i n g m e a n i n g , t h e

i n v e n t i o n s a p p e a r a s e x c e p t i o n s , a b b e r a t i o n s , c o n t a m i n a n t s . B u t w h a t i f

w e t a k e th e d r a w i n g s s e ri o u sl y , p o s i ti n g t h a t i f w e o n l y k n e w w h e r e t o

l o o k a n d h o w , t h e y c o u l d m a k e p e r f e c t s e n s e ? O n t h is v i ew , t h e i n v e n t io n s

b e c o m e a n o c c a s i o n t o l o o k a t t h e l e n s e s t h a t w e u s u a l l y l o o k t h r o u g h .

A n d i f w e s u c c e e d i n d o i n g t h at , t h e n t h e s e i n v e n ti o n s c a n b e c o m e c l u e s

t o th e p o w e r f u l s t r a t e g i es t h a t c h i ld r e n a n d a l s o a d u l t s b r i n g w i t h t h e m i n

c o n s t r u c t in g m u s i c a l c o h e r e n c e .

B a r b a r a M c C l i n t o c k , th e N o b e l p r i z e w i n n i n g b i o l o g i s t, p u t s i t t h i s

w a y :

So if the material tells you, 'It may be this ' allow that. D on 't turn it asid e and call it an

exception, an aberration, a contam inant. That's w hat's happ ened all the way along the line

with so many good clues (Keller, 1983).



T U R N I N G M U S I C TH E O R Y O N IT S E A R 37

A s L e o T r e i t l e r h a s s o e l o q u e n t l y p u t it: T h e r e is n o i m m a c u l a t e

p e r c e p t i o n ( T r e it le r , 1 98 9 ). E v e r y d e s c r i p t io n , e v e r y s e t o f s y m b o l i c r e p -

r e s e n t a t i o n s , t h o s e i n v e n t e d b y c h i l d r e n a s w e l l a s t h o s e a s s o c i a t e d w i t h

a c o m m u n i t y o f p r o f e ss i o n a l u s er s , a r e n e c e s s a r i ly p a rt ia l a n d t h e y a r e

s o in t w o s e n s e s : t h e y a r e p a r ti a l i n b e i n g

i ncomple t e

a n d t h e y a r e a l s o

par t i a l t o c e r ta i n a s p e c ts o f th e p h e n o m e n a w h i l e i g n o r i n g o t h er s . W i t h

r e s p e c t to a c o m m u n i t y o f p ro f e s s io n a l s , T h o m a s K u h n h a s p o i n t e d o u t,

f o r e x a m p l e , t h a t t h e p h y s i c i s t t h in k s w i t h t h e r e l a t iv e l y f e w b u t p o w e r f u l

s y m b o l i c fo r m a l i s m s s h a r e d b y t hi s c o m m u n i t y o f u s e rs , a n d t h e s e s h a p e

t h e t h e o r y a n d t h e s t r u c tu r e o f t h a t d o m a i n ( K u h n , 1 9 7 7) . S i m i l a r ly , P h il li p

M o r r i s o n h a s s a id o f m a p s , t h e c a r t o g r a p h e r s ' w o r k i n g n o t a ti o n :

E a c h m a p i s i n a w a y a

th ory

t h a t f av o r s c e r t a i n a p p r o x i m a t i o n s . P r o c e d u r e s l i k e s e le c -

t i on ,

simplification, smoothing, displacements to m ak e room , out-of-scale notation for

bridges, streams, and roads so narrow that they w ould become nvisible at true scale, enter

inescapably (M orrison , 1991).

A l l o f w h i c h m a y a p p e a r t o b e a n a r g u m e n t a g a in s t t e a c h i n g a n d l e a rn -

ing no t a t i ona l conven t ions , bu t t ha t i s no t a t a l l t he case . The i n t e res t i ng

q u e s t io n s a r e : w h a t a r e t h e a p p r o x i m a t i o n s a c o m m u n i t y f a vo r s, w h a t k i n d s

o f e n t it i e s a n d r e l a ti o n s i s t h e c o m m u n i t y p a r t ia l t o , h o w d o t h e s e d i f f e r

f r o m t h e k i n d s o f e n t it i e s a n d r e l a ti o n s t h a t o t h e r c o m m u n i t i e s a r e p a r t ia l

to , h o w c a n w e f in d o u t, a n d w h a t d i f f e r e n c e d o t h e d i f f e r e n c e s m a k e , t o

w h o m , fo r w h a t, a n d w h y ?

T o s t u d y c h i l d r e n ' s s p o n t a n e o u s p r o d u c t i o n s , ta k i n g t h e m s e r i o u s l y i n

s e a r c h o f a n s w e r s t o q u e s ti o n s s u c h a s th e s e , w e n e e d t o b e c o m e s o m e -

t h i n g l ik e c u l t u r a l a n t h r o p o l o g is t s : l i k e t h e a n t h r o p o l o g i s t e n t e r i n g a n e w

c u l t u r e , w e n e e d t o b e g i n w i t h t h e a s s u m p t i o n t h a t w h a t i s f o u n d t h e r e -

r it u a ls , m y t h s , m o d e s o f r e p r e s e n t a t i o n - n o m a t t e r h o w i n it ia l ly s t r a n g e ,

i n c o m p r e h e n s i b l e , m e a n i n g l e s s , th e y m a k e s e n s e to t h e i n h a b it a n ts o f t h a t

c u l tu r e . O n c e m a k i n g t h a t a s s u m p t i o n , t h e t a s k b e c o m e s m u t u a l a n d r e c i p -

r o ca l: w e m u s t l e a r n to u n d e r s t a n d o u r o w n b e l i e f s y s t e m s , o u r o w n d e e p l y

i n t e r n a l i z e d i n t u i t i o n s f o r m a k i n g s e n s e , e v e n a s w e l e a r n t o u n d e r s t a n d

t h e s e n s e - m a k i n g o f t h e o th e r . A s C l i f f o r d G e e r t z h a s s a i d o f t h e p r a c t i c e

o f a n t h r o p o lo g y :

. . . p r o g r e s s i s m a r k e d l e ss b y a p e r f e c t i o n o f c o n s e n s u s t h a n a s a r e f i n e m e n t o f d e b a t e .

W h a t g e t s b e t t e r i s t h e p r e c i s i o n w i t h w h i c h w e v e x o n e a n o t h e r ( G e e r t z , 1 9 7 3 ).

B u t h o w d o w e g o a b o u t s u c h s t u d i es , w h a t s o r ts o f q u e s t i o n s , s it u a ti o n s ,

w i ll e f f e c t iv e l y r e v e a l t h e m e a n i n g - m a k i n g o f o u r s o - c a l le d n a i v e i n fo r -

m a n t s w h i l e a ls o p er t u rb i n g , v e x i n g o u r o w n a s s u m p t io n s ? I h a v e f o u n d

t h a t t h e m o s t e v o c a t i v e s i tu a t io n s , t h e m o s t p r o d u c t i v e r e s e a r c h q u e s t i o n s ,

a n d t h e g r e a t e s t l e a r n i n g h a p p e n i n th e r e a l li f e o f t h e c l a s s r o o m - t h e

m o m e n t s t h a t a r i s e i n t r y i n g t o u n d e r s t a n d p u z z l i n g e v e n t s t h a t o c c u r i n



38 JEANNEBAMBERGER

t h e m i d s t o f w o r k i n g w i t h s t u d en t s. I n t h e s e m o m e n t s c a u g h t o n - th e - fl y ,

t e a c h i n g a n d r e s e a r c h , i n s te a d o f b e i n g s e p a r a t e a n d d i f f e r e n t k i n d s o f

e n t e r p r i s e s , b e c o m e a s in g l e , m u t u a l l y i n f o r m i n g o n e .

A N E X A M P L E , A N E X P E RI M E N T , A N O T H E R P R O B L E M

C o n s i d e r t h e f o l l o w i n g e x a m p l e : W h i l e w o r k i n g i n a c t iv e m u s i c c l a s s r o o m

s i t u a t i o n s , I w a s s u r p r i s e d t o s e e ( b a s e d o n m y a s s u m p t i o n s a b o u t w h a t

n e e d e d t o b e t a u g h t ) th a t 5 a n d 6 y e a r ol d s, w i t h o u t a n y o n e te a c h i n g t h e m

o r e v e n a sk i n g th e m t o d o s o, w e r e s h o w i n g m e w h e r e p h r a s e s e n d a n d

w h e r e a n e w p h r a s e b e g i n s , w h e r e t o s t o p a n d s ta r t a g a in . P u z z l in g o v e r

t h is u n e x p e c t e d h a p p e n i n g , i t s tr u c k m e a s o b v i o u s ( b u t p r o b a b l y f o r th e

f ir s t t i m e ) , t h a t i f a n y o f u s i n l i s te n i n g t o a p i e c e , c a n n o t t e l l w h e r e t o s t op

a n d s t a rt a g a in , w e s a y t h e m u s i c s i m p l y d o e s n ' t m a k e s e n se . T h is o f t e n

h a p p e n s i n l i st e n i n g , f o r in s t a n c e , to m u s i c o f a n o t h e r c u lt u r e . A l t h o u g h

h a v i n g l e a r n e d j u s t a s n a t u r a l l y to h e a r w h e r e t h e i r m u s i c s to p s a n d s ta r ts

a g a i n, t h e s a m e m u s i c m a k e s p e r f e c t s e n s e e v e n to t h e y o u n g e s t c h i l d r e n

i n t h a t o t h e r c u l t u r e .

T r y a l it tl e e x p e r i m e n t t h a t I h a v e o f t e n w a t c h e d i n m y M I T c l a s se s .

S i n g th e fi rs t, b i g p a r t o f t h e t u n e , D i d y o u e v e r se e a la s s i e t o y o u r s e l f

( u p t o a n d i n c l u d i n g t h e w o r d s , t h i s w a y a n d t h a t ). I f I a s k y o u , H o w

m a n y ' c h u n k s ' a r e th e r e in w h a t y o u s a n g ? , li k e m o s t p e o p l e , e v e n i f y o u

d o n ' t k n o w t h e w o r d s , y o u w il l p r o b a b l y s ay , t w o . T h a t is , y o u h e a r t h is

f ir st b i g p a rt o f t h e m e l o d y g r o u p e d in to t w o l a rg e r c h u n k s o r p h r a se s :

1 . D i d y o u e v e r s e e a l a ss i e g o th i s w a y a n d t h a t w a y ?

a n d

2 . D i d y o u e v e r s e e a l a s s ie g o t h is w a y a n d t h a t?

N o w i f I a s k y o u , W h a t i s th e d i f fe r e n c e b e t w e e n t h e e n d o f t h e fi rs t

p h r a s e a n d t h e e n d o f t h e s e c o n d p h ra s e , y o u w i ll p r o b a b l y s a y t h a t y o u

h e a r d t h e e n d o f th e f ir s t p h r a s e ( t h a t w a y ) a s re l a t i v e l y i n c o m p l e t e , l e f t

h a n g i n g , w h i l e y o u h e a r d t h e e n d o f t h e s e c o n d p h r a s e ( a n d t h a t ) a s

r e l a t i v e l y s e t tl e d , c o m p l e t e a n d r e s o l v e d . W h y i s t h is s o ?

Y o u r f ir s t h u n c h , l i k e m y s t u d e n t s ' , is p r o b a b l y t h a t t h e t w o p h r a s e s e n d

o n d i f f e r e n t p i t c h e s B u t , r e m a r k a b l e as it m a y s e e m , b o t h p h r a se s e n d o n

t h e s a m e p i t c h ( s ee F igu re 1 ).

M o r e o v e r , y o u r r e f e r e n c e t o p it c h w a s m o s t l ik e l y o n l y tr i g g e r e d b y

m y q u e s t i o n - y o u g r a b b e d f o r t h e n a m e o f a t h in g t h a t y o u a s s o c i a te



TURNING MUSIC THEORY ON ITS E R

39

D i d g o u e v - e r s e e a l ~ s - s i e g o t h i s w ~ g I n d t h ~ w ~ V ? D i d ~ o u

eY - er see a l as - sJe 90 t h i s wat j I nd t hat ? ,

Figure 1 Both phrases end on the same pitch.

wi th an sw ers t o que s t i ons l i ke tha t . 2 Bu t i n ac tua l l y s i ng in g and l i s t en ing ,

p i t c h a s a s e p a r a t e e n t it y , a s a p r o p e r t y c la s s , w a s a l m o s t c e r t a i n l y n o t

w h a t y o u w e r e a t t e n d i n g t o . T h e f e e l i n g o f t e n s i o n a t t h e e n d o f t h e f ir s t

p h r a s e , f o l lo w e d b y r e s o l u ti o n a t t h e e n d o f t h e s e c o n d r es u lt s f r o m t h e

r e c i p r o c i t y b e t w e e n a t l e a s t p i t c h a n d r h y t h m . Y o u w e r e r e s p o n d i n g t o

t hi s c o n f l u e n c e o f f e a t u r e s a n d t h e ir r e la t io n s w h i c h t o g e t h e r g e n e r a t e

s i tuat ional m eaning an d funct ion A nd i t i s i n t e res t i ng t ha t th i s r ec ip ro c i t y

i s r e f l e c t e d i n o u r m e t a p h o r s t h a t c o n f l a t e u p a n d d o w n i n p i t c h w i t h u p

a n d d o w n i n r h y t h m ( u p - b e a t, d o w n - b e a t ) . W h e n T u e s d a y is t h e 4 th o f

J u l y a n d its si tu a t io n a l m e a n i n g a n d f u n c t i o n is S u n d a y , t h e n M o n d a y

c o m e s A F T E R T u e sd a y.

T h e p r o b l e m i s th a t w e h a v e n o w a y o f t a lk i n g a b o u t a n d a c c o u n t i n g

f o r t h e s e p e r c e i v e d , s it u a ti o n a l c o n f l u e n c e s w i t h o u t t a k i n g t h e m a p a r t i n to

t h e i r s e p a r a t e p r o p e r ti e s . W e c a n s a y t h a t i n p e r c e p t i o n t h e m u l t i p l e p r o p e r -

t ie s o f e v e n t s a r e h i g h l y a g g r e g a t e d , m e r g e d , fu s e d . B u t p u t t i n g i t t h a t w a y

s t at e s t h e p r o b l e m r a t h e r t h a n s o l v i n g it. F o r t e r m s s u c h a s c o n f l u e n c e ,

a g g r e g a t e d , f u s e d , a l r e a d y i m p l y a collect ion o f sepa rate pro pert ies

w h e n , i n f a c t , a s y o u h a v e e x p e r i e n c e d i n th e L a s s i e e x a m p l e , th e s e p r o p -

e r t ie s s i m p l y d o n o t e x i s t a s s e p a r a t e e n t it ie s i n o u r f e e l i n g f u l , f u n c t i o n a l

h e a r i n g s . W e d o n o t p i e c e t o g e t h e r a h e a r i n g , p u t ti n g i t to g e t h e r o u t o f t h e

s e p a r a t e f e a t u r e s w e c a n n a m e - a p a s te - u p c o l l a g e o f , f o r i n s t a n c e , p i t c h ,

du ra t i on , acc en t , t im bre , r eg i s te r .

A n d i t i s n o t 0 n l y i n m u s i c a l e x p e r i e n c e t h a t th i s is t h e c a s e . C o n s i d e r

t h e e x p e r i e n c e o f g o i n g f a s te r . Y o u sa y , w h i l e s a il in g y o u r b o a t o r e v e n

w a l k i n g , N o w I a m g o i n g f a st e r. T h e r e i s n o a m b i g u i t y a b o u t it , y o u

e x p e r i e n c e th e c h a n g e a s ju s t t h a t. B u t t o e x p r e s s t h a t c h a n g e , a n d e s p e c i a l l y

to measure i t , y o u h a v e t o t a k e a p a r t w h a t w a s a n a l l - a t - o n c e k i n d o f

2 In fac t, the endings of the two phrases sound differentbecause of d ifferences n their

rhythm ic structure. Specifically, he first phrase en ds on a w ea k bea t with a shorter

duration, wh ile the se con dphrase ends on a strong beat and with a longer duration.



4 JEANNE BAMBERGER

t h i n g i n to t w o s e p a r a t e k i n d s o f t h i n g s t h a t d i d n o t e x i s t i n t h e m o m e n t ' s

e x p e r i e n c e - t i m e a n d d i s ta n c e . I n d e e d , fi rs t y o u h a v e m e n t a l l y t o c o n s t r u c t

t h e m , i n v e n t t h e m , f i n d th e m a s c o n s t it u e n ts o f y o u r e x p e r i e n c e . A n d y o u

a l s o h a v e t o c o n s t r u c t t h e re f e rence sy s t ems i n w h i c h e a c h o f t h e a p p r o p r ia t e

u n it s a n d t h e i r s y m b o l i c e x p r e s s i o n s a r e g i v e n m e a n i n g . M o r e o v e r , h a v i n g

d o n e s o, y o u h a v e t o m e n t a l ly c o n s t r u c t th e

re l t i onsh ip

t h r o u g h w h i c h t o

p u t t h e m t o g e t h e r - r a ti o . A n d t h e re s u l ti n g r a ti o i s n o l o n g e r tw o t h in g s b u t

a s in g l e t h i n g - v e l o c it y . D i s t a n c e a n d t im e , e a c h o f w h i c h y o u m a y h a v e

b e e n a b l e to e x p e r i e n c e s e p a r a t e ly , a r e n o w o n e - v e l o c i ty , t h e i n t e r a c t io n

b e t w e e n t h e m . A n d f i n al ly y o u m u s t c o m p a r e t hi s r e su l ti n g v e l o c i t y - th i n g

w i t h a n o t h e r v e l o c i t y - t h i n g - t h e v e l o c i ti e s b e f o r e a n d a f t e r t h e c h a n g e .

A n d t h a t r e s u lt y o u a r e a s k e d t o b e l i e v e in a s a r e p r e s e n t a t i o n o f y o u r

f a m i l i a r e x p e r i e n c e , g o i n g f a s te r .

W e c a n , h o w e v e r , i n t e rr o g a t e t h e s e e x p e r i e n c e d m o m e n t a r y c o n f l u -

e n c e s . T u r n i n g b a c k u p o n t h e m , ta k i n g th e m a p a rt , w e c a n l ib e r a te f r o m

t h e m e l d a n d n a m e t h e c o m p o n e n t p ie c e s o f t h e s e e x p e r i e n c e d c o n f lu e n c e s .

I n d o in g s o w e a r e a ls o , in a p r o f o u n d s e n s e , b r i n g in g t h e s e c o m p o n e n t s

i nt o e x i s t e n c e . A n d o n c e g i v in g n a m e s t o t h in g s , w e a l so g a i n a c e r t a in

p o w e r , t h e p o w e r t o p l a y w i t h t h e t h i n g s n a m e d , s h i f t in g o u r a t t e n t i o n a t

w i l l a m o n g t h e m a n d c o m b i n i n g t h e m i n n o v e l w a y s . T h e t r i c k i s t o b e

a b l e to s e l e c ti v e l y c h o o s e a m o n g t h e s e m u l t i p le r e p r e s e n t a ti o n s d e p e n d i n g

o n w h e n , w h y , a n d w h a t w e w a n t to u s e t h e m f or .

A s t e a c h e r s a n d r e s e a r c h e r s w e a l s o u s e n a m e d k i n d s o f f e a t u re s i n

a n e f f o r t t o i n t e r p r e t b e h a v i o r s , t o d e s c r i b e a n d t o d i f f e r e n t i a t e a m o n g

h e a r i n g s m a d e . B u t t h a t p u t s u s a g a i n i n t h e c e n t e r o f th e p a r a d o x : h o w

c a n w e a c c o u n t f o r h ea r in g s o f a n o t h e r b y m a k i n g r e f e r e n c e to t h o se

e n ti ti e s a n d re l a ti o n s e m b o d i e d b y o u r s y m b o l s y s t e m s , w h e n t h e y h a v e

n o t y e t b e e n c o n s t r u c t e d , w h e n t h e y d o n o t y e t e x i s t a s e n t i t i e s i n t h e

c o h e r e n c e - m a k i n g o f th o s e w h o m w e a r e tr y in g t o u n d e r s ta n d ? W h i le w e

m i g h t t h i n k o f t h e o t h e r s a s o u r n a i v e i n f o r m a n t s , t h e y m a y a l so b e

o u r a v i d a u d i e n c e s a n d c e r t a in l y t h e y a r e q u i te e f f e c t i v e l y m a k i n g s e n s e o f

t h e w o r l d a ll a r o u n d t h e m . S o w e a r e i n e v it a b ly l e ft w i th a p r o b l e m o f o u r

o w n c r e a ti o n : h a v i n g o n c e t a k e n a p a r t w h a t is e x p e r i e n c e d a s fu n c t i o n s a n d

f e e li n g s, w e a r e t e m p t e d to b e l i e v e th a t w h a t w e h a v e t h u s l e a r n e d to s a y i s

w h a t is b e i n g h e a r d a n d f el t, l e a v i n g u s to p u z z l e o v e r h o w t o p i e c e t o g e t h e r

w h a t , i n e x p e r i e n c e , i s n o t i n p i e c e s a t a ll . W h i l e I f i n d n o e a s y w a y o u t

o f th i s c o n f o u n d i n g s i tu a t io n , r e c o g n i z i n g it h e lp s t o t e m p e r c o n c l u s i o n s ,

m o s t o f a ll c o n c l u s i o n s a b o u t h o w w e l e a r n a n d w h a t t h a t m i g h t te l l u s

a b o u t h o w w e t e a c h . A n d t h a t b r i n g s m e b a c k t o t h e c la s s r o o m .



TURNING MUSIC THEORY ON ITS E R 41

I M P R O M P T U : A R E F L E C T IV E P L A Y G R O U N D F O R D E V E L O P I N G

M U S I C A L I N T U I T IO N S

D E S I G N P R I N C I P L E S 3

I m p r o m p t u i s a m o r e i n tu i ti v e, i c o n d ri v e n v e rs i o n o f M u s i c L o g o d e v e l o p e d

i n S . P a p e r t' s L o g o L a b b e g i n n in g s o m e 2 0 y e a r s a g o . 4 T w o v e r y b a s i c

p r i n ci p l es h a v e g u i d e d t h e d e s i g n o f b o t h M u s i c L o g o a n d I m p r o m p t u .

F i r s t , c o m p u t e r s s h o u l d b e u s e d o n l y t o d o t h i n g s w e c a n n o t d o b e t t e r

i n s o m e o t h e r w a y . S e c o n d ( b o r r o w e d f r o m H a l A b e l s o n ) , a n e d u c a t io n a l

c o m p u t e r e n v i r o n m e n t i s v a l u a b l e t o t h e d e g r e e i t c a u s e s i ts d e v e l o p e r s to

r e - t h in k t h e s t r u c t u re o f t h e r e le v a n t d o m a i n .

T h u s i n s t e a d o f s ay i n g , H e r e is t h is c o m p u t e r w i t h a ll t h e s e n e a t

p o s s i b i l it i e s , w h a t c a n I d o w i t h it ? I s a id , H e r e a re s o m e t h i n g s th a t

b e g i n n i n g m u s i c s t u d e n t s n e e d t o b e a b l e t o d o a n d t h e y c a n ' t d o t h e m

w i t h th e m e a n s t h a t a r e a r o u n d . I n sh o r t, h a v i n g t a u g h t b e g i n n i n g m u s i c

c o u r s e s f o r y e a r s a n d w r i t t en a m u s i c t e x t t h at l o ts o f o t h e rs w e r e u s i n g

( B a m b e r g e r & B r o f s k y , 1 9 8 8 ) , I g o t t ir e d o f h e a r i n g m y s e l f t a l k a b o u t

m u s i c , a n d a s k i n g s t u d e n t s t o li s te n t o m u s i c j u s t s o t h e y c o u l d t a l k b a c k ,

b e c a u s e i t d i d n o t w o r k .

S o I t o o k t h e p l u n g e: L e t ' s s e e i f a c o m p u t e r / s y n t h e s i z e r en v i r o n m e n t

c a n b e d e v e l o p e d t h a t w i ll

m e e t t h e s e u n s a t i s f i e d n e e d s .

M o v i n g i n o n

t h e p r o c e s s , I w o r k e d t o g e t h e r w i t h f o l k s i n P a p e r t ' s l a b a n d l a t e r m o s t

i n te n s i v el y w i th m y p r o g r a m m e r a n d f o r m e r s t u d en t , A r m a n d o H e r n a n -

d e z . M i x i n g m y m u s i c a l t h i n k in g w i t h A r m a n d o ' s f o r i n s t a n c e i n it ia l

i m p l e m e n t a t i o n o f id e a s , c r e a t e d s ta r tl in g s u r p ri s e s; m y h e a d w a s o f t e n

s p i n n i n g a s I c a m e t o s e e / h e a r s o m e v e r y b a s i c m u s i c a l e n t i t y o r r e la t io n -

s h i p i n a n e w w a y . F o r e x a m p l e , I r e c o g n i z e d t h a t m y n o t i o n s o f f a st a n d

s l o w w e r e s li m , i n d e e d - u p a n d d o w n in p it c h an d ra t e o f c h a n g e w e r e a ls o

m a k i n g f a s t a n d s l o w ; t im e - u n i t s b e c a m e h i er a r ch i c a l a n d h a d t o b e d i f fe r -

e n t i a te d f r o m p h r a s e b o u n d a r i e s ; a n d r e p r e s e n t a t i o n s h a d t o b e i n v e n t e d t o

s h o w a l l o f th i s. B u t j u s t a s i n s ig h t s h a p p e n e d , t h e y c o n t i n u o u s l y g e n e r a t e d

m o r e q u e s t i o n s , le a v in g u s w i t h n e w p r o b l e m s : H o w c o u l d w e i n v e n t w a y s

t o m a k e t h e c o m p u t e r t e c h n o l o g y b e r e s p o n s i v e t o t h e se i n si g ht s a n d n e w

i d e a s ?

3 Improm ptu in its present version, along with a text and projects, will be pu blished by

Oxford U niversity Press in 1996.

4 W hile Impromptu is easier to u se, more accessible han M usicLogo, it is constrained

by not giving the us er access to a re al com puter language. We ho pe in the near future

to em bed Impromptu in a L ogo-like environment. Users will then be able to evolve from

Impromptu's easy-to-use but limited technology to a truly extensible environment w here

possibilities for learning and for com posing beco m e topless.



4 JEANNEBAMBERGER

Musical Intuitions Three Premises

S o w h a t a r e t h e s e u n s a ti s fi e d n e e d s o f b e g i n n i n g m u s i c s t u d e n t s? M y s e n s e

o f t h e m d e r i v e s fr o m t h r e e p r e m i s e s a b o u t e v e r y d a y m u s i c a l i n t u it io n s

g l e a n e d f r o m t e a c h i n g a n d r e s e a r c h :

1. T h e k i n d s o f e l e m e n t s a n d r e la t i o n s n o v i c e s a t te n d to i n m a k i n g s e n s e

o f m u s i c a s i t u n f o l d s i n r e a l ti m e , a r e h i g h l y a g g r e g a t e d , s t r u c t u r a l ly

m e a n i n g f u l e n t it i e s s u c h a s m o t i v e s , fi g u r e s, a n d p h r a s e s .

T h e s e a re t h e u n i t s o f p e r c e p t i o n - t h e e l e m e n t s t h a t n o v i c e s h a v e

r e a d y a c c e s s t o , t h e i r f o c u s o f a t te n t i o n . W e d o n o t li s te n to n o t e s

a n y m o r e t h a n w e l i s te n t o l e tt e r s p r i n t e d o n t h e p a g e . F o r i n s ta n c e ,

i f y o u h a v e e v e r w a t c h e d k i d s p i c k i n g a t u n e o f f th e t a p e o n t o t h e ir

g u i ta r , y o u w i l l h a v e n o t i c e d t h a t t h e y r a r e l y g o n o t e - t o - n o t e . R a t h e r ,

t h e y l i s te n t o a s e l e c t e d p o r t i o n o f t h e p i e c e , a r e a s o n a b l e s t r u c t u r a l

c h u n k , h e a d i n g f o r w h a t t h e y ca l l a ta r g e t t o n e o r g o a l - n o t j u s t a n y

s t op p i n g p l a c e w i l l d o . T h e n a p p r o x i m a t i n g t h e g e n e r a l s h a p e a n d f e e l

o f t h e s e l e c t e d p o r t io n o n t h e i r g u it a r, l is t e n i n g a g a i n a n d a g a i n , t h e y

g r a d u a l l y m o v e i n o n t h e d e t a i ls . I t a k e th i s a s f u r t h e r e v i d e n c e t h a t w e

i n t u it i v e ly b e g i n b y h e a r i n g s t r u c tu r a l l y m e a n i n g f u l fi g u r e s a s m u s i c a l

e n t i ti e s , a n d o n l y w i t h f u r t h e r e f f o r t d o w e m o v e in o n th e n o t e s .

2 . T h r o u g h l i s te n i n g t o m u s i c o f o u r o w n c u l tu r e , w e h a v e b e c o m e m o s t

respons ive t o s t ruc tu ra l func t i ons such as s t ab i l i t y and i n s t ab i l i t y -

w h e t h e r a p h r a s e s o u n d s e n d e d o r is s ti ll g o i n g o n ; e v e n g i v e n a

c o n t e x t , w h e t h e r a n o t e s o u n d s a t re s t o r n o t.

A n d h e r e t h e n o v i c e a n d t h e e x p e r t c o m e t o g e t h e r a g a in : b o t h a r e

h i g h l y r e s p o n s i v e to c o n t e x t - t h e f u n c t i o n o f e v e n ts w i t h in t h e s i t u a ti o n

w h e r e t h e y o c c u r. O u r u n it s o f d e sc r i p ti o n - w h a t w e n a m e a n d n o t a te

- t e l l u s t h a t th e p i tc h , C , o r a m a j o r t h i rd a r e t h e s a m e w h e r e v e r t h e y

o c c u r. B u t th e m u s i c a l n o v i c e h e a r s n o t a t e d s a m e p i t c h e s a s d i f f e r e n t

i n r e s p o n s e t o t h e c h a n g i n g f u n c t i o n s o f t h o s e s a m e p i t c h e s w i t h i n

t h e p a r ti c u l a r c o n t e x t w h e r e t h e y o c c u r . A n d s o d o e s t h e p e r f o r m e r

w h o m w e d e s c r i b e a s p l a y i n g r e a l l y m u s i c a l ly . T h a t is w h y s tr in g

p l a y e r s , f o r i n s t a n c e , p a y c a r e f u l a t t e n t i o n t o f i n g e r i n g , b o w i n g , a n d

t o t h e s u b tl e ty o f i n t o n a ti o n - t h e s e a r e th e m e a n s t h r o u g h w h i c h

t h e a r t i s t p e r f o r m e r p r o j e c t s c o n t e x t u a l d i f f e r e n c e s a n d c h a n g e s i n

s t ru c t u ra l f u n c t i o n a m o n g i n s ta n c e s o f t h e s a m e n o t a t e d p i tc h . A n d i t

i s s t ru c t u r a l f u n c t i o n s t h a t g e n e r a t e f e e l in g s , i m a g e s , a n d a s s o c i a t io n s .

B u t w e s e e m , b y c o n v e n t i o n o r h a b it , to k e e p t h e l a n g u a g e o f s t r u c tu r a l

f u n c t i o n s ( h a r m o n i c f u n c t i o n s , r h y t h m i c f u n c t io n s ) a n d t h e l a n g u a g e

o f f e e l i n g s in s e p a r a t e r e a l m s o f d i s c o u r s e a n d c u l t u r e .

3 . T h o s e w h o p l a y a n i n s tr u m e n t k n o w a p i e c e b e s t as t h e f e e l o f t h e ir

b o d i e s ( li ps , a r m s , f i n g e r s ) o n t h e t e r r a i n o f t h e i r in s t r u m e n t . J u s t a s



TURNINGMUSICTHEORYON ITS EAR 43

w e m u s t m o v e s e q u e n t ia l ly in r e a l ti m e ( o n e s t ep o r o n e d a y a t a ti m e ) ,

s o p e r f o r m e r s m u s t p l a y a p i e c e s e q u e n t ia l ly a s i t u n f o l d s i n t im e . A n d

w h i l e t h e p i a n i s t m u s t p l a y o n e f i n g e r a f t e r t h e o t h e r , s h e k n o w s t h e

p i e c e n o t a s a s e q u e n c e o f s e p a r a te n o t e s o r a c ti o n s, b u t a s a s e q u e n c e

o f s h a p e s, f ig u r a l m o v e m e n t s , h a n d i n g s - w h a t I h a v e c a l le d a f e l t

p a t h t h r o u g h a p i e c e . F e l t p a t h s a r e , I t h i n k , t h e m o s t i n t im a t e w a y o f

k n o w i n g a n d a l s o h e a r i n g a p i e c e f o r t h e a rt is t p e r f o r m e r a s w e l l a s f o r

t h e n o v i c e w h o p l a y s b y e ar .

A s e v i d e n c e , tr y to s in g a s o n g y o u k n o w v e r y w e l l s ta r ti n g s o m e -

w h e r e i n t h e m i d d l e . F o r in s t an c e , t r y si n g in g A m e r i c a ( M y c o u n t r y

' ti s . . . ) s t a rt in g r i g h t u p f r o m o f l ib e r ty . A n d i f y o u d o p l a y a n

i n s tr u m e n t , r e c a ll w h a t h a p p e n s w h e n , h a v i n g l e a r n e d t o p l a y a p i e c e

f r o m m e m o r y , y o u f o r g e t s o m e w h e r e in t h e m i d d l e . I n b o t h s it ua t io n s ,

y o u m o s t l ik e l y h a v e t o g o b a c k a n d s t a rt o v e r a g a in f r o m t h e b e g i n -

n i n g o r a t le a s t f ro m s o m e m e m o r a b l e s t r u c tu r a l b o u n d a r y . A s f u r t h e r

e v i d e n c e i t i s a m u s i n g t o w a t c h m u s i c s t u d e n t s i n t h e tr a d i t io n a l t a s k

c a l le d t a k i n g d i c ta t io n - w r i ti n g o u t i n S M N , a m e l o d y p l a y e d to

t h e m b y t h e i n s t ru c t o r. T h o s e w h o p l a y a n i n s t r u m e n t a r e q u ie t ly , p r o b -

a b l y u n a w a r e t h e y a r e d o i n g s o, f i n g e r in g th e d i c t a te d p a s s a g e o n a n

i m a g i n a r y i n s t r u m e n t ( t h e f l u te p l a y e r u p i n t h e a ir , h o r i z o n t a l l y ; t h e

p i a n is t o n h e r d e s k i n f r o n t o f he r ; th e g u i t a r p l a y e r o n t h e p r e t e n d n e c k

o f h is i n s t r u m e n t ) : i n o r d e r t o h e a r t h e m e l o d y , th e y n e e d t o f e e l it o n

t h e i r i n s t r u m e n t .

T h e s e , t h e n , a r e m y b e s t h u n c h e s a t t h e i n tu i ti v e , g e n e r a t i v e p r i m i t i v e s

f r o m w h i c h m u s i c a l d e v e l o p m e n t b u il d s a n d g r o w s : t h e a b il it y to f o c u s o n

a n d h e a r t h e a r ri v a ls a n d d e p a r t u r e s o f f i g u re s a n d p h r a s e s , r e s p o n s i v e n e s s

t o c o n t e x t u a l f u n c t i o n s , a n d t h e f e e l o f a p i e c e i n t h e f i n g e rs . 5 A n d i f I

a m r i g h t a b o u t t h e m , it is n o t s u r p r i s in g t h a t s t u d e n t s, o f t e n t h o s e w h o a r e

b e s t a t i m p r o v i s i n g a n d p l a y i n g b y e a r ( a s w e l l a s t h o s e w h o a r e b e s t a t

i m p r o v i s in g w h e n m a k i n g a n d f i x in g m e c h a n i c a l g a d g e t s ), a r e b a f fl e d a n d

d i s c o u r a g e d w h e n w e a s k t h e m t o s t a r t o u t b y l is t e n in g f o r , l o o k i n g a t , a n d

i d e n t i f y i n g t h e s m a l le s t , i s o l a te d o b j e c t s - t o c la s s i f y a n d m e a s u r e w i t h n o

c o n t e x t o r f u n c t i o n a l m e a n i n g . F o r i n s t r e s si n g i s o la t e d , d e - c o n t e x t u a l i z e d

o b j e ct s t o w h i c h o u r u n i ts o f d e s c r ip t i o n r e f e r - t o m e a s u r e a n d n a m e

o b j e c t s in s p i te o f w h e r e t h e y h a p p e n a n d t h e i r c h a n g i n g s t r u c tu r a l f u n c t i o n

5 I find t quite telling hat when I have, on a num berof occasions, asked a car m echanic

who has just solved som e knottyproblem, Ho w did you do that? he says, Sm art hands.

And w hen, in differentcontexts I have asked a jazz pianist the sam e question, How did

you do that? , af ter he has just perform ed a particularly ngen ious improvisation, he too

says, Sm art hands.



JE NNE B MBERGER

- we are asking students to put aside their most intimate ways of knowing

figures, felt paths, context and function.6

But how can we give beginning students structural figures to play and

to play with that match their units of perception instead of our units of

description? Here is the paradox, again: how can we give beginning music

students ready-made meaningful structural entities that they can hear and

work with, when no te s a re necessary to make them? It was in an effort

to answer these questions and to satisfy what I believe to be the needs of

beginning students that I was drawn to the possibilities of the computer,

now coupled with a synthesizer. Even though I resisted taking the plunge,

it became necessary as a means for doing what I could not do in any other

way.

E n t r a n c e s a n d E x i t s

In designing Impromptu and in working on the projects that form the stu-

dents' working context, the following general question became of critical

importance: where can students most effectively

e n t e r

their study of some

domain, in this case the musical world, and how can they can most effec-

tively pr oceed- where and when should they exit to somewhere/something

else? Figure 2 shows a working model o f entrances and exits reflected in

the design of the projects and in their progression .

As Figure 2 suggests and following my first two premises, Impromptu

makes it possible for students to begin their music study at the mid-level of

structure - figures, phrases, funct ions. From the outset, students work with

these meaningful structural entities, using them

s i m u l t a n e o u s l y a s u n i t s

o f d e s c r i p ti o n a n d u n i ts o f p e r c e p t io n Structural entities, what we call

tuneblocks, are represented in Impromptu as pattemed icons. With the

computer connected through a MIDI interface to any synthesizer, the icons

play when selected (clicked on). Given a set of tuneblocks, students use

them right from the outset as the functional elements to experiment with

6 It is revealing that in our Laboratory for Making Things in a local public school,

we have found that children who are virtuosos at building and analytically debugging

complex structures using legos and other materials, are often the same children who are

having the most difficulty n their regular classroom subjects. This is not so surprising since

the emphasis in classrooms is primarily on learning and manipulatingsymbols.In the Lab,

children are learning to move back and forth between action and symbolic descriptions

by making working systems with materials in ~real space/time and making working

systems (music and graphics) in the virtual world of computer design. And in that moving

back and forth, the emphasis is on

confronting the differences between them

In this way

children recogniZewhat they know how to do so well in one world, while also recognizing

the transformations hat might be necessary for making sense of the other, symbolicworld

of school (Bamberger, 1991b).



TURNING MUSIC THEORY ON ITS EAR 45

M o v i n g u p a nd d o v n th e s t r u c t u r a l l a d d e r

r l o t l v i c e l e b o r a t i o n / d e v o l o p rn D n t

S e c t i o n s a n d s e c t i o n a l f u n c t i o n s

S i m i l a r i t i e s a nO O l l f e r e lq c e s a c r o s s

e v e n t s d i ; t a n c o d i n t i m e

F i g u r e s , p h r a ~ e ~ , f u n c t i e n ¢ - - ~ t a b i l i t y / i n B t a b i l i t y

A r r l v a l e e nd D e p a r t u r e s

P | t c h a n d m e t r i c v a lu es

I n t e r v a l s , c h o r d s

n o t a t i o n

Figure 2 Moving up and down the structural ladder

t h e d e s i g n o f s im p l e c o m p o s i t i o n s . A s s t u d e n t s l i st e n t o t u n e b l o c k s , g r a b

t h e m s o a s to a r r a n g e a n d r e a r r a n g e t h e n i n t h e P l a y r o o m , th e i r e x p e r i m e n t s

i n g e n e ra t i n g m u s i c a l c o h e r e n c e a re m u c h l i ke t h e w o r k o f a c o m p o s e r

i n s k et c h in g o u t a p ie c e . F ig u r e 3 s h o w s I m p r o m p t u ' s T U N E B L O C K S

w i n d o w .

S u c c e s s f u l l y c o m p o s i n g a tu n e t h at m a k e s s e n s e a n d t h a t y o u l ik e ,

inev i tab ly r a i s es que s t ions wh en s tuden t s try to acco un t fo r the i r in tu i tive

c o m p o s i t i o n a l d e c i s i o n s . D e c o n s t r u c t i n g t o e x p l o r e w h a t t h e y c a n n o t y e t

d o s o w e l l , t h e y c a n l o o k a t t h e g e n e r a l s h a p e o r p i t c h c o n t o u r o f t h e i r

b l o c k s ( s h o w n i n t h e G r a p h i c s W i n d o w i n F i g u r e 3 ) . A n d l at er , t h e y c a n

o p e n u p t h e t u n e b l o c k s , m o v i n g d o w n th e s tr u c tu r a l l a d d e r to l o o k a t



6 J E A N N E B A M B E R G E R

TUrld~l cks

H I N ~ 0 h H ~ l ~ I J i - J I I i i I I

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i i i i i I I i i I I I I i i | I I I I I t I |

: ~ R l l i l i l i l i l i l l l l t l i l t l l l l l l l l l l l l l i l i l N I I I I l J l i l i l i l t l l i l i l i I ¢

, , , , ,

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Figure 3

T u n e b l o c k s w i n d o w .

t h e m o r e d e t a i le d o n t e n t s o f t h e s e m i d - l e v e l s tr u c t u r a l e n ti ti e s - p i t c h a n d

d u r a t i o n v a l u e s ( s e e F i g u r e 4 ). 7

C i r c l i n g b a c k u p t h r o u g h t h e i r f a m i l ia r , m i d - l e v e l f u n c t i o n a l e n ti ti e s ,

h e a t i n g t h e m n o w i n n e w w a y s , t h e y g o o n u p t h e s t r u c t u r a l l a d d e r t o

l a r g e r s tr u c t u r a l r e l a t io n s - t o s e c t i o n s a n d s e c t i o n a l f u n c t i o n s , to m o t i v i c

d e v e l o p m e n t a n d t o c o m p a r i s o n o f e v e n ts t h at a r e d i s ta n c e d i n t i m e w i th i n

a p ie c e . L i st e n i n g to c o m p o s i t io n s o f m o r e w e l l - k n o w n c o m p o s e r s , t h e y

s e e k t o a c c o u n t f o r t h e a f f e c t a n d f u n c t i o n o f t h e s e l a r g e r s t r u c t u r a l r e l a t io n s

b y c i r c l i n g d o w n a g a i n t h r o u g h t h e s t r u c tu r a l la d d e r to t h e d e t a i ls .

T h r o u g h t h i s p r o c e s s s t u d e n t s l e a r n f r e e l y to s h i f t t h e i r f o c u s o f a t t e n t io n

a m o n g t h e m a n y p o s s i b l e k i n d s o f m u s i c a l e n t it ie s a n d m u l t i p le d i m e n s i o n s

a t d i f f e r i n g l e v e l s o f s t ru c t u r e d e p e n d i n g o n t h e i r q u e s t io n s , w h a t t h e y w a n t

t o h e a r , a n d w h a t t h e y w a n t t o a c c o u n t f o r. F o r i n s t a n c e , t h e y m a y i n it ia l ly ,

a s y o u p r o b a b l y d i d , h e a r t h e e n d s o f t h e fi rs t t w o p h r a s e s o f D i d y o u

e v e r s e e a L a s s i e a s d i f f e r e n t ( in r e s p o n s e t o a g g r e g a t e d fu n c t i o n ) . B u t

l a te r , t a k i n g t h e m o m e n t a r y c o n f l u e n c e a p a r t, l o o k i n g i n t o t h e c o n t e n t s o f

7 S t u d e n t s c a n s e l e c t t h e i r o w n s p e c i a l n a m e f o r a b l o c k ( e .g . , f a s t , s a d ) w h i c h

r e p l a c e s t h e p a t t e r n e d i c o n . T h e y c a n a l s o d e s i g n t h e i r o w n p a t t e r n e d i c o n , o r t h e y c a n

r e q u e s t a n i c o n t h a t s h a d o w s t h e p i t c h c o n t o u r o f t h e b l o c k . A n d a f u l l s e l e c t i o n o f c o l o r s

i s a l s o ava i l ab le .



T U R N I N G M U S I C TH E O R Y O N I T S E A R 7

M e l n d y B l a c k I n f n :

I I

Icon; ~ ~

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N o m - : l ,1 t N o m = - I c = n l T y p m - i t : ~ ~

~thuv P i l c h : ~ b . N

Nulrl~ur

5 3 4 5 6 5 4 2 2 3

i t c h e s :

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K e u l C I

I m t r u m e n t

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by

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N , w e I O C K 1 [ : l ]

Figure 4 The

c o n t e n t s o f b l o c k s

t h e b l o c k s , t h e y s u r p r i s in g l y d i s c o v e r t h a t t h e p h r a s e e n d i n g s a r e t h e s a m e

( in p i tc h ) . S a m e o r d i f f e r e n t d e p e n d s o n w h a t y o u c h o o s e t o o r a r e a b l e t o

s e l e ct f o r a t t en t i o n . T o a c c o u n t f o r th e d i f f e re n c e b e t w e e n t h e t w o p h r a s e

e n d i n g s , i t tu r n s o u t t h a t y o u h a v e t o s h i ft y o u r a t t e n t io n t o t h e r h y t h m i c

d i m e n s i o n o f t h e tu n e . B u t t o d o t h at , y o u h a v e to l i b e r a te t h a t s e p a r at e

d i m e n s i o n f r o m t h e m e l d , t a k i n g a p a r t th e a g g r e g a t e t h a t w a s y o u r i n it ia l

u n i t o f p e r c e p t i o n .

L e a r n i n g s e l e c t iv e l y t o m o v e t h e i r h e a r i n g u p a n d d o w n t h e s t r u c tu r a l

l adde r a l so he lps s tud en t s tow ards a c r it i ca l ab i l ity in the appre c ia t ion o f

m u s i c a l c o m p l e x i t y : to c o o r d i n a t e d e t a il a n d l a r g e r d e s i g n . A s t h e c o m -

pose r , Roger Ses s ions , u sed to s ay : the de ta i l s a r e genera t ing the l a rge

d e s i g n a n d t h e l a r g e d e s i g n i s i n f o r m i n g t h e d e t a il s , t o g e t h e r t h e y m a k e t h e

u n i q u e c o h e r e n c e o f a c o m p l e x c o m p o s i t io n .

G o i n g o n t o p r o j e c t s in v o l v i n g r h y t h m i c s t ru c t u re , s t u d e n ts c o m p o s e

w i t h d r u m b l o c k s t o m a k e p e r c u s s io n a c c o m p a n i m e n t s f o r t u n e s.

S t u d e n t s l i st e n t o a tu n e , k e e p t i m e b y c l a p p i n g t o t h e u n d e r l y i n g b e a t

h i e r a r c h y t h a t t h e tu n e g e n e r a te s , a n d t h e n m a t c h i n g t h e i r l iv e p e r f o r-

m a n c e , c o m p o s e a c c o m p a n i m e n t s t o t h e ir t u n e s u s i n g m u l t i p l e p e r c u s s io n

i n s t ru m e n t s . L a te r , m o r e i n t e r e s t in g p a t te r n s a r e c o m p o s e d : f ir st , a c c o m -

p a n i m e n t s t h a t f i t w i t h th e u n d e r l y i n g b e a t st r u c tu r e o f a t u n e , a n d t h e n

a c c o m p a n i m e n t s t h a t c o n f l ic t w i t h i t s u n d e r l y i n g r h y t h m i c o r g a n i z a ti o n .



48 J EANNE BAMBERGER

r ~ i

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I < ~ I ~:~ I :I'iI~IN :I:~I~II:I'~I~'~:i~i:N~'~I~I~I~I I~I~I~I~I~d'~III~:~INI~I~IiI~I~I~I~I ~INI I I I~

Figure 5 A f r i c a n d r u m p i ec e

P . I ;

. I . I

I I 1 .

I I 1 1 1

i : m

S t ud e n ts g o o n t o c o m p o s e m o r e c o m p l e x p e rc u s s io n p i e ce s , m o d e l e d f o r

i n st a n c e , o n A f r i c a n , B a l in e s e , o r ja z z d r u m r h y t h m s t h a t t h e y h a v e l i s te n e d

t o i n c la s s . T e m p o r a l re l a t i o n s o f r h y t h m p i e c e s a r e c a p t u r e d i n s p a c e - t o -

t i m e a n a l o g g r a p h i c r e p r e s e n t a t i o n s ( s e e F i g u r e 5 ) . T h e s e h e l p s t u d e n t s to

a c c o u n t f o r t h e r e la t io n s h i p s t h e y h a v e c o m p o s e d a n d a l so t o m a k e l i v e,

g r o u p p e r f o r m a n c e s o f t h e ir p i e c e s o n r e a l p e r c u s s io n i n s t r u m e n t s.

T w o o t h e r p r o je c t s e x t e n d t h e i r w o r k : p l a y i n g c a n o n s ( s u c h a s F r e r e

J a c q u e s , T h r e e B l i n d M i c e , o r r i d d le c a n o n s c o m p o s e d b y M o z a r t ) , a n d

h a r m o n i z i n g m e l o d i e s . S t u d e n ts w o r k o n b o t h o f t h e s e p r o j e c ts i n t e r a c -

t i v e ly in r e a l ti m e , t h u s c o m b i n i n g p e r f o r m a n c e a n d l i s te n i n g .

T o m a k e p r o je c t s m o r e e x c i ti n g , s tu d e n t s u s e I m p r o m p t u ' s M I D I i n te r -

f a c e to c h o o s e a w i d e r a n g e o f b o t h m e l o d y a n d p e r c u s s i o n i n st r u m e n t s

f r o m a s e l e c t e d s y n t h e s iz e r b y s i m p l y c l ic k i n g o n a n i n s t r u m e n t n a m e i n

t h e I n s t r u m e n t M e n u . 8 A n d s i n c e a ll m u s i c is s a v e d a s M I D I d a ta , c o m p o -

s i ti o n s c a n b e s e n t to a n y m u s i c e d i t i n g a p p l i c a t io n t o b e p r i n t e d i n s t a n d a r d

m u s i c n o t a t i o n .

ultiple Representations

I m p r o m p t u i n c l u d e s , a s y o u h a v e s e e n , s e v e ra l d i f f e re n t

kinds

o f n o t a t io n s .

T o h e l p s t u d e n t s l e a r n to m o v e u p a n d d o w n t h e s tr u c t u r a l l a d d e r, e a c h o f

t h e s e m u l t i p le r e p r e s e n t a ti o n s h a s b e e n d e s i g n e d t o s h o w d i f f e re n t kinds

o f e n t i t ie s a t d i f f e r e n t levels o f s t ru c tu r e . T h e b l o c k - i c o n s c a p tu r e t h e m o r e

a g g r e g a t e d f ig u r a l l e v e l o f a m e l o d y ; p i t c h c o n t o u r g r a p h i c r e p r e s e n t a -

t i o n s t r a c e t h e g e n e r a l p i t c h / t i m e s h a p e o f a b l o c k a s it p l a y s ; r h y t h m b a r s

a r e a s p a c e - t i m e a n a l o g r e p r e s e n t a t i o n s h o w i n g o n l y t e m p o r a l r e l a t i o n s .

A n d m o v i n g f a r t h e r in o n t h e d et a il s, o p e n i n g u p a b l o c k ( b y c l i c k i n g o n

s Improm ptu ma kes it e as y for instructors to m ap instrument nam es to the instru-

me nt num bers of individual synthesizers through its M IDI m enu. Imp romptu includes

pre-programm ed mappings for any Gen eral M IDI synthesizer and specific mappings for

most comm onlyused synthesizers.



TURNING MUSIC THEORY ON ITS EAR 49

B l o c k s

P i t c h C o n t o u r

R h g t h m B a r s

P i t c h e s :

D u r a t i o n s

e •

~

• O Q • I

I I I I I I I I I I I I I i

3213211111222232

4 4 8 4 4 8 2 2 2 2 2 2 2 2 4 4 8

S t a n d a r d

usic

N o t a t i o n

Figure6

Multiplerepresentations or H ot Cross Buns .

i t w i t h t h e m a g n i f y i n g g l a s s ) , s c a l e d e g r e e a n d l e t t e r n o t a t i o n f o r p i t c h

a r e d i s p l a y e d , a s w e l l a s p r o p o r t i o n a l n u m b e r s f o r t i m e v a l u e s . S t u d e n t s

a ls o h a v e th e o p p o r tu n i t y t o b u i ld b i g g e r b lo c k s m a d e u p o f s tr u c tu r a l ly

s i g n if i c a n t g r o u p i n g s o f t h e g i v e n t u n e b l o c k s . I c o n s f o r b i g g e r b l o c k s

r e p r e s e n t a n d p l a y t h e s e m o r e a g g r e g a t e d s t r u c tu r a l e n t it ie s . S t u d e n t s c a n

e d i t t h e c o n t e n ts o f b lo c k s b y c h a n g i n g p it c h o r d u ra t io n , a n d t h e y c a n

a ls o m a k e t h e ir o w n e n t i re l y n e w b l o ck s . 9

An E xamp le: Learning w ith M ultiple Representations

H o w d o s t u d e n t s u s e t h e s e m u l t i p l e r e p r e s e n t a t i o n s t o m a k e e x p l i c i t a n d

t h u s to l e a r n w h a t t h e y a l r e a d y i n tu i t iv e l y k n o w ? C o n s i d e r th e s i m p l e t u n e ,

H o t C r o s s B u n s , a n d i ts m u l t i p l e r e p r e s e n t a t i o n s a s s h o w n in F i g u r e 6.

S t u d e n ts b e g i n b y r e - c o n s t ru c t i n g t h e t u n e w i t h t u n e b l o c k s - a t a s k

t h a t i s i m m e d i a t e l y o b v i o u s f o r m o s t . B u t t h e a c t o f c o n s t r u c t i o n s i m u l -

t a n e o u s l y t u r n s i n to a p r o c e s s o f c o n s t r u c t i v e a n a l y s i s . L o o k i n g a t t h e

c o m p l e t e d s e q u e n c e o f b l o c k s o n t h e s c r e e n a n d l is te n i n g b a c k t o it, t h e

l a r g e r s tr u c t u r a l r e l a t io n s o f th e t u n e e m e r g e : t w o r e p e a t e d f i g u re s ( A ) ,

con t ras t (B ) , and re tu rn (A ' ) .

9 Standardmu sic notation is not provided within Impromptu n p art to en courage stu-

dents to learn to write music notationusing paper and pen cil, and also because, if they w ish

to compare, hey can send computeddata to any comm onmusic editing software or a print

out



5 JEANNEBAMBERGER

B

R e p e a t e d f i g u r e s o n t r a s t R e t u r n

F i g u r e 7

Structural relations

•

• - ' I I ' '

• • OO OO •

P | t o h C o n t o u r

U I i I I I I l i l l l I I

R h y t h m B a r s

F i g u r e 8 Revealing distinctions

W h i l e w o r k i n g o n t h e r e - c o n s t r u c ti o n o f t h e t u n e s t u d e n t s c a n c h o o s e

t o w a t c h m o r e f i n e -g r a i n e d r e p r e s e n t a t i o n - e i th e r p i t c h c o n t o u r o r r h y t h m

b a r s g r a p h i c s i n th e G r a p h i c s W i n d o w . W h i l e b o t h k i n d s o f r e p r e s e n t a ti o n s

s e e m t o fi t w i t h t h e b l o c k s c o m p a r i n g t h e m r e v e a l s d i s ti n c t i o n s t h a t a r e

h i d d e n i n th e m o r e a g g r e g a t e d b l o c k s r e p r e s e n t a t i o n .

F o r in s t a n c e t h e re t u r n t o t h e o p e n i n g f i g u r e a f t e r t h e c o n t r a s t i n g m i d d l e

i s p e r f e c t l y c l e a r i n t h e t u n e b l o c k s r e p r e s e n t a t i o n a n d i t i s a l s o p e r f e c t l y

c l e a r in t h e p i t c h c o n t o u r re p r e s e n ta t io n : t h e s a m e t h r e e - n o t e d e s c e n d i n g

c o n f i g u r a t i o n s ta n d s o u t b o t h a t th e b e g i n n i n g a n d a t t h e e n d . o f th e t u n e .

B u t lo o k i n g a t t h e r h y t h m b a r s t h e r e t u r n is o b s c u r e d . S p e c i f ic a l l y t h e

b o u n d a r y b e t w e e n t h e c o n t r a s t i n g m i d d l e f i g u r e a n d t h e r e t u r n s e e m s t o

h a v e d i s a p p e a r e d - t h e t u n e e n d s w i t h j u s t t w o e v e n t s i n s te a d o f th r e e .

I n d e e d i f y o u c la p j u s t th e r h y t h m o f t h e tu n e o r p l a y j u s t t h e r h y t h m

u s i n g a n I m p r o m p t u d r u m y o u h e a r th e s a m e e ff e c t - t w o e v e n t s at t h e

e n d a n d t h e b o u n d a r y b e t w e e n t h e f a s t e r m i d d l e a n d t h e r e t u r n s e e m s i n

t h e w r o n g p l a c e . W h y ?

_ _ [

• ooo o •

• • •

g i B e

T h e 3 n o t e r e tu r n i s c l e a r

Just 2

e v e n t s ; t h e r e t u r n h a s d i s a p p e a r e d

F i g u r e 9



TURNING MUSIC THEORY ON ITS E R 51

T h e b o u n d a r y i s o b s c u r e d b e c a u s e w h e n o n l y t e m p o r a l r e l a t i o n s a r e

r e p r e s e n t e d , t h e f a s t e r e v e n t s o f t h e m i d d l e f i g u r e run right on into the

return; t h e r e is n o c h a n g e to g e n e r a t e a b o u n d a r y a t t h a t m o m e n t . L o o k i n g

a g a i n n o w a t th e p i t c h c o n t o u r g r a p h i c s w h e r e p i t c h r e la t i o n s s t a n d o u t , it

b e c o m e s c l e a r t h a t i t i s t h e p i t c h d i m e n s i o n w h i c h i s c r i t i c a l i n c r e a t i n g

a n d a c c o u n t i n g f o r t h e p e r c e i v e d b o u n d a r y - t e m p o r a l r e l a ti o n s a l o n e w i l l

n o t s u c c e e d .

S e v e r a l i n s i g h t s r e s u l t f r o m t h i s c o m p a r i s o n a n d e a c h i s a n e x a m p l e

o f h o w r e f l e c t i v e l y m u l t ip l e r e p r e s e n t a t i o n s c a n h e l p s t u d e n t s l o o k A T

t h e i r i n t u i t i v e k n o w - h o w w h i l e a l s o b u i l d i n g o n i t . F i r s t , b y c o m p a r i n g

t h e t w o r e p r e s e n t a t i o n s , t h e p i t c h a n d t i m e d i m e n s i o n s b e c o m e c l e a r l y

d i f f e r e n t i a t e d . A l o n g w i t h t h a t , t h e r o l e e a c h d i m e n s i o n p l a y s i n g e n e r -

a t i n g b o u n d a r i e s i s a l s o r e v e a l e d : O n c e t h e t w o d i m e n s i o n s h a v e b e e n

d i s e n ta n g l e d , it b e c o m e s c l e a r th a t, i n t h e m o m e n t a r y c o n f l u e n c e o f o u r

l iv e l is t e n in g , t h e p i tc h d i m e n s i o n w i n s o u t i n g e n e r a t i n g t h e c o h e r e n c e w e

he ar . 10

S e c o n d , o u r h a b i ts o f g r o u p i n g i n v i s u a l / sp a t ia l p e r c e p t i o n a r e s i m i l a r to

o u r h a b i t s o f g r o u p i n g i n t e m p o r a l p e r c e p t i o n , b u t n o t n e c e s s a r i l y i n p it c h

p e r c e p t io n . A s t h e g e s t a lt p s y c h o l o g is t s h a v e t a u g h t u s , w h e r e e l e m e n t s

t h a t a r e o t h e r w i s e t h e s a m e a r e r e l a t iv e l y c l o s e r t o g e t h e r in space w e s e e

t h e m a s g r o u p i n g t o g e t h e r ; a n d w h e r e e l e m e n t s a r e f a r t h e r a p a r t in s p a c e ,

w e s e e t h e s p a c e b e t w e e n a s f o r m i n g a b o u n d a r y b e t w e e n g r o u p i n g s .

S i m i l a rl y , w h e n s o u n d i n g e v e n t s th a t a r e o t h e r w i s e th e s a m e , a r e r e l a t iv e l y

c l o s e r t o g e t h e r i n

time

w e h e a r t h e m a s g r o u p i n g t o g e th e r ; w h e n e v e n t s

a r e r e l a t i v e l y f a r th e r a p a r t in t i m e , w e h e a r t h e g a p b e t w e e n a s f o r m i n g a

b o u n d a r y b e t w e e n g r o u p in g s . T h e s p a c e - f o r- t im e g r a p h ic s c l e a r l y re v e a l

t h e s e s i m i la r it ie s b e t w e e n t h e t w o m o d e s o f p e r c e p ti o n .

A n d t h i r d , t h e c o m p a r i s o n l e a d s a g a i n t o s o m e i n s i g h t c o n c e r n i n g

t h e s tr o n g i n f l u e n c e o f c o n t e x t. F o r i n s t a n c e, e v e n t h o u g h t h e b e g i n n i n g

a n d e n d i n g b l o c k s a r e e x a c t l y th e s a m e w i t h re s p e c t t o t h e ir p i tc h e s a n d

d u r a t i o n s , T h e s a m e b l o c k , a s o n e t u n e b l o c k s p l a y e r r e p o r t e d , i s n o t

t h e s a m e a t t h e e n d . W h e n t h e b e g i n n i n g b l o c k is h e a r d fo l l o w i n g r ig h t

o n a f t e r t h e f as te r , a n d m o r e o n w a r d - m o v i n g m i d d l e f ig u r e, w e h e a r t h e

b e g i n n i n g c o n f i g u r a t i o n n o t a s a s ta r t- u p b u t a s a r e s o lu t i o n . T h e f i g u r e

i s i n f e c t e d b y it s n e w c o n t e x t u a l a s s o c i a ti o n , a c q u i r i n g i n t h e p r o c e s s

d i f f e re n t m e a n i n g a n d f u n c ti o n .

10 W hich is just the opposite of La ssie. Th ese findings sugge st em pirical resea rchques-

tions w hichsom e of m y students have been playing with: when is it the case that pitch wins

out in generatingboundaries, wh at make s the differences n the situtations wh ere rhythm

wins; and how wou ld you de sign experiments o find out?



5 2 JEANNE BAMBERGER

W h a t a b o u t m o v i n g i n s t i l l f u r t h e r t o a n o t a t i o n t h a t r e p r e s e n t s p i t c h

a n d d u ra t io n w i t h n u m b e r s ?

P ltC h~ |: [ 3 2 3 2 2 2 2 2 3 2

D ura tia n s : [ 4 4 8 4 4 8 2 2 2 2 2 2 2 2 4 4 8

T h i s r e p r e s e n t a t i o n i s c l o s e t o c o n v e n t i o n a l n o t a t io n . H o w e v e r , t o m a k e

s e n s e o f t h e s e t w o s e ts o f n u m b e r s ( w h i c h h a v e e n t ir e ly d i ff e re n t m e a n i n g s )

r e q u i r e s fi rs t m e n t a l l y c o n s t r u c t i n g t h e f r a m e w o r k , the reference systems

i n t e rm s o f w h i c h t h e n u m b e r s g a i n m e a n i n g . I f w e h e l p s t u d e n t s c o n s t r u c t

t h e s e r e f e r e n c e s y s t e m s , th e n u m b e r s g a i n t h e p o w e r th a t h as m a d e t h e m

s u rv i v e; i f w e f ai l t o d o s o , t h e s y m b o l s m a y b e u s e d b y s t u d e n ts a s j u s t

a c o d e f o r b u t t o n - p u s h i n g . I t i s a b i t l i k e p l u g g i n g i n t h e n u m b e r s i n

a n e q u a t i o n ( W = F * D ) w i t h o u t w o r r y i n g a b o u t w h a t r e a l w o r l d o b j e c t s t h e

n u m b e r s a n d t h e r e l a ti o n s d e s c r i b e d b y t h e e q u a t i o n r e f e r t o . 11

T h e s e s y m b o l s a n d t h e r e fe r e n c e s y s t e m s in t er m s o f w h i c h t h e y a c q u i r e

m e a n i n g , g a i n p o w e r b y b e i n g i n t e rn a l l y c o n s i s t e n t . H o w e v e r , f o r j u s t t h is

r e a s o n th e s y m b o l s o f S M N t e n d t o o b s c u r e th e c h a n g i n g c o n t e x t u l m e a n -

i n g t h a t t h e s a m e p i t c h o r d u r a t i o n c a n a c c r u e a s a m e l o d y u n f o l d s . A n d i t

i s f o r t h is r e a s o n t h a t s t u d e n t s w o r k i n g i n t h e I m p r o m p t u e n v i r o n m e n t a r e

i n t r o d u c e d t o t h e s e n o t a t i o n a l c o n v e n t i o n s o n l y w h e n t h e p a r t i c u l a r k i n d s

o f en t it ie s a n d r e la t io n s t o w h i c h t h e y r ef e r b e c o m e n e c e s s a r y a s m e a n s

t o w a r d s a n s w e r i n g t h e s t u d e n t s ' o w n q u e s ti o n s .

T h e f o c u s t h e n t u r n s t o a c o n s i d e r a t i o n o f d i f f e r e n c e s in t h e k i n d s o f f e a -

t u r es r e p r e s e n t e d b y t h e s e v a r i o u s r e p r e s e n t a ti o n s , a n d t o f in d i n g c o n g e n i a l

m e a n s f o r d e v e l o p i n g t h e

transformations necessary to moving meaning-

fully among them.

I n v e n t e d d r a w i n g s f o r r h y t h m s h a v e b e e n c r u c i a l i n

g u i d i n g th i s p r o c es s .1 2 B e g i n n i n g w i t h t u n e b l o c k s , u n i t s o f d e s c r i p t i o n

t h a t c l o s e l y m a t c h th e n o v i c e s ' s i n t u it i v e u n i t s o f p e r c e p t i o n , s t u d e n t s

m o v e i n t o m o r e d e t a i l e d b u t s t i l l b a s i c a l l y c o n f i g u r a t i o n a l g r a p h i c s t h a t

c l o s e l y m a t c h t h e i r i n v e n t e d d r a w i n g s - p i t c h c o n t o u r a n d r h y t h m b a r s .

O n l y t h e n d o t h e y m o v e o n t o t h e c o n v e n t i o n a l s y m b o l s t h at r e fl ec t , e v e n

d e p e n d o n , th e p r e v i o u s m e n t a l c o n s t r u c t i o n o f a n o r d e r e d s y s t e m o f r e la -

t i o n s - n u m b e r s f o r p i tc h a n d d u r a t io n . I n a p r o f o u n d s e n s e th e p r o c e s s o f

11 For exam ple, the M IT students in my education classes hav e all passed freshman

physics, are thoroughly familiar with the equation W=F *D , but fail to recognize its em bod-

iment in the pulley mechanism they are asked to design and build.

12 I refer, here, to drawings made b y children ove r the age of eight or nine which are

much the same as those made by college students with no formal music training.



T U R N I N G M U S I C T H E O R Y O N I TS E R

53

m o v i n g t h r o u g h t h e s e v a r i e d r e p r e s e n t a t io n s b ri n g s i n t o e x i s t e n c e k i n d s o f

e l e m e n t s , f e a tu r e s , a n d r e l a t io n s t h a t w e r e s i m p l y n o t c o n s t i t u e n t s o f th e

s t u d e n t s ' w o r k i n g m u s i c a l u n i v e r s e b e f o r e .

S o w e n e e d t o a s k : h o w d o m u l t i p le r e p r e s e n t a ti o n s a c t u a l ly i n f l u e n c e

e x p e r i en c e ? D o n e w l y e x i st e n t c o n st iu e n t s c h a n g e o u r i m m e d i a t e e x p er i-

e n c e o f a p i e c e ? D o m u l t i p l e re p r e s e n t a t io n s r e su l t i n m u l t i p l e h e a r i n g s ?

A n d c a n w e c o m p a r e t h e s e m u s i c a l q u e s t i o n s t o q u e s t io n s i n o t h e r d o m a i n s :

d o e s i n v e n t i n g ( o r r e - in v e n t i n g ) th e c o n s i t u e n t s d i s ta n c e , t i m e , a n d t h e i r

r e l a ti o n s h i p , c h a n g e o u r i m m e d i a t e e x p e r i e n c e o f g o i n g fa s t e r?

T o c o m e f u l l c ir c le , I w i l l a r g u e t h a t th e g o a l o f e d u c a t i o n ( f o r b o t h

t e a c h e r s a n d s t u d e n t s ) s h o u l d b e t o m a k e q u e s t i o n s li k e th e s e b e c o m e a n

i n te g r a l p a r t o f t h e l if e o f c l a s s r o o m s . A n d c r it ic a l t o m a k i n g t h a t h a p p e n

is lea rn ing to con s t ruc t , to unde r s tand , to apprec ia te , and to con f ron t the

d i f f e re n c e s a m o n g m u l t i p l e p o s s i b l e r e p r e s e n ta t i o n s o f p h e n o m e n a - t o

c h o o s e w h i c h k i n d s o f k i n d s a n d a t w h a t l e ve l o f d e ta i l y o u w a n t t o f o c u s

y o u r a t te n t i o n , a n d t o k n o w h o w t o d o t h a t m o s t e f f e c ti v e l y d e p e n d i n g o n

w h e n , w h e r e , a n d w h a t fo r.

A R E F L E C T I V E P R A C T I C U M

I n k e e p i n g w i t h m y d e s i g n p r i n c i p l e s , t h e n , I h a v e e x p l o i t e d c o m p u t e r

t e c h n o l o g y t o d o w h a t I c o u l d n o t d o i n a n y o t h e r w a y. I n s t e a d o f l o o k i n g

f o r w h a t t o d o w i t h t h e t r e n d y t e c h n o l o g y , o r c r e a t in g a r e c e p t a c l e h o l d i n g

i n f o r m a t i o n t h a t m i m i c s w h a t w e a l r e ad y h a d , t h e t e c h n o l o g y b e c o m e s

a r e s o u r c e t h a t e n c o u r a g e s s t u d e n t s to e x p e r i m e n t w i th , i n t e r ro g a t e , a n d

d e v e l o p t h e i r o w n i n t u i ti v e m u s i c a l k n o w - h o w . S t u d e n t s a re a s k e d t o w a t c h

themse lves a t work , to r e f l ec t on the p roces s , a s an in teg ra l pa r t o f the

p roce s s , it s el f. To do so , s tuden t s keep a log as they go a long - a log o f

t h e i r s p o n t a n e o u s r e s p o n s e s t o a h e a r i n g a n d o f t h e d e c i s i o n s t h e y m a k e

a lon g the way . The y a re a l so asked to try to a c c o u n t fo r the i r dec i s ions , and

t o p o n d e r h o w t h e i r d e c i s i o n s a re m a d e m a n i f e s t in t h e u n f o l d i n g s t r u c tu r e

o f t h e i r f i n a l c o m p o s i t i o n s .

S t u d e n t s a r e u r g e d t o k e e p i n m i n d t h a t t h e c o m p u t e r l it e ra l ly r e f le c t s

back , mi r ro r ing in every de ta il w ha t i t i s a ske d to do . Bea r ing th i s in m ind ,

a s u rp r is in g h e a r in g , a n u n e x p e c t e d t a lk - b ac k , c a n b e c o m e a m o m e n t o f

p o i g n a n t i n s ig h t , t h e tr i g g e r f o r a s t o p - a n d - t h in k - I w o n d e r w h y t h a t

h a p p e n e d ?

I t i s t h e s e c o n t i n u i n g i n v e s ti g a ti o n s i n to t h e i r o w n m u s i c a l i n t e l l i g e n c e

w h i c h b e c o m e th e g e n e r a t i v e b a s e f o r d e v e l o p i n g h e a r i n g s a n d a p p r e c i a -

t i o n s t h a t g o b e y o n d w h a t s t u d e n t s k n o w h o w t o d o a l r e a d y t o k n o w i n g

a b o u t a n d k n o w i n g why . G o i n g b e y o n d t h e i r i n i ti a l m u s i c a l i n t u it i o n s, t h e y



5 4 J E A N N E B A M B E R G E R

a r e a b l e to m a k e e x p l i ci t , f o r e x a m p l e , w h y t h e y h e a r p a r ti c u l a r g r o u p i n g

b o u n d a r i e s, w h y o n e p h r a s e w a n t s to fo l lo w a no t he r , w h y o n e p h r a s e

s o u n d s l i k e a n e n d i n g w h i l e a n o t h e r m o r e l i k e a b e g i n n i n g , e v e n t o a s k

w h a t g e n e r a t e s a b e a t a n d w h a t g e n e r a t e s w e a k a n d s t r o n g b e a ts . A s o n e

s t u d e n t w r o t e i n h e r l o g a t t h e e n d o f h e r w o r k o n t h e f ir st tu n e - b u i l d i n g

project :

A b i g q u e s t i o n i n m y m i n d : w h a t m a k e s a c e r t a i n s e q u e n c e o f n o t e s , b l o c k s - s o u n d l i k e

a n e n d i n g ? W e s e e th a t m o s t p e o p l e a g r e e o n w h a t t h i n gs h a v e e n d i n g s o u n d s , s o w h a t

makes it that way?

I n s e a r c h i n g f o r a n s w e r s t o q u e s t i o n s s u c h a s t h i s o n e t h a t t h e y h a v e p u t

t o t h e m s e l v e s , s tu d e n t s b e g i n t o h e a r a n d t o a p p r e c i a t e m u s i c a l c o m p l e x i t y ,

t o h e a r t h e d e t a i l s i n t h e l a r g e r d e s i g n . R a t h e r t h a n g i v i n g u p t h e i r i n tu i t io n s ,

t h e y l e a r n i n t h e s e r v i c e o f b e t t e r u n d e r s t a n d i n g t h e m .

N E X T S T E P S: A N E W P R O B L E M

W o r k i n g w i t h t h e s t u d e n t s h a s c o n f i r m e d m y i n i t ia l h u n c h t h a t if g iv e n

t h e o p p o r t u n i t y t o s t a r t w i t h w h a t t h e y k n o w h o w t o d o a l r e a d y , a n d

t o a c t i v e l y e x p l o r e a n d e x p e r i m e n t w i t h m u s i c a l m a t e r i a l s a n d r e l a t i o n s

t h a t p r o g r e s s i v e l y b u i l d o n t h e s e i n t u i ti o n s , s tu d e n t s c o m e t o u n d e r s t a n d ,

a p p r e c i a t e , a n d p e r h a p s m o s t i m p o r t a n t l y , t o c a r e a b o u t a n d b e m o v e d

b y c o m p o s i t i o n s t h a t p re v i o u s l y p a s s e d t h e m b y . A t t h e s a m e t i m e , u s i n g

I m p r o m p t u m y s e l f a n d i n w a t c h i n g t h e s t u d e n ts , I a m d i s c o v e ri n g a sp e c ts

o f m u s i c a l s t r u c t u r e a n d p a r t i c u l a r l y a s p e c t s o f m u s i c a l p e r c e p t i o n t h a t

h a v e m a d e m e r e - th i nk n o t o n l y s o m e o f th e c o m m o n a s s u m p t io n s o f

m u s i c i n s t r u c t io n , b u t t h o s e o f m u s i c t h e o r y a s w e l l .

T h e p r o b l e m I f a c e , n o w , is n o t w i t h t h e s t u d e n t s b u t w i t h m u s i c f a c ul ty .

S o m e y o u n g e r f a c u lt y i n m o r e p r o g r e s si v e m u s i c d e p a r t m e n t s h a v e g r e e te d

I m p r o m p t u a n d it s u n d e r l y i n g p r e m i s e s a s p r o d u c t iv e l y a d d re s s in g t h e

p r o b l e m s t h e y h a v e b e e n c o n f r o n ti n g : - s tu d e n t s c a n b e c o m e e n g a g e d w i t h

t h e i r o w n l e a r n i n g i n s t e a d o f b e c o m i n g d i s e n g a g e d , d i s e n c h a n t e d , d r i f ti n g

a w a y f r o m e v e n p a s s i n g t h e c o u r s e b e c a u s e i t s e e m s to t h e m i r re l e v a n t t o

w h a t t h e y c a r e a b o u t .

B u t o t h e r s s e e t h i s a p p r o a c h a s c h a n g i n g t ra d i t io n i n r a t h e r t o o d r a s t ic

w a y s . T h i s i s n o t s u rp r is in g : T h e c o m b i n a t i o n o f a c o m p u t e r e n v i r o n m e n t

w h i c h m a n y f a c u l t y s ti ll f in d s t r a n g e a n d i n t i m i d a t i n g , n o t a t i o n s t h a t a r e

d i f f e re n t f r o m t h e t h o s e t h r o u g h w h i c h t h e y u s u a l l y t h i n k a n d a c t , t o g e t h e r

w i t h a t e a c h i n g a p p r o a c h t h a t s tr e ss e s e x p e r i m e n t r a t h e r t h a n d r i ll , h a s t h e

e f fe c t o f p u l li n g t h e r u g o u t f r o m u n d e r t h e m . B u t h o p e f u l ly , g i v e n t im e

a n d t h e o p p o r t u n i ty t o p l a y w i t h th e n e w a p p r o a c h , m o r e w i l l c o m e t o



TURNINGMUSICTHEORYON ITS EAR 55

r e a l iz e , a s I h a v e , t h a t a p e r i o d o f c o n f u s i o n a n d d i s e q u i l i b r i u m c a n , w i t h

p a t i e n c e , g e n e r a t e n e w i n s i g h t a n d e v e n a r e n e w e d i n te r e s t i n t e a c h i n g .

R E F E R E N C E S

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M u s i c a n d T h e a t e r A r ts

M a s s a c h u s e t t s I n s ti t u te o f T e c h n o lo g y

C a m b r i d g e

M A 0 2 1 4 0

U S A