Dept. for Speech, Music and Hearing
Quarterly Progress and
Status Report
Beat theories of musical
consonance
Nordmark, J. and Fahlén, L. E.
journal:
volume:
number:
year:
pages:
STL-QPSR
29
1
1988
111-122
http://www.speech.kth.se/qpsr
STL-QPSR 1 / 1 9 8 8
BEAT THEORIES OF MUSICAL CONSONANCE
J a n Nordmark* and L e n n a r t E. ~ a h l h n *
C.
Abstract
Helmholtz' ' b e a t t h e o r y o f consonance h a s r e c e n t l y been g i v e n a new
lease o n l i f e , m a i n l y t h r o u g h t h e t h e o r y o f P l o m p a n d L e v e l t ( " T o n a l
consonance a n d c r i t i c a l b a n d w i d t h " , J.Acoust.Soc.Am.
3 8 , pp. 548-560,
1965) b a s e d on t h e c r i t i c a l - b a n d w i d t h concept. T h e i r e x p e r i m e n t s on p u r e
t o n e p a i r s a l l o w e d them t o c a l c u l a t e d i s s o n a n c e v a l u e s - f o r complex t o n e
pairs on t h e a s s u m p t i o n t h a t t h e r o u g h n e s s produced b y beats w a s a d d i tive. I n an a t t e m p t t o test t h e c o r r e c t n e s s o f t h i s assumption t h e
dissonance v a l u e s o f complex dyads a s w e l l a s t e t r a d s were r a t e d by
e i g h t e e n s u b j e c t s . There w a s a r e a s o n a b l e a g r e e m e n t f o r most d y a d s , b u t
f o r o t h e r d y a d s a n d f o r t h e t e t r a d s t h e m o d e l c l e a r l y d i d n o t work. A
comparison o f two- and f o u r - t o n e c h o r d s showed t h a t c o n s o n a n t i n t e r v a l s
m i t i g a t e t h e s h a r p n e s s o f s i m u l t a n e o u s d i s s o n a n c e s . The p e r i o d i c i t y
model f o r p i t c h p e r c e p t i o n a p p e a r s t o o f f e r a m e c h a n i s m f o r b o t h t h e
r o u g h n e s s and t h e p l e a s a n t n e s s o f i s o l a t e d chords.
Since t h e e a r l y d i s c o v e r y t h a t consonance is r e l a t e d t o s i m p l e
n u m e r i c a l f r e q u e n c y r a t i o s , many s u g g e s t i o n s h a v e b e e n p u t f o r w a r d t o
e x p l a i n t h e p a r t i c u l a r e f f e c t s o f t h e most common m u s i c a l i n t e r v a l s . The
best known, t h a t o f H e l m h o l t z , was made i n terms o f b e a t s . A s t w o p u r e
t o n e s are g r a d u a l l y s e p a r a t e d i n f r e q u e n c y , t h e beats become more and
more r a p i d a n d t h e s o u n d i n c r e a s i n g l y r o u g h , r e a c h i n g a r o u g h n e s s m a ximum, a c c o r d i n g t o H e l m h o l t z , f o r a f r e q u e n c y d i f f e r e n c e o f 30-40 Hz.
A t l a r g e r f r e q u e n c y s e p a r a t i o n s t h e r o u g h n e s s d e c r e a s e s , and t h e sound
becomes c o n s o n a n t f o r a l l i n t e r v a l r a t i o s . For complex t o n e s t h e s i t u a t i o n is d i f f e r e n t . Tones w i t h s i m p l e i n t e r v a l r a t i o s , f o r i n s t a n c e t h e
octave ( 2 : l ) r t h e f i f t h (3:2), and t h e f o u r t h (4:3) w i l l have h a r m o n i c s
t h a t o f t e n c o i n c i d e , and t h e r e w i l l b e b e a t s between few a d j a c e n t harmon i c s . I n c o n t r a s t , d i s s o n a n t i n t e r v a l s , s u c h a s t h e major s e v e n t h , w i l l
have n o o r f e w c o i n c i d i n g h a r m o n i c s , a n d many a d j a c e n t o n e s p r o d u c i n g
beats, H e l m h o l t z ' t h e o r y was w i d e l y a c c l a i m e d a t t h e t i m e , b u t n e v e r
g a i n e d t o t a l a c c e p t a n c e i n t h e s c i e n t i f i c and m u s i c o l o g i c a l community.
Criticism was v o i c e d on some i m p o r t a n t p o i n t s . F i r s t , many r e s e a r c h
w o r k e r s b e l i e v e d t h a t c h o r d s made u p o f p u r e t o n e s c o u l d b e set up which
were d i s s o n a n t w i t h o u t h a v i n g b e a t s b e t w e e n t h e t o n e s . S e c o n d , i t w a s
p o i n t e d o u t t h a t t h e 30 Hz f r e q u e n c y s e p a r a t i o n producing maximal d i s s o n a n c e o n l y seemed t o a p p l y f o r t h e medium f r e q u e n c y range. T h i r d , some
m u s i c i a n s f e l t t h a t s u c h a f u n d a m e n t a l m u s i c a l phenomenon a s consonance
c o u l d n o t p o s s i b l y b e b a s e d o n t h e mere a b s e n c e o f d i s t u r b i n g n o i s e
among t h e p a r t i a l s o f t h e tones.
*
Temporary a s s o c i a t e s , Music A c o u s t i c s Group.
STL-QPSR 1/1988
Recent p s y c h o a c o u s t i c s t u d i e s have g r e a t l y r e d u c e d t h e r e l e v a n c e o f
much o f t h i s c r i t i c i s m . I t i s a b o v e a l l t h e w o r k b y P l o m p & L e v e l t
(1965) w h i c h h a s made t h e b e a t t h e o r y o f c o n s o n a n c e t h e m o s t w i d e l y
a c c e p t e d today. Plomp and L e v e l t c o u l d c o n f i r m t h a t consonance f o r p u r e
t o n e s was a f u n c t i o n o f t h e d i s t a n c e b e t w e e n t h e t o n e s r a t h e r t h a n o f
f r e q u e n c y r a t i o s . The d i s s o n a n c e a s s i g n e d b y s o m e c r i t i c s t o c e r t a i n
combinations of pure t o n e s t h e a u t h o r s a t t r i b u t e d t o t h e high degree of
m u s i c a l t r a i n i n g t h e c r i t i c s had r e c e i v e d , which made them i d e n t i f y t h e
i n t e r v a l s h e a r d and c l a s s i f y them a c c o r d i n g t o t h e i r p r e c o n c e i v e d i d e a s
o f w h a t t h e c o n s o n a n c e v a l u e s s h o u l d be. L i k e H e l m h o l t z , t h e y f o u n d a
clear maximum i n t h e f r e q u e n c y s e p a r a t i o n c u r v e f o r d i s s o n a n c e , b u t one
which v a r i e d s y s t e m a t i c a l l y a c c o r d i n g t o t h e f r e q u e n c y range. Plomp and
L e v e l t c o u l d show t h a t i t w a s p r o p o r t i o n a l t o t h e c r i t i c a l bandwidth, a
p s y c h o p h y s i c a l c o n c e p t based on d a t a from s t u d i e s on a u d i t o r y masking,
l o u d n e s s a n d t h e ear's a b i l i t y t o h e a r o u t i n d i v i d u a l c o m p o n e n t s i n a
complex t o n e . A s a m e a s u r e o f t h e l i m i t f o r i n t e r a c t i o n b e t w e e n t o n e s
c r i t i c a l b a n d w i d t h i s c l e a r l y r e l e v a n t t o a u d i t o r y b e a t s . The maximum
v a l u e f o r d i s s o n a n c e b e i n g f i x e d a t a b o u t 25% o f t h e c r i t i c a l b a n d w i d t h
it w a s p o s s i b l e t o draw a n o r m a l i z e d c u r v e f o r consonance and d i s s o n a n c e
as a f u n c t i o n o f f r e q u e n c y s e p a r a t i o n between p u r e t o n e s . Using t h i s
c u r v e o n e may compute a t h e o r e t i c a l v a l u e f o r d y a d s of complex t o n e s b y
f i n d i n g t h e v a l u e f o r e v e r y p a i r o f a d j a c e n t h a r m o n i c s o f t h e t w o complex t o n e s and adding them up. The a s s u m p t i o n is t h e n t h a t d i s s o n a n c e is
t h e sum o f c o n t r i b u t i o n s from a l l p a i r s o f i n t e r f e r i n g harmonics.
F i g . 1.
Normalized c u r v e representing d i s s o n a n c e o f p u r e t o n e
i n t e r v a l s a s a f u n c t i o n of frequency d i f f e r e n c e i n u n i t s
of t h e c r i t i c a l bandwidth ( a f t e r Plomp & L e v e l t ) . The
g r a p h , i n a c c o r d a n c e w i t h Hutchinson & Knopoff, shows
t h e v a r i a t i o n i n d i s s o n a n c e r a t h e r t h a n consonance.
STL-QPSR 1/1988
The c u r v e w e computed f o r complex t o n e s c o n s i s t i n g o f s i x h a r m o n i c s
(Fig. 2 ) a g r e e s w i t h t h a t o f Plomp a n d L e v e l t w i t h s h a r p n e s s c l e a r l y
r e l a t e d t o consonance o f t h e i n t e r v a l s . The d i s s o n a n c e maxima are broader a n d n o t q u i t e i n a g r e e m e n t w i t h t h e common r a n k o r d e r i n g . I n p a r t i c u l a r t h e major s e v e n t h is n o t i c e a b l y less d i s s o n a n t a c c o r d i n g t o t h i s
curve t h a n earlier e x p e r i m e n t s and m u s i c a l p r a c t i c e have l e d u s t o
e x p e c t . I n v i e w o f t h e s e d i s c r e p a n c i e s we d e c i d e d t o e x t e n d t h i s method
o f c a l c u l a t i n g d i s s o n a n c e t o o t h e r c o m b i n a t i o n s o f complex tones.
1. Method
Stimuli
The s i g n a l s u s e d a s s t i m u l i i n t h e e x p e r i m e n t were g e n e r a t e d i n
real t i m e b y a c u s t o m - b u i l t d i g i t a l s i g n a l p r o c e s s o r c o n t r o l l e d b y a
p e r s o n a l computer. The s i g n a l p r o c e s s o r c a n be programmed b y microcode
to a c t a s a f i l t e r , g e n e r a t o r o r some o t h e r d e v i c e . I n t h e p r e s e n t
e x p e r i m e n t it was used t o g e n e r a t e a bank o f sine-wave o s c i l l a t o r s . The
two DA-converters used are 16-bi t s p e c i a l a u d i o c o n v e r t e r s . A s a m p l i n g
f r e q u e n c y o f 25 KHz was used t h r o u g h o u t t h e e x p e r i m e n t .
The s e t - u p was d e s i g n e d , b u i l t a n d p r o g r a m m e d b y o n e o f t h e authors.
A.
S u b j e c t s and p r o c e d u r e .
E i g h t e e n s u b j e c t s , m o s t l y s t a f f and s t u d e n t s a t t h e Dept. o f Speech
Communication a n d M u s i c A c o u s t i c s t o o k p a r t i n t h e e x p e r i m e n t s . A l l
r e p o r t e d a n i n t e r e s t i n m u s i c a n d a l m o s t e v e r y o n e had e x p e r i e n c e o f
performing music. About h a l f t h e g r o u p had some knowledge o f t h e t h e o r y
o f muslc, b u t o n l y t w o had s t u d i e d harmony s y s t e m a t i c a l l y .
The l i s t e n i n g c o n d i t i o n s w e r e chosen s o a s t o c o r r e s p o n d a s c l o s e l y
a s p o s s i b l e t o t h o s e i n t h e P l o m p a n d L e v e l t e x p e r i m e n t . The s u b j e c t s
judged e a c h s t i m u l u s on a 7 - p i n t scale, 7 c o r r e s p o n d i n g t o most d i s s o nant, and 1 t o m o s t c o n s o n a n t ( a c t u a l l y t h e r e v e r s e o f t h e Plomp and
L e v e l t scale). No d e f i n i t i o n o f d i s s o n a n c e was g i v e n on t h e i n s t r u c t i o n
s h e e t , a n d n o n e o f t h e s u b j e c t a s k e d f o r o n e . The l i s t e n i n g t o o k p l a c e
i n a sound p r o o f b o o t h a t a s e n s a t i o n l e v e l o f a p p r o x i m a t e l y 60 dB. The
s t i m u l i were a l l t o n e s w i t h s i x p a r t i a l s o f e q u a l a m p l i t u d e . They were
presented f o r a b o u t f o u r seconds, w i t h an e l e v e n second i n t e r v a l i n
which t h e s u b j e c t s had t o r e c o r d t h e i r r a t i n g s on a p r e p a r e d s h e e t . The
s t i m u l i were p r e s e n t e d on t w o s e p a r a t e o c c a s i o n s i n a d i f f e r e n t o r d e r . A
test s e s s i o n w i t h t h e f o u r t e e n d y a d s p r e c e d e d t h e f i r s t e x p e r i m e n t .
3.
2. Experiment 1
In t h e f i r s t experiment t h e s u b j e c t s l i s t e n e d t o f o u r t e e n i n t e r v a l s . T h e l o w e r t o n e w a s a l w a y s 240 Hz. T w e l v e were t h e i n t e r v a l s
STL-QPSR 1/1988
w i t h i n a n o c t a v e , and t w o w e r e minor n i n t h s , a n i n t e r v a l n o t p l o t t e d i n
t h e Plomp and L e v e l t study. One o f t h e s e w a s s i m i l a r t o t h e o t h e r t w e l v e
i n b e i n g b a s e d on complex t o n e s w i t h s i x p a r t i a l s , t h e o t h e r c o n s i s t e d
o f t o n e s where t h a t p a r t i a l i n e a c h t o n e which c o n t r i b u t e d most t o t h e
t o t a l d i s s o n a n c e v a l u e had been removed. The computed d i s s o n a n c e v a l u e
was t h e r e b y r e d u c e d from 2.35 f o r t h e two s i x - p a r t i a l t o n e s t o 0.68 f o r
t h e t w o f i v e - p a r t i a l ones.
The r e s u l t s a r e s h o w n i n F i g s . 2 a n d 3 . I n t h e f i r s t f i g u r e t h e
r a t e d d i s s o n a n c e v a l u e s have been added t o t h e computed c u r v e f o r t h e
d i s s o n a n c e v a l u e s f o r two s i x - p a r t i a l t o n e s o v e r a r a n g e o f j u s t o v e r an
o c t a v e . The s e c o n d f i g u r e p l o t s t h e r a t e d v a l u e s a s a f u n c t i o n o f computed v a l u e s .
o
r a t e d value
-computed c u r v e
frequency d i f f e r e n c e (sernitones)
Fig. 2.
Computed and r a t e d v a l u e s o f d i s s o n a n c e as a f u n c t i o n o f
frequency d i f f e r e n c e .
There is f o r most i n t e r v a l s a r e a s o n a b l y good a g r e e m e n t between t h e
computed a n d r a t e d d i s s o n a n c e . The r a n k o r d e r f o r r a t e d c o n s o n a n c e is
t h e t r a d i t i o n a l one, e x c e p t f o r t h e minor s e v e n t h , which r e c e i v e d t h e
same r a t i n g a s t h e minor s i x t h , which is commonly c o n s i d e r e d more consonant. The m a j o r s e v e n t h a n d t h e m i n o r n i n t h w i t h n o p a r t i a l r e m o v e d ,
however, were r a t e d c o n s i d e r a b l y m o r e d i s s o n a n t t h a n we w o u l d e x p e c t
from t h e c o m p u t e d v a l u e . B u t much m o r e r e m a r k a b l e i s t h e case o f t h e
minor n i n t h w i t h o n e p a r t i a l r e m o v e d f r o m e a c h c o m p l e x t o n e . Here, i n
s p i t e o f a d i f f e r e n c e i n c o m p u t e d v a l u e o f m o r e t h a n 70% t h e t w o n i n t h
i n t e r v a l s were j u d g e d e q u a l l y d i s s o n a n t . I t i s d i f f i c u l t t o r e c o n c i l e
t h i s r e s u l t w i t h a beat t h e o r y o f consonance.
STL-QPSR 1/1988
computed dissonance
F i g . 3.
Rated d i s s o n a n c e v a l u e s for dyads a s a f u n c t i o n
of computed v a l u e s .
Experiment 2
The n i n e t e e n c h o r d s i n t h e second e x p e r i m e n t were a l l c o m b i n a t i o n s
o f f o u r complex t o n e s w i t h s i x partials. A c h o r d was s o m e t i m e s used more
than once i n d i f f e r e n t a r r a n g e m e n t s ( i n v e r s i o n s ) o r w i t h d i f f e r e n t
t u n i n g . The m a j o r t r i a d o c c u r r e d w i t h a d d e d m a j o r s i x t h ( c h o r d 1 2 ) ,
minor s e v e n t h ( 8 1 1 0 , 1 5 , 1 6 ) , m a j o r s e v e n t h ( 7 1 1 4 ) , a n d o c t a v e ( 1 ,
1 8 ) . The minor t r i a d was used w i t h added m a j o r s i x t h (11, 1 3 , 1 7 ) , minor
s e v e n t h (41 9 ) a n d o c t a v e ( 2 ) . A l l t h e c o m m o n l y u s e d c h o r d s w i t h i n o n e
o c t a v e c a n b e f o u n d , w i t h t h e e x c e p t i o n o f t h e d i m i n i s h e d and t h e augmented c h o r d s . I n a d d i t i o n t h e r e w e r e t w o c h o r d s e x c e e d i n g a n o c t a v e ,
o n e a m a j o r t r i a d w i t h t h e t h i r d above t h e o c t a v e i n p y t h a g o r e a n t u n i n g ,
t h e o t h e r a n o n - t r a d i t i o n a l c h o r d (5). The n a r r o w l y s p a c e d c l u s t e r c h o r d
is, o f c o u r s e , a l s o a n o n - t r a d i t i o n a l o n e . The r e s u l t s c a n b e s e e n i n
T a b l e 1. ( T h e n o t e s a r e a b o u t o n e w h o l e - t o n e h i g h e r t h a n t h e a c t u a l
f r e q u e n c y i n o r d e r t o r e f e r t h e c h o r d s t o C r a t h e r t h a n B f l a t . ) The
j u s t m a j o r (1) and minor ( 2 ) c h o r d s were r a t e d p r e d i c t a b l y low i n d i s s o n a n c e , w h i l e t h e c l u s t e r c h o r d ( 1 9 ) a n d t h e i r r e g u l a r c h o r d ( 3 ) were
r a t e d c o r r e s p o n d i n g l y high. The minor s e v e n t h c h o r d s were g e n e r a l l y f e l t
t o be f a i r l y c o n s o n a n t , w i t h t h e e x c e p t i o n o f No. 10, where t h e n a t u r a l
s e v e n t h ( 7 :4) p r o b a b l y c o n t r i b u t e d t o a s l i g h t l y u n p l e a s a n t e f f e c t . The
tempered s e v e n t h c h o r d s were judged q u i t e c o n s o n a n t , t h e t h i r d i n v e r s i o n
( 1 6 ) e v e n m o r e s o t h a n t h e r o o t p o s i t i o n ( 1 5 ) . Among t h e o t h e r c h o r d s
t h e m a j o r n i n t h c h o r d w i t h o u t t h e t h i r d ( 6 ) s t a n d s o u t a s b e i n g con-
STL-QPSR 1/1988
t h a n even t h e f o u r t h f o r t h e kind of complex t o n e s w e are d e a l i n g w i t h ,
concludes t h a t t h e p s y c h o a c o u s t i c a l c o n c e p t o f d i s s o n a n c e h a s t o b e
modified s o a s t o i n c l u d e t h e m u s i c a l harmony a s p e c t . W e t h i n k , howe v e r , t h a t t h e r e i s n o t n e c e s s a r i l y a c o n f l i c t b e t w e e n t h e psychoa c o u s t i c d a t a and musical t h e o r y i n t h i s p a r t i c u l a r case. T e r h a r d t d o e s
n o t t a k e i n t o account t h e f a c t t h a t t h i s c a l c u l a t e d v a l u e is o u t o f l i n e
n o t o n l y w i t h t h e r a t e d v a l u e i n t h e s t u d y , b u t e v e n more w i t h e v e r y
e a r l i e r s t u d y where t h e i n t e r v a l h a s been r a t e d . The v a l u e i n o u r s t u d y
is, o f c o u r s e , i n complete agreement w i t h t h e t r a d i t i o n a l view. It seems
more n a t u r a l t o c o n c l u d e t h a t t h e model f o r t h e c a l c u l a t i o n s c a n n o t
account f o r t h e r e s u l t s a n d m u s t , i n some r e s p e c t s a t l e a s t , b e considered deficient.
Hutchinson & Knopoff (1978) have a l s o c a l c u l a t e d t h e d i s s o n a n c e of
dyads b a s e d on a v e r s i o n o f t h e c r i t i c a l b a n d w i d t h model m o d i f i e d by
means o f a c h a n g e i n t h e s h a p e o f t h e c r i t i c a l b a n d w i d t h c u r v e o v e r a n
extended f r e q u e n c y r a n g e . A s a c o n s e q u e n c e t h e d i s s o n a n c e v a l u e f o r a
g i v e n i n t e r v a l may d i f f e r b y a f a c t o r o f n e a r l y t e n o v e r a r a n g e o f
t h r e e o c t a v e s , which would seem t o make t h e model i r r e l e v a n t f o r musical
purposes.
I f t h e c a l c u l a t e d d i s s o n a n c e by t h e v a r i o u s proponents of t h e b e a t
t h e o r y a t times a r e s t r a n g e l y a t odds w i t h m u s i c a l e x p e r i e n c e , o u r own
computed v a l u e s l i s t e d i n T a b l e I seem t o j u s t i f y a more s w e e p i n g conc l u s i o n : performance o f music would be i m p o s s i b l e i f t h e b e a t t h e o r y ,
a t l e a s t i n i t s p r e s e n t f o r m , were c o r r e c t . Plomp a n d L e v e l t p o i n t o u t
t h a t t h e s t e e p n e s s of t h e c u r v e around t h e o c t a v e and t h e f i f t h (Fig. 2)
make t h e s e i n t e r v a l s much more s e n s i t i v e t o a d e v i a t i o n from t h e c o r r e c t
frequency r a t i o than o t h e r consonant i n t e r v a l s , such a s t h e t h i r d , which
rnay e x p l a i n why w e t o l e r a t e i m p u r e t h i r d s . But o u r n u m e r i c a l c a l c u l a t i o n s show t h a t even d e v i a t i o n s smaller t h a n t h o s e which n o r m a l l y o c c u r
i n musical performance imply a change i n d i s s o n a n c e which may be cons i d e r a b l y g r e a t e r t h a n a change from a chord which w e would unquestiona b l y c o n s i d e r being consonant i n t o a d i s s o n a n t one.
Even i f w e would b e p r e p a r e d t o s e p a r a t e c o n s o n a n c e i n t o t h e t w o
components s e n s o r y consonance and harmony, a s suggested by T e r h a r d t i t
is d i f f i c u l t t o s e e w h a t p o s s i b l e r e l e v a n c e t h e f o r m e r c o n c e p t would
have i f i t is based on t h e b e a t t h e o r y g i v e n t h e extreme s e n s i t i v i t y o f
consonance t o w h a t , a f t e r a l l , a r e c o n s t a n t l y o c c u r r i n g v a r i a t i o n s i n
intonation.
The e x p e r i m e n t a l r e s u l t s i n t h i s s t u d y merely confirm t h e inadequac y o f t h e b e a t model. The change from a s e v e n t h chord i n j u s t i n t o n a t i o n
(10) i n t o a tempered one ( 1 5 ) even r e s u l t s i n a l o w e r estimate o f
d i s s o n a n c e i n s p i t e o f a n i n c r e a s e i n c o m p u t e d v a l u e . I t is t r u e t h a t a
s l i g h t l y m i s t u n e d m a j o r t e t r a d ( 6 ) was j u d g e d s l i g h t l y more d i s s o n a n t
t h a n t h e chord i n j u s t i n t o n a t i o n , b u t t h e i n c r e a s e is still f a r smaller
than t h e computed change.
STL-QPSR 1/1988
Moreover, t h e s e are c e r t a i n l y n o t changes i n i n t o n a t i o n which would
be uncommon i n performance of music. There is a l s o t h e q u e s t i o n whether
a s l i g h t l y mistuned major chord can i n any s e n s e be considered d i s sonant. I n t h i s c a s e it is l i k e l y t h a t some s u b j e c t s r e a c t e d t o t h e outof-tuneness r a t h e r than what t h e y n o r m a l l y perceived a s dissonance.
S t i l l t h e most i m p o r t a n t f i n d i n g is t h e l a c k o f correspondance
between t h e r a n k o r d e r i n g f o r t h e computed and t h e r a t e d consonance of
t h e chords. T h i s p o s e s a problem. Plomp and L e v e l t have shown beyond any
doubt t h a t d i s s o n a n c e f o r c o m p l e x t o n e s i s r e l a t e d t o t h e p r e s e n c e o f
harmonics. What t h e n c a n a c c o u n t f o r t h e e m e r g e n c e o f d i s s o n a n c e when
harmonics are added t o pure t o n e s o t h e r than b e a t s among a d j a c e n t harmonics?
The p r e s u m p t i o n o f a d d i t i v i t y f o r t h e r o u g h n e s s o f b e a t s , t e n t a t i v e l y a d o p t e d b y Plomp a n d L e v e l t , seems t o b e s u p p o r t e d by o t h e r
s t u d i e s , such a s t h a t of T e r h a r d t (1968)r which shows t h a t t h e roughness
produced by amplitude-modulated t o n e s i n c e r t a i n c i r c u m s t a n c e s is addit i v e . Y e t , t h e f a c t t h a t i n o u r f i r s t experiment t h e s t r o n g e s t b e a t s i n
a dyad c o u l d be removed seemingly w i t h o u t a f f e c t i n g t h e p e r c e i v e d d i s s o nance i m p l i e s t h a t t h e r e can be no s i m p l e a d d i t i v i t y of roughness produced by b e a t s between a d j a c e n t p a r t i a l s .
But i f t h e i n t e r a c t i o n between a d j a c e n t components is n o t r e s p n s i b l e f o r d i s s o n a n c e , how can t h e harmonics e x c e r c i s e t h e i r i n f l u e n c e ?
W
e a r e l e d t o t h e c o n c l u s i o n t h a t harmonics a c h i e v e t h e e f f e c t of
d i s s o n a n c e n o t p r i m a r i l y from t h e i n d i v i d u a l i n t e r a c t i o n w i t h a d j a c e n t
harmonics, b u t b y c o l l e c t i v e l y i n t e r a c t i n g w i t h a n o t h e r g r o u p o f p a r t i a l s when t h e i n t e r v a l between t h e fundamentals is a d i s s o n a n t one.
Removing t h e fundamentals of two complex t o n e s , moreover d o e s n o t
leave j u s t t h e dissonance u n a f f e c t e d , b u t a l s o t h e p i t c h o f t h e tones,
which s t i l l c o r r e s p o n d s t o t h e a b s e n t f u n d a m e n t a l s . The l a t t e r e f f e c t
h a s been c a l l e d t h e r e s i d u e o r t h e p e r i o d i c i t y p i t c h phenomenon.
The term r e s i d u e was i n t r o d u c e d by Schouten (1940) t o i n d i c a t e t h a t
the hiqher harmonics, which cannot be perceived s e p a r a t e l y , a r e perc e i v e d c o l l e c t i v e l y a s one component ( t h e r e s i d u e ) w i t h a p i t c h d e t e r mined b y t h e p e r i o d i c i t y o f t h e c o l l e c t i v e waveform a n d e q u a l t o t h e
fundamental t o n e . I t w a s l a t e r shown t h a t l o w e r h a r m o n i c s a r e more
important t h a n t h e h i g h e r o n e s i n d e t e r m i n i n g t h e p i t c h . T h i s is what w e
would e x p e c t i f frequency r e s o l u t i o n is e s s e n t i a l f o r t h e p e r c e p t i o n of
low p i t c h , b u t i t is h a r d l y t h e p r o o f , a s is o f t e n maintained, t h a t t h e
hearing o u t of i n d i v i d u a l harmonics is n e c e s s a r y f o r t h e e f f e c t . P e r i o d i c i t y p i t c h may be p r e s e n t even when t h e i n d i v i d u a l p a r t i a l s cannot be
r e s o l v e d (Nordmark, 197d). I t is, f o r i n s t a n c e , d i f f i c u l t t o s e e how t h e
harmonics of complex t o n e s making up a minor second can be resolved. A l l
harmonics o f o n e t o n e a r e a t a d i s t a n c e f r o m t h e o t h e r o f a f r a c t i o n o f
a c r i t i c a l band - a c o n c e p t used t o d e f i n e t h e l i m i t s o f f r e q u e n c y
resolution.
STL-QPSR 1/1988
I t is t h e r e f o r e t e m p t i n g t o assume t h a t a l l t h e p a r t i a l s o f a t o n e
can a c t t o g e t h e r c o l l e c t i v e l y b o t h t o c o n t r i b u t e t o t h e s e n s a t i o n o f
p i t c h and, i n t h e p r e s e n c e o f o t h e r complex t o n e s , t o t h e s e n s a t i o n o f
consonance a n d d i s s o n a n c e . The r o u g h n e s s w e a s s o c i a t e w i t h d i s s o n a n c e
c a n be u n d e r s t o o d t o b e m a i n l y t h e r e s u l t o f i n t e r f e r e n c e on t h e b a s i l a r
membrane between t h e c o l l e c t i v e waveforms o f t h e c o n s t i t u e n t t o n e s , w i t h
o n l y a m i n o r c o n t r i b u t i o n f r o m t h e i n d i v i d u a l beats. A c o n s e q u e n c e o f
t h i s v i e w is t h a t we s h o u l d e x p e c t complex t o n e s made up o f i n h a r m o n i c
p a r t i a l s t o form less d i s t i n c t d i s s o n a n t i n t e r v a l s .
There are p r o b l e m s , however, w i t h a n y form o f t h e a d d i t i v i t y hypot h e s i s . A c o m p a r i s o n b e t w e e n t h e d y a d C-B, t h e r n a j o r s e v e n t h , a n d t h e
t e t r a d C-E-G-B ( 7 ) s h o w s t h a t t h e t e t r a d h a s a l o w e r r a t i n g (3.02) t h a n
t h e d y a d (5.35) w h e r e a s w e w o u l d , o f c o u r s e , e x p e c t t h e o p p o s i t e , a s
any a d d i t i o n o f t o n e s should o n l y i n c r e a s e dissonance. Furthermore, t h e
t h i r d i n v e r s i o n o f t h e c h o r d , where B is t h e t o n e j u s t below C ( 1 4 ) r was
judged much l e s s d i s s o n a n t (4.15) t h a n t h e m i n o r s e c o n d w i t h o u t t h e
t h i r d ( E ) a n d t h e f i f t h ( G ) , w h i c h a t 5.95 w a s t h e m o s t d i s s o n a n t d y a d
o f a l l . A s i m i l a r o b s e r v a t i o n h a s b e e n made b y K u n i t z ( 1 9 6 0 ) , who
p o i n t e d o u t t h a t a n y a d d i t i o n o f c o n s o n a n t i n t e r v a l s , s u c h a s E and G t o
t h e C-B m a j o r s e v e n t h , d e c r e a s e s d i s s o n a n c e .
Would t h e same b e t r u e i f we a d d a t o n e w h i c h f o r m s a d i s s o n a n t
i n t e r v a l w i t h one o r more o f t h e c o n s t i t u e n t s o f a chord? U n f o r t u n a t e l y
t h e experiments d i d n o t include examples o f s i m p l e a d d i t i o n s o f dissonances. H o w e v e r , a c o m p a r i s o n o f c h o r d s 9 a n d 1 9 , w h i c h a r e i d e n t i c a l
e x c e p t f o r o n e t o n e , g o e s s o m e way t o w a r d s a n s w e r i n g t h e q u e s t i o n . I n
t h i s case t h e c h o r d w i t h o n e d i s s o n a n t i n t e r v a l i s p e r c e i v e d a s l e s s
t h a n h a l f a s d i s s o n a n t a s t h e one w i t h t h r e e d i s s o n a n c e s . Can w e t h e r e fore conclude t h a t t h e addition of a tone o r tones t o other tones with
which t h e y form c o n s o n a n t i n t e r v a l s r e d u c e s d i s s o n a n c e ? We would t h e n
e x p e c t a major c h o r d t o sound more c o n s o n a n t t h a n t h e m a j o r t h i r d a l o n e ,
and t h e m i n o r c h o r d m o r e c o n s o n a n t t h a n t h e m i n o r t h i r d . The r e s u l t s
g i v e some s m a l l s u p p o r t f o r t h i s c o n c l u s i o n . The r a t e d d i s s o n a n c e v a l u e
o f t h e m i n o r c h o r d , f o r i n s t a n c e , was 2.28 c o m p a r e d w i t h 2.7 f o r t h e
minor t h i r d . There is a s i m i l a r d i f f e r e n c e f o r t h e m a j o r chords.
With d i s s o n a n c e a d d e d t o d i s s o n a n c e t h e r e i s , a s w e h a v e s e e n ,
sometimes a n a d d i t i v e e f f e c t . A t o t h e r t i m e s t h e e f f e c t a p p e a r s t o b e
t h e o p p o s i t e . One s u c h e x a m p l e w o u l d b e t h e a d d i t i o n o f a D a n d a n F
s h a r p t o t h e m a j o r s e v e n t h i n t e r v a l C-B. Our i m p r e s s i o n is t h a t t h e
r e s u l t i n g c h o r d is d i s s o n a n t , b u t less s h a r p and p e r h a p s more p l e a s i n g
and m u s i c a l l y meaningful.
What t h e n is t h e r e a s o n f o r t h e s e unexpected v a r i a t i o n s i n consonance and p l e a s a n t n e s s ?
There must o b v i o u s l y b e some o t h e r f a c t o r i n v o l v e d , one f o r want o f
a n e x p l a n a t o r y term w e may s i m p l y c a l l t h e m u s i c a l f a c t o r , and which is
responsible f o r t h e r e d u c e d u n p l e a s a n t n e s s o f d i s s o n a n t i n t e r v a l s when
c o n s o n a n t o n e s are added. I n some ways we have come close t o T e r h a r d t ' s
(1984) s e p a r a t i o n o f m u s i c a l c o n s o n a n c e i n t o t w o c o m p o n e n t s , s e n s o r y
consonance and harmony. The d i f f i c u l t y f o r u s w i t h h i s p a r t i c u l a r d e f i n i t i o n is t h a t s e n s o r y consonance, a s t h e p s y c h o p h y s i c a l b a s i s o f p l e a s a n t n e s s f o r a l l k i n d s o f s o u n d s , is c o n s i d e r e d t o be t h e mere a b s e n c e
o f r o u g h n e s s . T h i s d e f i n i t i o n is t o o n a r r o w f o r o u r p u r p o s e . To make
sense o f o u r r e s u l t s w e have t o i n c l u d e musical f a c t o r s even i n isol a t e d , s t a t i c c h o r d , f a c t o r s which T e r h a r d t would c o n s i d e r t o be p a r t o f
t h e harmony component and t o b e o f less i m p o r t a n c e f o r s i n g l e c h o r d s .
The d i f f e r e n c e i n o u r r e s p e c t i v e a p p r o a c h , however, is n o t o n l y one o f
emphasis. I t a l s o r e f l e c t s o u r d i f f e r i n g v i e w s o n t h e o r i g i n o f r o u g h n e s s and harmony. W
e t h i n k t h e p e r i o d i c i t y c o n c e p t is r e l e v a n t t o b o t h
roughness, a s d e s c r i b e d e a r l i e r , and t o h a r m o n i c e f f e c t s , s u c h as t h e
n a t u r a l n e s s and p l e a s a n t n e s s o f c e r t a i n i n t e r v a l s o r chords. No "explan a t i o n " c a n b e g i v e n a t p r e s e n t f o r t h e phenomena o f o c t a v e s i m i l a r i t y
or c o n s o n a n c e o f i n t e r v a l s w i t h s i m p l e o r c l o s e t o s i m p l e n u m e r i c a l
r a t i o s . B u t i t is a t l e a s t m o r e n a t u r a l f r o m t h i s p o i n t o f v i e w t o l i n k
t h e s e e f f e c t s t o r e l a t i o n s between t h e t i m e i n t e r v a l s g i v i n g rise t o t h e
p i t c h o f t h e c o n s t i t u e n t t o n e s i n a c h o r d t h a n t o assume t h a t t h e y are
e n t i r e l y due t o experience o r c u l t u r a l habits.
I n s p i t e o f t h e a t t r a c t i v e n e s s o f p e r i o d i c i t y as a unifying concept
f o r a number o f r e l a t e d phenomena, we must b e aware o f its l i m i t a t i o n s .
Consonance i n music h a s many meanings, some o f which are o n l y d i s t a n t l y
r e l a t e d t o t h e phenomena c o n s i d e r e d here. D i s s o n a n t c h o r d s , f o r example,
are o f t e n t h e m u s i c a l l y most p l e a s i n g , and c o n s o n a n t c h o r d s s o m e t i m e s
are d i s s o n a n t from a t h e o r e t i c a l p o i n t o f view.
The w i d e l y h e l d b e l i e f t h a t consonance e x i s t s a s a s e p a r a t e e n t i t y
t o b e d e f i n e d , measured, and e x p l a i n e d is a l m o s t c e r t a i n l y b a s e d on an
illusion.
Acknowledgements
The a s s i s t a n c e o f Kajsa Bacos i n c o l l e c t i n g t h e d a t a f o r t h i s s t u d y
is g r a t e f u l l y acknowledged.
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