Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Beat theories of musicalconsonance

Nordmark, J. and Fahlen, L. E.

journal: STL-QPSRvolume: 29number: 1year: 1988pages: 111-122

http://www.speech.kth.se/qpsr

STL-QPSR 1/1988

C. BEAT THEORIES OF MUSICAL CONSONANCE

J a n Nordmark* and Lennar t E. ~ a h l h n *

Abstract Helmholtz'

lease o n l i f e , consonance a n d 1965) based on

' b e a t t h e o r y of consonance h a s r e c e n t l y been g i v e n a new 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 lomp a n d L e v e l t ( " T o n a l 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, t h e c r i t i ca l -bandwid th concept. The i r e x p e r i m e n t s on pure

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 assumpt ion t h a t t h e roughness produced by beats w a s addi- t i v e . I n a n a t t e m p t t o tes t t h e c o r r e c t n e s s o f t h i s a s s u m p t i o n t h e d i s s o n a n c e v a l u e s o f c o m p l e x d y a d s a s w e l l a s t e t r a d s w e r e r a t e d b y 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 agreement f o r most dyads , 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 mode l c l e a r l y d i d n o t work. A comparison o f two- and four- tone c h o r d s showed t h a t consonant 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 mechan ism f o r b o t h t h e roughness 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.

S i n c e t h e e a r l y d i s c o v e r y t h a t c o n s o n a n c e is r e l a t e d t o s i m p l e numer ica 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 mus ica 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 beats. 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 a n d 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 requency s e p a r a t i o n s t h e roughness d e c r e a s e s , and t h e sound becomes consonant 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 o c t a v e ( 2 : l ) r t h e f i f t h ( 3 : 2 ) , a n d t h e f o u r t h (4 :3 ) w i l l h a v e 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 be b e a t s between few a d j a c e n t harmo- n 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 , such as t h e major seven th , 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 ime, 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 workers b e l i e v e d t h a t c h o r d s made up o f pure t o n e s cou ld be 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 requency s e p a r a t i o n producing maximal d i s s o - nance o n l y seemed t o a p p l y f o r t h e medium frequency range. Th i rd , some m u s i c i a n s f e l t t h a t such a fundamental m u s i c a l phenomenon as consonance c o u l d n o t p o s s i b l y b e b a s e d on 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 partials o f t h e tones.

* Temporary a s s o c i a t e s , Music Acous t i cs Group.

STL-QPSR 1/1988

Recent psychoacous t i c s t u d i e s have g r e a t l y reduced 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 sm. I t is a b o v e a l l t h e work 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 requency 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 some c r i t i c s t o c e r t a i n combina t ions o f 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 h igh d e g r e e o f mus ica l t r a i n i n g t h e c r i t i cs 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 heard and c l a s s i f y them accord ing t o t h e i r preconceived 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 requency s e p a r a t i o n c u r v e f o r d i s sonance , 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 accord ing t o t h e f requency range. Plomp and L e v e l t c o u l d show t h a t it 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 ca l bandwidth, a psychophysical concep 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 is 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 being f i x e d a t abou t 25% o f t h e c r i t i ca l bandwidth it w a s p o s s i b l e t o draw a normal ized 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 b e t w e e n p u r e t o n e s . Us ing t h i s c u r v e one may compute a t h e o r e t i c a l v a l u e f o r dyads of complex t o n e s by

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 harmonics o f t h e two com- plex t o n e s and adding them up. The assumption 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 curve representing d issonance 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 o f f requency 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 raph , i n accordance wi th Hutchinson & Knopoff, shows t h e v a r i a t i o n i n d i s sonance r a t h e r t h a n consonance.

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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 harmonics (Fig . 2 ) a g r e e s w i t h t h a t o f P l o m p 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 broad-

er and n o t q u i t e i n agreement w i t h t h e common r a n k o rder ing . I n p a r t i c - u l a r t h e m a j o r 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 c u r v e t h a n ea r l i e r e x p e r i m e n t s a n d m u s i c a l p r a c t i c e h a v e l e d u s t o expect . I n v iew o f t h e s e d i s c r e p a n c i e s we dec ided 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 combina t ions o f complex tones.

1. Method A. S t i m u l i

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 rocessor can be programmed by microcode to ac 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 exper iment 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 sampl ing f requency o f 25 KHz was used throughout t h e exper iment .

The s e t - u p was d e s i g n e d , b u i l t a n d programmed b y o n e o f t h e au- t h o r s .

3. S u b j e c t s and procedure . Eighteen 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 were chosen s o a s t o cor respond as c l o s e l y as p o s s i b l e t o t h o s e i n t h e P lomp 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 corresponding t o most d i s s o - n a n t , a n d 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 P lomp a n d

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 one. The l i s t e n i n g t o o k p l a c e

i n a sound proof boo th 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 ampl i tude. They were p r e s e n t e d f o r a b o u t f o u r s e c o n d s , w i t h a n e l e v e n s e c o n d i n t e r v a l i n which t h e s u b j e c t s had to r e c o r d t h e i r r a t i n g s on a prepared shee t . The s t i m u l i were p r e s e n t e d on two 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 rder . A

test s e s s i o n w i t h t h e f o u r t e e n dyads preceded t h e f i r s t exper iment .

2. Experiment 1 I n t h e f i r s t e x p e r i m e n t 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 . The 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

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w i t h i n an o c t a v e , and two were minor n i n t h s , an 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 similar t o t h e o t h e r t w e l v e i n being based 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 reduced 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 two f i v e - p a r t i a l ones.

The r e s u l t s a re shown 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 h a v e b e e n a d d e d t o t h e c o m p u t e d 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 over a range o f j u s t o v e r an octave. 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 as a f u n c t i o n o f com- puted v a l u e s .

o ra ted value - computed curve

frequency di f ference (sernitones)

F i g . 2 . Computed and r a t e d v a l u e s o f d i s sonance as a f u n c t i o n o f f requency 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 agreement 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 o n e , e x c e p t f o r t h e m i n o r s e v e n t h , w h i c h r e c e i v e d t h e same r a t i n g as t h e minor s i x t h , which is commonly c o n s i d e r e d more conso-

nant. 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 no 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 more d i s s o n a n t t h a n we would e x p e c t from t h e c o m p u t e d v a l u e . But much more r e m a r k a b l e is 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 emoved 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 more 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 is 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.

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computed dissonance

F i g . 3 . Rated d i s sonance 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 exper iment were a l l combina t ions

o f f o u r complex t o n e s w i t h s i x partials. A chord was somet imes used more t h a n o n c e 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 tuning. 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 (81 1 0 , 1 5 , 1 6 ) , m a j o r s e v e n t h (71 1 4 ) , a n d o c t a v e ( 1 , 18) . The minor t r i a d was used w i t h added major s i x t h (11, 13, 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 commonly 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 a n d t h e aug- mented 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 ,

one a major 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 pythagorean 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 chord (5). The n a r r o w l y spaced c l u s t e r chord 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 one. The r e s u l t s c a n b e s e e n i n Table 1. (The n o t e s are 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 requency 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 major (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 - nance, w h i l e t h e c l u s t e r c h o r d (19) 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 consonan 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 robab ly 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 consonant , t h e t h i r d i n v e r s i o n ( 1 6 ) e v e n more 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-

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than even t h e f o u r t h f o r t h e kind of complex tones w e are dea l ing wi th , 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 be

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 , how- eve r , t h a t t h e r e is n o t n e c e s s a r i l y a c o n f l i c t be tween t h e psycho- a c o u s t i c d a t a and musical theory i n t h i s p a r t i c u l a r case. Terhardt does no t t ake 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 va lue is ou t of l i n e

not 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 even more w i t h e v e r y e a r l i e r s tudy where t h e i n t e r v a l has been rated. The va lue i n our s tudy is, of course, i n complete agreement w i th 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 and mus t , i n some r e s p e c t s a t l e a s t , be con- s ide red d e f i c i e n t .

Hutchinson & Knopoff (1978) have a l s o ca l cu la t ed the dissonance 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 bandwid th model m o d i f i e d by means o f a change 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 an extended f r e q u e n c y r ange . A s a consequence t h e d i s s o n a n c e v a l u e f o r a g iven i n t e r v a l may d i f f e r by 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 oc taves , 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 dissonance by t h e va r ious proponents of t h e bea t theory a t times a r e s t r a n g e l y a t odds wi th musical experience, our 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 sweeping con- c lus ion: performance of music would be impossible i f t h e b e a t theory , a t l e a s t i n i ts p r e s e n t fo rm, were c o r r e c t . Plomp and L e v e l t p o i n t o u t

t h a t t h e s t eepness of t h e curve around t h e oc tave 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 dev ia t ion 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 impure 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 than those which normally occur i n musical performance imply a change i n dissonance which may be con- s i d e r a b l y g r e a t e r than a change from a chord which w e would unquestion-

a b l y cons ider being consonant i n t o a d i s sonan t one. Even i f w e would be p r e p a r e d t o s e p a r a t e consonance i n t o t h e two

components sensory consonance and harmony, a s suggested by Terhardt it is d i f f i c u l t t o s e e what 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 it is based on t h e b e a t theory g iven t h e extreme s e n s i t i v i t y of 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

in tona t ion . The experimental r e s u l t s i n t h i s s tudy merely confirm t h e inadequa-

c y of t h e bea t model. The change from a seventh chord i n j u s t i n tona t ion (10) i n t o a t empered one ( 1 5 ) even r e s u l t s i n a l o w e r estimate o f dissonance i n s p i t e o f an i n c r e a s e i n computed 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 ma jo r t e t r a d ( 6 ) was judged s l i g h t l y more d i s s o n a n t than t h e chord i n j u s t i n tona t ion , bu t t h e inc rease is still f a r smaller than t h e computed change.

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Moreover, t hese are c e r t a i n l y not changes i n i n tona t ion which would be uncommon i n performance of music. There is a l s o the ques t ion whether a s l i g h t l y m i s t u n e d ma jo r c h o r d c a n i n any s e n s e be c o n s i d e r e d d i s -

sonant. In 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 eac t ed t o t he out- of-tuneness r a t h e r than what they normally 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 c o r r e s p o n d a n c e between t h e rank order ing f o r t h e computed and t h e r a t e d consonance of t h e chords. This poses a problem. Plomp and Level t have shown beyond any doubt t h a t d i s s o n a n c e f o r complex t o n e s is 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 emergence o f d i s s o n a n c e when harmonics are added t o pure tones o the r than b e a t s among adjacent harmo- n i c s ?

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 by Plomp and L e v e l t , seems t o be 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 Terhardt (1968)r which shows t h a t t he roughness produced by amplitude-modulated tones i n c e r t a i n c i rcumstances is addi- t i ve . Y e t , t h e f a c t t h a t i n our 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 could be removed seemingly without a f f ec t ing t h e perceived d isso- nance i m p l i e s t h a t t h e r e can be no s imple a d d i t i v i t y of roughness pro-

duced by b e a t s between ad jacen 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 ad jacen t components is not r e s p n s -

i b l e f o r dissonance, how can t h e harmonics e x c e r c i s e t h e i r inf luence?

We a r e l e d t o t h e conclusion t h a t harmonics achieve t h e e f f e c t of dissonance no t p r i m a r i l y from t h e ind iv idua l i n t e r a c t i o n w i t h ad jacent harmonics, b u t by 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 par - 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 sonan t one.

Removing t h e fundamentals of two complex tones , moreover does not l eave j u s t t h e d i s s o n a n c e 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 t o n e s , which st 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 fundamen ta l s . The l a t t e r e f f e c t has 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 re s idue was introduced by Schouten (1940) t o i n d i c a t e t h a t

t h e h i q h e r h a r m o n i c s , wh ich c a n n o t b e p e r c e i v e d s e p a r a t e l y , a r e pe r - ceived c o l l e c t i v e l y a s one component ( t h e r e s idue ) w i th a p i t c h de t e r - mined by 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 and e q u a l t o t h e fundamental tone . I t was 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 than the higher ones i n determining t h e pi tch. This is what w e would expec 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 percept ion of low p i t c h , bu t i t is hard ly t h e proof , a s is o f t e n maintained, t h a t t h e hearing ou t of i nd iv idua l harmonics is necessary f o r t he e f f e c t . Peri- o d i c i t y p i t c h may be p resen t even when t h e ind iv idua l p a r t i a l s cannot be

reso lved (Nordmark, 197d). I t is, f o r i n s t ance , d i f f i c u l t t o s e e how t h e harmonics of complex tones making up a minor second can be resolved. A l l

harmonics o f one t o n e are a t a d i s t a n c e from 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 u sed 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

It is t h e r e f o r e t empt ing 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 a n d , i n t h e p r e s e n c e o f o t h e r c o m p l e x 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

can be unders tood t o be main ly 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 shou ld e x p e c t complex t o n e s made up o f inharmonic 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 problems, however, w i t h any form o f t h e a d d i t i v i t y hypo- t 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 rnajor 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 shou ld o n l y i n c r e a s e dissonance. Fur thermore, t h e t h i r d i n v e r s i o n o f t h e chord , where B is t h e t o n e j u s t below C (14)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 any a d d i t i o n o f consonant i n t e r v a l s , such as E and G to t h e C-B major seven th , d e c r e a s e s dissonance.

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 e x p e r i m e n t s d i d n o t i n c l u d e examples o f s i m p l e a d d i t i o n s o f d i s s o - nances. However , 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 are 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 some 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 is p e r c e i v e d as less t h a n h a l f a s d i s s o n a n t as t h e one w i t h t h r e e dissonances . Can w e t h e r e - f o r e c o n c l u d e t h a t t h e a d d i t i o n o f a t o n e o r t o n e s t o o t h e r t o n e s w i t h which t h e y form consonant i n t e r v a l s r e d u c e s d i s sonance? We would t h e n e x p e c t a major chord t o sound more consonant t h a n t h e major t h i r d a l o n e , and t h e m i n o r c h o r d more 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 conclus ion. 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 similar d i f f e r e n c e f o r t h e major 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 is , 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 would 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 chord is d i s s o n a n t , b u t less s h a r p and perhaps more p l e a s i n g

and m u s i c a l l y meaningful. What then 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 conso-

nance and p l e a s a n t n e s s ? There must o b v i o u s l y be some o t h e r f a c t o r invo lved , 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 call 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 reduced 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

consonant o n e s are added. I n some ways we have come close t o Terhardt ' 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, as t h e psychophysical basis o f p lea- 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 s e n s e o f o u r r e s u l t s w e h a v e t o i n c l u d e m u s i c a l f a c t o r s e v e n i n i s o -

l a t e d , s ta t ic chord , 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 ha rmony c o m p o n e n t a n d 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 approach, 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 on t h e o r i g i n o f r o u g h - n e s s and harmony. We t h i n k t h e p e r i o d i c i t y concep 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 , a n d 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 "expla-

n a t i o n " can be g i v e n a t p r e s e n t f o r t h e phenomena of 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 . But it is a t l eas t more 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 a s s u m e t h a t t h e y are e n t i r e l y due t o e x p e r i e n c e o r c u l t u r a l h a b i t s .

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 un i fy ing concep t f o r a number o f r e l a t e d phenomena, we must be 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. Dissonan t chords , 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 consonant c h o r d s somet imes 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 as a s e p a r a t e e n t i t y

t o be 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 based on an

i l l u s i o n .

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.

Refe rences Hutchinson, W. & Knopof f , L. ( 1 9 7 8 ) : "The a c o u s t i c c o m p o n e n t o f w e s t e r n consonance" I n t e r f a c e 7 , pp. 1-29. - Kameoka, A. & K u r i y a g a w a , M. ( 1 9 6 9 a ) : "Consonance t h e o r y P a r t I: Conso- nance o f d y a d s " J.Acoust.Soc.Am. 4 5 1 pp. 1451-1459. - Kameoka, A. & Kuriyagawa, M. (1969b): "Consonance t h e o r y P a r t 11: Conso- nance o f complex t o n e s and its c a l c u l a t i o n method", J.Acoust.Soc.Am. - 45, pp. 1460-1469.

Kuni tz , H. (1960): D i e I n s t r u m e n t a t i o n , Bre i tkopf & H a r t e l , ~ e i p z i g .

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Nordmark, J. (1978) : " Frequency and p e r i o d i c i t y a n a l y s i s " , pp. 243-282 i n (E.C. C a r t e r e t t e & M.D. Fr iedman, eds.) Handbook o f P e r c e p t i o n I V : Hearing, Academic Press , New York.

Plomp, R. & L e v e l t , W.J.M. (1965) : "Tonal consonance and c r i t i c a l band- width", J.Acoust.Soc.Am. 38, pp. 548-560. - Schouten, J.F. (1940): "The r e s idue , a new component i n s u b j e c t i v e sound a n a l y s i s " , Proc.Kon.Ned.Akad.Wetensch. 43, pp. 356-365. - Terhardt , E. (1968) : "Uber akus t i s che Rauhigkeit und Schwankungsstarke", Acustica 20, pp. 215-224. -

Terhardt , E. (1984): " 'The concept of musical consonance: a l i n k between music and psychoacoust i c s " , Music Percept ion 1 pp. 276-295. -