Beat theories of musical consonance

Dept. for Speech, Music and Hearing

Quarterly Progress and Status Report

Beat theories of musical consonance

Nordmark, J. and Fahle? n, L. E.

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



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C. BEAT THEORIES OF MUSICAL CONSONANCE J a n Nordmark* and L e n n a r t E. ~ a h l h n *

Abstract Helmholtz'' 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

lease on 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 Plomp and L e v e l t ("Tonal

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. 38, pp. 548-560, 1965) based on t h e critical-bandwidth concept. Their experiments on pure tone p a i r s allowed them t o c a l c u l a t e dissonance values- f o r complex tone 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. In an attempt t o test the correctness of t h i s assumption the dissonance values of complex dyads as well as t e t r a d s were r a t e d by eighteen subjects. There w a s a reasonable agreement for most dyads, but f o r o t h e r dyads and f o r t h e t e t r a d s t h e model c l e a r l y d i d n o t work. A comparison o f two- and four-tone chords showed t h a t consonant i n t e r v a l s mitigate 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 dissonances. 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 mechanism f o r b o t h t h e roughness and t h e pleasantness of isolated chords.

Since t h e e a r l y discovery t h a t consonance is r e l a t e d t o simple numerical frequency r a t i o s , many s u g g e s t i o n s have been 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 beats. A s t w o p u r e tones are g r a d u a l l y separated i n frequency, 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 frequency separations t h e roughness decreases, 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 tones t h e situat i o n is d i f f e r e n t . Tones w i t h simple i n t e r v a l r a t i o s , f o r instance t h e octave (2:l)r t h e f i f t h (3:2), and t h e fourth (4:3) w i l l have harmonics 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 harmonics. In contrast, dissonant intervals, such as the major seventh, w i l l have no o r few c o i n c i d i n g harmonics, and many a d j a c e n t o n e s producing 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 gained t o t a l acceptance i n t h e s c i e n t i f i c and musicological 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 could be set up which were d i s s o n a n t w i t h o u t having b e a t s between t h e tones. Second, it 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 nance o n l y seemed t o a p p l y f o r t h e medium frequency range. Third, some musicians f e l t t h a t such a fundamental musical 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 Acoustics Group.

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Recent psychoacoustic s t u d i e s have g r e a t l y reduced t h e relevance of 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) which h a s made t h e b e a t t h e o r y o f consonance t h e most w i d e l y accepted today. Plomp and Levelt could confirm t h a t consonance f o r pure 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 frequency r a t i o s . The dissonance assigned by some c r i t i c s t o c e r t a i n combinations of pure tones the authors attributed t o the high degree of

musical 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 heard and c l a s s i f y them according t o t h e i r preconceived i d e a s of what t h e consonance v a l u e s should be. Like Helmholtz, t h e y found 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 varied systematically according t o t h e frequency range. Plomp and Levelt could show t h a t it w a s proportional t o t h e critical bandwidth, a psychophysical concept based on data from studies on auditory masking, loudness and 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 components i n a complex tone. A s a measure o f t h e l i m i t f o r i n t e r a c t i o n between t o n e s c r i t i c a l bandwidth 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 dissonance being f i x e d a t about 25% o f t h e c r i t i c a l bandwidth it w a s possible t o draw a normalized curve for consonance and dissonance as a function of frequency separation between pure tones. 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 intervals a s a function of frequency difference i n units 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 graph, i n accordance with Hutchinson & Knopoff, shows the variation i n dissonance rather than 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 of Plomp and Levelt w i t h sharpness 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 ular t h e major seventh is noticeably less dissonant according t o t h i s curve than earlier experiments and musical practice have l e d us 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 of calculating dissonance t o other combinations of 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 by 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 by a personal computer. The s i g n a l processor can be programmed by microcode to act as a f i l t e r , g e n e r a t o r o r some o t h e r device. In t h e present experiment 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 audio converters. A sampling 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 programmed b y o n e o f t h e authors.

3. S u b j e c t s and p r o c e d u r e . Eighteen s u b j e c t s , mostly s t a f f and s t u d e n t s a t t h e Dept. o f Speech

Communication and Music 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 reported an i n t e r e s t i n music and a l m o s t everyone had experience 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 of muslc, b u t o n l y t w o had s t u d i e d harmony systematically.

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 c o r r e s p o n d 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 Plomp and L e v e l t experiment. The s u b j e c t s judged each stimulus on a 7 - p i n t scale, 7 corresponding t o most dissonant, and 1 t o most consonant ( a c t u a l l y t h e reverse 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 sheet, and none o f t h e s u b j e c t asked f o r one. The l i s t e n i n g took 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 about four seconds, with an eleven second 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 sheet. 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 session w i t h t h e fourteen dyads preceded t h e f i r s t experiment.

2. Experiment 1 In the f i r s t experiment the subjects listened t o fourteen interv 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

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