PERCEPTION OF TONAL CONSONANCE. R. Plomp and W. J. M. Levelt*

Transcription

1 IX PERCEPTION OF TONAL CONSONANCE R. Plomp and W. J. M. Levelt* Cononant chord, a ued n muc, are characterzed by mple frequency rato of the conttuent tone. Although th relaton between rato mplcty and cononance ha drawn conderable nteret n the pat, no unanmou opnon about t orgn ext. Recent development n hearng theory, equpment degn, and meaurng technque jutfed a new tudy of the phenonenon. It appeared that the expermental reult of th tudy were confrmed by tattcal analy of the chord of mucal compoton. * Preent addre: Center of Cogntve Stude, Harvard Unverty, Cambrdge (Ma), USA. 105

2 PERCEPTION OF TONAL CONSONANCE R. Plomp and W. J. M. Levelt One of the oldet dcovere n the feld of tone percepton, datng back to Pythagora, concern the ngular character of chord produced by a trng vbratng n two part wth length rato of 1 : 1, 1 : 2, 2 : 3, and 3 : 4, repectvely. Thee tone nterval were called cononance and on them the harmony of Wetern muc ha been developed, epecally o nce, n the Mddle Age, other nterval wth rato of 4 : 5, 3 : 5, 5 : 6, and 5 : 8 had been accepted a mperfect cononance. Nowaday we know that thee cononant tone nterval are characterzed by mple frequency rato of the conttuent tone. A uch, however, th change n phycal decrpton doe not throw more lght on the phenomenon of cononance, and opnon are tll dvded a to t orgn. Helmholtz' explanaton of the dfference between cononant and donant nterval on the ba of nterference between adjacent harmonc (Helmholtz, 1863) ha been oppoed by many other nvetgator who tred to explan the phenomenon n term of percepton of the frequency rato telf, concdence of harmonc, dfference tone or fuon (for reference cf. Plomp and Levelt, 1965). Th dvergence of opnon on the orgn of cononance, combned wth the fact that mot reearch on t date from before 1920, made t attractve to ubmt the phenomenon to a new nvetgaton n whch full ue could be made of more recent development n hearng theory, equpment degn and meaurng technque. In th tudy, a revew wll be gven of the mot nteretng reult of thee experment, a far a they are avalable now. To avod mundertandng, t may be 107

3 ueful to emphaze that the ole concern of our nvetgaton wa why cononance related to mple frequency rato. Although the concept of cononance rather vague and may be dfferent for mucan and laymen, th relatonhp alway nvolved. In our opnon t refer to the pecular enoral experence aocated wth olated tone par wth mple frequency rato. We hall ue the term tonal cononance to ndcate th charactertc experence. The tudy of th phenomenon wa the purpoe of our nvetgaton. Dmenon n the percepton of tone nterval One poble approach toward the queton why cononance related to mple frequency rato to determne n whch way tone nterval are categorzed or, n other word, whch crtera are ued n dcrmnatng them. Th mean, n fact, that we are lookng for the pychologcal dmenon n our percepton of tone nterval. If frequency rato one of the crtera we may expect to fnd a correpondng dmenon n the expermental reult. A, after Helmholtz' concepton, donance related to nterference, t of nteret to examne nterval contng of mple tone (nuod) a well a nterval contng of complex tone (fundamental plu harmonc). The bet way to nvetgate the pychologcal dmenon n the percepton of tone nterval to ue a nonverbal technque, n whch a ubject aked to compare h mpreon of dfferent nterval. Snce, n th repect, the method of tradc comparon very attractve, t wa ued n the preent nvetgaton. The ubject, then, can operate three puhbutton, each correpondng wth a dfferent tone nterval. By preng thee puhbutton uccevely the ubject ha to elect the two nterval whch ound mot mlar to hm and the two whch ound leat mlar. In th nvetgaton (Levelt, van de Geer and Plomp, 1966) 15 dfferent tone nterval were nvolved, gven by the frequency rato 1 : 2, 2 : 3, 3 : 4, 2 : 5, 3 : 5, 4 : 5, 5 : 6, 4 : 7, 5 : 7, 5 : 8, 4 : 9, 8 : 9, 11 : 12, 8 : 15, 15 : 16. To avod the nfluence of ptch a much a poble, all tmul had the ame mean frequency vz. 500 cycle per econd (cp). The component tone of the nterval were ether mple tone, or complex tone (wth harmonc up to 4000 cp, all wth equal loudne). The tmul were reproduced by a loudpeaker, placed n front of the ubject at a oundpreure level of 5560 db (normal ltenng level for peech and muc). Snce wth a et of 15 tmul 108 = 455 dfferent trad correpond, t wa mpo

4 ble to preent any ubject wth the complete et of trad. We therefore developed an ncomplete balanced degn contng of 4 block, each of 35 trad. Each block wa preented to 4 ubject (nonmucan) ; they all judged one block of tmul twce, once for mpletone and once for complextone nterval. A the block were o degned that all 105 poble par of tmul were judged once by each ubject, we obtaned 16 judgement of each par. On the ba of thee repone, for each tmulu par a 'mlarty ndex' wa computed, defned a the number of tme that a pecfc par wa judged more mlar than the other par. Th reulted n two 15 X 15 matrce, one for mpletone and one for complextone nterval. Any matrx element thu determne the 'pychologcal dtance' between two nterval, a large mlarty ndex correpondng wth a mall dtance and a mall mlarty ndex wth a large dtance. The mot utable technque for further analy of uch mlarty matrce that developed recently by Krukal (1964a, 1964b). For a better undertandng of the problem, and of the way n whch the data were handled, the followng bref explcaton may be of value. Suppoe that the percepton of tone nterval determned by three dfferent pychologcal factor. Then, each of the 15 tmul can be repreented a a pont n a threedmenonal pace wth orthogonal axe, the coordnate of the pont correpondng to the contrbuton or loadng of each factor n the percepton of the tmulu concerned. The mlarty matrx nform u about the relatve dtance between the 15 pont. It wll be clear that on the ba of th matrx the betfttng confguraton of the pont can be computed. The only crteron of Krukal' technque that the rank order of the dtance between nterval pont n the tmulu pace mut be the nvere of the rank order of the correpondng mlarty ndce. The degree to whch th crteron not fulflled can be expreed n a percentage of 'tre'. In our cae we do not know a pror how many ndependent factor are nvolved. Therefore, perhap a pace wth more than three dmenon requred to nonvolate the data. On the other hand, 15 pont can anyway be ftted n a 14dmenonal pace, meetng Krukal' crteron wth zero tre. Of coure, n the fnal oluton that confguraton wll be taken whch ha the mnmum number of dmenon tll acceptable. The betfttng confguraton were computed n all pace wth 10 or le dmenon. Th wa done by mean of Krukal' MDSCAL computer programme. On the ba of what Krukal conder a 'far' tre (10%), t appeared that, both for 109

5 mpletone and complextone nterval, a threedmenonal pace uffced to ft the data. For the nterpretaton of th reult t of nteret to look for 'common dmenon' n both pace. We mean by th that, n the mpletone nterval pace, a drecton mght be found for whch the projecton of the nterval pont on an ax n that drecton cloely correlate wth the projecton on an ax of the correpondng pont (ame frequency rato) n the complextone nterval pace. Computaton howed that both pace had two dmenon n common, n other word n each pace a plane wa found for whch the projecton of the correpondng nterval pont are near to each other (correlaton coeffcent of and for the two dmenon, repectvely)*. The projecton of the tmulu pont on th plane are not evenly dtrbuted over the plane but appear to be ordered along a horehoelke pattern wth nterval wdth (frequency dfference between the conttuent tone) a a parameter. Wthout gong nto further detal we may conclude that th relaton how that we have n fact only one underlyng dmenon, common for both mpletone and complextone nterval, namely nterval wdth. Furthermore, the queton of nteret whether there a dmenon n the percepton of tone nterval related to frequency rato, for we know that cononance depend on t. For that reaon n both the mpletone and complextone pace, a drecton ha been determned for whch the projecton of the nterval pont on an ax n that drecton ha a maxmum correlaton wth rato mplcty. Although other meaure are alo acceptable, the frequency of the lowet common harmonc wa taken a a crteron for rato mplcty (for ntance for the nterval 1 : 2 the actual frequence were 333 and 666 cp, o 666 cp wa taken; for 2 : 3 the frequence were 400 and 600 cp, gvng 1200 cp, etc.). A the dtrbuton of thee frequence rather kew, the logarthm wa ued n the computaton. It appeared that there exted an a pror correlaton between nterval wdth and rato mplcty, owng to the nterval elected; therefore, meaure had to be taken to elmnate th artefact from our reult. After correcton for t, we found that, n the complextone nterval pace, a drecton could be determned gvng a hgh correlaton (r = 0.914) wth rato * We are much ndebted to J. P. van de Geer for provdng the technque to compute the common dmenon and the dmenon for maxmum correlaton wth rato mplcty decrbed below. 110

6 mplcty, wherea n the mpletone nterval pace th wa not poble (r = 0.215). Summarzng the reult of th experment, we may conclude that tone nterval, whether contng of mple or complex tone, are dfferentated on the ba of nterval wdth and that, n addton, only complex tone are alo dfferentated on the ba of mplcty of frequency rato. Th concluon mple that, for mpletone nterval, tonal cononance a a enoral experence can only be related to nterval wdth and not to rato mplcty. Th mean that all explanaton of cononance n whch the harmonc do not play a role, have to be abandoned. Relaton between cononance and nterval wdth for mpletone nterval Knowng that, for mpletone nterval, cononance related to nterval wdth, t of nteret to focu attenton on th relaton. It wa nvetgated through experment (Plomp and Levelt, 1965) n whch ubject judged mpletone nterval wth dfferent nterval wdth and mean frequency on a 7pont ratng cale, 1 correpondng wth mot donant and 7 wth mot cononant. Some ubject aked for the meanng of 'cononant'. In that cae the term wa crcumcrbed a 'beautful' and 'euphonou'. Th jutfed becaue, a had been found earlerly (Van de Geer, Levelt and Plomp, 1962), 'cononant', 'beautful' and 'euphonou' are hghly correlated for nave ubject. The tone were reproduced by a loudpeaker at a oundpreure level of about 65 db. The experment were carred out for mean frequence of the nterval of 125, 250, 500, 1000, and 2000 cp. Each ubject partcpated only n one tet eon n whch he had to judge 12 to 14 dfferent nterval wdth around one of thee mean frequence. The nterval were preented fve tme n a random order, and only the data of thoe ubject were mantaned who gave uffcently content repone (r > 0.5 between the core of frt and lat ere). In th way, reult were obtaned for about ten ubject at each mean frequency. A an example, n Fg. 1 the reult for nterval around 500 cp are reproduced. At other mean frequence mlar curve were obtaned, thu ndcatng that there a clear mnmum n the cononance core followed by a broad maxmum for wder nterval. Th reult confrm the valdty of Helmholtz' aumpton that cononance related to nterference. Two tone very near to each other, gve low beat whch 111

7 \ \ \ V % f ' 11 tl N y Z *V r" * \ \ \ \ % \ \ v frequency dfference n cp * Fg. 1. Cononance ratng core of mpletone nterval wth a mean frequency of 500 cp a a functon of frequency dfference between the tone. The old curve correpond wth the medan, the dahed curve wth the lower and upper quartle of the core (averaged over 11 ubject). are evaluated a cononant. For larger nterval wdth, thee beat are o rapd that the ound obtan a rough and donant character whch dappear for tll wder nterval. However, t wa found that, contradctory to Helmholtz' vew on the matter, the frequency dfference for whch the nterval mot donant, depend on the mean frequency of the nterval; th alo the cae for the nterval wdth for whch cononance core ncreae no longer. The queton can now be aked whether thee data can be related to other properte of hearng. They can, ndeed. In recent year, many nvetgaton have been publhed n whch the concept of the 'crtcal band' play an mportant role. Th crtcal band, whoe wdth a functon of frequency, can be condered a the reolvng power of the hearng organ for ound of dfferent frequence (for more detal, ee the econd tudy). A comparon wth the preent reult how that tone nterval wder than crtcal bandwdth are judged a cononant, wherea maxmum donance occur for an nterval wdth around a quarter of crtcal bandwdth. Th upport the aumpton that donance due to nterference. 112

8 Cononance for complextone nterval In practce, tone a produced by mucal ntrument, uually cont of a fundamental and a number of harmonc. Th mple that, for nterval compoed of thee tone harmonc of one of them may nterfere wth harmonc of the other. Accordngly, the degree to whch th occur wll affect the cononance value of the nterval. Th nfluence can be llutrated n the followng way. From the reult of the precedng experment a tandard curve wa derved whch repreent cononance or donance of mpletone nterval a a functon of nterval wdth wth crtcal bandwdth a a unt (Fg. 2). Aumng that the total donance value of a complextone nterval equal to the um of the donance value of each par of adjacent harmonc, th total can be computed by ung the rghthand cale of Fg. 2. Although th aumpton rather peculatve, t not unreaonable a a frt approxmaton. It ue may be jutfed n llutratng how t predct, for complextone nterval, the dependence of cononance on nterval wdth and frequency rato. On th ba the curve of Fg. 3 and 4 were computed for complex tone contng of x harmonc. The frt fgure how n whch way cononance vare a a functon of nterval wdth, wherea the other graph llutrate how the cononance of ome nterval, gven by mple frequency rato, depend on frequency. The curve llutrate: (1) peak of the curve of Fg. 3 correpond wth t.o x crtcal bandwdth Fg. 2. Standard curve repreentng cononance of mpletone nterval a a functon of frequency dfference wth crtcal bandwdth a a unt. The cononance and donance cale are arbtrary. 113

9 o3 :1 r 3:4 56 4:5 V 2:3 J J I 3:5 \ \. \ u 5 V frequency n cp Fg. 3. Illutraton of the way n whch cononance of an nterval wth a lower complex tone ot 250 cp and a varable hgher one depend on the frequency of th tone. Both complex tone cont of x harmonc. The vertcal lne repreent nterval wdth after the equallytempered cale. 5O :2 y * ' =C~, 5:6 3:4 4:5 Y3 10' frequency n cp * Fg. 4. Illutraton of the way n whch cononance ot ome nterval wth mple frequency rato depend on the frequency of the lower tone. Both complex tone cont of x harmonc. mple frequency rato of the component tone; th how that, for complex tone, cononance related to thee mple rato; (2) more mple rato are repreented by harper peak, o the octave (1 : 2) and ffth (2 : 3) are much more entve to a devaton from 114

10 ther correct rato than are other nterval; th explan why n the equallytempered cale (vertcal lne of Fg. 4) the mpure thrd (4 : 5 and 5 : 6) are much more tolerable than mpure octave and ffth would have been; (3) the relatve heght of the peak of Fg. 3 and the curve of Fg. 4 ft rather well the rank order of cononant nterval a accepted by mucan; (4) wth decreang frequency cononance better preerved for more mple than for more complex frequency rato (Fg. 4), reflectng the mucal practce to avod thrd (4 : 5, 5 : 6) at low frequence. Stattcal analy of chord n muc The cloe relaton between cononance and crtcal bandwdth raed the queton whether n muc, too, we may fnd th relaton. A the experment howed, maxmum donance correpond wth an nterval wdth of about a quarter of the crtcal bandwdth, wherea cononance ncreae no longer for nterval wdth exceedng th crtcal value. Th ugget that epecally th range wll be of nteret n muc to dfferentate between more cononant and more donant chord. To check th aumpton tattcal analye of the chord n ome mucal compoton, the 'vertcal' dmenon of muc, were carred out. An llutraton may erve to explan how the analye were done. Suppoe we are ntereted n the occurrence of nterval wth c 2 = cp a the lower tone. We then ort out all chord contanng c 2 and a hgher tone multaneouly and determne the fracton of tme durng whch the nearet hgher tone eparated from c 2 by a dtance of 1 emtone, by 2 emtone, etc. It may appear that dtance of 1 emtone a well a of 15 emtone are rare, wherea a dtance of 4 emtone rather common. Of th 'denty dtrbuton' of nterval we can determne the 50% (medan), 25% and 75%pont, repectvely. Thee number repreent a good meaure of the wdth of nterval wth c 2 a the lower tone. By repeatng the procedure for other tone over the relevant frequency range, we can fnd how the 50%, 25% and 75%pont depend on frequency. A n practce mucal tone nearly alway cont of a number of harmonc wth ampltude comparable wth the ampltude of the fundamental, we are alo ntereted n denty dtrbuton n whch harmonc have been taken nto account. Th can be done, popularly ad, by plottng the note of the frt n harmonc n the core of the 115

11 muc and treatng thee addtonal note n the ame way a the orgnal one. In th way denty dtrbuton can be determned for varou value of n, premng that the addtonal note ft our tone cale. Th rather well the cae up to n = 10. Denty dtrbuton a a functon of both frequency and number of harmonc were computed, ung epecally developed equpment, for part of mucal compoton of J. S. Bach, A. Dvorak and A. Schoenberg. In Fg. 5, ome reult are reproduced graphcally, baed on the thrd movement of Dvorak' Strng Quartet Op. 51. The denty dtrbuton of the other compoton led to mlar graph. A comparon of the old curve wth the dahed curve correpondng wth crtcal bandwdth and a quarter of th bandwdth, repectvely, how that, for a number of harmonc repreentatve for mucal ntrument, all curve have about the ame hape. Moreover, the tuaton of the old curve demontrate that mot nterval have a wdth between crtcal bandwdth and a quarter of t. Thee reult ugget that crtcal bandwdth play an mportant role n muc. Apparently the regon over whch for mple tone the cononance mpreon trongly depend on nterval wdth ued for 'modulaton' between more cononant and more donant chord. We hould realze that th equally deep 'penetraton' n the borderland between pronouncedly cononant and donant ound a reult of many uch factor a the Wetern tone cale, the number of multaneou tone and the prmary nterval elected by the compoer. Concernng the latter factor, a comparon of the denty dtrbuton of three compoton howed that n modern muc the jutmentoned borderland penetrated more deeply than n older muc. Concluon The nvetgaton trongly ugget that the relaton between tonal cononance and mple frequency rato, a t ha been found n practce, a reult of nterference of adjacent harmonc. The fact that the Fg. 5. Reult of a tattcal analy of the chord of the thrd movement of A. Dvorak' Strng Quartet Op. 51 n eflat major wth n (= number ol harmonc taken nto account) a a parameter. The old curve repreent the 25%, 50% and 75%,pont, repectvely, of the cumulatve denty dtrbuton of nterval, plotted a a functon of frequency. The dotted curve correpond wth crtcal bandwdth and a quarter of th bandwdth. 116

12 n z ~ _ \\ A 2 S g <J j 28 " : o to K m a> n J 129 v, E \ X, ' 5» 'l S» _ '? f 1 f. " ' > v >: 220 p vl B 898 ~ x 913" 1077 f ' (a) n = 1 1 R 3 " A 1220 "" 1024 (c) [ "909" Ce) 456.* (g) _ > E n = 3 y y y n = * r n = 7! S : :? : J K 7! r,'! y (b) H h h : : ~ f a P S E 2 P 1 1 ' X ' ' "".. I I I I I V 7 < 4 ~~ t y I;I n = 2 V 7 (d) n = 4 7 ' (f) f = n = 6 7"T n 1 III ' o * frequency n cp * " ft y ~~ " V w $ o> I (h) n = 8 Fg

13 percepton of nterval contng of mple tone not governed by frequency rato, but only by nterval wdth, confrmed by the fndng that for nave ubject, the cononance value attached to th type of nterval a contnuou functon of frequency dfference. The expermental fact that frequency rato an mportant factor n dcrmnatng complextone nterval can be explaned by ung the hypothe that, n th cae, not only the fundamental nterfere, but that adjacent harmonc do too. Moreover, the experment howed that th nterference only occur for frequency dtance wthn the 'crtcal bandwdth'; the mot donant nterval correpond wth a frequency dtance of about a quarter of th bandwdth. Th relevance of crtcal bandwdth for the percepton of tonal cononance upported by the reult of tattcal analye of chord n mucal compoton. REFERENCES HELMHOLTZ, H. L. F. VON (1863), De Lehre von den Tonempfndngen al phyologche Grundlage fr de Theore der Muk. F. Veweg & Sohn, Braunchweg. GEER, j. p. VAN DE, w. j. M. LEVELT and R. PLOMP (1962), The connotaton of mucal cononance. Acta Pychol. 20, KRUSKAL, j. B. (1964a), Multdmenonal calng by optmzng goodne of ft to a nonmetrc hypothe. Pychometrka 29, 127. KRUSKAL, j. B. (1964b), Nonmetrc multdmenonal calng: a numercal method. Pychometrka 29, LEVELT, W. J. M., J. P. VAN DE GEER and R. PLOMP, Tradc comparon of mucal nterval. Brt. J. Mathemat. Statt. Pychol., n pre. PLOMP, R. and w. j. M. LEVELT (1965), Tonal cononance and crtcal bandwdth. J. Acout. Soc. Am. 38,