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COMBINATION TONES AND OTHER RELATED AUDITORY PHENOMENA

A DI SSERTA TION

SUBMITTED TO THE FA CULTY

OF TH E

GRADUA TE SCHOOL OF A RTS AND LITERATURE

IN CANDIDA CY FO R THE DEG REE OF

DOCTOR OF PHI LOSOPHY

DEPARTMENT OF PSY CHOLOGY

BY JOSEPH PETERSON

u rr N o o r r un Psvc a o wc xcu. Rm " wa Sm u n . Pvumun AS “o uo c 39 ,

1 908.

PREFA CE .

The first part of this mo no graph is devo ted primarily to a critical exposition of the important theories of combination to n and t t nt e n es a s a eme of th facts upo which they rest . This undertaking i nevitably leads to the mention of a considerable number of closely related phenomena whose significance for I e general theo ry is often crucial . n view of th conditions l n in the t t the t it n th t prevai i g li era ure of subjec , has bee ough expedient that this presentation should in the main follow h n n . n c nt nts c ro ological li es The full a alyt ical table of o e , t t t the n int n n oge her wi h divisio o sect io s , will readily e able he x readers who so desire to consult t te t o n special topics . The second part of the monograph report s cert ain experimcnta l t n t observa io s made by the au hor o n summation tones . In the pages which follow several physical terms are used which are commonly understood and need no defin i tion here . A few may be newto readers no t familiar with the literature l ” of acoustics . A pendular ( or pendu um ) vibration is a in - t n I t t n the simple , or s us form, vibra io . n several quo a io s ” ” n t and t n the n wor o e , some imes sou d , occurs where mea “ ” “ in t nt t n g is precisely that of he word tone . I errup io " ton e is used synonymously with i ntermittent tone . By transformation of the primaries is meant the process of superposi tion of vibra tions explained by Helmholtz in his m t t t n t n t he n n t n t n a hema ical de ermi a io of origi of combi a io o es . A c in t n t n o f r t r f n n omb a io o e may , cou se , be ei he a dif ere ce to e or a summation tone . and to n t the two t n Where p q are used desig a e primary o es , a p refe rs to the higher nd q to the lower tone . Frequently h and l are used for the higher a nd lower tones respectively : t e n no w h se are most commo ly used , especi ally by German ’ t the writers . N is some imes used for lower a nd n for the S t s it the e hiflier tone . ome wri ers have u ed for i nterrupt d and m for the interruptio n tone ; others have used these letters

I l l i v PREFA CE . i th n e reverse order . Since quo tations are so frequently made from various writers no exclusive use of any of these letters can t well be adop ed here . In n t n the t an t n i desig a i g oc ave of y o e , usage also var es . ” ” S t c e. . o t c and t o t s ome wri ers use , g , where hers use , s ill her

c . In t s the no w n C C z hi paper more commo usage , g , l ” 3 C c 1 . c c C n o t c n 2 . vib , , , , has bee ad p ed , bei g 8 d I have taken the liberty to change all other markings into this t in the he ex nt l t t t n . In t erime a sys em quo a io s p par , however, the n x e where Koe ig forks were used e clusively, I hav simply

th n t n . Ut Re etc . e en e. employed desig a io s used by Ko ig, g , a, s, , t h U n t e c 1 2 . z bei g same as ( 8 d . v ) In quo ting fro m the French and German where the trans t n o wn e n t t n la io is my , I hav used si gle quo a io marks t R t t E ter The heories of u herford , Waller , Hurs , mile E and t no t n n in t Kuile , wald, o hers , have bee co sidered his h . In th n on t e t t nt te paper e opi i of wri er, hey have co ribu d

5 e. n t n to the t t n n t n . o hi g subjec immedia ely u der co sidera io , , t in seco ndar t n n but as et co n cer a y audi ory phe ome a , are , y , cerned essentially with the prima ry phenomena of hearing . It in t nn t n to x t is a pleasure , his co ec io , e press gra itude and to c n a t n to t a k owledge oblig io my eachers , Professor J. R.

l n r. n a d D . . t on o o e A ge l J B Wa s , fr m wh m I have receiv d n t n and t t in co s a t advice pa ient cri icism . I am much debted to

r . n n n M . W V . D. Bi gham for ki d suggestions a d assistance . I also wish to thank my fellow students of the psychological t t f x labora ory, who served as subjec s o r my e periments C T ON ENTS.

A P RT . H STO C L N D I I RI A A Cac L.

: 1 mmy BSE V T S A N D a e B T N m m . mal 8 E O R A ION T s , COM INA IO T To N Es, BEATS , RESONANCE PH BN OM BN A , IN BRRU PTION m TON Es A ND so FDa .

PI E-HELM HOLTZ IAN DEV ELOPMENTS ’ a l i es e un r r scoveri N turall to . o h o ( ) Ea y D a y L ad T Y g s T e y, Becaum the First Difi erence Tone has a Frequency Equal

to that o f the Bea ts o f the Primaries . ' 6 But h i l eo hm ( ) P ys ca Th ry is Opposed to such a View. O s

( c ) The O hm- Seebeck Con troversy ’ HELM H OLTz s PosxTrON A N D CON TamurrON s

’ ( a ) Being a Physiolo gist he Defends Ohms LawBetter than did Ohm imself who was h si i t H , a P y c s ’ b e he ( ) H elmholtz s R sonance Theo ry Illustrated . T Resonance o nce n as l el un . M r t io w u b . o ull C p Vag e y H d y T Y g , J e and o thers

‘ ’ ‘ ’ ( c ) The Princ iple o f Resonance ; Perfect and Imperfect o A li tio n to Anal i in Rwona t rs. pp ca ys s the Ear ’ 4 Wh elmho ltz o ul no t Ac ce t oun s heo r e ( ) y H C d p Y g T y. Th Pla ce of Stimula tio n in Co chlea is Impo rtant ’ ( e) Helmho ltz s Explanatio n o f Beats and o f Combinatio n ‘ ’ ‘ ’ ones b T , ( r) Subjective, ( 2 ) O jective I 3 , (I) From Theo retical Co nsideratio ns he Disco vers the Summa tio n f terw e one hich he A r s e rs. H thus T , w a d H a Ob ’ ta ins Ano ther Objection to Yo ung s Theo ry ' ( 1 ) An Inco nsistency in Helmho ltz s own Position ’ Bo sA N U ET s ATTE PTE P Q M D SU PLEM ENT . ’ ’ BosAN QU ET s STU DY OP B EATs ( FA voa AELE To HELM N OLTz s Tq av ) ’ mmAPP S As O PP NT To H eLM E LT Ko EAR AN ONE O z s VIEW . a Sho ws tha t Primaries o f Lar e Inte r ( ) g rval Beat. Lawfo the itch o f and lower be ts which ma p ”W a , y give rise co beat no ees o f similar orders

V vi CON TEN TS .

’ ( 6) Koenig s Viewas to howthe Ear Experiences Beats and Beat To nes : Any Perio dic Fluctuatio n within Certain Limits o f Frequency may be Sensed as To ne i The nterrutio n ( c ) Experiments in Suppo rt o f th s View. I p ’ o ne enerate b nterru tio n o f a o ne b me ns T , G d y I p T y a o f a Perfo rated Disk K B WAs N OT TH E RST To P F R U CH ExPEruM ENTs O N IG FI ER O M S , HOWEV ER ’ ( a ) The Webers Experiment o n Rotating Tuning Fo rks ( 1 82 5 ) ’ ( b) Seebeck s Theo retical Investigation o f such Pheno mena ( 1 844 ) ’ ( c ) Helmho ltz s Experiment and M athematical Explanatio n

d ee e eats the Weber x riment with li h l D f ( ) B tz R p E pe , S g t y i

( e) Stefan Co ntinues the Experiment ; Hears the Varia tio n

o n d M athem tio T es an Develo ps atica l Explana n o f Them.

ater he a o ts the M etho use afterwar as we have L d p d d d ,

f o seen, by Koenig . Pro ess r E . M ach uses it at abo ut

the same time, and Hears Variatio n To nes . (f ) In the M eantime Radauhas also Explained the Obj ect ive

ri in o f ria i n o ne He fi ue name fo r O g Va t o T s. ( rst s s this these to nes)

eetz is a ain tim a o x erim n He Acce ts ( z ) B g S ul ted t E p entatio . p the x lanatio ns o f tef n and Ra au ari tion o nes E p S a d . V a T , en ha e th , d co me to be Regarded as Obj ective, and had b en f n l Rein o rced by Reso ato rs. Y et Koenig did no t App y this Test to the Interruptio n To ne ’ M M er er ment imil r to at o f ni he h A . . s x i K ( ) ay E p , S a th oe g ; Regards Beats o f Interference as Similar to those due to Interruption ( i) This Calls o ut a Statement fro m Lo rd Rayleigh ; Beats o f nterferen n er o a Cha n e o Phase the o thers do I ce U d g g f , no t

’ KOENro s N ExT ExPEmM EN T

( a ) Beats o f To nes Perio dically Variable in Intensity give rise to Co ntinuo us To nes when a Certain Frequency is e e Reach d . Rayleigh and Bosanqut Suggest that this does no t Pro ve Anything fo r Beats o f Interference b Koeni Antici tes this Ob ectio n and in Ano ther Ex eri ( ) g pa j , p ment n e vo rs to mit te the h n e o f hase o , E d a I a C a g P , Sh w vii CON TENTS .

h eat s ve rise when e uent eno uh ing t at the B s till gi , Fr q g , to Tones By an Experiment on the Wave Siren Koenig finds that a

Shif t o f Phase o f Upper Partials slightly Alters the To ne. ’ H erma nn re This is contrary to Helmholtz s theo ry. ’ x ri ents gards Koenig s Experiment as Inadequate. E pe m on the Ediso n Phonograph lead him to Co nclude that

no t h s h n e er se but the enera l o rm of the P a e C a g p , G F

' i still Wave is what maka the Difi erence. The quest on

Koenig Applim Wave Siren Experiment to Beats : Beats

o Zahm thin s this when fast Eno ugh Produce T nes. k ’ i is Co nclusive fo r Koenig s Theo ry. Rayleigh g vm Rea sons why it is no t

oen i hel to the eson nce o thesis as did o un but K g d R a Hyp , Y g , ’ Re hm Nowth t he fi n s ts R sult jected O s Law. a d Bea e ing fromLarge Intervals o f Pure To nes ( i ) he Supposes hm’ L that he has Additio nal Evidence against O s aw. But this is by no M eans Co nclusive ’ Ko en ig s Viewo f the Relation o f Beat Tones to Co mbina

io n o n ir elieves tha o mbin tio n o nes xist t t es. F st B t C a T E ; Pro ves their Existenc e by M eans o f Auxiliary Fo rks r u e x l ins Sum late Do bts that ther are such tones. E p a

( z ) Preyer Obj ects that it Requires the Pmence o f Upper rti ls which do no t xi His o wn x l n tio n Pa a E st . E p a a 26 — ( b ’ ll — ( ) Koenig s Rep ly ; the I nterval ( b a ) : 26 is to o Large. Koenig is fo rced to a S tri king Inco nsistenc y

SEOTTON 2. TH E O sJ Ec rrvrrr OP COM BINATION TON Es.

’ ( a ) Preycr s Experiment. Co mbinatio n Tones from Tuning Fo rks Fail to Affect Delicate Reso nating Fo rks ( 6) Lummer makes a M icro pho ne Reso nato r Respon d to the umma o f i S tion T ne o Pr maries fro m the Harmo nium.

Wien fi e ( c ) M . nds that a D licate Reso nato r is no t afi ected by Lo ud Diflerence To nes fro mQ uinc ke Tuba ’ d ermann s x eri men o u D er one n ( ) H E p t. L d ifi enc e T Tra s mitted thro ugh Telepho ne do es no t Afi ect a Delic ate Tun~ ing Fork CON TEN T8 .

li nin The Experiment o f Riicker and Edser. A De cate Tu g Fo rk is make to Carry a M irro r : Light Streaks Reflected Co m by this M irro r are Observed thro ugh a Telescope. binatio n To nes are Objective when Primaries are gen

h i true even o f nterme iate erated with a Siren. T is s I d ’ r o n oeni s o wer eat No te o f nter Diffe ence T es. K g L B I val Octave Fo und no t to Afi ect Reso nato r ; the same Ex eri is true also o f Beat No tes o f Upper Partials. p

fi rr r n t r esults e ments Veri ed with a M i o Reso a o . R N ga tive when Primaries were Independent in Origin Fo rsyth and So wter Pho to graph Co mbinatio n To nes from Siren To nes

h f i h e n to r einfo rces irst and eco n . c ae er wt so a K. L S , R R F S d Difi erence To nes o f Primaries fro m H armo nium and Triad Apparatus General Co nclusio ns : Co mbinatio n To nes f ro m Dependent

rim ries are Ob ective in art Others if Ob ective P a j , p ; , j

l r x remel Wea at al , a e E t y k The Questio n o f Objec tivity is Impo rtant in Co nnectio n with o ur Problem; Objec tive Pendular Vibratio ns need

n e o o nflict no Physio lo gica l Explanatio , and H nce do n t C ’ with Ohms Law

T N T PT N Es. SECTION 3 . LA ER EXPERIMENTS O IN ERRU ION To

M ATH EMATICAL TH EORY H AD N OTY ET EXPLA IN ED TH EIR ORIGIN . ’ nne x erim erf r e r n H ea ( a ) De rt s E p ent o n the P o at d Si e . e H rs ’ e M er e e in B ats ge into To nes. H R gards these ( aga st ’ Ohms Law) as Subjective ’ b ermann s x erimen The ather Ill- efine n er ( ) H E p t . R d d I t ’ rutio n o ne o ether with the No ise o f the o tatin p T , T g R g

is inall Obliterates the iven o ne Alto ether A D k , F y g T g . To ne Transmitted by Telepho ne is Interrup ted : same

esul The v r e R ts. Sa a t Too th d Wheel gives Similar ul es ts. He o nclues with oeni th t the Ear n R C d , K g , a Se ses Rapid Beats as To ne

’ o i t Acce ts this iewand r es esi es e h V g p V , U g B d that H lm o ltz s Theo ry Fails to Explain the Lo ud Intensities o f some o mbinatio n C To nes. He A ttempts to give a Theo ry to Co mbine the two Opposing Views TS ix CONTEN .

‘ v ODI C N Sac rron 4. FURTHER CRITIc IsrIs A N D M FI ATIO s o r HELM ' H o LTz s VI EW.

’ N M U LTIPLI ss DI PFICU LTI F R HELM H O Tz s HBRM A N ES O L WIN . ( a ) Hears a Second Difference To ne much Lo uder than the First ( 6) Difi erence Tones are Audible even when ears are filled with Wax o r Co t ton against Drum ( c ) To nes Co n ducted Indepen dently to each Ear Pro duce Dif

( d) Suggests that the Fibers are too Small fo r Reso nance with Lo wTo nes ( e) Dec ides tha t any Perio dicity within Certain Frequency Limits may be Sensed as To ne ; so Abandons Reso nance

heo r but still o l ifi ner es o f T y H ds to Spec c E gi Nerves . ' erm nn s M i le o ne Lawo f its Phase Re er (I) H a dd T v sals . ( 1 ) Experiments to Sho wthat Phase Changin g Tones may be Perceived ( the Too thed Siren )

h heo r tha Difi e o e from evers ls ( ) T y t rence T nes R sult Phase R a . ' ( i ) H ermann s M o dified Reso nance Theo ry ’ (i ) M ax M eyer Repeats H ermann s Experiments wit h Similar

esults hin s that ermann ear er Partials R ; T k H H d Upp , no t a M iddle To ne ‘ ' ’ ( l ) Wundt s Supplement to Helmho ltz s Theo ry ; Nerves may be

i e l timulate n o nsis n f D r ct y S d ; I c te cy o this View.

1 ross and o o win in o n o ndu in one to h ( ) C G d F d , C ct g One T eac Ear tha t such o nes ma ea t but o no en te , T y B , d t G era Difi erence To nes ( m) St umpf Disco vers the Interto ne ; Explains it on Basis o f

h sio lo ica A a ta ti i l elm P y g l d p o n , thus st ll Ho ding to H

’ Ebbinghaus s M o difi catio n o f the O hm- H elmho ltz ian View; Nerve Elements at First Undifi erentiated ; Partial Vibratio n o f basilar M embrane Fibers ; Frequency o f Stimulatio n Determines Pitch ; Explains Origin o f Dif ’ ference o nes without Ohms Lawbut f ils to see th t T , a a there is a Fundamental Co ntradictio n between the Co n cept ions o f Partial Vibratio n (which Requi res Perfect Resonato rs ) and Inelasticity (which Denies such Perfec

REC ENT ExPERIM EN Ts : TH EY Rsmovs CERTAIN IM C N EN TS O T .

p o RTA N T O EJ EOTION S To TH E O H M -H ELM H OLTZ IA N

’ e nate e nterrutio n o ne Schaef r and Abraham Reso th I p T ,

o f o m lete nterrutio n 2 o f Perio ica ll ari ( I ) C p I p , ( ) d y V ’ ’ - able Intensity ; also Hermann s Phase c hanging To nes. In Later Experiments they Reso nate the Variatio n To nes

e h in erru tio n o nes and fi nd , as Koenig had do n , t at t p T are most Intense when H igh Fo rks are used and when

hese t e the Variatio n to nes are Sca rcely No ticeable. T sults Suggest that Interruptio n To nes are really ( in part at least ) Difi erence To nes o f the given To ne with the Variatio n To nes

Zwaa rdemaker b nterrutin 6 times er seco n a o ne y I p g 4. p d T Transmitted by telepho ne had Heard a Powerful Inter ” rutio n o ne whi however he e ar e as Ph ical p T , ch, , R g d d ys in Origin Schaefer and Abrahammake a M o re Extended Study o f the Intensity Relations under Co nditio ns Similar to those o f Zwaardemaker ; the Interruptio n To ne was really no t intense and well-defi ned ; it was Strengthened by Reso nato rs ; therefo re its Explanatio n is no t a Physio lo g

ical Pro blem

S ECTION 5 . INTENSITY RELATION S.

T E TI N H Q U ES ON OF I TENSITY RELATIONS . ( a ) Unimpo rtant to Earlier Writers M M fi b . a e n ( ) A . y r ds that LowTo nes may Obliterate Higher Ones

( c ) M ayer and Koenig Hear Difi erence To nes fro m Primaries

whose re uenc ies are above the imi f Au ibili F q L t o d ty . ( d ) Hermann Definitely Raises the Intensity Obj ectio n to ’ H elmho ltz s Theory ; Hears a Second Difi erence To ne

o uer an he irs m f h e L d th t F t. Stu p ad Observ d the same Pheno meno n

’ ( e) M eyer s Theo ry Devised Especially to meet the Intensity Diffi culty ; M eyer Divides Co mbinatio n To nes into Three

in s. A rief utline o f i h I K d B O th s T eo ry. t is the o nly Theo ry that Explains why a LowTo ne canno t be Oh literate b i he t i has o fil d y a H g r. Bu t n t yet Ful led the ur se o f i ein o ts i. e. to x l in nten it el io n P p B g ; , E p a I s y R at s. xi CONTENTs.

ws P OMBI AT O TON Es. SECTIO N 6 . LA O C N I N ’ f om in ion o ne NO Pl ce fo r In ( a ) M eyers Laws o C b a t T s. a

ter M e er hm u es t e ma be Difi erence The lat . y S gg d . y

’ Krueger s Recent Experiment : Primaries used were o f Inde pendent Origin and Upper Partial Tones were Eliminated e i A i e n e o f nter ls by I nterfer nce P pes. W d Ra g I va uh e ul s : umm tio n ones and nte me i te St d d. R s t S a T I r d a

Die rence Tones H eard Frequently. Lawo f Die rence ne e ts Of Wi e nt rv ls o un to be Due to To s. B a d I e a F d e o n here no i ver Die renc T es. T are Subj ect ve O tones

’ Helmho ltz s Explanatio n o f Subjective Co mbination Tones ’ efecti e as ho wn b ennet t s u a D v , S y D S bj ects who h d Neither Tympana no r fi rst two Ossicles ’ " ’ h er su lement u e Sc ad s pp , S gg sted by Helmholtz s Expla nation o f Obj ect ive Co mbination To nes ; Experiments to u o r his ie He x l i Au ible m S pp t V w. E p a ns d Su mation on s as Difi erence o nes o f er T e T Upp Partials . ’ ’ Meyer s Review o f Schaefer s Article ; Objects to the Sup plement Schafer in Reply Admits Explicitly that bot h Summation o nes and nterme i te ifference o nes xist but are T I d a D T E , ’ too Wea to be ear n er Or in r o n itio n k H d U d d a y C d s . Are Subjective and Object ive Co mbinatio n To nes Expli cable on the same Physical Principle ? Statements by Helmho ltz and Lo rd Rayleigh Co nsidered ' A Sugges ted M od ificatio n o f Helmho ltz s Theory

XP E M E AL : H PART II . E RI NT T E Exp LA N ATION OP SU M MATION TON Es As DIPPEREN CE TON ES OP UPPER PARTIA LS SH OWN I MPOSSIBLE

Old Experiments Repeated ; Those no t Repeated r o e to n es ate P lit Of x lainin umm Pu p s , I v tig oa ibi y E p g S atio n To nes as Difi erence o nes entirel o r in a rt o f er Partia ls I o T ( y p ) Upp . I the use o f uxi i r M A or s. his Metho d , l a y F k T etho d has its Dan ers which are Po inte o ut and o nscio usl Avo e g d C y id d. There

The Efi eet o n the Generamrs o f Beats of Inaudible Difi erence

' ‘ Tones ; most Evident when the Interfering Difi ermee l ones C N N T O TE S .

b r r m rf o nso nan hi are o th o f the Fi st O der. In I pe ec t C ces t s

i i n r i r n m rfe t atio n . I e Co nd t o is Neve Poss ble. Illus I p ct Co nso nances the Beating o f Diff erence To nes always Afi ects

P i llustra io n o f ets o f h ee P im o n rimar es. I t S T r r ary T es

whi a ll eat as a esult o f nterference f Difi eren o n ch B R I o ce T es. Proo f o f the Cause o f the Beat ing ; Difi erence To nes Audible when the Primaries are Stro nger Beating less M arked when a Difi erence To ne o f two To nes Inter n feres with a Third given To e ( an Auxiliary To ne) . Here

mo re atisf o ril t h the Beats may be S act y Lo ca ed . (T e Causal

o i ere Co nnec ti n s les s close. ) But th are still Do ubts abo ut

in he e h Lo cat g t B ats. Cases in wich this is Especially True. Illustratio ns

When the i hest o f the hree o nes i . e. the Auxiliar is H g T T ( , y) very weak the Beating may be due So lely to Presence o f sum

i e i mat o n To n . Obj ec t o ns : ( I ) Beats may be f ro m Lo wDif

n e n r fere ce To ne, ( 2 ) o r du to Ge e al Difiiculty in Lo cating

A im le x eriment to llustrate the o rce o f thi Beats. S p E p I F s Obj ec tio n Co nditio n Under which these Obj ect io ns have less Fo rce : La rge

n erv betwen Auxiliar o ne and i her Primar llus I t al y T H g y. I

n liar o ne e in t ratio s. Y et Auxi y T B ats Pla ly. Such Cases,

f e eem o n i e the Presen f m io e there o r s t ca t ce o a u mat n o n . , I d S T A Possible Obj ectio n : but it will no t co me f ro m Dangero us

Quarters and will mean no thing . The Impo rtant Questio n : A re Summatio n To nes Really Difi erence To nes ? Riicker and Edser Showed that Obj ective Summa tio n To nes are no t Are the Lo uder Summatio n To nes due to Upper Partials ? Schaefer w i le E and Others have Recently Tho ught so . T o Poss b x

h h — t h t an planatio ns o n his view. ( I ) a ( ) d

— = h r o r n h — mt = h t ( 2 ) u( h t ) + , + The First Viewis Analo go us to the o ne which Derives the Seco nd ut is os ibilit is Re Difi erence To ne f ro m the First . B th p s y and was in ee earl is ute b futed by an Experiment , , d d , y D p d y

e ntensit elatio ns var with Difi erent Hermann and o th rs. I y R y ’ n The m ro babilit o f Pre er s Pitches o f Primary To es. I p y y . ar e ntervals o f Primarie Explanatio n : It Invo lves too L g I s. It is Dis ro ved b Ex erM entS in which Primaries are Sub p y p— — stituted fo r a b and h t in the Equat io n a h ( h t ) N T T CO EN S .

t h + . In such Cues the Summa tio n Tone sho uld be

e o e eve e Lo uder but it pro v d t b the R rs . The Second ViewCo nsidered ( I ) Auxiliary Fo rks ShowExistence o f P rti ls onl u to e ifth 2 ubstitu n a rima r a a y p th F , ( ) S ti g P y Tone fo r the First Upper Partial Rep resented in the Expres sio n 21 — 10 it is Pro ve that the er Partial in Seco nd , , d Upp

Difi erence Tone is inefi ective. This would also ho ld fo r Summatio n To nes if they are Difi erence To nes Nowit is also Pro ved by Tests with A uxiliary Tones tha t Lower Orders o f Partials than tho se Required fo r Explain ing the ’ - ummation one as oeni x l ine it ive no beat t o ne. S T , K g E p a d , g Several Experiments make this Co nclusive A more Direct Dispe o f the Validity o f this Second View: Sev tio n o n were i eral Summa T es D rectly Audible. Pro o f : Sev e e eral Obs rvers S lected To nes to M atch them. On the B asis o f the Fact tha t if a M istuned Uniso n Bea monce per secon d o f e Prim ries ill ea t wice he the Seco nd Partials th a w B T , t fo un tha t these umm i ir hree es etc. it is t on Th d T Tim , , d S a Tones Actually Heard ca nno t be Explained as Die rence

General Co nclusion : Summa tion To ner not Explicable a: Dil Ierence To ner Tabular Statement o f Chief Obj ections to the Ohm-Helmho ltz ian

Tabula ted Statemen t o f Eflic iency o f Theo ries Co nsidered . Appendix : Recent Investigatio ns Co nsidering the Possibility o f Resonance in the Ear

COMBINATIONTONES AND

OIHERRELATEDAUDITORY PHBNOMENA

R PA T I .

S R C AN D R C HI TO I AL C ITI AL .

SEC I . EA RLY B SER ATI N S A N D H E IES CO TION O V O T OR , M

B IN ATIO N N ES EATS RES N A N E H EN EN A TO , B , O C P OM ,

N TER U PT N N ES A N s T I R IO TO D o FOR H .

’ It is well known that up to Helmholtz s day dili erence tones to n t m were supposed origi a e fro rapid beats . We shall see t hat this view had taken its rise naturally from the fact that t he vibra tion-number o f the noticed difference tone equals ex l t t O f the n t n a ct y he number of bea s ge era i g tones . ‘ in t n t n o in 1 Comb a io o es were disc vered 7 4 5 by Sorge , who nt n t o appare ly heard , usually , o ly hose which c rresponded in l he pi t ch to wha t is no wca led t first difference tone . Krueger 2 suggests that So rge heard this tone only in cases where it is reinforced by coi ncidence with one or more of the combi nation in the a O f the t e tones of higher order, as c se fif h or th maj o r

The pitch O f the first combi na tion to ne was estimated an the t n o n t t n I 3 octa ve to o high , by I alia vi li is , Tar i i ( 7 5 4 ) . ’ i tin This tone is somet mes called Ta r i s tone . Its t rue pi tch was determined theoretically by Chladn i in fo r i nterva ls a wfi ch may be represented by n : I . Wilhelm Weber

‘ - m al . o m o itio n 1 V m ad i n ed i s . . e A Sarg , o rx C p , 745 47 " ’ n h l tudies I I I i bins tionsta e i . S . O I . 2 . Za r fl eer e der Co rn . P , XV . o , p 07 ! lmh lt o a m er . x3 also He o S ensa ti n: r third ed. o f Krueg , g . as. 9 9 ; z . ] , J SE H ETE S O P P R ON.

also t o n the t t t n and r d specula ed pi ch of his o e , ag ee l o — n s i with Chladni . He a s remarked what in ma y case s t rue — that double clangs whose vibration ratio does no t depa rt to o

the t u t n n Verhiiltnissen i . e. much from re rela io s (wahre ) , , S t n n n n the ff n to n lightly mis u ed co so a ces , should give di ere ce e “ r ( weakened) of the nearest consonant interval . Kruege thinks that the tone Weber had in mind is the interto ne ! ( Zwischenton ) later discovered by Stumpf Weber is led to the conclusion that when the vibrati o n ratio lies as nea r to as it does to it should be possible ’ sometimes for o ne to hear a Tartini s tone a third lower than e r t n and o m the second octave below th lower prima y o e , s e r n e t times a tone two octaves below the lower prima y . I v s iga ’ ” to t t s . tions have failed thus far, he said, discover bo h of he e On the previous page he states somewhat definitely the view

nn n n . n e. . ra earlier a ou ced by Thomas You g Whe , g , 4 vib tions of the o ne wave-series f all into the ca r in the same period O f n O f the t th will x n d time as 5 vibratio s o her, ere be e perie ce the ca r in r o ne n n n by , eve y such period , i crease (A schwelle ) and o ne in nt n t O f t n n t cu decrease i e si y each o e , or, whe hese re r to o to t t n l rapidly be perceived separa ely , a deeper o e wi l be m n n in t e t heard ( e pfu de ) , which is equal pitch o th one which would result if the drum were actuated by vibrations 4 times slo w I n t o s the fi t - as ( 4 mal a gsamer) as h e of rs wave series . Such a tone is two octaves lower than the o ne generated by the fi t - rs wave series (p . 9 In the nt n Blein mea ime Baro , whose work was brought to ’ n t x n V o n t Weber s o ice by Ale a der Humbold , had been work 1 in t t n d n the n nt t n a g wi h vibra i g cor s , usi g fu dame al o e ( 2 5 6 ) and the other prima ry tone varying from this pitch to 5 1 2 n e n vibratio s . He had heard a s co d combination tone with the interval 2 and with three other intervals between the

’ ’ W r ’ W. ebe eber i Tartini sch T n n , U d e e 6 e , Po endo rfI s A nna) . def h s“ gg P y ,

. 1 82 . 2 16 . XV , 9, p ' ue er i r o . c 1 K g , p t , p . 9 1 . ' m C. Stu f To n h s c o lo i I I . 1 p , p y g e, 8 0, p . 80. , 9 4 '

W. W r i ebe o . c t 2 1 , p , p . 9 .

Le a ro n Blein Ex o e ’ b , p s de q uelque: expérieneer ”o wed “: M 1 etc . , 1829 .

J E H ETE OS P P RSON.

"5 n origin . Helmholtz erro eously adopted this idea though it n n t nt t t t t r was i co sis e wi h his ma hema ical heo y, as we shall see

later . n S n t C n Hei rich cheibler, a silk ma ufac urer of refeld a d the n t n m t m m t a ts i nve tor of a o o e er , ade a ore careful s udy of be , n from resonated tones Of tu ing forks . His work is repo rted 1 6 fi he t R e . t o n s by t schoo l eacher, o ber He rs c clusively e tab lished the fact that the number O f beats O f the mistun ed ( verstimmt ) prime equals exactly the difference O f the double n n vibrati o ns Of the generati g to es . The point Of fusio n o f t nt n t r n t n at 1 6 s n wth bea s i o a u i a y se sa io he placed per eco d, i ’1 7 o ut taking the pitch into consideration . Scheibler based his

' investigation o n the results O f H ii llstro m and o n the assump t n t t t n the nt O f o n io ha all bea s , beyo d poi fusi , go over into ’1 8 combination tones . But this View had long since been expressed by de ls " n t m n . In t to Grange , a d la er by Tho as You g his repor the S t n 1 00 n x t t t Royal ocie y , Ja uary , 8 , You g e plici ly s a es the ” he ff n in t t . t t two n heory The grea er di ere ce pi ch of sou ds, “ t the m the t t a t t h he wri es , ore rapid bea s , ill las , like t e dis tinct pufi s O f air in the experiments already related they co m municate the idea of a continued sound ; and this is the funda ” ° n 2 me tal harmonic described by Tartin i . This View has S ince ’ been known as Young s theory . th t it t m This eory , hough la er beca e widely accepted by he t n no t to t t . S. acous icia s , did appeal physicis , G Ohm . He m fin the t n n had de ed o e physically as a si us vibration .

n to t fin t n the m t er Accordi g his de i io ear, says Hel hol z , p ceivas endula r vibra tio ns a lo ne a s sim le to nes a nd reso e p p , lv s a ll o ther perio dic mo tio ns o f the a ir into a series o f p endula r

n e . 1 S ensa tio ns o f To , p 5 4 d .

A r hun . Ro eber Un te suc en des H errn Scheibler iiber , d . so . hl g g Sc i ge, ’ Schwebun en der tiis n n O S se Po . A a l . I I . 8 g , , , 1 . if . an d g g XXX , 34, p 333 492 ff . rue r c t ut s e o . i . . 1 6 . B ee uo ta tio n f ro m Me g , p , p 9 q yer a bo ve. K , ” I id b . , p . 197 . ” de le ran e N o uvelles resea rches sur la n a ture G g , et la pro pa ga ti o n dc ’ so n M is cel h -m t . at a h . so c . riv . Ta uri nen sis T , . I . 1 p p , , 75 9 . ( I ha ve no t yet seen this a rti c l m s lf e y e . )

Wo rks Dr Y un b o . o k V Peac oc o l. I . 18 f g , y , , 5 5 , p . 84. Stum f i o . c t . . 2 0 Ohm , p , ; , o . A nna l. Bd . p p 4 . 1 P gg , 47 p 5 3 . T NE A D LA T PHEN OM BNA COMBINATION O S N RE ED . 5 vibra tio ns hearin the series o sim le to nes which co r es nd , g f p r po ’ 23 he sim le ibr t o n with t p v a i s. ’ e ao - t x i t This is th called Ohm s law . I is e pressed n his R : I n x m t t way by Lo rd ayleigh t is fou d by e peri en tha ,

- n n to t i . e. n o r whe ever accordi g heory a simple [ , si us form , e l t n nt the e n n t n ca p ndu ar] vibra io is prese , corr spo di g o e n be but n the m t o n s nt t n the heard , whe ever si ple vibra i is ab e , he

t n nn t h . t t r o e ca o be eard We are, herefore , jus ified in asse t i d t n e in t t t n an r t i . . g ha s mple o es vibra io s of a ci cular ype [ , the n o b a simplest form of vibra tions , si us vi r tio ns] are indis nn t solubly co ec ed . This law was discovered by ’ Ohm s defin ition of a tone gave rise to the well-known dis “ u t a d S n ensaio be ween Ohm n A . Seebeck. eebeck was o t ’ to n t t n always able recog ise upper par ial o es , where Ohm s t to x In o t law required hem e ist . her cases where he did hear the th t th eore ical upper part ials , they were weaker tha n e theory n required . He co cluded that the defini tion of a simple tone as to o t d a nd t no n given by Ohm was limi e , tha t o ly pendular t n but th t n t vibra io s , o er vibra io al forms , provided hey were no t to o t the en widely separa ed from p dular, were capable of t n in the ca r the ns t n n m t n exci i g se a io of a si gle si ple o e , which ,

t . o n t t however, had a variable quali y He c sequen ly asser ed that when a musical tone was compounded of several simple t n t the nt n t o f the n t t n o es , par of i e si y upper co s i ue t tones went to the n t n t o f the ri t n t it and increase i e si y p me o e , wi h which fused , that a t most a small remai nder exci ted in the ca r the sensation no t n o f an upper partial tone . He did t formula e a y deter n t n n the t n mi a e law , assig i g vibra io al forms which would give the impression of a compound Helmholt z took up the defense o f Ohm . The diffi culty we experience in hearin g r t to n t no n o uppe par ial es , he wri es , is reaso for c nsideri ng them to be weak ; for this di ffi culty does not depend on their n t but n nt ff nt t n inte si y up o e irely di ere circums a ces , which could no t be properly estimated until the advances recently made in

lmho lt o it . . He z, p . c , p 5 6.

lei h The r n I . 1 . 1 . Ra y g , o y o f S o ud, , 894. p 8 “ n I 1 XI L 1 na] L X 1 ft L 8 . hm Pa . A 8 t 6 k O . g e . 43 . 5 3 : , 44. Seebec , P031 .

A nnal LX. 18 it L I I I . 6 . and 68 K. s , 43. 449 ; X . 35 3 3 - H . ho l cit . elm tz. o p . pp . 5 8 9 . 6 J E H ETER OS P P SON. the the t t the physiology of senses . He says ha for some of t m t t n the n the fi t bes usical quali ies of o e , loud ess of rs upper

partials is no t far inferior to that of a prime tone itself . There is no di ffi culty in verifying this last fact by experi h t the t n no ments o n t e to nes of strings . S rike s ri g of a pia or n and t t o ne its n a n mo ochord , immedia ely ouch of odes for instant with the finger ; the constituent partial tones having t his n n t n t n and the t l ode will remai wi h u al ered loud ess , res wi l “ disappear? In this way we can convince ourselves that t he o r first and second upper partials are by no means weak . F tones no t produced o n strings this a po sterio ri proof is no t so to n t n t to m the easy co duc , because we are o able ake upper t B n t n n p ar ials speak separately . ut eve he by mea s of a resonator we can appreciate the intensity of these upper part ials by producing the corresponding no te o n the same or some other n t nt nt ts n t the n t i s rume u il i loudness , whe heard hrough reso a or, ” 28 Ml t it m to t t t the . agrees wi h ha of former his , see s me , is an argument merely to establish the fact that in the complex t t o n tit nt n wave here are physical , or objec ive , c s ue s correspo d

ing to upper partials .

Seebeck might well agree thus far with Helmholtz . H is

nt nt n t t the ea r n n t n co e io is ha , u like a physical reso a or, is u able

n n t i e. to t t t t t . apprecia e hese par ial physical vibra io s as dis i c , ,

o n z t m x t in r m t nn . M t a aly e he , e cep a ve y i perfec ma er ore o the nt in t t n t nt o ut t t n t poi dispu e , he , Helmhol z poi s ha whe n fi t n n and the tt nt n fix n it a to e is rs sou ded alo e , a e io ed upo , it can n n the t n t it be perceived eve whe lower o e , wi h which had “ I n . n n c o n former occasio s fused , appears polypho ic musi t its o wn t n t m n proper, where each par has dis i c elody , a pri cipal means of clearly separating the progressi o n o f each part has always consisted in making them proceed in difi erent rhythms and o n difi erent divisions of the bars ; or where this could no t n at an t n t in be do e , or was y ra e o ly par ly possible , as four t it an nt t to par chorals , is old rule , co rived for his purpose , t m the c let three par s , if possible , ove by simple degrees of s ale ,

and let the fourth leap over several . The small amount o f

” 8 c . I bid ., p . 5

id . 8 . I b . , p 5 b ” d . c I bi ., p 5 8 . M BINA TI T NES A ND RELA TE HE M ENA CO ON O D P NO . 4 alteration in the pi tch makes it easier for the listener to keep the identity of the several voices disti nctly in In the case of compound tones whe re all the part ials start together and nt n t the t t n t and a t co i ue wi h same rela ive s re g h , all cease the a ti it ttl n t t an n nt the i s me me , is li e wo der ha a alysis i o var o us n t nt ns t n In t ffi t . n co s i ue se a io s is more di cul such cases, eve w tr n ca r it the t n ith a ai ed musical , requires applica io of a co n s le unt t to a The in iderab amo of atten ion m ke the analysis . fl n the t co n t n ue ce of upper par ials of a mpoud musical o e is , “ h b no n n t . v t e o moreover, y mea s u fel They gi e c mpoun d

l n . tone a brighter and higher effect . Simp e to es are dull When they are compared with compound tones of the same t n n to ti t the n n n to pi ch , we are i cli ed es ma e compou d as belo gi g t a higher Octave than the simple tones . I is very easy to n n t e make a mistake of a Octave . This has ha ppe ed o th most n t celebra ted musicians a d acous icians . Thus it is well known th t t n t n i t and t t a Tar i i , who was celebra ed as a violi s heore ical

n t t n t a t to h. musicia , es ima ed all combi ational ones n Oc ave o hig to t n in t n n the The problem be solved , he , dis i guishi g partials of a compound tone is that of analysing a given aggre gate o f sensations i nto elements which no longer admit of n l t in n a a ysis . We are accus omed a large umber of cases where n t ns ff nt n in i f n t t the se sa io of di ere ki ds or d f ere par s of body , xi t ut n to n s t t t t n t s n e s sim l a eously, recog i e ha hey are dis i c as oo as he and to t ttent n t y are perceived , direc our a io at will to any o ne of them separately . Thus at any moment we can be t n t see t separa ely co scious of wha we , of wha we hear, of wha t a d tin t in fin in the we feel , n dis guish wha we feel a ger or great to e t s nt t t So n , whe her pres ure or ge le ouch , or warm h . also i

the fi n . n n to in eld of visio I deed , as I shall e deavor show t a d t n t m n wha follows, we re dily is i guish he i dividually from o t and t t t an nn t t n each her, ha his is i a e facul y of our mi ds . “ The matter is very different when we set to work at n investiga ting the more musical cases of perceptio , and at more co mpletely understan ding the conditions under which the above nt n (fist io ctio n can nn t the in me io ed or ca o be made , as is case We then become awa re that two 8 E re n JOS PH ” aso . different kinds or grades must be distinguished in our beco ming n co scious of a sensation . The lower grade of this co nsc ious t t h est n ness, is ha where t e influence of the sensation in qu io makes itself felt only in the conceptions we form of ext erna l t n nd and t n a t n n t . s hi gs processes , assis s i de ermi i g hem Thi can take place without our needing or indeed being able to as certa in to what particular part of our sensations we owe this t n I a or hat relatio of our perceptions . n this case we will s y that the impression of the sensa tion in question is perceived s n e l nd y th tical y . The seco nd a higher grade is when we im mediately distinguish the sensa tion in questi o n as an existi ng h part of t e sum of the sensations excited in us . We will say r e i then that the sensation is pe cei v d a nalyt ca lly. The two t cases mus be carefully distinguished from each other . Seebeck and Ohm are agreed that the upper partials of t n nt t ackno l a musical o e are perceived sy he ically . This is w edged by Seebeck when he admits that their action o n the ea r the the un x changes force or quali ty of so d e amined . The dis pute turns upon whether in all cases they can be perceived analytically in their individual existence ; that is whether the ca r n n o n t o r t x whe u aided by res a ors o her physical au iliaries , which themselves alter the mass of musical sound heard by the can t n and nt n t nt n observer, by mere direc io i e sity of at e io dis tin uish t and in t the cta th g whe her, if so wha force , O ve , e ” et he x in the 3° t c . t t n Twelf h, , of prime e is s give musical sound . In an xt n nt o n the nt n e e ded argume foll wi g se e ces quoted , Helmholtz points o ut that wi th respect to a ll our senses we perceive synthetically until byaccident or by consciously directed experiment and attention the contents of our perceptions are o n t t he e further analyzed . He c cludes , therefore , ha t ear do s the t n t n n actually take up separa e pe dular vibra io s of a cla g , and that by practice and special training we may become able n t n n t n t to perceive them as i dividual o e se sa io s , whereas wi hout

such practice they are unanalyzed .

It n t t n t e. . is ge erally agreed ha physical reso a ors ( such , g , t the as the Koenig cylindrical resona ors , Helmholtz spherical

t t n n t n the n - resona ors, u i g forks) ake up o ly pe dular form

'

id . 62 fl. I b ., pp H N M NA COM BINA TION TONES AND RELA TED P E O E . 9

n the an n t n t s i ti n vibratio s of air, d each reso a or o ly ho e v bra o s ,

r h n its o wn n t . mo eover, w ose periods lie very ear a ural period t t e n t the l t The more nearly perfec h reso a or, more c osely mus the period of the air wave correspo nd t o its o wn in order to e t n . n n e. . n n to e n s be ake up Tu i g forks , g , respo d o ly fr que ci tz t an i t t e very near their o wn . Helmhol sugges s ns ruc ive x eri ent in t nn t n to t t an to p m , his co ec io , show ha a pi o is able analyze the vowels of the huma n voice i nto their pendular t the an t th t the vibra ions . Raise dampers of a pi ofor e so a all t n can t t n n the a in a ther a rt s ri gs vibra e freely , he si g vowel f , , o a the he he n loudly t ny no te of piano, directi ng t voice to t sou d ing board ; the sympathetic resonance of the stri ngs directly - e he o e h . n n o i to e t t e echoes t e same a On si gi g n , same is

rev-echoed The experiment does no t succeed so well if the damper is removed only from the no te o n which the vowels n t o f t e e are su g . The vowel charac er h echo arises from th re-echoing of those upper partial tones which characterise the

. tt and vowels These , however, will echo be er more clearly when their correspondi ng higher stri ngs are free a nd can vibra te t I t n the t the t . n t t sympa he ically his case , he , in las resor , mus ical effect of the resonan ce is compoun ded of the tones of a nd t t n o several stri ngs , several separa e par ial tones combi e t produce a musical tone of a peculiar This experiment may also well be used to illustra te Helm ’ n n l n the holtz s view of to al a a ysis i ear, if we thi nk of ea ch pia no string as communica tin g wi th a nerve which mediates

the n n t n n t n . ca r the n correspo di g o e se sa io The , like pia o , by n the n fi t the u mea s of basilar membra e bers , akes up vario s a n o consti tuent pendular vibra tions of y c mplex wave . These n n t n the n n n co stitue t vibra tions, he , call up correspo di g se sa t n a re n t ti l ex io s , which , however, usually perceived sy he ca ly, n a d o cept in ca se of special traini g n directi n of attention . The n th t fi t t t t reso ance eo ry , hough rs clearly s a ed by Helmhol z , was a conception that flirt ed before the minds of Thomas n nn Mul and Y ou g . Joha es l er, n t n n a As has already bee sugges ed , reso a ce m y occur in

1 [ bids p . 6 c . ’ ' o hn . MeKeo dric k in Schaef er s h sic ia V ol. I I . t o . 1 1 . J G , P y n s , w, p 79 1 0 J H T N OSEP PE ERSO .

different degrees . Thus a system may show free or forced vibrations . The period of a free vibration depends o n the constitution of the system i tself ; the vibration ' is made by the system when disturbed from the position of equilibrium a nd t ha t to . t n n he t n s lef i self A forced vibra io , o t o her ha d, a period determined solely by the external force acting o n the he t o n t . So n t xt n t the sys em lo g as e er al force ac s , forced vibra i F t a nt n but n . co i ues , a free vibratio quickly dies away ur her, vibrating system of o ne degree of freedom may have the amp li t nt n l tude of i s moveme s reduced by damping . Dampi g wi l o n xt n t n and it n n t o n a so e i guish a free vibra io , s i flue ce is fel

t n n t to n sm. forced vibra io , whe here is an approach isochro i n t n x t in an o ne t a Now , whe a forced vibra io is e ci ed y par of t he t n and rati n sys em , all t o her parts are also i fluenced , a vib o the x t o m tu n o n the of same period is e ci ed, wh se a pli de depe ds "8 n t t the st co s itution of the systemas a whole . If a par of sy em n t n m t t is especially affected withi a cer ai li it of ampli ude , i is in the position of a system having o ne degree of freedom

acted o n by a given force and independent of the natural period . Resonance usually occurs when there is an approximate equality

of periods between the vibrating body and the resonator . In m he tu t n so e cases , t ampli de wi hin which reso ance is possible n in t and m n s may be co siderable ; o hers, very small ; uch depe d o n n o n the in as regards delicacy of res a ce , degree of damp g n n - ti that may be called into play . Tu i g forks are suscep ble t t t n to no twitstand of sympa he ic vibra io a remarkable degree , in the difli cult tt n t in t n t g y of se i g heir mass mo io , because hey admit of a long accumulation of minute impulses ; but for this reason there must be precise agreement between the pitches of the t It t t t set wo . forks is also observed ha , if a fork is hus n t nt o n n n agoi g, i co inues s u di g for a co siderable Now if the analysis of tone is eflected in the cochlea o n the n n n x t to find in t t n pri ciple of reso a ce , we shall e pec ha orga a

rather complex diflerentiatio n of structure . Each constituent pendular vibration must be taken up by a certain part of the

i h r un V l . I . revi sed editi n 1 8 6 . 0. Cf . Rayle gh, T eo y o f S o d, o , o , 9 , p 7

k c t . 1 1 . n th rinci le o f resonance see als r Mc Kendric , o p . i ., p 77 O e p p , o Lo d

un l I . 1 8 . Ra lei h Theo r o S o d V o . 6 y g , y f , , 9

1 3 T JOSEPH PE ERS ON.

”1m reso nato r. After this point of blending was reached a fur ther increase in the velocity of the disk did no t change the c har t he n Ex r d e a t n t . t t c er of se sa io reme veloci ies , of cou se pro uc such violent agitations at the mo uth of the resonator a s to ” n x nt n re der e perime i g impossible . With the aid o f two competent subjects Professor M a yer o btained results which he co nsidered more reliable than t hose which he had previously published in the A merica n J o urna l o f ien t 1 n c c . t t n S e for Oc ober, 874 These resul s are succi c ly give 37 in the following table :

S N D

0 s . 011 64 . 395 ee

Uta 1 28 .02 22 see.

Uta 2 5 6 .0142 see.

So l. 384 1/ 102 .0098 see.

Ut. 5 1 2 .oo 76 sec .

1 . s c Mio 640 / 1 5 2 006 5 e .

So h 768 1/ 166 .o o60 sec .

s Ute 1024 1/ 180 .005 5 ee.

S n he t n N the n o a represe ts t o e used ; , umber of d uble vibr

e. he t n D the t o n the n t o n i . t io s ; , dura i s of residual se sa i , , reciprocal of D is the number of interruptions per second t e

n he - n t s quired for a co tinuous tone . L is t number of wave le g h contained in the separate impulses into which the sound had

n in to the o nt n n t n e. . bee divided order produce c i uous se sa io ; g ,

64 2 5 2 . 5 From his results M ayer constructs a curve illustrative o f t F the dis the law which he seems to have es ablished . rom ” “ the x nt find cussio n of the curve of e perime s , he says , we that the law connecting the pitch of a sound with the duration t n x t of its residual sensa io may be e pressed hus ,

in t o n o f n the t o n the in which D equals, frac i s a seco d , dura i of residual sonorous sensation correspondi ng to N number o f ”88 vibrations per second . “ hic a a z ine th Series LIX. 1 8 . . Philo so p a l M g , 4 , , 75 , p 35 3 “ id . . I b . , p 35 5 “ 6 I bid ., p . 35 . BI A N T N S AND ELA TE H N M ENA 1 COM N TIO O E R D P E O . 3

Helmholtz did no t hold that the ear is a perfect reso nator sympa thetic vibration in the cochlea is no t of a very high order n n i tun n r s the t n of sensi tive ess, for, u l ke i g fo k , vibra io soon e n n to act n it a subsides when th impi gi g wave ceases upo ; nd, n t the a n furthermore , i dividual par s of basilar membr e or th t n t to the th i cochlea , ough vibra i g mos easily periods of e r o wn n l n t n to near t i freque cy , a so respo d less s ro gly periods he r This sympa thetic vibra tion is still sensible for the ”3" o v n o i nterval f a semitone . This is of course equi ale t t e n t saying tha t the vibrati o n in th cochlea , a sweri ng o external

to a nd no t t . vibra tions , is a degree forced wholly sympa hetic a t n t the t n t On this b sis , he , he mee s objec io urged la er by several ” t t t so t t n n n cri ics , ha small s ruc ures are i capable of respo di g sympathetically to so slow vibrations as those of the lowest “ audible tones . That the ca r can perceive such phase variations as are rep ent the tin two n n t n x n res ed by bea g of eighbori g o es , is e plai ed , not the the x but e from physical form of comple wave , from th fact tha t the areas o f the basilar membra ne affected by the two t o n t t es respec ively overlap , so hat each vibration series is “ periodically strengthened and weakened . A thi n stri ng st retched o n a so unding board o n which are placed two vibra t n he t h be ing tuni ng forks of early t same pi c , may seen to vibrate m n the n . t t in same fashio A s re ched me bra e , somewhat re th m o f the a r sembli ng e dru c , may be made to carry a small t t to the n s iff stylus . This s ylus is made draw vibra tio s of a

em n o n t t n n . n t m n t n m bra e a ro a i g cyli der Whe his me bra e , he , t t n two t n t t e is set into sympathe ic vibra io by o es ha beat, th undulati ng li ne drawn by the stylus shows that periods of strong t vibra tion alternate with periods of almos entire rest . Similar

n Dr. o tz tt curves have bee made by P li er , who a ached the writ n to the t n the o m o f a i g stylus audi ory bo e ( c lu ella ) a duck , nd then produced a beating tone by means of two organ pipes of

' ti ns 0 ne . 1 c . S ensa o ] To , p 44 “ mann A rc /t it : . d . Ph sio lo ie LIX 1 8 1 Her . . E . L , , I g y g , " 9 . p 5 15 ; Max

M er Z it hr cho ! XV I 18 8 . ao . e se . . s ey . I P y ., 9 . p

H i 1 6 d . Cf . elmholtz . o p. c t” p . 4 “ ’ E us ’ i c f . arti c l bbin ha E o I b d v p . 166 ; e on g xpla na tion f Beats. by Bentley w h l mer o . a P . 66 . “ erner A . l s c o W . ] y ” XV“ p 1 J SE H p s rsasozv 4 O P .

the m n t . x nt s H early same pi ch This e perime shows , say el t t t n the t n the ts o f t wo hol z , ha eve audi ory bo es follow bea he tones . Generally this must always be the case when t pitches of two tones struck differ so little from each other and t t the t n the t t a t the from ha of proper o e of sympa he ic body , th latter can be put into sensible vibration by both tones at t t o no t o nce . Sympa he ic b dies which do damp readily, such

t n n - n e nt two x t n t nes as u i g forks , co s que ly require e ci i g o which xt n tt in t in to w is differ e raordi arily li le pi ch , order sho v ible

and the t t t w. Fo r s beats, bea s mus , herefore , be very slo bodie

n t n etc . the diflere oe o f readily damped , as membra es, s ri gs , , n h x t n t n t an n nt e ts t e e ci i g o es may be grea er, d co seque ly th bea may succeed each other more rapidly . This holds also for the elastic terminal formations of the

t n . auditory nerve fibers . Jus as we have see that there may be ’ “ th t C t a visible beats of e audi ory ossicles, or i s arches may lso be made to beat by two tones sufliciently near in pitch to set ’ the same Cort i s arches in sympathetic vibration at the same t n the nt n t t n n n the time . I f he i e si y of audi ory se satio i nerve fibers involved increases and decreases wi th the intensity he t t o n the t n t the n t n of t elas ic vibra i s , s re g h of se sa io must also increase and diminish in the same degree as the vibrations o f e n n t n the n In th correspo di g elas ic appe dages of erves . this ’ case also the motion of Corti s arches must still be considered as compounded of the motions which the two to nes would have t n produced if they had acted separa ely . Accordi g as these moti o ns are directed in the same o r in Opposite directions they n o r n t t n will rei force e feeble each o her by ( algebraical) addi io . It is no t till these motions excite sensation in the nerves that any deviation occurs from the law that each of the two to nes a nd each of the t wo sensations of tones subsist side by side without

Now it becomes very evident that Helmholtz could no t ac e n t t t n cept th theory of Thomas You g, ha bea s whe they become nt nt n t n rapid enough pass over i o co i uous o es . As the two

“ ’ While he speak s in terms o f o rti s arches it i s to be understoo d tha t C . s t r h se nl m acco rding to hi la e vi ew, t e o y edia te the vibra ti o ns o f the basilar mem His ar ument o f co urse a li s a l wl rs. e e ul e l to th bran e fibe g , , pp q y ese fibers. “ 1 66 e H elmholtz, o p. c it., p . . T N ND REL E a o n wa I COM BINA TION O ES A A T D m n . 5

n the n primary to nes diverge, the sectio s of basilar membra e affect ed by these tones also separa te farther and farther and

t t to . t n nt finally cease al oge her overlap The bea s , co seque ly , n t n dependen t o n this overlappi g of vibra i ng sect io s , must grad all in in nt n t t in in n and u y dim ish i e si y as hey crease freque cy, ’ o t t n at a certain point must run ut . Wi h high o es more bea ts can be heard per second before this point o f disappear h t n the it t n ance is reac ed ha is case w h low o es, because high

- n t n t n e. . t n to es give more bea s for a give separa io , g , a semi o e , ’ t n t t n c n to t han do low o es . Bea s , he , ac ordi g Helmholtz s view can occur only when their generat ing tones lie near to gether in pi tch ; but the term generati ng tones here may mean n t the primary to es , upper par ials of primaries , or combination ' t t nt to n t th t o n m t tones . I is impor a o e a Hel hol z s theory t e la ce o stimula tio n in the th t n the ma nner h p f cochlea , ra er ha h n tl the t n i . e. t e n n t n ( , freque cy) , is direc y de ermi i g co di io of “ It the pitch of the experienced to ne . is o n this very point that ’ Ebbinghaus view di ffers fun damentally fro m tha t of Helm ’ t m t nt nt in t ho ltz . I f his i por a poi Helmhol z s theory is kept n t to n t n he in mi d , i is easy u ders a d why t view that bea ts go over int ff n t n un t n o di ere ce o es ( Yo g) should be hrow o ut of court . The stimulati o n to which the first combination tone ( h t ) an swers must be much farther up the cochlea than the st imula t n n n to the m t n h a d t io s correspo di g pri ary o es , n . Think what an excursion the vibra tion f o r this combi nation tone would have to make over the basilar membran e fibers as the primary t n s ! It t nt o e diverged migh well be gra ed by Helmholtz , so t t two nt in n far as I see , ha i erfer g pe dular vibrations in the basilar membrane could give rise to pendular vibrations of t o ffi nt m n t o her peri ds . A su cie a ou of asymmetry might well But t ne t n be found there . hese wvibra io s would not give rise to newt0nc n t n n t t t in th se sa io s , u less hey s ar ed up , e liquids the h t n u t of coc lea , similar vibra io s which co ld be aken up by ‘ the bers o tha t erio d. t t th fi f p To illus ra e , suppose e tones c ‘ and d and t n t , of vibra io s respec ively, are given and that the sections of the basilar membra ne thrown into I 6 J SE H s r n O P r easo .

t t t n t to . N o w sympa he ic vibra io by hem overlap , a degree , o n nt t in the n new s accou of asymme ry membra e , period of i t o n e Su t n t e. . 2 v bra i migh be g era ed , g , 5 6 p pose the sections o f these stimulations are in the region rep ro nt o n the se ed accompanying line by R and S .

X R S

The newvibration series of 2 5 6 per second would of course

t n in t n o f the m n the iber also , he , be his regio me bra e , whereas f s s m a thetic to tha t re uenc t in the n X y p f q y are loca ed, say, regio , and the nerves correspo nding to these fibers are the o nly o n es

n t t m t the sensa tio n which , whe s imula ed , will edia e of tha t ‘6 tone . the nt a t : n t new Now , poi m de above is his u less his vibra tion-frequency can communicate its perio d to the surrounding t t it can t n the m t t fi a t X fluid so ha be ake up by sy pa he ic bers , d o o n l n . A n o n such t e wi l be se sed such a process is impr bable . ’ n t to m t o siti ve nt t This leads us a urally Hel hol z s p co ribu ion , new x n t n the o in o n t o n t n th a e pla a io of rig of c mbi a i o es , e prin n t ciple o f which we have already a ticipa ed . He had shown why o n his o wn theo ry of cochlear action combinati o n to nes canno t result from rapid beats ; no whe was to explain how t ca n o and in t x n t n hey be pr duced , his very e pla a io he found , new n to t as we shall see , reaso s believe tha beats do no t pass over into co mbination tones . In the explanation of clang ’ analysis and the operati o n of Ohms law we have had to ” n t m o t and n t nt he enu cia e , says Hel h l z , co s a ly apply t pro posi t o n t t s t mo t o n the and o t t i ha o cilla ory i s of air her elas ic bodies , d n tin t n o pro uced by several sources of sou d ac g simul a e usly , are always the exact sum o f the individual motions producible by each source separately . This law is of extreme impo rtance n it the in the theory of sou d , because reduces consideration o f compound cases to those of simple ones . But it must be o h “ Thi s suppo siti o n o f the generati o n o f newf requenc ies in the basilar mem

n i s o nl h o theti c al and is made so lel f o r ur o ses o f ex l bra e y yp , y p p p an ati on . I t ’ illustra tes clearly why o n H elmho ltz s theo ry one co uld no t suppo se tha t bea ts n n s give rise to co mbin a ti o to e . M BINA TI T NES A ND REL T HEN M E A l CO ON O A ED P O N . 7 served tha t this lawho lds strictly only in the ca se where the vibrations in all parts of the mass of air and of the sonorous elastic bodies are of infinitesima lly s ma ll dimensio ns; that is to nl n the t t n n t the t s say, o y whe al era io s of de si y of elas ic bodie are so small that they may be disrega rded in compariso n wi th the n the a nd in the n whole de si ty of same body ; same way , o ly when the displacements of the vibrating particles va nish as compared with the dimensions of the whole elastic body . Now certainly in all practical applica tions of this law to sonorous o the at n ver small and n n h b dies, vibr io s are always y , ear e oug to being infini tesi ma lly sma ll for this law to hold wi th great n n th n s t exact ess eve for e real so orous vibrations of mu ical ones , a nd by far the greater part of their phenomena can be deduced

t . from that law in conformity wi h observa tion Still , however, there are certain phenomena which result from the fact that this law does no t hold with pe rfect exactn ess for vibrations of t t t ver sma ll elas ic bodies , which , hough almos always y , are far l s l. ne t n from being infinitesima lly ma O of hese phenome a , st the n co m with which we are here i ntere ed , is occurre ce of “ bi na tio nal Now it may be -shown that co mbina tio na l to nes must a rise whenever the vibra ti o ns a re so la rg e tha t the square o f the displa cements has a sensible influence o n the “ mo tio ns? As a simple example Helmholtz develops mathematically the system of waves which would result from the motion of a single heavy po i nt under the i nfluence of two pendular wave series . Let us represent the mass of a heavy point able to oscillate in the di rection of the axis of X and let the force which restores it to its position of equilibrium be

3 h o n bx

Now if we suppose that two systems of sonorous waves act upon it with the respective forces

sin t an d ~ sin t the t n t n o f p q ( q c) , equa io of mo io bec mes

d’x

! 2 m . —an ba o sln t 2 + + f p + q “ 2 S ensations a] Tone. p . 15 . “ n i M Appe d x XXL. p. 4 12. 1 8 T N JOSEPH PE ERS O .

F om th t n m t t n the ri r is equa io Hel hol z ob ai s , besides p ma ry

t n and n t o n t n and a rt . o es p q, various combi a i o es upper p ials th 2 2 and n r to be o f Of ese p, q, p q, p q may be co side ed the fi t 2 2 2 2 rs order ; 3p , 3q, p q, p q, p q, p q, of “ h he fi r r t e n and o . t t t seco d order, so n Of hose of s o der the to n the t t nt n the in e ( p q) will have grea es i e sity, if tensities of the primary tones are nearly the same ; the tone ( p q) will be much weaker and the tones 2p a nd 2 q will di l ” be heard with fli cuty as weak harmonic upper partial to nes . nt n t the o m n t n t n n o t The i e si y of c bi a io o es , accordi g t his deter n t n t n t n to the o th in mi a io , mus remai propor io al pr duct of e tensity of the generating tones ; and hence must increase more a t n t r pidly ha he primary to nes . t n u The previous assump io [which , of co rse , is a purely arbi trary o ne] respect ing the magnitude of the force c alled nt ct o n n i o a i , amely ” a x bx

n X n ts n It n no t ts implies that whe cha ges i sig , cha ges merely i

n but its t . n t t n c n sig , also absolu e value He ce his assump io a hold only f o r an elastic bo dy which is unsymmetrically rela ted to po sitive and negative displacements . It is only in such that “ the the nt can a flect the t n d co m square of displaceme mo io , an ” binational tones of the first order arise . n n m t the n As is well k ow , Hel hol z supposed u symmetrical structure of the outer drum membrane of the ear to be a most m n n the n t n n t n t favorable co ditio for ge era io of combi a io ones . B t nt u t n it to ut a more impor a circ ms a ce , as appears me, ” "2 “ nt n the when the to nes are powerful , he co i ues , is loo se nd formatio n of the joint between the hammer a anvil . I f the n n the n the handle o f the hammer is drive i wards by drumski , n B anvil and stirrup must follow the motion u conditionally . ut

“ ’ H ms n m cla lm lt a s i s acco rdin to i llstrii o en ture. H i ll Thi s, as He ho z s y , g

’ s will b rememb red did no t ex lain their o ri in i n thi s wa ho wev r trcim, it e e , p g y, e . ‘ s t 2 1 when th si ns are chan ed are the tones ac t a The tone p 2g, o r q p e g g , ully

st ds f r h hi h r heard. Wherever these letters a re used, p an o t e g e and q f o r the

lo wer to ne. H elmho lt it . 1 v z , o p . c ., p 4 3 . The abo e determinations co nsti tutin a ppen di x , g

XI I . a ear A nn l B ed in o . a . d. 1 8 6 . , pp P gg , 99, 5 “ S ensa ti n n . 1 8 1 o s o f To e, p 5 b, 4 3 b . " i 1 I b d., p . 5 8 b .

20 J SE H ETE O P P RSON.

c a where is some co nstant . If we no wassume for the simplest periodic function which expresses an alternate shutt ing and n n n m ope i g, a ely — 0 = A ( I sin

and n to n t nt t t s m to al co sider p be co s a , ha is , suppo e be so sm l and the n x t t the t o u i flu of air so copious , ha periodical loss hr gh the n n no t nt t the be f ope i g does esse ially al er pressure , q will o the form — q = B ~ ( I sin 2mt ) where = E cAp .

In this case the velocity of the moti o n of sound at any the fi t m t place of space lled wi h air, us have a similar form , e so that only a tone wi th th vibrational number n can a rise . But t n t n t if here is a seco d grea er ope ing of variable size , hrough which there is sufli cient escape of air to render the pressure p “s n t n n t nt the a ir periodically variable , i s ead of bei g co s a , as o u the o t n n t t the passes t through her ope i g, ha is, if p is of form — = ~ p P ( 1 sin 2 1rmt ) then q will have the form q cAP ( 1 sin 2 1rnt ) ( 1 sin 2 1rmt ) cAP [ 1 sin 2wrnt sin a l

cos 2 1r ( m n ) t V; cos 2 1r ( m

n in t n to the two t n n and m t wl He ce , addi io primary o es , here i l the t n m n and m n t t the two n be also o es , ha is, combi a tio na l tones of the first order . In reality the equations will always be much more compli cated than those selected f o r showing the process in its simplest h m. t n n n n t e the for The o e will i flue ce pressure p , as well as tone m; even the combinati o nal tones will alter p ; and finally the magni tude of the opening may no t be expressible by such a t fo m simple periodic function as we have selec ed r . This will a n no t m the to n m n and m n m— n to occ sio erely es , , , , b t rt and the be produced , ut also heir upper pa ials, combina

Thi s then seems to re uire vibra tio ns tha t a re n ite and no t in n , , q fi , fi itely ’ a v small as H elmho ltz sa ys bo e ( see no te immedi a tely preceding thi s) . t t in hese upper par ials , as may also be observed The complete theory of such a case beco mes l t and n the nt y complica ed , he ce above accou of le case may suffi ce to show the nature of the

s are no wgenerally agreed that there are ao t es as Helmholtz here de ermines theoretically . ctive combi nation tones because they have

Helmholtz assures us tha t he himself heard them only strenghtened them wi th resonato rs but has also “ ti n n t t l o las c membra es respo d sympa he ical y t them . t t t n in th hough , ha eve cases where e primary tones

nstruments t m n in - he wi h a com o w d chest , t force o f the combinational tone is generated I arra nged the po rt vents in the i nstrument t t o ne the two n t he says , so ha of ge era ors th the l mo air by bel ows ved below by the foot , t wa s o wn th genera or bl by e reserve bellows, pumped full and then cut o ff by drawing o ut

s n- t and t n n t t the pres io s op , I he fou d ha com weaker tha n for the usual ar portion which the resonators

crating tones have no mechanical co n nt wi th reso nators is possible In such ation tones seem t o have no corresponding h t n in t e . t vibra io s air They are , herefore ,

tz t no t , however, subjec ive is a good term . ‘ tones are o bjective in the se nse that they ng pendular vibra tions externa l to the a na l in the co chlea t t t , whe her hese vibra ions take i in xt n to the t t o r in the or g e er ally ear al oge her middle ear . ’ It lo s he nt n fol w from t developme s already give , he says , that “ - 1m . 1 ao c f . 11130 . 1 a . , pp 4 9 , p 5 7

X. [ . 1 c 1 b s ecia l A nna l I 18 6 M , p 5 3 , 5 7 ; e p l y Po g g . C , 5 , p . 5 39 . “ i M d” p . 1 5 7 c . 2 2 ETE N JOSEPH P RSO . we do no t necessarily have to seek the cause of combin ation to nes in the manner of sensory receptio n ( Empfindungsweise) the t n but t t th two s t by audi ory erve, ha wi imul aneo us to nes of suffi cient strength there can correspond to the combin atio n tones actual vibrations of the drum and of the ossicles of the ear . These vibrations are exp erienced in the usual way by the n nerve apparatus . Co sequently the combinati o n to nes wo uld no t m t x t n but have a ere subjec ive e is e ce, would be object ive , even though they can originate only in the vibrating pa rts of the In cert ain cases we have found that their orig in is xt n o t c r entirely e er al t he a . ‘ In the mathematical investigation of wave movements in ’ ’ the nt n t o ne air, co i ues Helmhol z , as a rule concerns o ne s self only with the terms of the equations which co ntain the fi rst power of displacements ( Elongationen ) of the air part icles nd o ne the m a disregards higher powers of such displace ents . If the terms which contain the second power of displacements ne fin the n : I E n e are retained , o ds followi g ( ) very poi t of th m in the t o n n n to the n air ass, which vibra i s correspo di g i dividual m r to n t n n m nt new pri a y es are s ro g e ough , beco es a ce er of sec o nda r t m n to the n y wave sys e s , which correspo d harmo ical over r 2 tones o f the respective [prima y] tones . ( ) Every point of the air mass where the vibrations of the two given prima ry to n m t n o ffi nt n t o es si ul a e usly reach su cie mag i udes ( Gr sse ) , be comes the center o f newseconda ry wave systems which corre o m n t n to n and t t spo nd t co bi a io es , from hese arise bo h dif ference and summation to nes of the fi rst and of higher Thus from purely theoretical considerations Helmholtz co n the m n t n t n cluded that , besides co bi a io o es already discovered, there exist others with vibratio n-numbers corresponding to to n n m summa tio n t t a t h etc . n h , , These es he a ed o es , x h t 2 t h n to t etc . in contradistinctio hose e pressed by , , , n which he called differe ce tones . m t n t n in t r Though sum a io o es are mos cases ve y weak, Helmholtz succeeded in hearing them and in thus verifying his U n the n n . n theoretical determinatio s si g polypho ic sire , he

’ nn l I 3 6 . Ueber Co mbin a ti o nstiine, Page. A a ., C x-o 1 5 » P 5 37 “ 8 I bid ., p . 5 3 T H M E COM BINA TI ON TONES AND RELA ED P ENO NA . heard no t only the first summation tone but also those of the n n to 2 and z in seco nd order correspo di g p q, q p his equa n t h the u t n t n the tion . The followi g able s ows s mma io o es of second order which he heard

( 2 X 5 -l- 4 ) I 4

h t t t t t a nd x nt o b Helm ol z hus , bo h heore ically e perime ally , ta i t to nst the n t h ins a third po n urge agai You g heory, t at beats n t t t go over into combi a ion ones . His hree arguments agai nst t i t t n t : 1 t x h s heory , he , are hese ( ) The heory does no t e plain 2 the origin of summation tones . ( ) Under certain conditions co mbination ( both difference and summation ) tones exist o b ‘ ectivel n n ent the ca r j y, i depe d ly of which would have had to ’ t the t nt e t n ga her bea s i o a n w o e. ( 3 ) This supposition ’

i . e. n t nn t n t the co n [ , You g s heory] ca o be reco ciled wi h law fi a ll t x nt t t the n n rmed by o her e perime s , ha o ly to es which the ear r n to en t n the hears , co respo d p dular vibra io s of Befo re we consider t he various objections that have been to t s nt let n n urged he e poi s , us co sider for a mome t a certa in ' incons istency in Helmholt z s o wn sta tement of the origin of M n s . m n t n e n t t n t o t n i . . combi a io o e ul iple c bi a io al o es [ , t o nn t in n t hose of higher rders] ca o ge eral be dis i nct ly heard , except when the generatin g compoun d tones contain audible harmonic upper partials . Y et we cann ot assert that the com b ti n t n nt a t t b ina o al o es are abse , where such p r ials are absen ; ut in tha t case they are so weak tha t the ear does no t readily recogn ise them beside the loud generati ng tones and first dif In th fi t t to n ferentis l. e rs place heory leads us co clude tha t t t t and in the n they do exis in a weak s a e , ext place the bea ts of t i t x t n impure intervals also es abl sh heir e is e ce . In this " we al stro t n the lti co m case , as H l m sugg es s , co sider mu ple “ X . a t . o A nna l CI . P gg . s . p s “ ne . l 6 c . 16 ab s i Sen satio n: 0] To , p s ; p 7 . On the q ue t o n o f the uni versali ty ’ ’ o f hms luv see elow . li . O b . pp 43 ” A I . 8. Pm. M XX V" p 43 3 SE H ETE 4 JO P P RSON.

n t n t t : he fi s nti e bi a io al to nes o arise hus t r t differe al to n , o r co mbina tio na l to ne o the rst o rder n t o n wth the f fi , by combi a i i n t n t n t t diflere a es ge era i g o es hemselves , produces o her nti l to n , or co mbina tio na l to nes of the seco nd order; these a ga in pro duce new n t the n r t and ff nt the fi st o o es wi h ge e a ors di ere ials of r rder, and so ’ This arrangement does no t at all follow fro m Helmholtz s t m t t s n and in co n ma he a ical deduc ion give above , is , moreover , “ tradi io t - t ct n o well known empirical facts to be considered la er . e he 2 . m m t to n . t t The e q p, g , deduced a hema ically fro t n and n t n n t n primary o es , p q , u der cer ai assumed co di io s , is no t formed by a combination of a diflerence tone of the first ’ t o ne the r n to he e ua t on order wi h of prima y tones, accordi g t q i (1 (p q) z q p ; neither is it dependent upon a sub t ne jective upper part ial to ne z q . Let us suppose that ei her o of these conditions of origin may be t rue and see where theo ’ reticall n n o n t o wn m t n y we shall la d , eve Helmhol z s assu p io of transformation due to asymmetry of the ear drum. Sup he m t he on and the pose that in the vibration of t dru o t t es p q, t n 2 2 in c n n the m etr o es p , q, p q arise also o seque ce of asy m y E nt n t z no r can t ac of that organ . vide ly ei her q p q reac b k o n that sa me membrane so as to pro duce with o ne of the p ri z o e n e ma ry to nes the combination to ne q p f th seco d ord r . t n t t n t it at in Such a o e , herefore , mus origi a e, if does so all , m t t t n still arther in t h so e asymme rical s ruc ure lyi g f , where bo the primary tones and the tones corresponding to z q or p q Bu f n n . t n o will operate , i a se se , as coequals such a successio origins in difi erent structures is absurd when applied to the ea r where probably only two successive structures of suitable asym

and nn n x t . But m metry ( the outer i er membra es) e is , aside fro t n t t the to n e our assumption, i seems plai ha all es deduc d by Helmholtz must spring directly from the effect of the prim a ry

n f o r 2 z e. . t ct to es ; p q, or q p , g , is jus as much a produ

t o n z . of the equa i as is q, or p q

' t t an nt t ff nt This s and , suppor ed by argume qui e di ere from

d . i . 1 H elmholtz, o p . e t . , p 5 4 n fference tones a arin much lo uder than Such f ac ts, I mean, as seco d di ppe g

the first. t n in 1 1 b M . n in was ake 88 y R. H . Bosa quet his the B ea ts o f Co nso na nces o f the F orm ( where some whole Bosanquet shows ma themat ‘ ’ Ellis (who favors the bea t-tone view ) er haza rdous that in the drum t here are six summation-tones and six ro duced by direct transformation of the pri

efi ect t m to the out r i . e. of er s up f r h o der [ , ’ ar double integrat ion process of Helmholtz s 1 2 Sensa tio ns o To ne 4 , f ] is to show tha t those resultant sounds which higher orders ca n become great independ depend o n terms of lower He o ne of the chief objections urged against that the tones derived from terms of ro duced with the greatest from terms of lower orders it is no t here mai ntai ned that nt the n t e the has proved his poi , prese wri t r is of at the true so lution of this diffi culty must be sought

n the t-t n t line ; for eve t hough bea o e heory, urged so n t t he ffi t by Koe ig, be accep ed , di cul ies are probably

of no less import ance to the theory of co m the nt t e the two poi raised above , whe h r no t in principle ective combina ren or har e ultimately a lack of symmetry in the of a heavy poi nt vibrat ing under the in s i t n o n ry tone . Th s ques io will be c sidered

Helmholtz had explai ned beats o n the pri nciple of inter

“ - - a ine 01 Series XL 1 88 1 . 3 0 6 and 2 06 . Cf . Plcilo eo phia l Ma g z , 5 . , . pp 4 3 49 5

- . 1 . o n the questio n raised . p 49 9

t n . 3 d. Helmho ltz. S ensa io s 0] Tone, p 5 3 “ Bosanq uet. o p . cit ” p . 497 . “ 1M 1» 495 2 6 SE H p a ra JO P mozv.

erence in the basi r br n S f la mem a e. uch interference is po ssible o n in e ly case of small intervals . Th heating of imp erfect consonances he attributed to the presence of upper partia ls or " difi erence t n n m to t th of o es ; i so e cases , bo h of ese con dit ions . o t t o n the n t He laid m s s ress prese ce of upper par ial tones . s n t t n t n nt r a Bo a que , accep i g his ge eral poi of view, ea ly m de an " investigati o n of beats of various intervals . M ost of his ex eriments m t t n the n o n p were ade wi h o es from e harm ic organ, in which he had a series of tones separated for the most part ’ m n fi by single com as . By mea s of tubes properly tted into the and o nn t n t the n t t ear c ec i g wi h reso a or, he was able prac ically to shut o ut all vibrations of frequencies other than those un der

consideration . t n t Af er co siderable prac ice he succeeded , as did also his i lo ca tin the bea ts th the o e e n M r. tt n wi l wr n frie d , Parra , g to s ’ “

o the interv s use . t S f a l d I quo e his o wn words . uppose he n t C : c n n and x e t mistu ed oc ave was sou di g, I e amined th lower note wi th the resonator ; sometimes it appeared loud and

a o t m t n w . n steady , t ther i es as if bea i g po erfully O removing

- nt o the c ar the the resonator attachme fr m , lower note was

to t . x n t n always heard bea powerfully The e p la a io was simple . When the nipples of the resonator-attachment fitted tightly into the n t n the ca r but the n o ears , o hi g reached u if rm vibrations of h n But t the t e resonator soundi g C . if here was slightest loo se n tw n the n and the t ca r the ess be ee ipple passage of ei her , secon d n c the m n t o n o t in and ote ( ) of co bi a i g , gave rise to the sub e i e difi erence t n nt n j ct v o e , by i erfere ce of which with the These e t C I explain the beats o n that note . b a s a re therefo re

To avoid the introduction of all sorts of transformations ’ n o n the tn the m nt dependi g grea ess of displace e s , Bosanquet 7 8 used notes of moderate strengt h . The notes employed were " 1 and es ec i all ch . 10 . 1 if . S ensa tio ns o f To ne, p . 5 4 a , p y , p 79 “ ‘ s ut On the ea ts o f o n so nances o f Th F rm ’ R. H . M . o an e e o h zl B q , B C ,

hi th eries XL 1 88 1 . 20 8 . P l . M a g ., s S , , , p 4 ” 1 . I bid ., p . 43 ’ H c riti ci s s in this onn c ti o n oeni s ex erimen t s e e c e K g p s, oo n to be c on sidered . ’ Ko enig s tones were so lo ud tha t vari o us c o mbin a ti on to nes might be in troduced

r uh transf o rmation o f the rimaries on the rinc i le ex lain b H th o g p p p p ed y elmho ltz.

28 SE H ETERS N JO P P O . vibrations as o n the relation of the amplitude to the thi ckness of the According to Koenig beats are produced no t o nly in case ’ m nt t t r but are so of s all i ervals , as Helmhol z s heo y requires, al

n the nt . heard, with less intensity, whe i ervals are large As the prima ry tones diverge from unison at C ( 64 ) the beats t the t gradually increase in frequency . A place where hey cease t n to be perceived distinctly ( about 1 2 or 1 3 per see. ) hey begi to pro duce the effect of a rolling which is constantly accelerated until the interval reaches about the fourth ( 2 2 beats per Beyo nd the fourth to about the fifth there is a co n fused n ro nflement t n h n to rumbli g ( ) , always very s ro g, w ich begi s é e h clear up ( d bro uiller) as th interval approaches t e sixth .

o ne n m n t . Here agai hears a si ple rolli g, hough very rapid This rolling diminishes somewhere between the sixth and the nt nt the to n the seve h u il , when upper e reaches frequency of ut 1 1 6 1 1 t n o ne n to n 1 2 a abo to 8 vibra io s , is agai able cou t nd 1 0 ts t separate bea . These bea s gradually grow slower until

he t t . n n at t oc ave hey disappear These phe ome a , Koenig goes

x the t two eries o e o n to n s b a ts. e plai , are resul s of f I f we ' nt the nt rv n zn the o ne set t h represe i e al by , of bea s , t e louder n n o O to n the nt o es , gradually i creases fr m , as i erval diverges

m n on to the t . n the in er fro u is oc ave These Koe ig calls f io r, o e A t he t he or l wr beats . t same ime t other series is decreasin g

in f n n to O . n t t t the su eri o r reque cy from This series co s i u es p , er n or upp beats . The confusio heard near the fifth is t hen x n the two t to easily e plai ed ; for series of bea s cross , so speak , A t the n n 2 . t t t x t a freque cy of / his place hey e ac ly coincide. When the frequency of the in ferior beats is much less tha n n/ 2 o ne hea rs these beats easily ; when their frequency is co nsider t n n 2 the t ably more ha / superior bea s are heard , less easily ,

however . ’ n he t 2 n As rises above t oc ave , or , similar phenomena are n n experienced though with decreasi g i nte sity . U nder favor able conditions Koenig succeeded in hearing beats with intervals

iiss . tass n un der St e u t e n . s . oeni eber den Urs r S o w. W K g , U p g , eid . A nna l ’ f r m Ellis translati on in H elmho lt 8 1 . uo ted o 1 o . c it XII ., 8 , p 337, q z , p ., p . 5 28. ELA TE H M E A R D P ENO N .

z d zez ) or ( O ) . These beats are of

’ to K n th n t oe ig s view , ese beats , whe hey t n ed as continuous o es . These t- t n the t or bea notes . Designa i g lower bea s ' b m a nd the n t y upper by m, Koe ig gives hese their determination :

' m= n — hn

— ( h 1 ) n

"0 the t n some whole number . With excep io of a i s t the t n nt m and d scu sed la er, o es represe ed by ly primary beat-tones that arise from generati ng to his o wn 8 : 1 1 and the tones ' ’ m correspond to Helmholtz s first and ’ h = 2 n in cases where , Koe ig s lower ’ s Helmholtz s second order of differ

corresponding to p a nd so o n . ’ ates in Helmholtz s " Zahm says that a in the upper beat-notes of ’ is no t t rue of Helmholtz s mathe explanation that he gave of com istake becomes evident if Helm ” a st ep further . We shall later

’ d A o ust ue . . e iq , p 94

seen, ha ve no ref erence to the

tone, as a ma tter o f f ac t. o f ten

a tho ro ugh study o f

gave

He olt im : t . lmh h second o rder a) q and p 24 . z

s f . u ra 2 . ne p + 2q and a c s p , p . 3 ° SE H ETE N 3 JO P P RSO .

Suffi consider more in detail the matter of combination to nes . ce it here to say that any beat-note which correspo nds to a com t to n t n no t t o the bina ion e , as heoretically determined , eed r uble h existence t i . e. t e followers of Helmhol z , , so far as mere of

t n n n t n nt n t . such o e is co cer ed , aside from ques io s of audible i e si y It would be wro ng to infer that both the upper and the

- ts t n at the t . lower bea , or bea to es, are always audible same ime “ in t m he the nt r a This is fac seldo t case . If we divide i e v ls examined into groups ( 1 ) from to ( 2 ) from o t o to 1 5 an d to ( 3 ) fr m o ( 4 ) fr m 5 , so

n he t and t-t n xt n tt t n o , t lower bea s bea o es e e d over li le more ha the and the ts and t lower half of each group , upper bea bea Fo r t tones over little more than the upper half . a shor dis n in the the t t t o t ta ce middle of period bo h se s of bea s , or b h t-n t and t t-n t t t a h bea o es , are audible , hese bea o es bea wi h e c

By means of a method for which Koenig is indebted to and Desa ins n to M M . Lissajous , he e deavors make clear his

and t-t n t t nin k idea of the origin of beats bea o es . Of wo u g for s m o ne r o n o ne its n m and the he akes car y of pro gs a s oked glass, o n o ne its n t t it can other of pro gs a s ylus , so placed tha record e he m h o n th glass t combi ned move ent of t e two forks . The

' fo rks are then actuated by an electric curi ent . In this way a m n n t t the o co pou d curve is co s ruc ed , form of which is shown t vary according to the vibration ratio of the interval represented “ n t t the x n the by the forks . Koe ig supposes ha ear e perie ces

— nd n t t n the t-t n s — beats a , whe hese are fas e ough , bea o e di rectly from the form of the objective waves as represented by o wn : the t the these curves . Here are his words bea s of n nt the n n harmo ic i ervals , as well as of u iso , should be deduced h m t n o n and directly from t e co posi io of waves of s u d , we should assume that they arise from the peri o dically alternating coin idences x m the n t n t n and the c of similar ma i a of ge era i g o es , of x m x m ma i a with opposite signs . The similar ma i a of these n nt n he n n t x harmo ic i ervals , as i t case of u iso s , will ei her e actly t x n n t o n in he coincide, or else here will be ma ima of co de sa i t

- i s in a tio n T ne . 2 0. Ell , S ens s of o , pp 5 9 3

“ - ni it . 6 . Koe g, o p . c ., pp 9 7 HE M E A RELA TED P NO N .

lyin g between two successive vibrations of the t n tl n o ne and t o n o e , sligh y precedi g sligh ly f llowi g ut in both cases the effect on the ear will be the n n n fluctuation ) is no i nstantaneous phe ome o , gradual increase an d diminution o f the ia According to this view not only pendular i ntensity fluctuation wi thin certain n car as t one . Koe ig leaves to the of the actual process by which the has e th sis . He , how ver , put for

ntal facts in support of his view .

first experiments o n this subject which Koenig t o n nt t r hose periodically i errup ed so unds . Fo devised large brass disks wi th circles of holes

ameter. t t 1 He used hree such disks wi h 6 , 2 4 l s t y . The e disks when ro ated before made to interrupt the tone at regular

' o ns t n r , he , could very easily (by va y a i e increased or decre sed n frequency . t n n - no t n t u i g fork , reso a ed , was held the rota ti ng disk and the t one was apertures in the disk through a tube h of t e apertures . The observer

side of the disk . when the tone of a tuni ng fork t series of bea s were heard , some

nterference. When these beats nt n ey gave rise to a co i uous tone . ib nt t 1 2 8 ti ( 5 1 2 d . v . ) was i errup ed mes ’ hea rd no t only the tone c but also the B t no ll ut his was t a . Two d t n to the s also appeare , wi h freque cies equal sum and the difference of the interruptions and the vibration num h In the s n n t ber of t e given tone . pre ent i sta ce hese tones were ’ ’ g ( 5 1 2 1 2 8 384 ) and e ( 5 1 2 1 2 8 These " mmen an er Time a nn I I . 186 . Ue er den usa kl wei . A a l. L . b Z g Z . P ge , C V , p ’ Ellis a Quoa d from transl tio n in S ensatio ns 0] To ne .

M BI TI T N ND L TE H N M ENA CO NA ON O ES A RE A D P E O . 33

theoret ically investigates the

periodically n times per second . t n t t n to deduc io s , ha , accordi g ere should result from such fluctua but also tones correspondi ng to ” to S did not l . eebeck fo low up n This , however , was do e by Helm ho ltz who used for the purpo se his double siren . In 1 863 Helmholtz describes his experiment in the first ” o e n n en the tra a edition of his T n mpfi dug . I quote from nsl tio n of Ellis of a later edi tion : The lower bo x of my double 1 siren vibra tes strongly in sympa thy with the fork a when it n and he is held before the lower Openi g, t holes are all covered , n the i the but no t when the holes are open . On putti g d sk of siren in rota tion so that the holes a re alternately opened a nd

- r the n n he t nin . cove ed , reso a ce of t u g fork varies periodically he f d m the n I f n is the vibrational number of t ork , an umber the bo x o n the t n t of times that a single hole in is pe ed , s re g h the n n n t n the t and of reso a ce will be a periodic fu c io of ime , consequently in the simplest case equal to 1 sin a mt. Hence the vibra tional mot ion of the air will be of the form

sin am t ) sin 2 arnt sin 2 wrnt

cos z a ( m n ) t V2 co s 2w( m

the t n m n and m— n y we hear o es + , or t 111 s l is rota ed slowly , will be very mal ,

the l t . On t t nearly same , wi l bea ro a ear separates Beetz had been experimenting wi th In 1 85 1 he reported to the Physical lts of a repeti tion of the Weber exp eri

had a rrived at a result different

I I P0 . A nna l L I fl . 43 X , 35 3

186 . ci ted 3. p 5 97. by K. L. Schaefer in

I I !” 1905 . 1» 5 32 . J E H ETE 34 OS P P RSON .

o t t t n the Fo r th to ne o f the fr m ha ob ai ed by Webers . him e tun n - i g fork did no t disappear ; it only became weaker . I ” i t n t at the t a a e t ne as d s i c ly heard same ime , he s ys , high r o well as a series of beats which coincided in number with the - h n t n t e tun . umber of half revolu io s of ing W Weber, in tt to t t t t the the a le er Bee z , sugges ed ha reason why higher ’ note had not been perceptible in their [ the Webers ] experi ments was that they had employed a slower and mo re noisy ’

. t t x n ti lathe A sa isfac ory e p la ation was s ll wanting . ” t t n t the x nt n t wo La er Bee z agai ook up e p erime , usi g 1 2 nd 1 02 forks of 5 a 4 vibrations per second . When these o t t t tw t n forks were r a ed abou elve imes per seco d, Beetz fo und that the pitch of the lower fork was raised about three- fourths t d the t n an . of a o e , higher abou a half tone He hea rd a ga in

the ts two r t n the . Th he bea , for eve y revolu io of fork e p no meno n no t to nn ct t the t an , he holds , is be co e ed wi h r smission he t n the to the o ne of t vibra io s of fork air, for hea rs the rise ’ in the t t tt n o ne o n pi ch jus as well , or be er, whe lays e s head ’ o n the lathe and stops one s ea rs ” n no n t t nt The phe ome , says Bee z , is hus e irely o bj ective, and consists in a real increase of the rate of vibrati o n of the ’ I in t n n fork . t is fac o ly a other form of Foucault s pen dula r experiment . The vibrations tend to continue in the same plane t t in t t t it as ha which hey were produced ; hey are hus , as were, t n tt to t and o t ra smi ed a hicker bar, so pr duce a higher one . The amplitude of the vibratio ns a t the same time beco mes sm a ller ; it n n and gradually , however, i creases agai , reaches a minimum — r x n . in tn t t [Que y, a ma imum Tra s a foo o e] every ime t hat the t n to its n s t o n to o ne difi erin fork re ur s origi al po i i , or g from e t it by It is thus that th bea s are produced . If the t n n the n t o n t n t it and fork ur s o ly slowly, pla e of vibra i ur s wi h , in this case the fundamental tone is heard above without beats ; . o n n o ct t n and th tur ing m re quickly , obje ive bea s soo arise , e tone “ M th eri s II . 1 866 . . hil. a . S e P g , 4 , XXX , , p 5 35

i n ro ti render Stimm abeln P A nn l er Tii e o . a . I I I fi b d e . 1 866 g , gg , CXXV , .

. F ster transla ted thi s a rticle into En li sh hi if . . M th 0 o l. a . Seri es 49 G C g , P g , 4 , ff I I . . XXX , 5 34 f r m en tle and Sabine A ns ur uo ted o . J o . o s ch X I . o l. V . 8 Q B y , f P y , , p 4 7

- M a . th Series I I . . 6 c f . Beet hil . . z , P g , 4 , XXX , pp 335

6 SE H newsman 3 JO P .

t n a n nt no o ne t tu th o mula he co sta quantity . If w subs i te is f r

a in the fi t o ne t the x u e s m a th for rs , ge s for e c rsion of th y p et ically vibrating bo dy

’ a sin 2 r n ( t sin z avu( t 0)

' ' a 1r n n 0 a r n n / 2 cos ( ) ( t 1 ) / 2 cos 2 ( + ) ( t

But each of these expressio ns represents a single pendular v ibra

' ‘ t n the o ne n t n t n n and the t io , havi g a vibra io ra e of , o her ’

t n n . t t n S n n tha his a ra e of By ac ual observa io , tefa fou d t lower and higher tones corresponded in pitch to the demands of this “ In n t r in the 1 a seco d ar icle which appea s same periodica l , Stefan describes some experiments in which he interrupted the 0 t n t n n - the t 1 2 o e of a u i g fork by me hod already described, which t n m t t ro Koenig used la er . We lear fro his repor that P fesso r 1 03 rn M t t the t E st ach had , a abou same ime , also employed this t t t the t n method . Bo h inves iga ors heard o es correspondi n g to ’ ‘“ m n and m n m t t n of Hel hol z s equa io , which Rada u i t e m nt t n t t R had n h ea ime de ermi ed heore ically . adaucalls i e these tones va ria t o n to n s. t t t the t and he Bee z , s imula ed by resul s t hypothesis of St efan and R t n the x nt adau , akes up agai e perime s with a View to test ing this Finding the hypothesis well borne o ut t t t n t t n wi h ro a i g pla es, he ur ed again to rotating forks to determine whether they to o gave the to nes to be expected from ’ n t the Stefan s formula . He fou d hat lower tone as observed x t to the corresponded appro ima ely calculated tone . The higher to n o t an the t e , h wever, was always much higher h heory re ff n tw n the quired . The di ere ce be ee observed and calculated

bin . c it . . 88 c f Bentley and Sa e, op , p 4 ; . also Uber einen akusti schsa ’ - un r ha ir. A ha . Wiss u Wi ersuch S its sbe . d. d d . . s en ath u M . na t miss V , g , . KI,

1 66 6 . LI I I . , Abt. 2, 8 , 69 6

I Ab 2 866 if . I bid ., L V ., t. , 1 , 5 97

r Sup a , p . 3 1 . ’ Ueber die A enderun des Tones und der Far e dut ch Bewe r g b gun g , Pou. 86 A nna l . I I . 1 1 8 fi . , CX , , 5 ”‘ M o niteur S cientifi q ue, 1 865 , 430 ff .

’ Ueber den Einfluss der ewe un der Ton uelle auf die Tonhah P t B g g q e, o u. '

nna l . 186 8 fi . A CXXX , 7, 5 7 M INA TI T E A E E H M ENA CO B ON ON S ND R LA T D P ENO . 37

t values became very large with rapid rates of rotation . Bee z t the ff n t n n found , however, hat di ere ce became rifli g whe he took his observations with a resonator ha ving an opening 5 mm. he nst 2 mm in t . t o n t t i ead of 5 . diame er Wi h such a res a or observed values coincided very closely wi th the values computed ’ ‘ h c b St n . t t 2 6 y efa s formula Bee z used ese forks , 5 , ‘ ’ a = 0 c = 1 2 and t t 1 and revo l 44 , 5 , hree ra es , 3 u

t n n . t t n two io s per seco d He ook, also, some observa io s for

6 and t n . In t very low forks, 4 7 7 vibra io s almos every case , ’ Beetz s observed va lues are larger t han the calculated values . ’ ’ In t t t t t n an n a x n » his las paper, Bee z accep ed S efa s d Rada e pla a tio n of the All these studies have been made by men interested in si m and t t t t va ria o t nes phy cs , pri arily , hey have rea ed hese ti n o e r as actually existing o bjectively as p ndula r vib a tio ns . Stefan and Beet z had actually used resona tors in determining their t M r v x n ti n the n m n n at pi ch . o eo er , while e pla a o s of phe o e o fi st ff n nt r t o ne can r di ered co siderably, or were e i ely wi hheld , no t help being struck with the steady advance to ward uniformity of opinion as theoretical dete rmi nations go hand in hand with t the the accumulation o f empirical data . The ou come of whole e he process is a brilli ant achievement . Y t in t face of all this, n t n the interru tio n o r n t Koe ig, also a physicis , co sidered p , i er mittence t n an t n to the o n n t t o e , objec io res a ce hypo hesis , wi h m o ut nt t t n ts o b ect i it l , appare ly , es i g for i j v y

o M . n o n n t In 1 r . M 87 5 P fessor A ayer, e gaged a o her prob t n the t nt t n t lern, bu usi g same sor of i errup io appara us as Koenig m n i n t l not t n n n t nt e ployed , i c de al y iced his phe ome o of resul a b n n the tones and descri es it pract ically as did Koe ig . Whe n t o ne its n n t the t disk is stat io ary, wi h of ope i gs opposi e mou h nt t t the ca r x n o f the resonator, it is evide ha will e perie ce a sim ple sonorous sensation when a tuni ng-fo rk is brought near the h t . n n t e mouth of the resona or O revolvi g perforated disk , two t n n t n —o ne t addi io al or seco dary o es appear, sligh ly above ,

it . 8 . t abin e c . Ben ley an d S , o p. , p 4 9 “ wa vibration c Non e o f these f o rmula , i t i s true, sho o rresponding to thi s

a whose re uenc n umber i s e ual to the n um er o f in terru ti o ns but tone t , f q y q b p ; we shall see tha t two ex lana tio ns ma be iven o f its o ri in nei ther o f whi ch p y g g , is eontradicto ry to the reso nance hyp o thesis. 8 J SE H ETE N 3 O P P RS O .

he t s in t o her slightly below the pitch of the fork . A n in crea e velo city of rotatio n causes the two secondary soun ds to di verge et t mthe n t the t n nt o n r n y fur her fro o e of bea i g fork , u il , eachi g t n t the two o n n o se ra t a cer ai veloci y , sec dary sou ds bec me pa ed m t m xt at the mo m nt fro each o her by a ajor si h , while same e t nt o n n n o f the o t a resul a s u d appears , formed by a u io f rk wi h nt the upper and lo wer of the seco ndary sounds . The resulta 1 08 is the lower second o ctave of the note given by the fork . t n n th t t t n o f the the On fur her i creasi g e veloci y of ro a io disk , two n o n and the t nt a nd the ea r seco dary s u ds resul a disappear, experiences only the sensation of the simple soun d pro duced the f the x n t v by fork , whose beats at this stage o e perime ha e blended into a smoo th continuous ’ M t t m nt at the o n n t t t the ayer s s a e e , c clusio of his ar icle , ha beats from this rotating disk in the interruptio n of a cont inuous so und are like those due to the interference of two tones n early in t o ut m R the in : equal pi ch , calls fro Lord ayleigh follow g

The difference between the two kinds of beats is conside ra ble . If there are two vibrati o ns of equal amplitude and slightly

ff n in n nt co s 2 1m t and co s 2 7 n t di eri g freque cies , represe ed by , 3 , the resultant may be expressed by

2 co s 1r n n t co s 1r n n t ( l , ) ( l 2 ) , and ma t n n y be regarded as a vibra io of frequency V2 ( 1 and o f m tu 2 o s 1r n n i c t . n a pli de ( 1 , ) He ce , n passin g t the m tu n n nt hrough zero, a pli de cha ges sig , which is equivale to a cha ng e o f pha se o f if the amplitude be regarded as always po sitive . This change of phase is readily detected by measurements in drawings traced by machines for compounding h observati o ns . If a force o f t e above character a ct upo n a t o 11 n sys emwh se natural frequency is ( 1 , ) the effect pro d ed m he t t uc t . t m t is co para ively small If sys e s ar from rest, the m o r t at fi t but t t successive i pulses co pe a e rs , af er a ime the t m n to t the ff t la er i pulses begi des roy e ec of former ones . The t t n n to o f n n a d n grea es respo se is give forces freque cy l n , , and o f n n n t o a force o f frequency V2 ( 1 , )

1“ Thi s i s r l s - lle in rr n H p obab y the o c a d te upti on to e. e seems to reg ard thi s ’ as diff er n ne. f ch f r w tone a e ce to C . S ae e s vie . 86 bel w. , p , o “

h l th ri s LI X. 18 6 i . M a . Se e 8. P g , 4 , , 7 , p . 35 A D E T COMBINA TION TONES N R LA ED PHENOMENA . 39

the t n n t n On o her ha d , where a si gle vibra ion is re dered e n t an ta t intermi ttent by th periodic i terposi ion of obs cle , here is no such change o f phase in consecutive revivals . If a force h a an t the ff t of this c aracter ct upon isochronous sys em , e ec is indeed less than if there were no i ntermittence ; but as all the m t n the i pulses o perate in the same sense wi thou a ny antago ism , ns nt tt nt t n respo e is powerful . An i ermi e vibra io or force may be represented by

2 1 o s 2 a m! 2n d c ) cos ,

he the t n and the f re in which n is t frequency of vibra io , m the n s t queney of intermitte ce . The amplitude is alway posi ive , n t n a nd varies between the values 0 and 4 . By ordi ary rigo o metrical transformation the above exp ression may be put in to the form — 2 cos 2mt + cos 2w( n m) t + co s 2 sr ( n m) x; which shows that the intermitt ent vibration is equivalent to

three t n n n n m and n m. simple vibra io s of freque cies , , This is the expla nation of the secon dary sounds observed by

Fro m tones that are periodically interrupted Koenig went x n to periodically variable tones . In this e perime t Koenig hoped by a sort of synthetic process to p roduce bea ts similar to those due to interference of neighborin g ton es and by increasing their f t t t d n requency to prove ha hey pro uce a co tinuous tone . He 1 used a large disk with seven ci rcles of 9 2 holes each . The holes of each circle varied periodically in diameter from 1 to 6 n ho s n o mm. The seve circles of le were arra ged t vary period ica ll the t n 1 2 1 6 2 2 8 6 and y from ou side i ward , , 4 , 3 , 4 , 4 96 I in t imes respectively . n blowin g aga st these circles through in t n the d in t t n en a tube 6 mm . diame er whe isk was ro a io , Ko ig — heard no t only the tone correspondi ng to 1 9 2 the number of h to the t n— but n the t t n was t oles revolu io also , whe ro a io fas n the o ne in n n to the n e ough , each case correspo di g umber of — . e 1 2 1 6 2 e he t n in m t 4 . tc . t varia io s dia e er , , , 4 , , as case may n the t t n tin it be . Whe ro a io was slow he heard a bea g ; as “ - l h r un hil IX 1 . 3 8 c f . also Ra ei h T eo o So d V P . M ag ” " 880, pp 7 9 ; y g , y f , ol.

L. $94» pp . 7 14 . 0 ET N 4 JOSEPH P ERSO . in the ts eca and o ra d th creased, bea b me more m re pi until ey 1 1 1 finally blended into a second pure tone . Zahm says that the forego ing experiment wo uld seem to be conclusive as to the true nature of beats and beat no h o v r co n iz ed that n t so t w . Koe ig was sure of is, h e er He e g if the complex vibra tion thus produced were actua lly like that t n two nt n t n o x t to ea resul i g from i erferi g o es , we sh uld e pec h r in this case no t only the beat-tone but two other to nes co rre spo nding to two primary tones which would pro duce similar ‘

t n . To t : In t s bea i g quo e shor , if a series of 96 isochronou impulses of which the intensity increases and decreases 1 6 times nt x ct two to n 1 t o ne represe e a ly es which give 6 bea s, o ught to hear the two primary tones in question ( which wo uld here be ‘ the t n and 1 0 f o rmin the nt 1 bu e o es 88 4 , g i erval t thes n n tones are never produced . The reaso for this differe ce be tween the beats and the separate isochronous impulses of periodic intensity ought to be sought in the fa ct that the com pound of two to nes a and b near uniso n is equal to a to ne o f medium pitch ( a b) / 2 of which the intensity no t on ly in creases and diminishes periodically but undergoes a cha ng e of si n n n t the g o ce duri g each bea , as formula

sin a + sin h - 2 co s

a b 2 nt the nt shows , where cos ( ) represe s periodic i ensit y o f the to ne ( a ’ n M r S tt t n Whe . po iswoode repor ed Koe ig s Lord Rayleigh did no t consider it at all convincing aga inst ’ do no t t t 5. e. t t t n t o n Helmhol z s heory , , ha bea s ge era e c mbi a m the t nct o n t n t a tion tones . He ade same dis i i be wee hese be ts n t t n n and those due to interfere ce ha Koe ig did , co sequently he t nter erence t t could no t see how bea s of i f , wi h his change o f

co uld generate a tone sensation . Bosanquet raised the m - oeni cit . 1 0 1 3 and Pa . A nna l. L I I . 18 6 . 1 8 K g , o p. , pp 4 4 , ge , C V , 7 , pp 77

- s m t . . c f . al o ah o . ci Z , p , pp 334 5 m Zahm it . . , o p. c , p 335 m

i it . 1 . Koen g , o p. c , p 43 m — r 8 2 P o c . M us . A rt , 1 78 9, p . 1 8. n ' S upm, p . 38. developed it more fully in his article re e w t t l x n H sho s , by a ma hema ica e pressio t t he t nt oped by Koenig abo ve , ha t resul a ro duce a tone having oscillations of is defined by a pendulum-vibration half the difference of the frequen cies of t t r in he says , is wha is ac ually hea d ” t than two commas apart . I f his held intervals the primary no tes would no t at would be heard would have

the note heard would

which would beat very rapidly . the note 5 is no t heard at all in the supposing tha t in some unexplai ned way

2 i . e. the difi erence is ( p q ) / [ , half of to n t primaries] gave rise a o e , as Then the speed of tha t note does no t

fi - t rst beat no e , which has

seems fully to have appreciated the force of these for he immediately takes up experiments to meet

two t n 0 and . vib I f o es of 8 96 d . are so unded “ t n a t x 0 he says , hey ge er e a tone of / é ( 8 96 ) 88 vibrations wi th an i ntensi ty i ncreasi ng and diminishing 1 t an a 6 imes , d t each passage from o ne beat to a no ther there c n n t t the xi o f n the is a ha ge of sig , so ha ma mum compressio of first vibra tion of the following beat is half a vibra tion behi nd the maximum compression of the last vibration of the preceding x To meet this case he performed two e p eriments . “ lai M a 1 ri - t 1 1 Se es XL 2 1 . P g " 5 . , 4 3 m Wh ere the two vib ra s —t ti on s co fl and co : ( pt ) , having eq ual ampli tudes r: om ( are c bined o n the same recepti ve mec hani sm.

‘ ‘ ia the r s l n esulta nt di p aceme t. ’ e er di Z mmenk ev eler Tane n L I I o en U e naa lan o . A al K ig , b g , P gg C V - 1316 . w1 77 m. ’ . 3 d p 5 3 , quo te f rom Ellis translatio n in S ensat io n a] Ton , p . 5 34. 3 4 JOSEPH PETERSON.

In the first he divided a circle o n a large disk into 1 7 6 equal t n h t he fiv e ints m n t in . n t par s, u beri g e par s their order I po 1 five l inc ea s n in , 3 , 5 , 7 , 9 , he drilled holes , gradua ly r i g

to and t n in . e diameter 5 mm . hen dimi ish g He did likewis in the o nt 1 2 1 1 1 and 20 and n the nts 2 2 p i s , 4 , 6 , 8 ; i poi 3 , 5 , d d a s o wn 2 2 and 1 an so o n . n w 7 , 9 3 ; Whe such a isk bl t e upon thro ugh a pipe with the diameter of the larges op n ing, in addition to the tone 88 and the very powerful tones o f the 1 t the to n 0 and u but t period 6 bo h of es 8 96 co ld be heard , hey and o n n the n s the were very weak , , accou t of rough es of deep " ‘ n ffi to I he e a s the t t . n t t w o e , di cul observe his case phas ’ 1 20 t t . n to o nt co n same hroughou , says Ellis Koe ig, , appare ly e sid red it so . To imitate the cha ng e of pha se Koenig divided each o f two concentric circles o n a disk into 88 parts and dispo sed the holes which were to represent the successive beats alternately on each and 1 6 s to circle . As 88 holes periods give 5 % hole each n to two to t and o n the period , Koe ig ok periods ge her, pierced e n o nt 1 2 6 and o n the first circle th divisio al p i s , , 3 , 4 , 5 , sec o nd 1 0 1 1 t n o n the fi t 1 2 1 1 1 6 , 7 , 8, 9 , , , hen agai rs , 3 , 4 , 5 , 1 6 1 and o n the n 1 1 8 1 20 2 1 2 2 and , 7 seco d 7 , , 9 , , , , so “21 z h i t t . t e o n for h The si e of h les, his case also, gradually an increased d diminished to represent beats . When these circles of holes were blown upo n a t the same time through two the m t the t n n and o n e pipes of dia e er of larges ope i g, placed th m o ne and the t sa e radius , circle from above o her circle fro m wt n at t n o f the t a t belo , he each revolu io disk here were cre ed o o n m r n 1 6 t in nt n t 88 is chr ous i pulses , va yi g imes i e si y, which changed sign o n each transmission from o ne period of intensity In t x nt the two t n 0 a to the other . his e perime o es 8 nd 96 he t x m nt e r were more distinct than in t firs e peri e , where th ci cles “22 of holes were blown upon from o ne side only . ’ ” ’ In reply to Bo sanquet s objection) that Ko enig s beat-n ote

l i h s : I n assin thro uh ero the am li tud ch Lo rd Ray e g ays p g g z , p e an ges si g n, which i s eq ui valen t to a cha ng e o f phase o f i f the ampli tude be always ’ u r . i a . 8 regarded as po s ti ve, s p , p 3 ’ u rien e A uti u 1 1 Ex e d co s e . Cf . Fig . 4 , Quelq es p c s q , p 44.

’ - n 2 2 c f . also uel ues E erie A Pa . A n a l. L I I . . x nc es d co s gg , C V , pp 3 3 ; Q q p s

- h o kin s o f i sk s sh wn in Fi s . 0 n . 1 tiq ue ( were the tw d d are o g 4 a d pp 44 5 . “ r 0 1 . S up a, pp . 4 , 4

4 J S’EPII T 4 O PE EIWON.

t n s n n o e , i cludi g both the evenly and unevenly number en te in a ll g era s cases , quite i ndepen dently of the t ns t t t n the e e i y of hese o es , stro ngest and nea r st qua l the ff n h 54 di ere ce of p ase , while the diflerenCe 0 t n the t be wee o hers , both as regards intensity and ac When unevenly numbered partials only are cc the di fferences o f phase 54, and 94 give the same qua as do also the differences o and 54 ; but the form er and t t n the acu er ha la tter. n t the He ce , al hough quality of tone p r1n c1p a o n the number a nd relative i ntensity of the ha m m un the n n co po ded , i flue ce of di fference o f tone [ ph1 by any means so insignificant as to be entirely negli; in n t t t he difler he may say, ge eral erms , ha t ences in t 1 relative intensi ty of the harmonic tones compound1 those differences in the quality of tone which are 1 i in t t d nt i the mus cal s rumen s of i ffere fam lies , or in h

utteri ng different vowels . But the alteration of ph these harmonic tones can exci te at least such ( ill quality of tone as are observed in musical instrun in ff nt n n the e same family , or di ere voices si gi g sam “ t n u n diflerences A ready apprecia io of s ch mi or , u t n Rayleigh , req ires a series of no es , upo which a n w be executed , and they may escape observatio n t a t t t 1 si gle no e is av ilable . To me i appears ha are in harmony wi th the view that would ascribe tl ’ n an n t n ha from Ohm s law, involved i y recog i io of p c Kendrick t n f ro m to secondary M hi ks, nt t t the in n b of this experime , ha flue ce of phase is , ” 8 so absolutely negative as Helmholtz supposed . ’ n t n n it no t edly Koe ig s me hod , i ge ious as is , would reproduction of the tone exactly corresponding to th An n s n t d n f wave form . y co clu io , herefore , raw periment described must take this fact into conside t the t t t rt Hermann, who ado p s view ha bea s

1: an Elli s in Helmho lt o . cit Wied . A m t , p . 39 1. tr s. by z, p . m i h c it Vol . IL . 5 Rayle g . o p. , . 9 4 9 ’ ’ s h sio lo V ol. . 1 1 6. Schaet er P y gy, p 7 COMBINA TION TONES AND RELA TED PHENOMENA . 4S

’ t n t-t n t t n x bination o es or bea o es , holds ha Koe ig s e perimen t with the wave siren is inadequate as a proof that phase cha nge t o n the of upper par ials has an eflect quality of the tone . The t the t t compressibili y of air, he holds , makes it al oge her imp ro b able that the air vibra tions take the exact form of the curve o n ” the s n) n the n t wave ire He co siders sire , hough a useful x m nt nt nfit o n th e peri e s , e irely u for use e t t n i herefore , ook up exp erime ts with Ed and came to the conclusion that not the itself but the effect that is produced by such artia ls o n the posi ti o n of the maxima in the and o n the amplitude of vibra ti o n accounts 1 3 1 11 he n n t sou d . Herma n had already con er to explai n certain pheno mena of beamand ‘ we must a scribe to the ear the po wer of to ne sensa tio n to every so rt o f perio dicity Accordin g to this supposi tion the o I t ds t be formulated di fferently . t is bes drop alto gether a nd simply to say that the no t only by the amplitude of the partial nd h the o t the a c iefly , by f rm ( Ges alt ) of ermann fin t t t ds ha , o her thi ngs being equal , sha rpest when the cha nge fro mma ximum i o b o r s m st a ru t . So at t ( vice versa ) p , leas , I ’ In an tz t to y case , Helmhol s heory seems

o n this po in t . t n to the x nt n he e perime s of Koe ig, we may well t t n in no n resul s he ob ai ed were way co clusive . such a thi ng as an interruption tone is by no t r t t the - shed . Physical heo y shows ha ao ca lled t objectively bot h when they are produced

nunte h sia l LV I . 18 P . d. gesa y n , 94, p 474.

V . 8 f . ; also L I ., 476 5

5 “ “ Der S chall in miter saw g leicha Un eth i c 6 4 JOSEPH PETERSON.

n n o d nt tt n co t a n n as a co seque ce of peri ic i ermi e ce of a ns t to e, and when they result from such intensity changes as a re pro e he t n n - duc d by t rota io of a sounding tun i g fork . Indeed both t t o ma be trea o of hese cases , as physical he ry shows , y ted t e x m nt t n gether . From th e peri e al resul s of Koe ig it is no t so certain that the case of periodic intensity fluct uation p ro duced by perforations o f varying diameter may be classed with the other cases of periodic intensity changes or intermitten ce ; fo r the t n t n no t nt n a nd in this case varia io o es are me io ed , the ia “ terru tio n t n mn nt) n o wv p o e is very pro i e Koe ig has , h e er, by no means proved that beats of interference of neighboring

t-t n n th it nt t e tones produce bea o es, eve ough be gra ed ha t th n a n t beats of inte sity variati o n are c pable , whe heir frequency

- s fli ient n nt n t n n t n . is u c , of produci g a co i ued o e se sa io M athe ma tical theory shows that in the ca se of beats fro m interference

h n o hase. E n t o n there is a c a g e f p ve h ugh Koe ig, by a n in e io us t n to m t t t n g n me hod , has e deavored i i a e his cha ge of phase t nt n t t t o n and n in case of the bea s from i e si y fluc ua i , u der those n t t n the so - nt t n to n it co nditio s has s ill ob ai ed called i errup io e , is no t at all cert ain that he actually o btained the co nditio n s that It t an n t o n t the he sought . is s ill ope ques i whe her ear per ” ceives intensity fluctuations occurring with changes of phase)

’ L h f er s e rimen ts re e bel B f . . ae x e o rt d w . 88 . ut c . K Sc p p o , p Since the abo ve acc o un t was wri tten so me recent a rticles o n the perc epti on

o f h -di ff erenc have a eared in the Philo so hic al i r p ase e pp p M a g a s ne. Lo d Ra yleigh

N F ruar 1 0 re rts ex erimen ts in which n in o . e o wo to es o f 74 ( b y, 9 7) p p t n early

u h wer co nduc ted se a ratel o ne to each ear o f th s b eq al pitc e p y e uj ec ts . U n der such c ondi tio ns the loc a ti on o f the so urce o f so und seemed to fluc tuate f ro m right

f wa s bein on the side o f the mo re advanced h ase which o f c r to le t, al y g p o use ’ c mi n ls f N . f o r M ar h tran s tted to es t s f e R lei h a o c . o o th s hi t d. ay g ( 75 , ) e ubj ec ts

f wo tele hones chan in the ha se b mean s f o ears by means o t p , g g p y o a c mmuta to r

The results ree th tho se o f the rst seri es f a rr n emen t. a wi o x a g g fi e perimen ts. Lo rd Rayleigh c onc ludes that we are able to tak e acco unt o f phase-di ff erence ” a t the two c ars.

I n N o . 6 A ril 1 0 f h m ma a i n o t e sa e e L. T. Mo o re and H 7 ( p , 9 7) g z , . S . F ry re o rt ex erimen ts bearin o n the sam esti n p p g e q u o . They tran smi tted o ne to ne to the two ears b mean s o f Y - h y a tube. W en o ne o f the branch tubes was ma de slightly lo nger than the o ther so that the wave reached the c o rrespondin g car in later ha se than in the o ther e r th s n was l p a , e o u d oc a ted on the side o f the sho rt

u r t be. These esults theref o re a ree with tho se o f Lo rd R l i , , g ay e gh .

These results a re o f co urse in x , , e plicable o n the Helmho ltzi an theo r as o n y, m s o thers r ’ o t . Lo d Ra lei h think s tha t the uhold Ruth y g y p erf o rd s theo ry . Thi s wave siren to ascertai n he o f the t n ec t t nature o e, to the ff m t not be a i r a ive , is But the results obtai ned seem are no t so nega tive in their

to show exp erimenta lly that n t ns t n t ge era e se a io s of o ne .

arge scale , Koen ig of the compoun d he required dimen inverted a nd care metallic cyli ndrical hoop that it could be rota ted o n of the curve is that the ities would no wgive lesser tting o ff the current more than the inst rumen t as used by Koenig had so that he could examine a num ‘ ° n nt and t 3 conve ie ly compare resul s . a to o thed rim is rotated befo re a sli t fixed over t n and n t a t direc io , of a le g h least equal to the the the t l h t of curve , sli wil be periodically s or cco rding to the law of the curve ; and i f the t t n in the st sli , a mo io air mu be gen

the same law . And this motion must as that produced by the simultaneous simple tones withou t any admixture of v nta t n n ad a ge of his arra geme t, then,

f s s is in the c o to ne ana ly i o rt ex and no t in tbe ear,

' Certa inly in these experiments the intensi ty dltl er ne co nduc tion were no t sufi ci en tly co n trolled, and f ai th in the resul ts until f urther experim&ts alo n g ha ve been made. ’ uel aes E i es d A i . 6 x enc m ae 1 . Q q per cas q , p 0

he the Concerning t views, held by n t h n n Koe ig, hat t e ear is capable of recog izi g as n periodicity wi thin cert ain limits of freque cy , periodici ty with frequency 1 2 8 is also tt er tone to be heard same n n 2 is also a periodicity of / , n e n s theory is co c r ed , such que ’ " nt n ed by Ohm s law , co i ues Lord to n ts may compel us abando this law , ber that there is no thing to take its

11 experiments showed tha t at a certain the t nd the t n bea s a o es , which he sup

t t the . by hem , are heard a same time is is no t necessarily contradictory to his

in mi nd that Koenig still held to the as did You ng and ot hers before Helm '

e t . O n t t n p Ohm s law his view, he , as t n above , waves sligh ly divergi g from

‘“ lica tion Koen ig says o n this matter multaneo us perception of separa te beats results from their succession is no more the newhyp o thesis than with the old t as accepted by Helmhol z] , for we can very besides the general excitement of the basilar t t the t t each separa e bea , par icular par s of hose proper to nes correspond to the period

Ra ylcieh ibid p 469 ” . l en it h Firmi ng . Zur Lehre v d. Voca klang , Ze se r I . B io lo gic, XXXIJ N . F .,

“ ’ f M ax s t x in an C . Meyer a ttunp t a e pla in g o ther manner o f ana lysis, pa ge

’ t s “ r nc e Aea 1 uo Ex ie i ue . Q e pi s d ast q , p 37, trans. by Elli s in S ensatio ns aI ° 5 JOSEPH PETERSON.

the tr n n nd x t ast n of impulses , are more s o gly shake , a e ci e l i g ” t n n th vibra io s givi g e perception of sound . It will be reca lled that Helmholtz had to accoun t for all beats of wide intervals by the presence o f upper part i a ls or of difference to nes . Now Koenig suppo ses that he has a dditional n ai F x evide ce ag nst this view . ro m his appro imately pure to nes

t n o the n t n t n n - s ob ai ed fr m reso a ed cla gs , of large u i g fork , he had heard beats that would require the presence of upper t it nt n n to par ials , which would be e irely u reaso able suppo se were “5 nt . m n he t t prese Now , by ea s of t perfec ly pure ones ( as he 1 “ o the n no t n t b t so supp sed) of wave sire , he heard o ly bea s u al ’ t-t n n to z r bea o es, which accordi g Helmholt s view would requi e he t n t E . . e e . two to n prese ce of upper par ials g , gav s 3 and 5 ( 1 1 and 2 X 8 ga ve to ne 1 ( 3 x 8 2 3 But no ne o f these tones are inex " plicable from the fo rmul a of Helmho ltz and B o san quet) A nd these fo rmul a resulting from the development of lower he t o n no t x t t o . In t powers ly do e haus p ssibili ies hese cases, to o the t nt to n do no t de end u o n u er a rtia ls , resul a es p p pp p , we er b ti e r o b e t e h th sujec v o j c i v . Even if no to ne had before

n o n n to e. . it no t an bee heard corresp di g 3q p, g , is argu nt n t the o t t t t n x t me agai s p ssibili y ha such o es e is , that Koen ig no whears such a to ne ! Such tones ( if indeed Ko enig was no t t m t n in the t t i ac ually is ake pi ch , as o hers had been n rega rd to combination tones befo re him) are extremely rare wha tever theory is adopted ; and so far as the mere existence o f these t n o n n t nt no difli cult o es is c cer ed , hey prese y a t all to Helm ’ t o to an r ho lt z s theory . Las ly we c me impo ta nt point in co n ’ e i t t t n to m o tz o s n ct o n wi h his objec io Hel h l s p ition . H a wis Ko enig to expla in the perceptio n o f these to nes ? It is well known that by means of his manometric flames and his com x fin t o n n to t no w ple curves , he ds periodici ies corresp di g hem ; , if he endeavo rs to make a mathematical or theoretical state ment of these peri o dicities will he no t actually obtain formula like tho se o f Helmholtz and Bosanquet ?

Y et when Koenig co mes to acco unt f o r summa ti on to nes he po si ts the

s nce o f arti als o f still much hi her o rder ! Cf . belo w re e . f p p g pp 5 3 .

u r B ut see no te . 8 s a uo ted f ro m Ra lei h p 4 , p , q y g .

u r 18 s . Cf . p . p a

S3 Josem ”transo m

no t m but n are si ple are very rich i upper partia ls.

e. . t its two t ls g , wi h series of pa r ia

6 s

shows that the fifth partials will give a bea t~ tone 5 ) which is equal in pitch to the summation to : The fourth with the partials

6 1 2 1 1 1 3 , , 9 , , 5 , 8, 2 , 8 1 2 1 0 4 , , , 6 , 2 , 2 4, 2 8,

the t n t-t n he t gives o e 7 as a bea o e of t seventh par i als , In general the summation tones tha t a re audible n “ plai ned by the formula n ( a b) a bf whet stand fo r the upper and lower primary tones respec n o is some wh le number . In the French t e-publica tion in 1 88 2 the above o b

t . n n in s ill appear Here , however, Koe ig has e closed l the statement concernin g his proof of the existenc e 01 t n t n m n o f the x an io o es by ea s bea ts of au iliary forks , ‘ 1 0 3 he has added the followin g long note . I ha po rtant observation to make o n all those remarks c di fferential and summation tones which are found i t o n the in the A nna len t whi publica i of memoir , hose in nt but placed pare hesis , which I have reproduced l New researches which I have made o n bea ts of h i ntervals si nce the publica tion of that memoir haw stra ted that even in formi ng very wide mistuned intc mary t ones beat distinctly wi th a feeble auxiliary t n but the x i t n to i v the : o e ; , au il ary o es used d sco er e to i di fferential and summation tones , to o feeble be h

o f he n t i . e m n t t n . all har o ics o es of i ferior bea s , , of beat-notes of the primary tones ; and some of then e were harmonics of o ne of th primaries . They 01 n t t t t t n v ecessarily , o produce bea s wi h hese o es of C n nt th t ti e were harmonics . o seque ly from a m “ — Ko en ig does no t seem to ha ve used the f o rmula li d M : result he o f ten hu to po sit the p reseuee of very hi gh upper pu'ti l M INA TI N A ND HEN M ENA CO B O TONES RELA TED P O . 5 3

to in nt the x t n observed by me ceased prove, my judgme , e is e ce of co rresponding tones interfered with ( alteré) by the auxiliary ” 2 t . forks , as I had previously held hat they did A single excepti on sho uld be made in ca se of the auxilia ry he U1 d Si 1 0 . t t n t 8 an of 44 v s . which wi h primary o es 8 ( ) 3 ( 5 ) demonst ra ted by the presence of beats the existence of a feeble t ne v n t in b t the t n in o 7 , which, howe er, was o h g u o e of ferior

- t i . e. the he a t n t o f 8 and 1 . n t be s ( , lower bea o e ) 5 I deed ‘” tables given above show t hat the bea ts as well as the lower bea t-not es of the first period can often be heard directly up to the neighborhood of and it is con ceivable that by the x n o e t aid of an au ilia ry to e , n should perceive hem a degree further 1 4 : 1 After these considera tions there remai n only the two tones

M i, produced by the i nterval and Ré“ U1 : from the i nterva l 8 M ia ( 4 of which the existence may be regarded as really proved ; because t hey have been o bserved

he . t n no t n wn n directly by t ea r These o es , havi g sho any actio o n o n t n t nn t the res a ors , as I have already i dica ed , ca o have origin which Helmholtz attributes to differential and summation “ n he to nes ? but, bei g very feeble , while t primary tones in the int l h th t nt n t t erva s w ich form em have a grea i e si y , hey may x n the t n o f the n n be e plai ed by ac io feeble harmo ics , produced i e the m to n s n to t th ear by pri ary e ; for accordi g Helmhol z , every t n n t o n very strong o e , eve hough simple , sh uld produce harmo ics in the n in n l n nt the as orga of hear g, pri cipal y o accou of ym t t o f the t n n and t o n met rical s ruc ure ympa ic membra e , par ly account of the loose articulati o n of the hammer wi th the ” anvil) ‘ t n a t nt no After wha precedes , I k ow prese of experiment by which o ne ca n prove with a ny certainty the existence of ’ differential and of summation tones . In consequence of this discovery he has revised note III

“ ’ I t will be reca lled that Koeni g s experimen ts o n beats lead him to the

to n t: wi i ts twelf th e. conclusio n tha t o ne e bea th , g ., even tho ugh there are no

s upper partials pre ent. 1- ’ ' s es on eat: in 8 1 “: J A cm h u I . e., hi tabl b W u q . “ ’ This does no t at all fo llo wf rom Helmho ltz s View. “ h tude o f ni c en t i s ec uliar a tti o e a e 6 b w. See my o mm t on p K g. p g 5 , elo S4 JOSEPH PETERSON.

of the co nclusions at the end of the chapter to read The existence of differential and summation tones ca nno t at ’ present be demonstrated with certainty by any exp e riment . The original of this note ( given in a footnote) reads : The differential and summation tones which are produced by the n n two n t n the r t n co curre ce of very stro g o es , because vib a io s the tt to nfin t m n t t t n m n n of la er cease be i i ely s all , co s i u e a phe o e o ’ - which is i ndependent o f beats and o f beat tones . ” 7 ’ the W . Preyer very early raised objection to Koenig s x n t n mm t n t n t t in e pla a io of su a io o es , ha some cases it requires the presence of a great number of overtones which apparently o t t t the the nt n n . 6 . are prese He ci es case of i erval 49 5 2 8 v . d of which he has heard the to ne This o the n the 2 d t w uld require prese ce of 3 par ials , of a nd “5 8 i th t n n 1 2 1 v b. nd ( . a 2 6 . Wi u i g forks of 9 5 d . v the mm t n t n 8 [ 3 says Preyer, su a io o e 44 is heard clearly, even tho ugh both forks have been damped [ to eliminate upper Now in this case it is very improbable tha t the t t n t m n the s seventh par ial o es , af er da pi g of fork , would be

- lo ud enough to produce an audible beat tone . In conseq uence ’ ffi t t t in the n w of di cul ies of his sor way of Koe ig s vie , Preyer

to t it x n t n . refuses accep as a probable e pla a io He is , however , n n t t t to n t i of the opi io ha hese es , hough correspond n g in pitch

mm t n t n to o to h. On the to su a io o es , are loud be suc suggestion o f A unn t t t t ma x G . pp , herefore , he shows ha hey y be e p lained

ff n t n the seco nd o rder 5 . e. ff n as di ere ce o es of ; , di ere ce t o n es n m the ti n ff n t n o n an arisi g fro ac o of a di ere ce o e upper part ial . Thus instead of by mea ns of the thirty-second partials of the the nt r to th tones of i ervals eferred above , e summa tion tone can be acco unted for as follows — z x 5 2 8 ( 5 2 8 496 ) 3 2

l 3 I ‘m ) P~ 7, T NES ND ELA TE COMBINA TION U A R D PHENOMENA . SS

Every term of these acoustical equations is easily proved [ to ’ Preyer s general formula then is expressed thus — nb 1 ) b — n the fi t t n n r n t n 2 a nd the O ly rs over o e is ecessa y ; , he , fo rmula becomes simply

1 5 9 — a = b 2 b ( b ) a + ,

b m t t n 2a the t n When, however, beco es grea er ha summa io r n n he third tone must be explai ned as a diffe e ce to e of t order, — [ nb ( n 1 ) a ] 1 ) b

n n n the fi t t n or, whe we co sider o ly rs over o e ,

b — a b — z a = a b ( 2 ) ( ) .

Si nce thus far nei ther I myself no r any o ne else has ever heard the summation tones when the first o vert one was no t ’ the s t n it n t a t ame ime very plai ly audible , is a ural , he says , to conceive of the summation tones as di fference tones accord ing to the above n in t n and o n n Now Koe ig ur , good grou ds , objects to ’

t M . Preyer s explanation . To quo e : Preyer cites in favo r of his views tha t o n sounding to gether free reeds o f 496 and b = 1 z the n 1 02 = i . . v . 2 8 . v ib 5 d 3 33 , he heard sou d 4 d 64 , a nd he thinks that we ca nnot assume that the reeds had the

- i . t n t a nd . v b th t hir y seco d par ials , d If e sound 6 and no t the t o f 1 really observed was 4 , oc ave 3 or 33 , we might be really astonished that the thirty-seco nd partials were sufi ciently stron g in these tones to produce it : but the expla na M t n tion proposed by . Preyer is absolu ely i admissible , for 496

b. n n t n 2 . vi and 5 8 d , eve whe hey have co siderable force , give 2 t no t et the to n C to 3 bea s , which do as y allow deep e , be a t a t a t an t to n t xt he rd , so h y ra e such e mus be e remely weak . Now the octave of 5 2 8 ( or is the 33d harmonic of But o this excessively weak sound . tw primary sounds of 3 2 ib n n xt e d . v . n an d , eve whe e r mely powerful , ever pro “ Rab" 1 8 6 fi rst su ested thi s ex lana ti on then i nde en l ( 5 ) g g p , p dent y G .

s. t um f T h ie I L m R. Fa ri an d o ther C . St on olo 18 0 . 3 . App , b . p , f m g , , 9 , p 54 ’ Wied. A n a l I I I . 188 . 1 . I ha ve no t seen Pre er s ea r , XXXV , 9, p 35 y lier 5 6 JOSEPH ra n som

o un i e in w h 11 v b. eco n ann duce a s d of . The s d m r hic

M . Preyer thinks the so und might have been pro duced is equally opposed to all that has been directly observed when two m n t a the o ct a e pri ary tones sou d ogether . Thus he m kes v of

. ff r nt to n 2 i . e d a 8 t 6 . vib. a 5 ( , produce wi h 49 d i e e i l e

ib . nd t n n 0 the 0 . v a t t 6 roduce wth of 5 6 d , he makes his o e 5 p i

t t 2 8 i . e. 1 8 a new ff nt a to n welf h of 5 ( , , 5 4 ) di ere i l e of But these two sounds of 496 and 2 X 4 96 64 ) give the beat note 64 and no t 5 60 ; and if the so und 5 60 rea lly x t it t 1 8 2 60 e is ed, would give wi h , 5 4 X 5

0 6 the at-n 6 and a nt 6 but X 5 6 9 ) be ote 9 , lso more fai ly 4 4 , no t

It nt t n to t en t fin is i eres i g no te tha Ko ig, when he fails o d objective upper partials for this explanatio n of summatio n t ‘” h o nes , falls back for support upon t e subjective upper partials determined theoretically by Helmholtz and Bosanquet . u m t n to n w eo -e ua l t the The s m a io es are , ho ever, q wi h upper d no de enden n m n t n in t an t t . par ials p t. upo he Koe ig, he , is the peculiar position of accepting some of the values of x o b ’ ta ined from Helmholtz s equation and of rejecting others t hat stand co -equal with them! He would be far more consi stent to go back to his earlier view and grant the existence of wea k m n t n to n n t n . co bi a io es , if o hi g more

EC 2 N N IO . S TION . THE OBJ ECTIVITY o r COM B I A T To mas

m o t it m m ha t n b Hel h l z , will be re e bered , d dis i guished e tween combinatio n tones that a re o bjective and tho se that a re e 1 .mbjectiv . He had found that combinatio n tones fro m pri mary to nes o f instruments having a common windchest were reinforced by resonators and that they were able to set in

sympathetic vibration suitable membranes att uned to them . Bu o n t even these he had found to be mostly subjective . K e ig in fi t o e n a a h r n his rs publication n th seco d ry phenomen of ea i g, “ Tr l ans a ted by Ellis, S ensatio ns o f To ne, p . 5 36 , f rom a f oo tno te in ' ’ ue E - uel s x erien A t 1 2 8 . Q q p ees d eo us iq ue, pp . 7 “ u r f . s C p a , p . 5 3 .

oeni f o r h n s tion clearl in mind. K g, one, a d o t kept this di tinc y HEN M E A TONES AND RELA TED P O N . 5 7

n t n d in the existence of combinatio o es , combination tones no r the beat-notes de 2 rein forced by resonators . san t the quoted from Bo que , how by use r be t o ut t n but t t to so nato , shu all o es ha and t t in so o n t was attuned , ha d i g all bea s n s t u v n t to to es di appeared , h s pro i g hem be sub

er to o o n t i m to t y , , worked h s proble as whe her ‘ the ones are objective . He constructed for t nin xt in a : 1 0 - u g forks of e raord ary delic cy f 7 36, ’ 3 3 ib. c 1 2 6 82 an d . v 5 , f 36 , g 7 68 d ‘ the ra tios They were o n the slightest excitement that they could ’ t n t t nigh o ly . The hree lowes forks had the follo wing partials : ‘ ’ F the xst t t n the 2 d e ork f had upper par ial f s ro g, also , ’ and the 3d f weak. ” w 2 ’ n Fo rk (9 had the t and d c a d g strong . ‘ “ 1 Fork f had the st f strong . ’ ‘ ’ fo rks e and a ve 0r 6 a ave e The f g f nd f g , “ ‘ 8 g and gave or 9 and that shown by their m a kin g t cally . But we see tha which existed already st rongly ‘ the forks f an d c were sounded sep a ‘ ’ the c t t t l . also made forks f , vibra e sympa he ical y results did no t prove the object ive existence of the

duplicates . On the other ha nd the forks giving di fferential tones — or 8 o r 5 3 =a ‘ ’ — — ( z e or q 6 z 3 f o r 5 a =3 i — — ’ ’ ¢ =f W 9 S =4 f I a ot q — ’ f I =I o r 6 — ' ” f a z e o r 8 — ‘ ' f e z a o r 8

L II . 18 6 22 1 . C V , 7 , p .

79, I I . 3 5 JOSEPH PETERSON. utterly failed to produce the slightest effect o n the fo rks having ” the m sa e pitch . By a similar test Preyer satisfied himself that no ne of these

t m t n to n . no t in forks gave objec ive su ma io es He did , fact, fin sub i n ec e m i . e n o hea a n t v m t n t n . t eve d j su a io o es , , he did r y mm ” t n t n . t su a io o es Perhaps, says Preyer, hey might be made audible after properly a rming the fo rks by means of o x il he resonance b es wh e sounding . But t observation would no t be ’ While Preyer s general conclusion is that combinati o n tones a re t n in t a t ex eri subjec ive , he ack owledges a la er r icle , from p m nt t m nt n t t t o n n e s which he here e io s , ha appropria e c ditio s may be set up externally to the ear and made to genera t e corn 7 n t m n s o n t n . M e co n bi a i o es bra es , he ays, may supply such io n dit s.

One o f the experiments to which Preyer referred wa s that 8 ’ o Lummer n in m t n L er f . i umm O , which , Hel hol z s prese ce , made a microphone reso nator respond perceptibly to the sum m t n to n m n t the m n m a io e of pri aries ge era ed by har o iu . ’ In the m Lummer s x m nt M n sa e year as e peri e , . Wie o h ta ined only negative results . He found that a very delicate reso nator was never sensibly a ffected either by the loud di ffer ence to ne of the primaries generated by two Quincke tubes ” an ( Lippenpfeifen ) or by a tube d a telephone .

. m nn in 1 1 t an x L Her a , 89 , repor s e periment bearing o n the m o m o f t t m n t n to n sa e pr ble objec ivi y of co bi a io es . He admits that instruments with common windchest for bot h pri t ff n t n and t mary to nes give objec ive di ere ce o es , herefo re holds ’ that Lummer s experiment is no t to the po int since he used the t n t t m m n m. nn har o iu The ques io , as clearly s a ed by Her a , is, whether primaries of independent so urces generate object ive ‘ i ’ c o n t n . n t ann tw ombinati o es I ser ed, says Herm , o t ele

' ‘ ’ N o te tha t so me o f these a udible diflere nti al to nes are intermedi ate di f

f erence to nes .

- n To ne . 1 2 . F ro m Elli s, in S ensa t io s 0] , pp 5 3 W A r eber Co mbin a tio n stiine ied . W L VI II 1 Pre e U 88 . 1 y , , , XXX , 9, p 33 . a s ’ m fi ndl. ob . lan anal se Verbena . B erI h eber ein e e . s U p j K g y , p y . Gesellsc h

66 . 1886 . p . ‘ ’ Tonsti rke Wied nnu Ma x Wien Ueber Mess un der . A a l I I 1 , g , , XXX , 889.

60 mm ” re aso n.

duced . They decided to ma ke a ca ref nl test in whi n ed not m o d irectl a l e be e pl ye d y t a l. The used a tunin fo rk as resona to r y g , a Koenig tl f v o ib. 64 d . Since this inst rument is rela tively 1 x it n e e c e by reso anc , they used a very delicate m eth te t n t it set e i g whe her was in motion . Fo r this 1 mirro r wa s att ached to one o f the p ro ngs and a sj fo rmed by which the M ichelson inte rference bands duced. A movement of the prong amounting to h: length of light ( say 1/ 80, o o o of an inch ) would len gth of the path of o ne of the interfering rays b A len gth. periodic vibration of this amplitude we ” the n to e n ba d disapp ar . The ba ds were sometimes s t and by a odium ligh , somet imes by an elect ric ligh were watched by an observer through a telescope . ment of o ne hundred thousandth of an inch could detected . They first experimented with combination tons he the n . e t im maries generated by sire , i . , where pr o n n x nt s to I a c mmo wi d supply . The e p erime seem conducted with the greatest of care . By accurate 1 the revolutions made by the siren disk they could p ro bination tones of pi tch equal to that of the reso nal M any experiments were performed with di ff erent f l the but in the rimarh of primaries , every case where p t n the t the intervals no t grea er tha oc ave , fork was p affected when the difference tone had a frequency i o ts wn . The experimenters test ed also for the first diffe r of an i nterval greater tha n the octave A cc ’ Koenig s rules there are no beat-to nes of pitch int t the ex eri1 between the primaries . The resul of p n the m t n s h f1 unmistakable . Whe pri ary o e reac ed t t the ff n n to 6 a such ha di ere ce , 5 , correspo ded 4 vibr resonating fork was disturbed as usual . The 6 ra ther feebler than in the other experiments with n but t smaller tha the octave, here was absolutely ' e t t the ff n t as to th objec i ve reali y of di ere ce one. I A EN .

t e t in when h required pitch was o b a ed, ” 1 8 i a s t w lost . ’ for Koenig s lo wer bea to to ne wi th inter

5 7 6 ( 5 7 6 2 x 2 5 6 64 the pitch of the

The results were always nega tive . We " n t t n o n s ti ut t ega ive ha po i ve res l s , hey say ; a long time o n two occasions to get evidence b t nt note , u e irely

summation tone , Al wi ndchest . t their sum 64 When the pitch the experiments left in the saw them no shadow of doubt as to the a note corresponding in frequency to the

x n t n t n will be remembered , e plai ed summa io o es ' the t-t n o f upper part ials . Usually bea o es r harmonics are equal in pi tch to the primary tones

8 . he o f t t . t . t heir upper par ials This is rue , g , of

1 2 , 4 , 6 , 8, 0 6 1 2 1 3 , , 9 , , 5 .

= 2 the a t n the 4 , lower prim ry o e ; 9 r 1 2 8 the f t the y ; 4 , irst upper par ial of 1 1 0 the f t t-t n 5 5 , irs such bea o e

als that we could expect to hear by itself . t t f nt t the hough , is somewha dif ere wi h fourth

1 1 3 , 6 , 9 , 1 2 , 5 , 1 8, 2 1 2 1 4 , 8, , 6 , 20, 24, 2 8.

’ e the t-t n t be H re bea o es , if such here , of the fourth 6 2 ETERS JOSEPH P ON.

t s o u r m n nt sum t n upper par ial sh ld be mo e pro i e , pre ably , ha

e t-to n the xt e t c e n th bea e of si h upp r par ial , which orr spo ds m e r nd e o e t n t n . ck r and dse th t n t th sum a io o e Rii E fou o e 7 , e he mmat n t n to x t o ct and d now i . . t , su io o e , e is bje ively, deci ed the t n m 2 0 o u t ac to test also for o e 5 ( fro This gh , ’

to n x n t n to t n t n . cording Koe ig s e pla a io , be s ro ger ha 7

- he i e. n t o n n to t ff n to ne . Whe speed corresp di g his di ere ce [ , tt n t n the n s 5 ] was a ai ed , here were occasio al flickers of ba d , it n t x t n But so tha t is possible that it has a objec ive e is e ce . e t n the ff t n t t u b o n th o her ha d , e ect was less ha ha prod ced y m t n to n e the m i . . n n su a io e [ , The ba ds ever disappeared an n n t t m t n the for y co siderable le g h of i e , as hey did whe forks n to the mm t o n to n and the x m nt no respo ded su a i e , e peri e left doubt in our minds that the greater effect was produced by the ”m n summatio tone . n t t ff nt t t m n e A o her sligh ly di ere es was ade , beari g o n th m to t mm t n t n o same proble , as whe her su a io o es are due t upper o n t ma partials . N o t all primary t es have ra ios that ke it pos sible to explai n the summation tone as produced by p a rtials h o f the m i . e. t e t n sa e order, , equa io — a + b = n ( a b)

o t the m does n always apply . I f pri ary tones have the ra tio “ b = 2 d — b = e . t n a an a . g , he + 5 7 . The l o th partial of the upper note beating with the 1 sth of the lower note ( 1 60 would i ndeed have the same f re ne the mm t n t n but it to que y as su a io o e , appears be absurd to suppose that so improbable a combination should produce m t appreciable results . If we assu e tha any pair of partials can t t to n the n m hus produce objec ive es , u ber of combinat ion tones will be so great tha t the fork ought to have been disturbed frequently when the note of the siren was being raised to the r i a tter t h equ red pitch. A s ma o f fa c wen the C o f 64 vibra

tio ns was assed so tha t a ll the a rtia ls were hi her tha n the p , p g

itch o the reso na tin o rks no such dis turba nces were p f g f , ever o bserv ed except when the dif erence o r summa ti o n-to ne o f the ri ma ri wa r ue es s o d c d. uttin t e r h p p P g, her fo e , all suc fantastic

. I m “ 3 'o P 35 HE M ENA P NO .

the x t t t e perimen may be regarded , as a es summation tone ca n be pro duced when it cannot " o partials of the same order . n o o f he m the , for the summation t ne t pri aries of 1 6 t n the n the n , hey fou d , by disappeara ce of ba ds , o s tu previ us case , did ac ally that it was no t the third t n w t m l t st fo r o e , ho ever, hey ade a so a e part ial was found to shake the bands a of ro tation made it equal to the pi tch of but the ba nds did no t disa ea r whe ea s pp , r wiped o ut by the summa tio n to ne when Both by this means and by the to nes ( one 2 5 and the other 2 7 ) d themselves tha t the effect of the l n and t t the ance cou d be dist i guished , ha tone was no t due to the partial of the

n mirro r-reso na to n t t mea s of a r, co s ruc ed by Pro n 6 the to respond to a vibration freque cy of 5 7 , rs demonstrated the objective existence of summa ro m primaries generated by a double siren wi th a ere made wi th primaries and In every case the

d the mirror . " t n R and Edser impor a ce , say ucker , of our results by an inst rument of totally "1 8 o rom that fi rst empl yed . also perfo rmed in which the primary t n b t the t by u i ng forks , u resul s were whatever was produced [ o n the reso nat that if objective combin a tones are produced in such cases they are much less intense ” e t than those generated by th siren . Tes s were also made t t n and t t n o n In the fi t wi h reed o es wi h o es of rga pipes . rs c the lt n t n in the n t t ase resu s were u cer ai ; seco d hey were nega ive . " - i n I bid, pp . 35 2 4 ( talics mi e) .

W ” D. 354

T ES AN D RELA TE PHEN M ENA 6 COMBINATION ON D O . 5

r nt no t t b t t . fo ceme was grea u unmis akable Schaefer foun d , to o t t no n nt s n he , ha rei forceme was pos ible whe t primaries en t o t t were g era ed fr m separa e wind chests . He fur her agreed ’ with Helmholtz s results in finding tha t the degree of strengt h en ing of the combination tones by resonators is always small in comparison wi th the intensity wi th which the tones are no r mally heard without resonators . The greater part of their 22 nt ns t n t no t t . i e i y, he co cludes , is , herefore , of objec ive origin From the abo ve experiments bearin g o n the question of the objectivi ty of combi nation tones it may safely be i nfe rred that if combi na tion tones from p rirna ries generated independ t a ll t xt ently exist object ively a , hey are e remely weak . Lord t t n to the Rayleigh ass ures us, however, ha owi g want of sym “ metry due to condensa tion and rarefaction in the air the formati o n to some degree of octa ves and combination tones is a ma thematical It is a t once evident that the question of the origi n of the — seconda ry phenomena o f heari ng is closely connected wi th is — indeed only a nother phase o f that which we have just been

he n t n t n e. . d us in . t n isc s g If combi a io o es , g , could all be show t o ectively the question as to their cause might well n be f o r physicists to a swer . It 18 an appreciation o f this fact that has caused several acousticians in recent years t o invest igate rather carefully the question o f object ivity wi th

- reference to various so called resulta nt tones . We shall return no wto the subject of interrupti o n tones an d of varia tion to nes and see what effect the question of object ivity has had nt t t t l t tm n o n the developme of heir heore ica rea e t.

‘ ‘ e f er Wei ter emerkun xu meiner neuen Erkliirun K. L. Scha , B g g ,

- — a . A rc hie L XI I I . 1 00 8 o f . . 6 . r w , XX . 9 . 73 7 , pp 75

o S o I L 18 6 . . Theory f und . , 9 , p 45 9 66 ETE JOSEPH P RSON.

ATE xp an s s ON N TERRU PTION SECTION 3 . L R E m m I

TON ES .

We sawthat the earlier studies of the periodi ca lly va riable t n t and t t co me to o es were made by physicis s , tha hey had a rather genera l co nclusion that if a to ne n is interrupt ed p eriod icall m t in a ta n t m o t to n es o r y imes cer i period of i e , her , c r e spo nding to n m and n m sho uld arise as actua l pen dular vibrations in the a ir; and that in some cases these to nes had n t to t t he a i f a t r actually bee loca ed as heir pi ch by t d o reso n o s, though the question of objectivity had no t been explicitly con i e ed S n t t n t t n r s d r . everal i ves iga ors had also o iced a o e co r e s o ndin to m t the f the m t t p g , hough results o a hema ical deter

minatio ns no t o a t n . t n us o e did sh w such vibra io The o e , m r

nt n n n t t . over, had appare ly ever bee reso a ed objec ively 1 1 nn t t the n In 887 H . De er ook up problem where Koe ig ’ ennert s o n n m t and r had left it . D w descriptio of e hod esults follows : On a disk with three circles which were ea ch divided ’ int t n t t o 96 equal par s, he so co s ruc ed holes of equal di ameter that o n the first circle 4 parts that were perfora ted regula rly alternated with 4 unperforated ones ; o n the seco nd circle 3 per f o rated t t n o n o the r par s , wi h 3 u perf rated o es ; n thi d 2 per f o ra ted 2 t n t t . o h , wi h u perfora ed par s Therefo re n t e fi rst circle were twelve groups of 4 ho les with 4 blank spaces be t n in the n 1 6 o o f s wee each group , seco d gr ups 3 hole with 3 n tw n h o and o n the t 2 bla k spaces be ee eac gr up , hird 4 groups 2 t 2 n of holes wi h bla k spaces between each group . Now if the circles were blown upo n [ through a tube] during the ro ta t n o f the o ne t t t n t io disk , heard , wi h slow ro a io , bea s which, h m t m t t o n nt t n . wi ore rapid ro a i , erged i o o es The beats of the t fir t nt nt t n t n t the hird circle s we i o a o e , he hose of second, and t t s he fi In e h las ly ho e of t rst . ev ry p ase of the investiga t n the t n t in the t n 1 : 2 t at e io o es s ood rela io 4/3 , so h when th t the to n m the t a to c o e lowes of es fro bea s was equ l , n hea rd, 3 the t n C h o o n to the 6 h s besides o e , whic c rresp ded 9 ole , also ’ he t l 2 t hree tones in the series c c .

1 ’ ' H er usti sch - . Denn t Ak e h si l uh r o . n ters , p y U c ungen, A rchie f ur Ohrenheil

kunde I . 1 88 . 1 1 ff . , XX V , 7, pp 7

I bid ., p . 1 81 .

6 8 r r JOSEPH s easo n.

t n t the fi st to uc lo wto n the ber of revolu io s wi h r , prod e a e of am t h h o nes s e pi ch as that of the first disk . The two ig er t o f o ff nt n the ra t B the use were, c urse , di ere , beari g io y of the remaining disks seve ral exp eriments were p erf o rmed t m t t r ta t n o ne n ce as wi h si ilar resul s . Af er a apid ro io o t i s, the s s t t the t n c mes o ut disk gradually lo e speed, ha upper o e o F r more clearly as the lower o ne falls in pitch . in ally the lo we e and o n t the nt t n o n ceases ly bea s of i errup io s are hea rd . Very a t t o n n o ut the t n a s in the r pid ro a i always drow s high o e , x I previous e periments . t is clear from this change o f the lo wer t n nt t nn t t the ei es ra o e i o bea s , argues Herma , ha ear perc v pid ‘ beats as a co ntinuo us to ne . “ . t a t t the m t m t t t ese ex erimmts W Voig , abou sa e i e ha h p in n to n t t t o n he s were progress , e deavored co s ruc a heory t ba is t m ti t m n t o n t nt t of ma he a cal de er i a i s, which should ake i o acco un two points inwhich he tho ught the theory of Helmholtz was ’ 1 n and o t r n to o i t s sa tis insuffi cient . ) Koe ig he s had show V g factio n that every periodic fluctuation in the air was sensed by “ o the t heo to n . a n t the ear as a e This , of c urse , Helmhol zi ry 2 Di erence to nes t o un do no t denied . ( ) ff , as Voig himself f d , e a lwa ys ha ve the intensity tha t H lmho lti s theory dema nds . m tio n t n t t an t sense On the assu p , he , ha y periodici y may be d and tin the tu n r t n u as a tone , rejec g dis rba ce of vib a io by s per ’ t o t n t t t d t t lm o t positio n, Voig b ai s resul s ha a mi bo h of He h l z s ’ ’ - o n R d n t t s . combinati o n to nes an of Koe ig s bea e eally , how 7 t t the Ohm- e m ever, as Krueger sugges s , Voig gives up H l x t n n holtz theory of analysis without e p lici ly recog izi g it . Voigt finds from theoretical co nsiderati o ns that the summation to ne nt o f the t the fi t the o urt sho uld no t exist for i ervals oc ave , f h , f h

and t t its x t n in the the a o and the third , ha e is e ce case of m j r

d the tw t st n . sixth , an elf h is very que io able

‘ - 86 . I bid q pp . 3 7 ’ wier ein f ac her Tiine Wied A n X Ueber die Znsammenlt lang z e , . naL, L , 1890,

- 6 2 6 0. pp . 5 ‘ This sta tement must be understo od as applying to the ra tes o f vibration

16 to er second . thin a udible limi ts, e. g . , p

' ' l u S tudien I I . 1 01 . 26 . P l . , XV , 9 , p 5 rONes A ND newED 6 COMBINATION T PHENOMENA . 9

T E r1 1 N S SECTION 4 . FUR H R Cm c s~1s A N D M ODIFICATIO ’ o r H s uwo erz s 1aw n V .

‘ ’ nn no w t t t t t Herma decides ha Helmhol z s heory, beau iful as it is is inadequate to exp lain the empirical facts that have

n t . n D nn t a nd m n t t bee ga hered Koe ig, e er , hi self have fou d ha the beats of periodic i nterruption produce sen sat ions of co n tinuous tone ; Koenig found that the diflerence tone correspond in g to ( p q ) is heard only fo r intervals no t much wider than ’ half ( i ) an octave . Voigt has pointed o ut that the intensity relations of combi nation tones do no t agree wit h the demands of the Helmho ltzia n theory ; and Hermann himself had never heard summation tones a nd had never found any o ne who could 2 t n n t e o o t n In hear hem eve u der h m st fav rable circums a ces . ’ addi tion to Voigt s observati o n o n i ntensi ty Hermann has him ’ ’ self observed that the fo rks e ze o n resonance boxes give the n Th s si di fference tone F o n very gentle soundi g . i mple fac t ’ ‘ in itsel is suicient to dis ro ve the H elmho ltz ia n f, he says , ff p ’ e n 8 th o ry Of co mbi a tio n to nes . B m n t r t n to ff r n ut Her a n has still o he objec io s urge . Di e e ce t n un can n n the o es , he fo d , easily be heard eve whe ears are ‘ t ff t t t n o r fi t wax m n t t the s u ed wi h co o lled wi h a co pou d , so ha ’ drum can no t funct ion as Helmholtz s theo ry demands that it u the n ti n o m n t n t n t n sho ld for ge era o of c bi a io o es , by ra sforma

t n the e . F t o m nn n th t o n io of primari s ur herm re , Her a fou d a ha ving the tones of two tuning fo rks conducted o ne to each ca r “ t s t l o t t a nd n t n to n . hroug h pipe , he s i l heard b h bea s combi a io es t i t t the t n t the t n In h s case he supposed ha o es , hrough media io he n the t act to th in h ca r but t t of t bo es of head , bo h ge er eac , ha - the i s difference tone origi n in drum is mpos ible . t t nn t Aside from hese diffi cul ies , Herma also sugges s that ' Helmholtz s resonance theory is improbable from the fact that n to t t l it requires fibers hardly mm . lo g respond sympa he ical y

“ ’ a Theo ri e der Co mbinati on stiine P i en A rchie LI . 1 8 1 . , fl g , X X , 9 . pp 499 -5 1& ’ 00 l bid ., p . 5 . ' 1 Ibid, . p . 5 2 . t a- U M . pa S 3 . ‘ n t x rimen r and o o dwin fl o w. 8ee the i erestin g e pe t by ( I on G . p . 75 bel ° ETE N 7 JOSEPH P RS O . to to n t n t u t n es of less ha for y vibrations per second . S ch a hi g he believes to be contradicto ry to what we know o f sympathetic “ n vibrati o . in the t n n t the v wo f m Now , face of his evide ce agai s ie Hel t m nn to n n t t n ns hol z , Her a decides aba do i . No hi g remai ’ t n but to t n to the n t cti n o f he , he says, re ur old a ural dedu o

ff n to n o . e t h e r the o er m t i . O a scri be to t e a w di ere ce es fr bea s, , p Of respo nding with the sensa tio n Of to ne to every kind of ’ 7 erio dicit within certa in re uenc l mit In t n e p y f q y i s . his wcreed he ho ped to find peace o f mind with the difi culties enumerated

above . But while we are compelled to put away the reso nance o t the n the fi n t e hyp hesis , pri ciple of speci c e ergies of nerves m n n ff t nt n . x nt h e ai s u a ec ed, he co i ues No e perime s ave y t made

the applicati o n of this pri nciple to to ne analysis impro bable . o t n in the n in o t We are f rced, he , case of heari g, as we are her n to n n the x n n ho w t n t n se ses , aba do hope of e plai i g a cer ai o e x m t n n e clusively or preferably sti ula es a certai erve . Now with this view that the ear perceives as tone no t only n t o n but a n o t n t n pe dular vibra i s , also y peri dic vibra io , wi hi t n n t m nn t o t cer ai freque cy limi s , Her a o k up a s udy of other a possible to nes. I f we construct the resultant of two curves of equal ampli tudes representing frequencies of m and n we obtai n a curve t t n s t t the x t o n t t its m ha approaches si uo i y, wi h e cep i ha a plitude fluctuates perio dically from zero to the sum of the amplitudes

of the curves m and n . The period of this resultant curve corresponds to the arithmetical mean ( m n ) / 2 of the p ri d t m nn the maries a n is, herefore , by Her a , called ‘ “o n Koenig referred to it as the so n mo ye . If no wm and n are the vibration numbers of the prima ry tones respectively in

2 1r n and t e t n n n seco ds , if hes o es are co ceived as cosi us curves t m nn n t the t heir for ula , begi i g wi h opposi e phase , is

a mt a cos nt cos , ‘ hi lf H elmholtz an ticipa ted this o bj ec ti o n mse ; see abo ve, p . 13 . ' 1 bid 1 - ., pp . 5 3 4. ’ ’ H ermann eitr e z ur der Lehre vo n der lan wahrnehm n , B a g K g u g , Pfi i g er s

- h LV I . 1 8 . A rc ie , , 94, pp 46 7 99 . ' I bid ., p . 486 . ’ t 1 Quelques Expériences d A cous iq ue, p . 43 . ELA TE HE M ENA ‘ COMBINA TION TONES AND R D P NO . 7 whe re m is grea ter than n ; or by simple trigo nometric co n n versio , — m a ( cos mt cos nt ) 2 a sin t si n 2

In this case ( it: n ) 2 is the vibra tion number of the middle tone and sin ( 11: n) 2 represents the intensity of the fluctua

t n o f the to n and nt h n n l . io e, evide ly will c a ge sig periodica ly " this c ha n e o si n nt As Lord Rayleigh assures us , g f g is equivale ’ ha n e o h e t n to c as . t n a g f p The middle o e , he will period It nt icall n its i . e. n 2 n . y cha ge phase , , i every seco ds is evide ‘ that between each two phase reversals of this middle tone only a relatively small number of vibra ti o ns can take place . The general expression o f this number is obtained by dividing m n m n and t V; ( ) by ( ) is , herefore ,

n as Herma n shows . ‘ It is evident that this middle tone wi th changi ng phase wi ll no t act o n a resonator of high degree of resonance where the cumulative c flect of a great number of impulses is im

I e. . the n n t nt . n t n n p or a such a case , g , as reso a ce of a u i g the n i l t o f fork, seco d series of mpu ses wi h a reversal phase

wu t n to ne t the eflect the fi t . f t o ld e d u ralize of rs The ef ec , if ’ t the t n t n any here be , of middle o e , for his reaso , would be the n t o grea test in cases where reso a ors would so n be damped , e It i . . n n . , where reso a ce is of a low degree will be easily understood nowwhy the middle tone is no t suscep tible of nn t rein forcement by physical reson ators . Still Herma though that he had occasionally heard this tone in ca ses of interference n nt o f primaries formi g simple i ervals . By diffe rent arra ngements of teeth in the toothed siren Her mann found ( 1 ) that periodic vibration may be perceived as a tone even though there is a regularly recurring change of no t t to o t phase which does occur wi h grea frequency . For ex a t c n n its t 1 ample , periodici y ha gi g phase wi h every 6 th vibra l t n tio n is sensed as a tone . The imi s of to al perception in

“ ’ u r 8 H S . ma nn s r hie . o f also er ure a F e/ s A c LVI . p , p 3 ; fig , l u , , p . 485 . 7 3 Josef a ” ransom

auch cases were fo und to be a t abo ut four vibr

n h a : each cha ge of p ase . He found ( 2 ) th t in c n in nterm cha ge of phase , as well as periodic i may arise whose vibration number equals the nu

s r in he ex l i d erence reversal . Acco d gly p a ned il ing fro m this peri o dic fluc tua tio n o f phase a nd a t ’ e middle ton . The period of the fluctuation i the difler e m n an e enc , d, moreover, the namb tions of the middle tone wi thi n each phase be for wide int ervals tha t it renders the existence 0:

tone in such cases impossible . He was thus abl t ns to x n a o t t of his precarious assump io , e plai n h which he held as contradictory to the H elmho l Now how shall we explai n the fact that a 1 be sensed as a tone even though the change 0 as often as once in every four vibrations [ see ( 1 st imulation of the nerve endi ngs is efi ected enti vibra tions we should suppose that a chan ge W e the tonal pe rception at all . O n th other ha nd is eflect ed by the presence in the cochlea of p a the effect of phase change ought to be muc h gn ’ r nn t u t He ma , ho gh he had rejected Helmhol z s n t n t th to t the fere ce o es al oge er, decides posi t n he ain imperfect reso ators in t ear . Si nce t hat oppo sed to resonance it is natural that the should only seldom succeed in gettin g any sympl f in the cochlear resonators . Thus he accounts l

this tone is seldom heard . o n nt t the Now , however, he is c fro ed wi h that caused Helmholtz to reject the view that ra|

i nto conti nuous tone . This view is opposed to n n th t n r of nerves and reso an ce , i a each e ve is I et n t i difi its o wn resonator . To g arou d h s l assumes that each resonator operates no t direct} but upo n the nerve cell a nd thus in directly upon tha t eac h o f these cells is functio nally co nne “ But howwill he explain the f ac t tha t the in tensi ty o i l

tone so metimea exceeds tha t o f the fi rst ? A COMBINA TION TONES AND RELATED PHENOM EN . 73

reso na to r. It tt th n wh re in the ma ers no t , e , e the series vibra t n t t to the n n t io sympa he ic given to e is , for every reso a or com munica tes with the only nerve that can mediate the co rrespond ing ton e . 1 3 In 1 896 M ax M eyer reports experiments o n the toot hed a nd also o n the perfora ted siren that confi rm the results of the t nt t similar experiments made by Herma nn . Bu M eyer poi s ou the fact that these experiments do no t necessarily suppo rt the o a - an n middle tone the ry of Hermann . The ph se ch gi g vibra tions produced on the siren are a ll of approximately equal ampli~ t s s t the l t n r n 0 and ude , wherea hose of midd e o e va y betwee t n xi M n t t on t t a cer ai ma mum value . — eyer fou d ha fas er ro a tion of the siren the phase chan ging tone gradually gave place to th t t n i n e e varia ion o es wh ch finally survived alo e , while th s former disappeared altogether . M eyer sugge ts that what Hermann heard and interpret ed as the middle tone may have been upper partials of the primaries which in so me cases co rre ’

d o . s o n t n the t n . E the p harmo ics of supposed middle o e . g , ‘ ‘ ‘ l t n c : e m t n u t midd e o e of g is , which igh have bee s gges ed by the upper part ial It is important to note that Hermann did no t seem abso lutel t n t t the t n in t An y cer ai himself ha he heard o e ques ion . y o ne who kn ows the force of imagination in trying to hear a n weak tone of a cla g, especially when the pitch of the expected t l n n i n n o par ia is k ow , w ll feel i cli ed to d ubt under the cir ‘ ’ cumsta nces that Hermann actually heard the middle tone . ‘“ Krueger says that the existence of such a tone becomes more improbable as experimental condi tions appro ach the actual co n ’ ditions demanded by Herma nn s theory . “ ’ n t in 1 8 t m nt to t Wu d 93 sugges s , as a supple e Helmhol z s th r a n t t diflerent the t eo y, hypo hesis qui e from heory of Her nn n nn and t ma . With Koe ig, Herma , o her, he agrees that ’ Helmho lt z s view ca nnot explain cert ain secondary phenomena in n o nt o ut nn of hea r g, such as have bee p i ed by Herma . Cross

” ’ M Me er Ue er Combina ti o nrtfine Zeit schri t . . s chol . IX. t 8 6 y , b . / I P y , , 9 ,

“ 165A, 196 6 . ” I hil . Studien I t m . 2 0 P , XV ” o , p 7 . “ und h - W t. il S ien V I I . 2 . . ud 6 W . t 1 P , L pp 4 5 ca n n 74 JOSEPH n o . " and Goodwin heard bea ts from so ft tones in t n pi ch, eve when o ne tone was sep a rately ea r t n n t , precau io s bei g aken aga inst the condi

~ n the . S bo es of head uch results , ho lds Won n fi e a i f each erve ber, however stimulat d, c n

particular sensation . Bea ts of wide interval r in x o n th t hea d, are e plicable is heory . Th v r n the as be o e come , says Wu dt , by simple vibra tio ns o f bo th to nes ca n so meho wdi rectl bers o the a udito nerve t t in to fi f ry , wi hou go g he n n e t reso a ce apparatus in th ear . In case t n t n nt n t l io of his ki d , such i ermissio s of s imu i no t in a condi tion to produce a tone could St n t n n co di ions ecessary for a to e sensa tion . This supposition is opposed to that of

( I ) it retain s the resonance hypothesis as for. holtz and also his expla nation of bea ts an d t Fo r any mechanically comprehensible inte n n t t t stimulati reso a ce hypo hesis , his direc heari ng is no t in opposition to the reso nanu must stand to complete it ; the hypothesis d It does no t hold to speci fic energies of nerves n t t t remembe O both of hese poin s , i will be rt the po sition just opposi te to that of Wunl attuned to repo nd to certai n objective vibral m t t t n at the nerve fiber by sy pa he ic vibra io , i cannot the objective vibra tion directly st imuli Ewald showed that animals from which I t n xt t ti t to so! labyri n h has bee e irpa ed , s ll reac cases where the tactile stimuli were supp| Even though he attributes the response to gl t no t x l the n di lation, hat does e c ude ki d of m

t n e. . t assumed . While o her erves , g , hose t l t exci table by so low i ntensity of s imu a ion, ” meric cad o A rts and S cience Free. 0] the A an A . f , “ R Ewald h sio l Untersuchun en iiber do : . , P y . g usuall - Wiesbaden rS a 2 . I t has been ctaves . I O , , Q . p 9 y ’ r wald s animals reac ted to tactile stimula tion. eve . that E

6 t o s es: s re 7 s r xso n.

' undt s o t n t t a contradict W p si io , al hough Wund t akes c n

t n a rt the t . hall there! from a cer ai p of resul s We s , amin e a li tt le more carefully the results and the co ndi

this experim ent . A n effectual mea ns of making audible very a n a l tions is to close the ca r with a bit of beeswax an d p t stem of the fork ligh ly a gainst the wa x. In this 1 vibrations are transmitted to the membrana tymp a ni t small amount of air enclosed wi hi n the mea tus, as is cle the fact that the sound of the fork is hea rd o n touching long after it cea ses to be audible o n touchi ng its st en h n in t c t pi nna of t e ear . He ce his ase here is no con du the middle ear or inner ear thro ugh the bo nes o f ti

[ this cannot yet be asserted beyond question] . N o ww that the vibra tions of a fork could be heard lo ngs touched to the wax in the ear than when held aga inst tl We therefore took two small tonometer forks ma kil n t t nt and l t he beats per seco d , s ruck hem very ge ly , he d n t the t t e i he agai s ee h ; loud beats w re heard n t ears . forks were then held in this position until the beats ha d to n t and the ceased be audible, whe hey were removed, t n each was ouched to the wax closi g the two ears. I the two n t nt but tin t in the o es were heard , fai ly dis c ly, i t a nd n n t r wh ch hey were held , accompa yi g hem we e f ai n to n i the to ca r as seemi g wa der n head from ear , is e t n t th case wi h bi aural beats . The bea s were c nt the e t cou ed by subj c . The experiment was varied slightly as follows : 4 only was closed with wax ; the other was imm ersed in t x nt t basin of wa er. The e perime was hen repeated na the ff n t t o ne n t with di ere ce ha fork , i s ead of being ta t the t to the in its ear was ouched marble bas , vibratio : o the n a r t e transmi tted t e closed c hrough th water . T; ” ex results were obtained as before . The pen menaj elude that aerial vibrations actin g upon the ear a re nj mitt ed through the skull or bo ny parts of the head fi ” to the t ear o her. U o n t t the lts the x nt t e p his par of resu of e perime , h n PHEN M EN /l COMBINA TION TONES AND RELA TED O . 77

t But e d . th x nt o n an largely bases his heory e perime ers go , t s t in n t n o ut hei r re ul s agree ge eral , as hey poi t , with those of x nt n in 1 1 ca rs n similar e perime s by Thompso 88 . The bei g c wax o t five t n t losed with , a brass rod ab u fee lo g was held ligh ly a gain st the wax in each . When the stems of fo rks struck by two t nt s n t the t end the assis a s were pres ed agai s far her of rods , e t n in the n m n an v ry loud o es were heard ears , u acco pa ied by y

diff nt t n . o ne the ere ial o e I f, however , of rods was removed t he and t t in t the tt from ear pressed igh ly aga s head, or, be er, n t the t t ff nt n aga i s ee h, a loud di ere ial to e was heard a t once in he ns the t t ear aga i t which rod was placed . I f bo h rods were in t the t the f held aga s tee h or head , dif erential tone was heard ”

in t . D f n t n t bo h ears i fere ce o es were , herefore , readily pro duced whenever bone conduction was made possible so tha t bo th h e to nes mig t o p ra te in the sa me inner ea r. These results were co nfirmed by several other di fferent tests which need no t be

' The experimenters conclude that while H elmho ltz s ex a n t n t t t the t pl a io of bea s may be par ly righ , produc ion of beats is in part due to the condit ion resulti ng in the sensorium itself

when two interfering tones are sensed . They o fi er no explana t n the n n t n t n t t fi io of origi of combi a io o es , hough hey nd, co n — ’ crary to their exp ecta ti o ns based upo n Helmho ltz s explana — tion of asymmetry in the tympa num that such tones arise even when it is impossible for both generati ng tones to opera te t o n t t n toge her ei her ympa um . t n o n the n difleren e These resul s , beari g heari g of c tones, ’ certai nly contradict the very fundamentals of Wundt s sup ’

l nt to t t . On the t n t p eme Helmhol z s heory o her ha d , hey are in agreem ent wi th the modification of this theory suggested

1 0 . below , page 4 As early as 1 890 Stumpf poi nted o ut a phenomenon which ha d it e to no t n nt n ust n a h h r bee me io ed by aco icia s, nd which ’ seems in a measure to conflict with Ohm s law as i nterpreted “ a t t o t n - by Helmholtz . ( ) If I ake w o es about a semi tone he e 1 l l n t . rt in he m . and a apa t idd e regio of scale ( g , g o n the v o lin s St the two t n i ) ays umpf, I hear primary o es , but also , and t t t n h over above hese , a hird o e w ich lies between them, 8 1 m s r ason 7 0 e e .

r so mewhat nearer the lower than the highe . 1 o n a nd t k n t te nd has a very so ft col ri g, wi h ee a r i t n wh ats whf withi n the ca ; t is this o e ich be , t t o a r to r to nes remain consta n . The w prim y t en — t n is 0 judgment, no iceably weak ed , more ha

two tones are sounded a t the same time . “ t a t ( b) If I take tones tha lie farther ap r , ‘ d not I] ! the a e. . an gion of sc le ( g , g I do bu n two and t two st tone , t o ly primaries ; hese t r t n the tt nti n ms to bea . If, howeve , I ur a e o t m t i s to the a t to o ne of he , h s alway seems be be “ c o n the t h n t two t n ( ) If, o her a d , I ake o e -t n tbs nearer together than a musical semi o e, so im s te the difference limen for simultaneous to fi ut to d t t n . It tone , an tha bea i g is dif c l say

between the primaries . n n St tt t t ex As is well k ow , umpf a emp s o nomen on of the so -called i ntertone ( Zwischer basis of an assumed physiolo gical accommodat pri nciple all the fibers affected by a certai n give n t n n n to t t to n and no the se sa io correspo di g ha e , i t of ts o wn period . I f this supposi ion be no t m expect to hear a number of tones of different i nte h n n t n . e si gle pe dular vibra io Now , he says , t It a r d in t : ( ) , desc ibed above , is pro uced his way 7 in t n the n n ? g sec io s of basilar membra e , correspo d In t t t n . n n t be o es , overlap his co di io here wi ll 0 f and nerve iber equally affected, more i ntensive t fi by both forms of stimula ion. This ber wil the n t n the nt t n fi se sa io of i er o e . This ber may constrai n the neighboring fibers or cells in the

fi n . n u t own speci c e ergy The ervo s struc ures e. — fall i nto three groups t he upper and the IOWCJ the m t n the nt t to e pri ary o es ; i ermedia e , th is outside tones will be weakened by the loss of t

” - um f n r h I I . 1 8 0 . St To c o lo ie 80 1 translat C p , p y g , , 9 , 4 , - A nt w a ho ! t o 6 . tchener . l o . s c 6 Ti . ] P y “ XV“ o s. 7 ” ‘ ’ This c anno t be translated sn iddle to ne. Tha t wo uli

' ‘ whi I s Hermann s mittel to n ch ha ve thu transla ted. D c o usmnTION TONES AN RELA TED PHENOMENA . 79

o he nt t n a nd the nt n the tions cont ributi ng t t i er o e , i erfere ce of

two modes of stimula tion will affect this tone and make it beat .

In ’ the case of ( b) the overlappi ng is to o slight for the p ro the n duction of a n intertone . Where primaries are very ear e t n t together in pitch th three o es simply fuse toge her. ’ t f s ti n the n Ebbin ghaus accepts S ump descrip o of phe omena , “ an nt t n an d wrongly supposes that i er o e, such as is described, “ is to be expected o n the Helmholt zian view . The Zwisch ento n is precisely what the Helmholtz theory does no t exp lain ; it is precisely what we should no t expect from that But it is to be noted that Ebbi nghaus himself does no t attempt

to explain how such a tone is produced . He has no exp l ana

t t n n . fl n tion . His heory admi s of o e We shall brie y co sider

’ Helmholtz s theory cannot explain the beats of wide inter ‘ : i t x a n the n vals, he says t canno e pl i origi of interruption ’ t n t n t n t n o es , which o es he s ill co siders , wi h Koe ig and Her 25 n o t a nd to man , t be subjec ive , arise from a rapid beati ng

e n t n . t to n x of th give o e He objec s , moreover, a e planation of combination tones which makes these tones so di fferent in n t t n the origi from bea s as Helmhol z co ceived matter . Her ’ “ ann fi t n nn x m s modi ca io s are u ecessarily comple , so Ebbi ng the n haus proposes followi g . The general theory of analysis as defined by Helmholtz is I n n . t t h good is i co ceivable , however, hat t e nervous elements ’ a re from the first as closely specialized as Helmholtz s theory ma o n sup poses . We y c ceive of the cells of the cochlear nerve a t fi t non- Ea as rs specialized . ch cell may be able to mediate e ev ry to ne se nsa tion . The close associa tion of each cell with n t ti a reso a or of a par cular period , however, causes it gradually

” E in ha us Gm ds fi e der s c h l i , o o e 1 02 . t . A ll bb g g P y g , 9 . p 3 7 the page ref er ’ ences to Ebbin ha ua Ps c ho lo ie refer to the rst e g y g fi di ti o n . The sec ond edi ti on a which I did no t ha ve when h ( ws) . t e abo ve was writt en c on a , t ins no thin g ’ necessi ta ti n an cha n es in Ebhi n haus theo r as sta g y g g y ted here. In the la ter edi ti on Bbhin ha us ha s ho wever l f ’ g , , e t out hi s obj ec tio n tha t Helmholtz s theo ry ‘ ’ ca nno t ex lain the o ri in o f in terru ti on t n ’ p g p o es, but he obj ects to Helmho ltz s

ex lana ti on o f Ma ts and some rel a ted heno men p p a, p . 334.

entle and Ti tchener o eit B y . p . ., p . 69. ” Ebbi n haus it - , o . c . 1 2 . But se n g p ” pp e ote 2 a o ve. 3 3 3 , b ” I N -t note, 9 3 1 7 . 80 se T Jo m PE ERM N .

to acquire a special physiologica l habit for 3 cc! t t n It o o t st of s imula io . comes t resp nd mos ts o wn s n it a re period of i re o ator, though still c n e peri o ds . These oth r periods of stimulation m in two ways : ( 1 ) by the weak response o f its to tones o f nearly the same frequency as its ‘ t in t n - a t i i par ial vibration . Every s gle o e w ve s r k membra ne sets into sympa the tic vibratio n no t c irectl a ttuned to it but a lso to a certa in extent d y , , —, which are a ttuned to its harmonic undertones til into part ial vibration by the fo rmation of no dal e 0 i . . t n 6 0 t n v n t i 3 g , a o e of vibra io s is gi e , w ll vibra tion the fibers co rresponding to the periods 3 1 0 t N o o r Ebbin ha us it h a l 2 e c . w c is d m , , f g , p b the re uenc o the sti mula ti o n n a ll y f q y f , he ce when stimulated together by part ial vibra tions wi h . t e e n same tone ( e. g o n correspondi g to 600 E t n to n a ti case ) . ach cell he comes respo d rel e its o wn to the t th octave of period , less easily wel owbea tin x n o n the n N , g is e plai ed pri ciple I t ns in the n fi it of vibra io basilar membra e bers , as t E two nt n t n th hol z . ach of i erferi g o es is l I strengthened and weakened . n case o f wide i ference is possible through partial vibrations l spo nding to harmonic undertones o f the given ’ diflicult'y of Helmholtz s theory of bea ts is o w t t t t i n jus how hese bea s are media ed ( . e

t t an ho w t he st n . media e hem d ) , is que io The ” s n no t the nt i beats , ays Ebbi ghaus , are i ermed al fi but t tl t t t e bers , hose direc y correla ed wi h h The tones mediated by these fibers are then

periodically in i ntensity . When fast enough I n difi eren e Koenig held , may be se sed as c tones we seem to have the anomaly of two tones

fi . no t the same ber or cell Perhaps , for highet of a period to o frequent for the na tural peril t n t t t e Very rapid bea i g, as bea s ha are rapid nouj

1 I bid ., p . 3 7. M BIN A TI N waves AND E TE H N M E A 8 ; CO O R LA D P E O N . to c n ut m et n o es , m s be ediated by fibers s into partial vibratio ; for in such cases fibers set into whole vibration will be to o far

apart to overlap . The beating in these cases may have a period more natural to the cell thus a fi ected through partial vibra tion t t t e t In the t n th t han is ha of th objective one . such a case o e a ‘ "a is media ted is a difi erence tone . Somewhat i nconsist ent wi th this view is the o ne expressed o n the u n o previo s page of his book, where Ebbi ghaus seems t take the view t ha t bea ts are experienced immediately from the he t t x w . form of objec ive comple ave This is , of course , a f nt ns t n t t an d n to no n dif ere co idera io al oge her, such respo se pendular forms of the objective wave requires in the fibers a condition that is direct ly opposed to such delicate elasticity as t par ial vibration requires . The above exp lanation was o n the b t t n n he t n asis of par ial vibra io by mea s of t forma io of nodes . Y et Ebbinghaus says this : The above tone waves a and b will

in . h t t t i . e t e t t erfere wi h each o her, , ampli udes of heir n t n and t r t t t n t i dividual vibra io s , he ewi h heir in e si ies , will period n a n n t t n er ica lly st ren gthe d weake . These fluc ua io s difi very much a ccording to the intensity a nd pitch relations of the n t n two tones . U der cert ai n condi io s they have the rhyt hm ” — h t n t 2 t h and o n . , u der o hers , so The small struc tures of the basilar membra ne vibra ting sympathetically should o t t t t t the n n , however, be hough of as s ruc ures of ature of tun ” t n n n the n ing forks , as has ac ually bee do e si ce Helmholtzia

t h . But t th set nt t n xt n eory , al hough ey are i o vibra io by e er al impulses only when these impulses in a measure correspond to th own ti n nu t o t t n eir vibra o mbers , hey have , d ub less , s ill o ly slightly elastic power and ca n co ntinue their movements no co n si er ter the essa t o n o t e o b e e d able time a f c i f h j ctive impuls . can t e o n nt t d They , h refore , accou of heir acquired isposi tion, no t remai n uns fi ected by external impulses of their o wn and n the nt n t period ; , accordi gly, above me io ed ampli ude ” I bid ” p . 323 . ” - a: = wer h higher to ne, t lo tone. “ s f Y et on p. 3s t he says a proo o f the po ssibili ty o f pa rtial vibration o f ‘ the ba sila r membrane fibers that the deeper fibers o f the contra oc ta ve o f the

r th welf th v th piano . p odnee eir t , yea e en eir f o urteen th pa rtial tone as a

- ’ n est splendid a f ter clang o f the tone. 82 T JOSEPH PE ERSON.

fluctuations of the objective to ne waves will be ta ken up mo re or less faithfully by the portio ns of the basilar membra n e o n the a t ve s o w whole correspo nding to them . If they recur rel i ly l ly n d e uenc as we hear them as beats ; i case of more rapi fr q y, ” rattling ; with still greater frequency as roughness) A nd this o n t n es in t a dif roughness , of course , grea er freque cy merg o

- ference tone . ’ t at the Ebbin haus s ex an There is , herefore , basis of g p l a E n aus ma ti o n a fundamental conflict . However bbi gh y x n th nt t n t t nte renc attempt to e plai e i er o e , he mus posi for i rfe e a high degree of i nelasticity of the basilar membrane fibers. an t n t t o nt a d ct r to Such assump io , hough , is direc ly c r i o y o ne n t n t t t t o n h of his mai presupposi io s , ha of par ial vibra i , whic n requires highly elastic fibers in the cochlea . His assumptio of partial vibratio n of the basilar membrane fibers for the ex n t nt s o ntra pla ation of bea s of wide i erval , is likewise c dicted by that of inelasticity to explain how beats can give rise to dif ference tones . We shall turn back no wto the consideration of a f ewmore recent experiments o n the so -called interruption and the va ria t n t n It t t t t s— n io o es . will be recalled ha hose wri er Koe ig, nn n t E n s — n o flerin t ri s Herma , Wu d , bbi ghau who had bee g heo e to replace or to supplement the Ohm-Helmholtzian view had all conceived of the interruption to ne as inexplicable o n that I t n n t t n view . t was to hem a evide ce ha beats whe frequent eno ugh generate tones . ” and tt t e In 1 90 1 K . L . Schaefer O o Abraham ook up th ‘ study of interruption tones with a viewto testing whether t they existed objectively as pendular vibra ions . To interrupt the t D nn t t t o f the tone they used me hod employed by e er , ha A stopping up certain holes of a siren . large wooden disk 4

in t . It a ed and I . t mm . thick 5 cm diame er was used was perfor

in t . by a circle of holes 5 mm . diame er Of these 44 were tightly closed in such an order that 4 Open holes alternated regularly with an equal number of closed ones . The disk was rotated wi th a constant speed by an electric motor and a cur

i ha s t . . 23 . Ebb n g u, o p . c i , p 3

’ ' ' un stii ne P luer A rchiv L I I I tudien iiber Unterbrech r . 1 01 S g , / g , XXX , 9 , - 2 1 1 . pp. 207

3 4 JOSEPH rereason.

e In en th interruption tone 5 f . four similar experim ts they varied the scheme respectively as follows

O O O O O O X X O O X X

O O O O O O X X X X X X

O O O O O O X O X O O X

O O O O O O X O O X O X,

and in o t t to the e . es e each case g resul s similar above , , b id s ‘ the to ne 6 0 c ) also the interruption tone 5 f) which in e r n o t n t ve y case was rei f rced wi h a reso a or . ‘ U n to o t n t t n nt t o n to n s si g a hed sire , hey ob ai ed i errup i e n o fi t n rei f rceable to a still higher degree . The rs sire of this n t t t s cm in t and o n ts ki d ha hey u ed was . diame er had i edge 1 0 t t h 8 ee h equally distant from o ne another . When t e disk

was rotated the teeth were lightly to uched with a visi t ing card . n r n nt to t t s s Now whe eve y i h o h was removed hey heard , be ide

the n t n o ne n n to I . T pri cipal o e 9 , also correspo di g his , as t t t n t n n n t n m t n s a ed , was s re g he ed by mea s of reso a ors eve ore ha o i i nt t n t n m was p ss ble w th i errup io o es fro perforated disks . U n n o f 1 00 t t fi t fi own si g a sire ee h , of which every f h was led d , ‘ t t n n the in t n c an nt r t n hey ob ai ed , whe pr cipal o e was , i e rup io ’ l to n a . t n n e This also was s ro gly rei forced . ’ They decided no wto study Hermann s phase cha ngi ng to nes in i a s milar way . They found in the laboratory two of Her ’ mann s too thed disks arranged to reverse the phase of the tone e n 24 times in every 1 80 vibrations . On th o e disk a too th was n t n nt and t th t two missi g be wee every seve h eigh h space , so a spaces fell to gether ; o n the other a space was missing between every seventh and eigt h to o th so that two teeth occurred to N o win t m n t . n a b ge her hese cases , as Her a had alre dy o “ the t n 1 80 the o ne n n served , besides o e , correspo di g to 2 4 was tt the to n also plainly audible . This la er is e Hermann thought e But ha e was caused by th periodic phase reversals . Sc ef r a nd A bra ha m f o und tha t it was stro ngly reinf o rced by mea ns of e t physica l r so na to rs . Various o her cases of the supposed phase n t n t n t i t changi g o es , cases hat had bee s ud ed bo h by Hermann

M t t t m t . F m and by eyer, were es ed wi h si ilar resul s ro these

“ ' ’ 04 n hi LV I . 1 8 ma f . Pflaig er s A rc v, , H M N P ENO E A .

’ t it n u, may be co cluded , say the cha ng e o f phase in and o f o n f o r the genera tio n o f a pa rticula r so rt those cases ( first observed by Hermann ) ging principal tone is a ccompa nied by a bration-number corresponds to the number t o be regarded as a simple tz t a ls ein f acher Unterbrech ( di eser le ere .

en ist ) . too k up furt her experiments along the same first a paste-bo ard di sk ( Pappscheibe )

2 2 t . row of 4 holes of cm . diame er was held close to the circle o f holes as connected with the ears of the observer tubes ( H o rschlauschen ) so that a very It could be detected . was found that o ut blo wing the ho les it produced corresponded to the number of

the resonator per second . This it was as low as the contra -A ;

- rceptible even to contra E. It was easily t n t reso nators . To dis i guish hese tones from tones pro duced when a rota ti ng disk i nterrupts t m t n Scheibenton called he disk o es ( e ) . They the m t the of course , sa e pi ch as interruption n the o n o n t d whe holes are bl w up hrough a tube . that t he va ria ti o n tones ( corre nt n n t , whe is a one interrupted ble when t he tone of a loudly soun d t t a ro a ing disk . They decided to o these t nes . The disk used in this m t was I . hick . It contained a

r 86 I . in t ci cle o f holes each cm diame er . The disk was t n t m e rota ted some imes by ha d , some i es by el ctrici ty . Fo r the n t n the t n to nt t the ge era io of o e be i errup ed by disk, t hey used

a l the z o -E nn p rincip l y Be ld delma series of forks . By means

' ’ Abrah m tud r r h a u. a S ien fiber nt e b c nm L. Sch ef er 0 U e K. , p tone, Pfi flg er s - Ar hi XXV II I 1 02 . 1 . c e, LX n 9 , pp 475 9 86 ETE JOSEPH P RSON. of movable weights these forks ca n be made to give a continuous series of frequencies from the lowest audible tone to t ha t of the ’ to ne a . The variatio n tones heard in this series of experiments are F t n in t 8 . t give a able, page 4 3 rom his table it appears tha t n o t t o n t n a t the t e t h hey ever heard b h varia i o es same im , houg

t n t t t d x c t in the t I . The his is o s a e e pli i ly repor , believe higher variation tone is in general more easily perceived t han the tt n t s and lower . The la er was ever heard wi h fork below fl , neither o ne was heard with forks below e. The to nes heard were reinforced by several ki nds of reso nators . As proof t hat these variation to nes reinforced were no t disk tones the ex perimenters fo und that they were strengt hened o nly when the o rk at the s t the was directl o o site f , held oppo i e side of disk , y pp e o the mo uth o f the r s na to r. ' In view of these facts Schaefer and Abraham suggest the hyp othesis that the so -called interruption tone is really a dif ference to ne of the intermitted tone and o ne or both of the a variation to nes . The formul — ( m+ n > n = — — n ( n m) show that it may possibly be a resultant of the coincidence of r n h two difle e ce t n . s t e x nt ackno wl o es Thi idea , as e perime ers F is no t nt n t t . o r edge, e irely origi al wi h hem before Koenig

nt t it an nt t o n to n . M . M had i erpre ed as i errup i e , A ayer , hear in t to n t t t it t a d g his low e , sugges ed ha migh be reg rde as a resultant sound formed by the union of the sound of the fo rk ” ° d he o the n 3 t the an t n i e. wi h upper l wer of seco dary sou ds , . , o iflerence to n it he o n t n . D of t variati es es , will be remembered, es l n were then frequently called r uta t tones . “ Koenig had remarked that interrupti o n tones are weak t o t o the wi h low f rks , while wi h high loud f rks , where variation t n a no t at t o es are sc rcely , or all audible , hey are very promin ent . The observations of Schaefer and Abraham agree with the t e ' 8 8 n nt tt n t n sults o f Koe ig . The i ermi e ce o es for Koenig were “ r n r A merica n J o una l o f S c ie c es a nd A ts, CI X., 1 875 .

’ - E er ce A c ti ue . 1 8 1 . Quelq ues xp ien s d o us q , pp 3 40 ’ f r in N l h lo ic M en h n I L. Schae e a e s P sio des sc e s I I . 1 K . g y g , , 905 , p . 5 34.

88 JOSEPH PETERSON.

t t n Zwaa rdemaker ntl ea The resul s ob ai ed by , appare y r ct and a d ing somewhat slowly o n Schaefer Abraham , c lle o ut fro m them an extended series of experiments published fo ur yea rs “ s to t t t t e i t later . They de ired es more a ccura ely h n ensity t the nt t o n t n and the nc n rela i o ns of i errup i o e of pri ip al to e, n m to t c d aa rde r under conditio s si ilar hose des ribe by Zw make . In their experiments they were able to vary within wide limits the pitch of the principal tones and the frequency of interrup

n s t n s t as ra tio n . tio . They u ed o e of as high pi ch vib s From more than 200 diflerent tests in which the tones and the n n t t t a interruptio s vary withi wide limi s , bo h absolu ely nd with t to t t t n d the l n re l : respec each o her, hey ob ai e fol owi g suts t n h t n t in The primary o e , w ich was heard very dis i c ly the tele n n no t nt t n d o r ent pho e whe i errup ed , weake e very much i rely disappeared as soo n as the interruption process began . On the the ss ca t d n other hand, more or le compli e cla g appeared in its tt n nt n d o ne two ract r st place . This la er cla g co ai e or cha e i ic partial tones whose vibration numbers were dependent upo n the pitch of the given pri ncipal tone and upon the frequency o n t o n c n e of the interruption . Under special c di i s a o fus d to ne n t t rr was heard whose freque cy equaled hat of the in e up tion. These results seem to show that Zwaa rdemaker ha d no t de o n scribed the phenomena carefully enough. The t es obtained were in every case reinforced by reso na—tors . Schaefer and Ab raham conclude as indeed Zwa a rde t t x nt o ne— maker, wi h far less suppor from e perime , had d that the so -called interruption tone exists object ively as a p endular vibration capable of afi ecting physical resonators ; tha t its ex

no t . planation, therefore , is a physiological problem This tone consequently o flers no diffi culty to the Ohm-Helmholtz ian law It n the of to nal analysis . is perceived upo same principle as “ t o ct t n that upon which o her bje ive o es are perceived .

’ ’ hre vo n den so en annten nterbrechun stiin n Zur Le g U g e , B rud e s A ns-cl.

- der Ph sik I II . 1 0 . 6 100 . y , X , 9 4, pp 99 9 ” ’ 1 N a l s Ph sio lo ic I I I . 1 I bid ., p . 000, and ge y g , , 905 , p . 5 36; Fro m their vario us statemen ts they seem to co n sider the interruption tone ’ ' as pa rtly a di sk to ne and partly a diflerenee tone o f two sets o f primaries so

l nnec ted as to make it xist o b ec tivel clo se y co e j y . Bo th o f these consti tuents o f the n t interrupti on to e then are Obj ec ive. M BIN A TI T NES AND RELA TE HEN M ENA 8 CO ON O D P O . 9

N E A . SECTION 5 . I NTE SITY R L TION S

We come next to o ne of the most di fficult of acoustical It the t intensit rela tio ns problems . is ques ion of y which n n i It n t but t t o w . s we mus co s der is a lo g s ory, we mu be brief . l t in nn t n t t s i n the We shal ake up , co ec io wi h his que t o , other troublesome o ne as to wha t co mbina tio n to nes are actua lly i ea r . o h d The tw problems are closely allied . Th s is evident — i f we admi t what may become clear soon - that there are b n t n t n no t n no c n t com i a io o es which are heard u der rmal o di ions . The earlier writers paid but li ttle direct attention to the ’ intensity relations of combination tones . U ntil Helmholtz s f n t n n 1 In time only three or four di fere ce o es had bee heard . ’ Helmho ltz s work it is no t always clear whether he actually In ens t ns the n te . S a io o To ne heard to es of which he wri s his f ,

. f r n t n t 1 e. n o the xt page 5 5 , g , he co siders dif e e ce o es up si h ’ n In o ne t l x t t t o rder, i clusive . place he e ls us e plici ly ha he a n t n no t nl the fi t m t n t n but he rd , from sire o es , o y rs su ma io o es ’ t nt 2 and a . l t also hose represe ed by p q a p These as , he s t n t n in n r says, were very weak. That umma io o es ge e al are t t t the t t ex n an d in weak, he s a es bo h as resul of ac ual perie ce a t r ed o n x t connection with his heo etical d ucti . He sta tes e plici ly tha t multiple combination tones ca nnot as a rule be distin ctly t t n t n th t n n heard , but ha i cer ai cases ey make hemselves k ow ‘ by hea tin g wi th other tones . he m t n In 1 M . M t t t 876 A . ayer made i por a discovery tha nt n t n t sounds of considerable i e si y, whe heard by hemselves, may be completely oblitera ted by lower sounds of suffi cient in the t n n t t no n n ten sity . On o her ha d he fou d ha sou d eve int n can i n t t the n t n when very e se , d mi ish or obli era e se sa io of ”5 a concurrent sound which is lower in pi tch . This phenom ’ en n ow n n ff t no t n but l o , n well k ow , a ec s o ly Ohm s law a so some

’ r . CL Hi llstriim, sup a , p 3 . ‘ S Os , 9“ 1 3. ‘ - 1 6 . . Ms Senutio ns o f To ne, PP' 5 5 : p 0 3 thems ties ca n . indeed , no t yet be a pplied to the determi nati on o f the rela t ive in tensities o f combina tion to nes ;

s v minati on o f the re uenc ies o f s ch is o f we er in the deter u tones . it ue, ho , f q ‘ I bid., ma d. ‘ ‘ A. M. M e Res srehes in Aco usti hil Ma th Seri es I I . 18 6 ey r. e cs} P . x“ s , . 7 , J E H T OS P PE ERS ON .

’ o bj ectio ns urged aga inst Helmho ltz s theo ry o f co mbination

’ We ha ve a lrea dy co nsidered Ko en ig s m alts as to the ques tio n o f wa t com ina o to - h b ti n nes ( bea t to nes ) are audible. While he did no t ra ise the intensity questio n so directly as it has since been ra ise re uentl o k the rela tiv intens es o f d, he f q y sp e of e iti the t-t n s and en a o e it will be reca le t o ex a in bea o e de v r d, l d, p l the mo re easily perceptible summ a tio n to nes as result ing from ‘ upper partials . La ter he a ltoget her a bando ned a belief in the ex ten o o is ce f combinatio n t nes.

In 1 8 . a t 94 A M . M yer repo rts an interesting experimen wh o - ich he perf rmed with bird c a ll whistles . These whistles gave to nes beyond the upper limit o f a udibility : With t hese ” w tl - i . di eren n t to n e. fi ce t es ha his es , he says , bea es [ , o ] ve been obtained when the vibrat ions o f either whistle alo ne were in audible .

- t t n s e n ta ined b r. Bea o es , he adds , have al o b e ob y D Ko enig a nd myself in Paris with t uning forks whose frequencies

he r. n an ici te surpass t limit o f audibility. D Koe ig t p a d me in the production of these beat-to nes by severa l Hermann in 1 89 I definitely raised aga inst the Helmholtz ian theo ry the object ion no t only that diflerence tones were in many ca ses heard louder than they should be accordin g to that t o r but t t r t e he y , also ha difle ence ones of th seco nd order were in o m s t o f fi s the s e ca es much louder than hose the r t . He c ites z z ca se of the maj o r third c ze which gives the difi erence tone ‘ 2 ’ 20 e r t n t . o ne as o e g ) ve y dis i c ly This c e al n , he ffi nt to i o the t o f the tz an says , is su cie d spr ve validi y Helmhol i 8 theo ry of co mbination tones. Stumpf had i n deed sho rt ly ‘ o before this o bserved the same phenomenon . He f un d that g 2 2 co mes o ut mo re clearly when c and e are so un ded in a piauis “ n t o n l simo than it does whe hey are s u ded loud y .

‘

r . 1 5 . S uf a , pp 5 ’ n th Pr uc n f at-t n Tw f re . er e o d tio o e o es f A l d M May , O B ro m o V ibrating ’ o dies whose Fre uenci es a re so Hi h as to be Se aratel l n a udible B q g p y , Rep . o f

i n . . the B rit . A sso c . [ o r the A dv . o f S c e ce, 1894, p 5 73 ‘ ra . . S up , p 69

h 18 I m st be no ted ho r n s o lo i I L 0 . 2 . t u weve To p yc g e, , 9 , p 49 , , tha t o n louder so unding o f the first di ff erence ton e bec o mes so lo ud as to interf ere with F on the principle disc o vered by Mayer that lower tones may obliterate higher a s.

2 H T 9 JOSEP PE ERSON.

. t . t n s t t e t 3 Objec ive II , o e ha aris from he fa ct that an unsymmetrical body is forced to vibrate synchronously with two

- or mo re wave series . ’ So M t far as I see, eyer prac ically agrees with Helmho ltz s explanation (with certain necessary modifica ti o ns) of ca ses ( 2 ) and He admi ts that summation tones are sometimes hea rd in these two cases but no t in case ( 1 Both kinds of o bject ive n t n t n tt nt t to the o t combi a io o es are of li le i eres psych logis , he ’ - - sa . M so t o n t t n a ys eyer s called wave reduc i heory, he , pp lies pro perly only to the group of combination to nes usually ca lled

subjective . This group of subjective combin ati o n to nes evi dently includes the prominent co mbinatio n to nes which ha ve no n n t n t n —th correspo di g objec ive pe dular vibra io s , ose whose in ’ t n t m t t n o f e si y Hel hol z s heory is supposed i capable exp lain ing . Thi s makes possible a good deal of ambiguity as to which group shall claim certain tones ordinarily admitted to be sub ’ t n t t nn t ject ive . All weak o es which his heory ca o explain can

n to m t i . e. to 2 and easily be give over Hel hol z , , groups ( )

( 3 ) of the divisions just given . ’ M t r t it tt t to x n eyer s heo y, hough a emp s e plai the o rigin of ’ o n the t n t o n t n a ly subjec ive , combi a i o es , is , however, theory n o f how the ear analyzes to nal cla gs . All to nes origi na ting in the xt n to the t t middle ear, or e er ally ear al oge her, must co me ’

to the nn t . t t i er ear as objec ive They mus , herefore, all be n t treated alike . The pri ciple of his heory is this : When a wave impinges o n the ear the movement of the sta pes corre s in n to the t the spond ge eral objec ive form of wave , whether it

is a complex wave or no t . This produces certain forced move nt the n C t and n me s of whole orga of or i basilar membra e . The membrane is crowded downward near the stapes to make room e the n n the e for th liquid displaced by i ward bulgi g of fen stra . xt n n the nt s n The e e sio of moveme is, of cour e , depe dent upon the force exerted against it and for the peri o d o f time that this n in the o f t n s t xt force acts . He ce case low o e i e ends farther he up the cochlea than in t case of high tones . In the former

’ h wever tha t he abso lutel re s H lmho I t must be no ted, o , y j ec t e ltz s theo ry o f Th roc ss o f anal sis o f resona to rs in the ear. e p e y the tones that come into the

r o s inner ear is entirely dil erent f o the tw theo ri e . M I A TI T ES A ND A TE H M ENA CO B N ON ON REL D P ENO . 93

the n t n and case i ward bulging lasts longer . Now if a high o e a lowo ne opera te at the same time the lower portion o f the organ of Corti will evidently be stimulated more frequently t n the o t o n t r the h ff t n ha p r i far he up coc lea , which is a ec ed o ly by nt - the less freque vibration . Whenever the two wave series act upon the fenestra in the same direction at the sa me time the resulting i nwa rd motion will be greatest and the stimula tion

of the organ of Cort i will ext end farthest up the cochlea . Different sections of the orga n will consequently be stimulated with difi erent periods of frequency accordi ng to the form of h is the x t a ectin he . tc comple wave hat is fi g t ear Now , pi determined entirely by the frequency o f the stimula tio n of the

o r an o Co rti . t o n t n t t t g f Each sec i of his orga , herefore , ha is stimula ted will mediate a tone correspondi ng to the frequency its t m ti n of s i ula o . ’ It is to be noted here that M eyer s theory affords a n easy x n t n the t no t x tin e e pla a io of fac , which probably o her e is g m rely m t n x n . t t t n a physiologica l heory ca e plai ; viz , ha a high o e y

t t b o ne the t no t t . be obli era ed y a lower , whereas opposi e is rue A low tone though very weak cannot be obliterated by a higher

t n t t . o e , even of grea in ensi ty t t n t n the By a very laborious process , par ly de ermi a io of o the t and t th t f rm of objec ive wave , par ly ma ema ical calcula t n M n find n the o f n io , eyer has e deavored to , i case several give nt t t n o uht o nd t l i ervals , wha o es g t be heard , a wha shou d be “ the ampli tude and frequency relations of them . In he nt The results are not very satisfactory . t i erval of

the fi t e . the n f h , . g , where primaries have equal give ampli t the the f n t n 1 udes , calculated results would make dif ere ce o e ) m st t n the w 2 n xt in nt n t and the o in e se, lo er primary ( ) e i e si y , It t t . upper one ( 3 ) compara ively very weak is , of course , but just to say t hat the relative intensity of tones ca nnot be

determin ed solely by the ratios of their amplitudes . In reply to the cha rge that the tones actually heard do no t correspond exactly wi th his calcula ted results M eyer suggests that a co n sidera tion at the sa me time of a ll the various aspects of the complex p rocess is impossible ; that results can be obtained from o o ne ti to e the considera tion f on ly of the variables at a me , th

” - Sec esz eeia l l ean kn . 06 1 L M XV L ( 8 5 23 fl. p ly ] 1 3 . 77 ; . 9 ; SE H ETE 94 JO P P RS ON.

’ he t i n t t . m a t eglec of o hers Very well , Hel holtz s m thema cal a t o n n n t a s lm c lcula i s are made u der very similar co di ions , He i n holtz himself admits . Without co nsidering then the quest o ’ to t M t is fl and to t as whe her eyer s heory super uous , as whe her — his tri-partite division of co mbination tones is unnatu ra l such questions will be more in place when we shall ha ve seen his — newboo k let us co nsider his laws for subject ive combination “ o n t es .

’ 6 M svs w N - SECTION a s La s o r COMB I A TION To m3s.

These laws do no t express a ll the diflerence to nes which o ne might possibly hear in every possible co mbination of o h e ti e t n but o i i ff n c v t d flerences . e. c j o es , merely h se [ , di ere e tones] which o ne is mo st likely to hear in those co mbinat i o ns ” which correspond to relatively simple ratios .

1 n the t the t o n t no t difler . Whe ra io of vibra i ra es does

e . 1 1 : 8 o n o ne d ff much from ( . g , or 99 99 9 ) ly i er he — t n . It x h ence to e is heard is e pressed by t formula , h where h is t e higher to ne and t is the lower . In this ca se a

n nt to n as St m . mea or i er e is also heard , described by u pf

2 n the t he n he n : n 1 . Whe ra io of t freque cies is of t form flerence he di t n n n o 1 i e. h t t t . w o e correspo di g ( , ) is al ays t t A o s ro nges . few of those als appear whose numbers corre

o nd to n 1 n . e 1 nd s 2 etc . . the t n a p , , ; g , o es gives 7 , n l the . t n a l 6 , 5 If is a ra her small umber, we really hear

u he : 2 1 . t n m n to 1 e. . t t n o es fro dow g , o es 4 5 give 3 , ,

. t t n nt n nt 3 O her ra ios of small umbers , represe i g i ervals n an t o m n t n t n nt less tha oc ave , give c bi a io o es represe ed by — — h a h — t he t n the h t 2 t . t nt , , 3 If i erval is less ha — t t n t i t n the fi t and the fi fth , h is s ro ges ; if t lies be wee f h

2 : h the t n t . octave , is s ro ges nt t n the t do no t the fi t 4 . I ervals larger ha oc ave give rs ere h h difi nce tone ( t ) which would be between t e primaries. As a rule only o ne difi erence tone is easily not iceable in these e lles r e cases . It is fo und by ta king th sma t difi e enc between the larger number of the rati o and any multiple of the smaller ” h 1 1 1 1 n e . . t e to n . umber, g , es give 3 x 4

” ’ m ntar Lab . o urse ns f . A udi to r ensati o ns in the Ele e A . f our. C y S y C , of — - /n V I 01 e hri t XV I . . 2 . Frye ah, X ., 1905 , 293 3 ; Z itsc f , , pp 3

6 J SE H ET N 9 O P P ERS O .

t crimination . M eyer has indeed heard these intermedia e dif d : 8 n t n m e. . the nt s a n fere ce o es hi self ( g , 5 from i erval 3 ) ‘ ’ In his review of Schaefer s theory of subjective combina t n t n n t n to t t sub ective io o es, however , I u ders a d him say ha j “ difi erence to n n t n the no t exist and es lyi g be wee primaries do , that subject ive summati o n tones are explicable as difference he o n nt d t ns o n th tones . Perhaps t vari us appare t co ra ic io ese ’ at po ints are due to his use o f the term subjective . Wh ever stand he may take as to subjective summation tones a nd inter ifi erence o n c n n no t a c o unt mediate d t es , they a appare tly be c ed he for o n t pri nciple of his theory . 6 A a o n n nt an a s to few years g , Krueger , i depe de of y bias ’ t n to o an xt n n t t n the no m n heory , u der k e e ded i ves iga io of phe e a resulting from two simultaneously soundi ng to nes of inter

n zn n d t n n th vals varying from to n z4 . He use u i g forks wi t n t d t adjustable weights . The o es were conduc e hrough p ipes from the reso nance boxes of the forks to an adjoining room h t i t where t e observer was seated . Upper par ials were elim na ed with co nsiderable success by means of a number of side p ipes n i o e n ne n perpe d cular t th mai o . The le gt h of each p ipe was o ne-fourth that of the wave of the overtone which was thus to n t nt t t t be elimi a ed by i erference . The forks were ac ua ed wi h n o m o n t n as much u if r ity as was possible . The gr u d o e was l 2 “ usually c ( 2 5 6 ) o r o ne of its octaves 6 or c . Occasio nal trials with other ground tones lead to no perceptibly divergent ’ results .

nine well trained o bservers o ne t em n Krueger had , of h bei g t r t n t o a violinist wi h ve y acu e a aly ic p wers for high tones .

Occasi o nally Krueger himself served as observer . r n the o st t The article is ve y long . O ly a few of m importan n the nt of the results can be give here . Krueger divides i ervals fi t xt n n : n to n 2 n studied into three periods . The rs e e ds from ,

he n n : 2 n to n z n the t n : n to n : n. t seco d from 3 , hird from 3 4 e he S t n t n t o n n nt l i . . t umma io o es were s udied ly i cide a ly, ,

‘ ’ t h - Zes schrif t f ai r Psy c o !” XL, pp . 1 86 7 . ’ ’ L I . 1 . 6 . P u er s A rchiv . 00 fi g , XXX , 9 , p 5

ud I I . 1 1 . m t . f . Phil S t 0 n te. reu r so understands hi oo C . 20 o K ge ” XV , 9 , p 5 ’ n h ud 1 r r h n Zweikliin e P il S t . X I 00 Feli x K uege , Beobac tu gen an g , . , V .. 9 ,

- — pp . 307 3 79 and 5 68 6 63 . c o st ume TI N T ES A ND ELA TE HEN M ENA O ON R D P O . 97 o b not a to tu t but ntl servers were sked s dy hem , freque y called ion to t i tt nt . n t a e hem They were heard , however, all hree s but o e t in the fi t the int period , were l ud s rs where erval is less

t n an t . n t e t ha oc ave The violi is who s rved as subjec , was able to t a t n hear hem t almost any time . The mos favorable i terval for the hearing of summati o n tones proved to be in the prox im the n r t t ity of major seve th . K ueger feels sure hat hese to nes su ma tio n t n f o r the m i are really m o es , pri ar es proved to be pract ically free from overtones . The pitch was deter o f t o t n l min ed by mea ns a on me er . Occasio al y ( as was found to be the case with difference to nes as well ) the summation tones ’ I t he would seem to be subject ively raised or lowered . n third period ( intervals between n : 3n and n : 4n) the summation to nes t the t o n t and n no were o o high for ome er, he ce could t be t r M obi s the n t n D . u t accura ely measured . , violi is , freque ly S t n ca lled attention to them even here . umma io tones as a rule a ppeared to wa rds the end of the clang as if in the period of l o er e t ha e oudest soundi ng f difi enc ones they d be n obli terated . e e t n e n n r e Low difi renc o es wer , i ge eral , hea d earli r in t an i ne In t the dun and t t . g, for a shor er ime , h h gher o s bo h o f the periods in which the intervals were greater than an oct ave e m a d : n n the r difi eren t . fi st ce n i . m to n n to ( , 3 4 ) , o e was dili erence t n it l n te intermedi a te hea rd. This o e , wi l be o d, was i h be en the r r es St nt ackno l in p tc twe p i ma i . umpf has rece ly w edged that he is co nvinced of the existence of in termediate ’ f t n s in in dif erence tones . These o e are always very weak ’ i t o f x nt t t tens ty . The resul s Krueger s e perime s show ha from e o n l n two m t n t t th comp u d c a g of si ple o es here resul , besides t n t n ut five f n t n difi erent the summa io o e , abo dif ere ce o es of orders whose pitches may be determin ed from the follo win g rule : Find first the difi erence o f the vibra ti o n numbers of the primaries ; then conti nue to find the difference between the two s t t n smallest numbers resulti ng after each uccessive sub rac io . The “ n t t e eren e t series of difi ercnces obtai ed represen s h difi c ones . ce t n D the n Representing the first dili eren o e by , , seco d by D, , the l n : the t in t n et c. , Krueger gives fol owi g formula for de erm a io : o f the pitch of all but D5

' e XI 5 . 3 8 eit ltr . t ica l 0 6 . Z sr . I Po " XX X, 9 5 , p ' - 23 . F. Kruger. o p. cit . pp. 3 8 SE H ETE N 9 JO P P RS O .

’ ’ n n n n he n D, ( bei g t higher primary to e) . n D2 i ( D, ) Fo r intervals smaller than the oct ave ’ = n — = n — n D2 D, 2 . — n Ds i (D, D, ) i ( 3 or D n the nt n the fi t D, , , whe i erval is less tha f h , D D n the nt r tw n the fi t , 2, whe i e val is be ee f h h t e octave . ’ :t — = i n — n D, (Da D, ) ( 4 3 ) for intervals up to the fifth ; or D D nt n the o urt and , , for i ervals smaller tha f h ,

D D nt r tw n he t and fi t . , 8 for i e vals be ee t four h f h — ’ h h i (D2 D3 ) i ( 4n 3 n ) between t e fift and the t oc ave, or e D2 D3 for intervals between the fifth a n d th xt and D D nt tw n the xth major si h , , 2 for i ervals be ee major si 9 In no the difi erence t n 2D the and the octave . case was o e , of

- 1 0 first over tone heard . I e o Z i hen o ne n nt rt nes ( wsc t ) were freque tly observed. This phenomenon occurred no t only between primary to nes of small nt but tw n r and difi erence t n o r i ervals also be ee a prima y a o e , t two difi ere n n n nce t . nt t be wee o es The i er o e , as Krueger de it no t to fin t and t n o ne scribes , does seem be so de i ely dis i ct ly a t ’ o ne as would suppose from Stumpf s description . l l ’ Helmholtz s theory had been objected to o n account of the t t t it no t x n the t nt h fac ha could e plai bea s of wide i ervals , suc

as Koenig had observed . While Stumpf early observed these n m n n phe o e a which Koe ig had described , he remarked that there are two kinds of beats easily distinguishable to the pra ct iced — t o f the t n and ear, bea s higher primary o e , deeper beats co n 1 2 nected with the lower tone . When overtones of the pri

‘ - i . 26 . 1 b d ., pp 3 7 ” VI I . 1 1 . 222 . Phil S tud . 0 . , X , 9 , p n I w in tro s ec ti o n s it i s alwa s mo re o r l ss b n my o n p y , e , a ea tin g mass an d the

s m wh t rer to th r I f primary to nes are drawn o e a nea ge e . o ne o f the primaries is ‘ ’ s dd nl sto ed the o ther at o nce mak es a little um to ts u e y pp , j p i no rmal pitch.

in ham who h a s a well-trained ear tells me tha t hi s M r. B g , , experience i s similar

to mine in this respec t . ” ' ’ um f b r di Ermittel . von bertrin n . St Ue e e O e Wied . A nna l L I I . C p , , ., V ,

if . 1 896 , p . 660

1 0° SE H ETE JO P P RS ON. combination tones have been heard by subjects who had lo st from bot h ea rs the tympanic memb rane and the first two ” s . t M r n st d su a o sicles I have myself, wi h . Bi gham , udie ch case in which the subject readily heard both the first a nd the se n t n is r t co d difference o es of tuning forks . While it t ue tha ’ cases of this kind do no t prove that Helmholtz s theo ry is wro n t at an t t t t na t o n tones g , hey, y ra e , show ha subjec ive combi i m in he m may be due to causes other than asy metry t drum . Hel ’ t s x t o n he t n to sa he hol z e plana i of t origin of such o es is , y t

t n t . leas , i comple e “ In nt . S s su rece years K L . chaefer has propo ed a pplement he tz n t r t co n to t Helmhol ia heo y of subjec ive mbi at ion tones. Helmholtz had made a very simple mathematica l sta tement concerning the origin of o bjective combination tones when the prima ry tones were produced by generators with a commo n

e . he n n . t . t tt windches ( g , polypho ic sire ) He admi ed that his m r treatment o f the case was ve y imperfect . A co mplete statement of the co nditions would give more resultant tones than those obtained . The essential con dition the n t n o f t nt t t n s i for ge era io such resul a objec ive o e s this . There must be exerted o n the air escaping through o ne of the n n a nd s n the n e periodically ope i g clo i g holes of wi d ch st, a periodic change in pressure due to the escape of air through in th nn the other hole . Now Schaefer sees e i er ca r a co ndition fi n t nt and n to the h ful lli g his requireme a alogous case of t e siren, when the tones m and n fall o n the ca r the movem ent of the n t the n ma t stapes agai s oval wi dow y, for prac ical purposes, n to two n ra he argues, be co ceived as equal such orga s ope ting n n separately against that window . Co ditio s similar to those in the wind chest o f the polyp honic siren will then be pro duced r in the liquid o f the inner ca . There will be a periodic ampli tude fluctuation of the vib rating bodies set into mo tion by t to the n t o the primary ones . This will give rise combi a i n to nes

h m ROIe o f the T m anic Mech a ’ f W. D. in a ni sm C . B g , y p in A uditi on.

h XIV 1 0 . 22 . s c o l. Rev . . P y , , 9 7, p 9 u un der su . L. h a f er Eine ne e Erkli r C K . Sc e , g bj ombina tionstiine auf ’ ’ ' H mh sch n R sonan zh these P ii er A r h rund d . el o ltz e e o s c ie L I I I 1 G yp . fl g , XV , 899. M BI A TI N TUNES J ND ELA TE PHENOM KNA ‘01 CO N O R D .

’ n m n m e o ltz et c . rm , , , as is shown by H hn h s dete i nation 20 e n in of obj ctive combination tones . As evide ce favor of his view he cites the rather close correspondence between the o b e jectiv and subject ive co mbination tones . He himself under to o k an invest iga tion of the first difi erence tone with intervals t n n a i t a t . t n n grea er ha oc ve Besides o her strume ts , he used for gene ratin g the primary tones both tuning forks and the bar

mo nium i . e. he t t the x t n t s t and ; , es ed for e is e ce of bo h ubjec ive t nte i t n n difi erence t n objec ive i rmed a e ( zwische liegende ) o es . In E n o t n t no ca ses were they audible . ve pr perly u ed resona ors did the o b ecti e di eren n no t make audible j v fi ce to es in quest ion . He co ncludes from his experiments tha t bo th subjective a nd o b ective intermedia te di e ence to nes either do no t exist a t a ll j ff r , o r that a s o o sed to o ther di e ence to nes the a re a t least , pp ff r , y to o weak to be perceived under the o rdina ry co nditi o ns o f hea r ”1 n and difi erence t n i g . Koenig M eyer had heard o es inter ” d t n the S me ia e in pi tch betwee primaries . This chaefer does in t as n no n . x to t de y He e p la s hem , however, bei g due upper p artials thus : — — 8 3 x 3 2 x

t o n 2 0 t t t He admi s page 5 , ha he himself wi h Professor Stumpf ha d the t n difi erence t n the nt heard o e 3 as a o e from i erval 5 : 2 . i t n unt o n the n e Th s o e , he says , is acco ed for same pri ciple , i . . , = o n 4 x 2 5 3 . Summati tones when they have been o un fo r difi eren e heard , are t be acco ted also as c tones resulting e n t f rom th prese ce of upper par ials. ' ” M ax M in S e eyer a review of chaefer s article , d nies the a lleged co rrespondence between objective and subject ive com Riicker and Edser t o u n t n . n t bina tio o es , he poi s , had proved the objective exist ence both of summation tones and of inter ” ” Recentl Schaef er re orts an ex erimen t in uds : A n n l a . d . Ph y p p ( D ysik, w mo a I I 1 0 . 2 in hich he de ns tr ted tha t membran s XV “ 9 5 . p 5 7 e vibra ting to mb a ti two tones gi ve rise to o bj ec ti ve co in o n to nes . It is no t c lear whether he

e n s a l thi s to the inn er membran es o f the a m a to pp y e r, as p art o f hi s exp lanation

i s o f subjecti ve comb na tio n to ne . ” P a e/ s A rc hie L V I I I . 12 . l t , X ” p 5 ” I P M L I I . 18 . 3 ls a r6 C . a . A a o Zeitschr XL . . s cho ) g e ” C V , 75 , p 94, ; I P y ” , - M rm r86 7 . “ ’ ’ 1 A reth a L I 1 00 fl. l , XXX " 9 . 49 1 02 J EPH OS PETERSON .

mediate diflerence tones when the primaries are produced from n t t mm n he n t . n t t n ge era ors wi h a co o wi d ches O o her ha d, he s the n n t ays , correspo di g subjec ive combination tones do not “ exist M eyer has elsewhere expressed the convict ion t t s t nt t t n a re t n ha ubjec ive i ermedia e o es audible . He hinks o w

that he was wrong . M eyer ridicules the idea o f analogy between the ear and o the wind chest o f the harmonium or siren . The Helmh ltz ian ’ to to it a bso formula , he says , is applied a case here which is lutely inapplicable ( den sie absolut nicht As to vibrating bodies in the air ( from which Schaefer had drawn an

n o M t a n t no r an ex eri a al gy ) , eyer says , Nei her y heory y p ence establishes the assumption that a body vibrat ing in a Flussi keit t the m t n m in an liquid ( g ) wi h si ple o e , is y per ceptible way influenced by the pressure fluctuations due to the M m the n t n n in the n n m m. o e surrou di g ediu uch ore , ge eral and n n t n x t t t two o ra t n well grou ded co vic io e is s , ha b dies vib i g in the same fluid under normal conditions remain mutually un

In t t n the A na l. d . Ph sik t e his la er ar icle i y , to M m nt n the x m n k r ferred above , eyer e io s e peri e t of Riic e

and Edser s as a po itive proof against such an assumption . x m nt t t x t t These e peri e ers , however, s a e e plici ly tha they lay ‘

t o n t r n t t n o n t t s t . less s ress hei . ega ive ha heir posi ive re ul s The resonati ng fork and mirror may no t have been delica te enough for the detection o f weak vibrations that may actually ’ n n M have bee prese t . Of course eyer s phrase in a ny per ce tible so a r a s ex eri ments h e p way saves him here , f p av e g o n . ’ M t t m nt n n n t r eyer s s a e e co cer i g heo y, however, should be com t the t t m o R n o n . pared wi h quo a io fro L rd ayleigh p . 6 5 above ’ ” In to M S t t ct reply eyer s review , chaefer admi s hat obje ive summati o n tones and subject ive intermediate diflerence to nes x t but t t t to o to e is , says ha hey are weak be heard under normal x m nt n t n iiblichen n n n e peri e al co di io s ( Versuchsbedi gu ge ) . He

eber mbin a ti o ns n n - U Co to e I u . is ert . erlin 1 a s 8 6 . 13 , g D , B , 9 , p . l ' l ’ P ii er s A rc hiv L I . . } g , XXX , p 49. ” id I b . . , p 5 4 .

” ' ’ P ii er s A rc hiv L I I I . 1 01 fl fl g , XXX , 9 , 73 .

JOSEPH PETERSON.

there is an essential want of symmetry between con densation nd t n and the t n n a es a rarefac io , forma io i some degree o f oct v “ and co mbinatio n tones is a mathematica l necessity) This statement makes it all the more probable that the two \/ cases develo ped by Helmholtz can both be wo rked o ut o n the ’ n o a t n It s em to same pri ciple of superpositi n of vibr io s . e s me that Helmholtz is no t right in supposing that objective combin ation to nes can be deduced from the siren even where the “5 t n in nitel m ex tl vibra io s are fi y s all , for, as he himself plici y t t no m n t n t n nt t a s nd s a es , co bi a io o es will arise u il here is eco n n t t a sufli greater ope i g of variable size , hrough which here is h l a cient escape of air to render t e pressure p periodical y vari ble, ” r th n s instead of being constant . Fo e pri ciple of superpo ition to take eflect it is necessa ry only that the generating to nes t and t t t t ac should somewhere be closely associa ed, ha a his p l e, n to the t t v a t ns wherever it happe s be, ampli udes of heir ibr io sho uld have a finite ratio to the mass vibrating in co mm o n with co nn t n s no t t n the two tones . N owwhere such ec io doe ob ai a r the n t n t n fi within externally to the c , co di io s cer ai ly are ful lled n t x n at ns the the ear . Eve hough we e clude all co sider io of m n t m find co n t n in the me bra es he selves , we a favorable di io he o hle t n t t in t s s the fluids o f t c c a . The objec io ha he e fluid vibrations are very small is easily met by the fact that the m ass e l . he s s r the a t n t t i . t o m of vibr i g s ruc ure ( , fluid ) is al ve y s a l, fi so that the proportion still may easily be nite . It is gratifying to note that thi s view is by no means co n h tradicto ry to what Lord Rayleigh has to say . To quote : T e production of external or objective combination-tones demands the coexistence of the generators at a place where they are in o tn t : t te co n strong . [He adds a fo o e The es ima s for densation of sounds just audible make it highly impro bable that the principle of superpo sition could fail to apply to so unds o f that order of This will usually occur only when the generating sounds are clo sely associated as in the po lypho nic In -t the c n t n siren and in the harmonium. hese cases o di io s are the t m n ued especially favorable, because limi ed ass of air i cl d

u V ol . I I . 1 8 6 . . Rayleigh, Theo ry o f S o nd, , 9 , p 45 9

hi ra . 1 . See q uo ta ti on f ro m m, sup , p 9 C M INA TI N T NES A ND RELA TE HE M ENA ‘0 O B O O D P NO . 5 in the instrument is necessarily a flect ed by both which o f l n n t t the t n the is, course, equiva e t to sayi g ha propor io of amp litude to the mass aflected is a finite one. et e not the two l o ne in n W—h h r or cases are physica ly pri cip al and we shall leave this question he re with physicists a l e he t te nts l o t we m y stil , o n th authority of t s a me of He mh l z i to the n e o f and of Lord Rayle gh just referred , apply pri cipl i n to the u the nn the i superpos tio liq ids of i er ear, where v bra i e n s n su t ons o f th primary tones are i clo e relatio . We may p t n s t pose hat all subjective combinatio tones ari e by his means . Let us see then what combi nation to nes might be heard accord ’

t n . . ing to Helmhol z s determi ation given in Appendix XII , pp

- en io n e 1 2 S sa s o To n . n te x 4 3 , of t f The seco d rm ( 2 ) series for i t gives 2 — and 4. p q. p q

The third term ( x3 ) gives

2 i 3 : : 3P’ ? qr P 2 4

s is as h t i the Thi as far Helm ol z has carr ed deduction . When it is carried farther the fourth term ( x, ) gives

1 2 i " 2 4p , 44, 3? Qt ? 4: P i 34:

The fift h term ( x, ) gives

i 2 2 i 5 9s 4? i q, 3? 9: ? 34, P i 49 and so o n . In general the ith term ( in ) gives i ) i ( 1 2 1 t . s, 0 i 1: 0 4. 0 34 " 2 i — 2 i i 1 p ( ) q, p ( ) q.

' Helmholt z s assumption upon which be based his equation t n n an t e but the of mo io is of course o ly arbi rary o n , results sho w that the hypo thesis which explai ns combina tion tones as resulting from superposi tion o f the prima ry tones accounts for tones which ma y have occasionally been heard by Koenig and ’ others and which have been regarded by some of Helmholtz s opponents as i nexplicable o n the basis of his mathematical “

L 1 . e I . Lo rd Rayl igh. Theo ry o f So und , . 896 , p 45 9 “ l am indebted to Pro f esso r F R. M lt n . 0n o , of the Universi ty o f Chica go , f o r a f urther inte ra tio n of the H s g elmho ltzian eqnatim. x06 J SEPH O PETERS ON .

t r . It to x t n heo y is , of course, be e pected tha o ly a few of all of these resultant tones are a t any time audible and that those ’ n to the t t heard will belo g lower orders . The fac tha occa ll sio na n diflerence e . eri t n a . ex y a seco d o e ( q p ) , g , is p enced louder than a first ought no t to be urged to o st renuously ’ n t t t no t at an t nt n w agai s Helmhol z s heory, , y ra e , u il we k o more o f the exact structures in the inner ca r which are concerned in t a t e at o — e s th n his purely m h m ic physical stat ment . Period wi i h t e t t st n e. . t ha to mos prac iced or mo usual ra ge , g , ough per ps be experienced as relatively louder than periods less frequently x e t he e perienced . The second difler nce one of t m ajor third

. . h o nl he ts ( 4 e g . is considerably louder than t e first y w n i itch is ne r he mi le e i r e i a t p a t dd o f th o rd na y sca l . This s t leas t o f the t t t l t rue forks a my command . Pure ma hema ica trea ment can be applied to the operation of anatomical structures ’ n t n An t l t t o ly with cau io . y cri icism of He mhol z s heory of n t n o n the the th to heari g, he , which is based failure of eory explain the intensity relations of combination tones a s actually experienced must take account of such obvious fa cts as those

here indicated . When the vibrations are so large that the displacements ’ ’ a flect the fo urth power of x in the equation k an bx cx ‘ dx n t t , a o her series of ones will arise , some of which will coinc ide with some of tho se determined above where only the e s co nd power of it was considered .

108 JOSEPH PETERSON. o n s o w otat n u e tat n the l er r ion. A t a certai medi m sp ed of ro io nt i nterruption tone is relatively more prominent . The i er

mitred tone is clearly audible ev en on very rapid ro t atio n . This tone was always easy to locate because it was produced from o ne o f the four circles of holes in the disk representing a major lls the o r . . S ea ch d A good way to hear what K . L chaefer dish to ne is thi s : Rotate rapidly a metallic disk perfora ted with o wand a o f he . N circle large holes, holding the ear near t circle

then touch the circle of holes with the corner of a piece o f paper. The a and o d es a t n p per is , of course, set into vibration pr uc o e ’ e a in t to the to n r s to l t qu l pi ch disk e . The former se ve oca e ’ the tt una e a e . t o n to the l r The disk one , I f u d , is audible id d car I is o s itut . t this tone which Schaefer regards as partly c n t in the ao - a nt d d nt g c lled i erruption tone . I produce goo i er ruptio n tones with paste-board disks perforated wi th a circle o f o es va n d a in t o n u n the h l ryi g perio ic lly diame er, by bl wi g po ho les ( while the disk was in ro ta ti o n ) thro ugh a fla t tube as ‘ Wlt lt as the reatest a The inte r tion g di meter of the holes. r up tune t s ro duced is ut ess r l ective as Schaef hu p do b l pue y obj , er has s o wn the n h . i crease in the si z e o f the holes bein g p ra ctically equal to an addition o f o ther circles o f ho les o f equal period as illustra tul in the acco m am' in fi re p g gu .

0 0 0 0 0 0 0 0

.4

0 0 0 0 0 0 0 0 M INA TI N AND ELA TE HEN M ENJ 1 0 CO B O TONES R D P O . 9

t the t t n in n wi h s em of a vibra i g tun g fork. Whe the forks e n a t the t b en w re of early equ l pi ch bea s were clearly audi le , ev when the ton es were very weak . The shift in the appa rent ‘ ir t n o f the un r o R d ec io so d , desc ibed by L rd ayleigh, seemed to to th one em hasis nt n t t t n b in me be ra er of p , or i e si y, bo h o es e g t i i hea rd in he r proper locations cont nuo usly . After the tones could no longer be heard to beat when the forks were held n t the to the one e w agai s p of head , or against th ax in o ne ear e in the the a nd th other aga st teeth, y were still hea rd to bea t n n t the wax o ne in c ar It t o whe held agai s each . is in eresting t n t in t nn t n t t it t i o e , his co ec io , ha is prac ically imposs ble for a lowtone to obli tera te a higher o ne when the to nes are thus n t t t u the wax in the commu ica ed separa ely hro gh ears . This lt t n t nt t th resu , if my observa io s are subs a ia ed by o er tests , is ’ most easily explicable o n the basis of a theory like M eyer s or ’ e t hat of Kuil s. In my o wn case it took considerable practice to hear inter mediate dili erence tones and summation tones from tuning s but e ti fi t t u n t S fork , I fe l sa s ed ha I s cceeded i bo h cases. ome o f the other students in the labo ra tory heard the summation tone aft er very little The important experiments no t repea ted in this work are 1 t t n i ( ) hose , as has jus bee said , wh ch were devised to prove ’ t he object ivity of interruption tones ( Schaefer and ‘ Abraham) ; 2 ) those devised to produce phase-cha nging ' t n n nn er a l a o es ( Koe ig, Herma , . ) nd those which proved t ha t these tones are strengthened by physical resonators ( Schaefer a nd Abraham ) ( 3 ) those experiments which have est ablished with cert ai nty the object ivity of combination tones in cases where the primary tones are in close mechanical co n noet c a xt n to the c a r Riick er and Edser s i e er ally ( , r a l. ) — ( 4 ) ela borate tests o n a great number of i ntervals both consonances — and dissonances to det ermine the general laws for the occur enc in ti n t n n M r e of comb a o o es ( Koe ig, eyer, Krueger) and 5 ) ex e i nt no t et i out t t an o ne p r me s , y carr ed sa isfac orily by y , study e t s t t n n tio n o ing th in en i y rela io s of combi a t nes . s t o TE JOSEPH PE RSON .

The historical treatment has been made as brief as it was tho ught well to make it in view of the fact that no a dequate sta tement of results already o btained has hitherto been given h n nt l w in t e E glish language . The exp erime s which fo lo are limited to the investigation of those exp lanati o ns of the summa tion tone which make it simply a difference tone of so me sort . In t t t n the t no t n o th a t the his limi a io , fac has bee overlo ked problem of intensity relations is probably the most imp ortant e of all the present day open questions in th field o f a co ustics. n t no t t n e 1 Fo r two good reaso s his problem is ake up her . ( ) n tun n t t r and ss The ( Koe ig) i g forks of his labora o y, po ibly of n t n t xt n x erimen a y o her laboratory , are i adequa e for e e ded e p ts t no bearing upon this question . N o t only are hey t variable in t to an xt nt but ff nt r t in he t pi ch y e e , di ere forks va y grea ly t quali y

- “ ( in the no n technical sense) of their tones . A nd this is not S o f the to t ff and s s l t all . ome forks seem be s i er les ea i y ac u d t t n t n t an l . ated ha o hers , heir o es disappear ear ier It is s t to an n r in practically impo sible , herefore, secure y u i fo mity - n the prima ry tones used . The various wind i struments are in x nt even more unsatisfactory e perime s o n intensi ty . With such inst ruments it is next to impossible to get smoo th feeble M x M t to in n t n . 2 a E o es ( ) eyer is abou publish, glish , a com " lete ta t nt and t fi t o n n t r p s eme jus i ca i of his o w heo y of hea ring. This theory was devised pri ncipally to exp lain the intensity ' relations o f combination tones where Helmholtz s theory is b t n to to t . It ut t said fall shor is fair, he , leave o M eyer the

nt n t . n n l x ns wh in i e si y problem This seco d reaso a so e plai y, the t n o f nt n t ti the historical sketch , ques io i e si y rela ons did not receive as full treatment as it merited . The method followed in these experiments is la rgely tha t of ’ t n m n x r t nes detect ing feeble o es by ea s of au ilia y o , a s they i t u t were called by Koenig . Th s me hod was sed wi h co nsider ‘ n n and so n us able confide ce by Koe ig , has al bee ed to some 5 It to d sc d extent by Lo rd Rayleigh . seems be i re ited by Krueger

W c zo llt o wsk a A Stud o f ertai n Phenomena See Dr. A . Co n n the y , y C cerni g ’ XIV f P R 1 06 . 8. Limit o Bea ts, ry . et a, 9 , ” p 37 “ bo ve. f no te n . 1 a C . , p 9 , “

o n ra . 2 . See q uo ta ti . sup , p 5

I I . Theo ry o f S o und, V ol.

1 1 a H T J OSEP PE ERSON.

n three n t n to n a re n in such ratio me . Now whe ge era i g es give t two rst ff n t n nt t ese rima r tones that heir fi di ere ce o es i erfere , h p y r n so m n e are all heard to beat ve y plai ly . Here are e i t rvals that I have found very good to illustrate this point :

‘ ‘ Ol 1 e t t 6 Utfi M ifi S , 4 b a s wi h 5

“

t the t t s . Us, : So ls z U , also oc ave of he e

t nt t s m ht b Several o her i ervals as good for his purpo e ig e a dded. ifi erence t n t t n d a fi r e . In s . some cases a seco d o e may bea wi h t, g , — ‘ Here 2 X 4 5 = 3 ; 7 n nt o ut t t a Lord Kelvin, i deed , poi ed his fac s ea rly as 1 n t t the t an c t d 87 8 . He fou d ha bea s of imperfe ly tune : : m t the r t n a the chord 3 4 5 were so e imes ve y las sou d he rd , as e n n the nt n vibration of th forks died dow , whe i e sit i es of the ” n at the end t t o three sounds cha ced o be sui ably prop rt ioned. When intervals o f this ki nd are sounding loudly the dif ereaee to nes are easily heard beating ; but the primaries and the differ ence tones are so closely interco nnected that the whole system

of to nes beats beautifully . This last statement is less true when three p rim a ry tones are given in such relations that o ne of these tones beats with a

ifi erence t n t n the t two . S d o s e. . or summa io o e of o her upp e, g , t o l n n and t t that U , and S , are sou di g loudly, ha while o ne hear:

“ ’ o e M i th t n to these tones n holds , wi a very feeble o e one s ear.

“

M t n . I In such a case the tone is bea s very plai ly n ca ses where the so -called auxiliary tone is very weak I simply hea r a beating t n t n t t n t a t the h and no tone . The bea i g is he loca ed dis i c ly pitc n o n r tt nt n and of the auxiliary . O ly ve y careful a e io ( fo r me with the auxilia ry fork sounding so loudly that it is a udible as t in the o th a t nes a tone) is beating also loca ed er prim ry o . The in the two t n n a is a lwa s heat g of lower primary o es, whe he rd, y o f the same frequency as that ( louder beating) of the auxiliary

” h e said that the no te i s sli The mark indic ates, as as b en . ghtly l at ch ed.

s imm by putti ng a piece o f wax o n o ne o f the prongs . It i aterial which note is hi h f o rk s when thus wei hted do no t vibrate lo n fla ttened, except that g g g .

This is the 7th pa rti al o f Uta.

X. 18 8 . 603 cited Ra ro c . Ro . S o c . o Edin . I lei h o cit P y f , , 7 . p . by y g , p. . M INA TI N T ES AND ELA TE HEN M ENA “ CO B O ON R D P O . 3

o t ne . The fo rks may be interchanged and o ne of the lo wer ‘ ' o ne the x l and t un s used as au i iary , herefore , so ded more d o he e an t t . m the fe bly held ear The results are uch same,

. e t r ll he t b t he 3 . t n t to a u t t , all h ee o es may s i be heard ; highes t ne t n t n m t f o r t t d t n t o , wi hi cer ai li i s , always , me , bea s mos is i c ly . Rapid bea ts especially a re mo re easily hea rd wi th high tones

t n t n . l x t ha wi h low o es This, of course , wou d be e pec ed from ’ Helmholtz s explanation of beats . What was said in the last paragraph about loca ting beats l in the t n to t n a so lower o es , applies especially o es of suc h rela t io ns that the interval between the middle of three primaries a nd the upper tone ( the auxiliary sounding feebly at the ear) o is considerably smaller than an octave . The f llowing inter v l t t and in the x m nt : als are il us ra ive , were used e peri e

A uxilia ry Fork

’ Fan : La S a l. ” Uk zfl is Ref

. Uls : S o I. 5 1‘s

’ ' Rt st d sb ( 2 : 3) F a . ( S ) ” S ain t/ l. T U )

“ (h ul a. Fe.

' ' 0mm. 7 ( t s )

When o ne makes the auxiliary fork feeble eno ugh in these cases and hears it o nly as a series of beats and no t as a co n tin uo us t n and t n t n s to o e , , fur hermore , whe his seems i such ca es be the nl t n o ne ca tt t n the t o y bea i g, n hardly avoid a ribu i g bea s to the auxiliary tone with a summation tone which is present

no . t x n t n but t audible The plausibility of his e pla a io is , of ’ t n th n the t x nt course , s re g e ed by resul s of Krueger s e perime he already referred to . His subjects actually heard t summa tion tones directly . But against such an explana tion of the beats of the auxiliary it fork, may be objected that these beats are no t due to the presen ce of a summation tone at all but that they are explained l t t e s the t l . t t n t n t ne a i y from fac il us ra ed above ; viz , ha ge era i g o s are o s a i t t t e f n t n o r so cl ely ssoc a ed wi h h ir dif ere ce o es , result he the ant t t n t tt t t . E. . in s , ha whe la er bea former bea also g , the first case of those given above it may be urged that a di ffer

t‘ ‘ ' Vve hasre ssot the k sa in the labo ra to sy so I eould ti ot use the lo wer oeta va ol thia interval. l ‘ 4 JOSEPH PETERSON . — ence to n e 4 ( fro m 9 5 ) interferes with the p rimary tone a nd t t the bea ts a re lo cate ri nci a ll in the to ne the 4 ha d p p y 9 , higher o f the two prim e It In addition to this obj ecti o n a noth r ma y he ra ised. ma be r tha t in en ral bea ts cann t be lo ca ted wth y u ged , g e , o i

an ree f ce ta int . his seems to som extent to be true y deg o r y T , e , ’ from the results o f Krueger s exp eriments . A very simple ex periment may prove tha t the ass ertio n ma de in this seco n d oh

e i ha o . a e . two ks f ct o n s un s . r o li j go d gro d T ke, g , fo ke t l tl e d tc t wo U o s . e ress o ne th pi h, as , f rk D p s igh y by metho x ks a r e n b . N owwen t ese two o r e so un plai ed a ove , h h f ded n e n e o together we ha ve o ne bea ti g to n . If a y s c nd to ne is so un utan s th th bea tin to n the seco on ded sim l eou ly wi is g e, n d t e

nl to t also . seco n to n as is plai y heard hea This d e, is t rue of the x r he so un so ee t a t o n the au ilia y above , may ded f bly h ly e he a n o a l b a ts are heard . T be ts of this to e seem t terna te with those of the first in a fashion that may be rep resent ed in this way Bea tin g to ne

Sec on d to ne

This is very no t iceable when the beats are slo w . Whether this influence of a beatin g tone o n other tones is o r is not merely 1 3 a psychological o ne ( and it probably is) is a question that has but tt n o n e s n e no men li le beari g th pre e t use of th phe o n . The mere fact that there is such an eflect pro duced by bea ting tones is o ne that must be recko ned with in all studies in which the locali z ation o f beats is impo rta nt . These objections make it doubt ful whether the bea ting awo ciated with the auxilia ry forks above proves the exi stence of ’ l m t n t n . n t r as t t a v su ma io o es Koe ig s heo y, as we l o hers h t ha e n n ha s no the fi t diflerence to n o f bee adva ced , place for rs es “ nt n x n the ctav . w large i ervals approachi g or e ceedi g o e No ,

I wll kn o wn flu ua ti n in si t f t is e e . tha t a c t ten o li ht will , g , g y g subj ectively aff t o h which ti l h ec ther li ts are b ve con stan t. g o j ec y The abo ve exp eriment. the". ' ’ f o re nee no t a r H , d be reg ded as a flecting elmho ltz s physi o lo gica l tho ry o f tonal

s ana ly i s.

“ ' I t i s true tha t wi th auxilia ry f o rk s nea rly a ttuned to the first di flerenee to ne oeni hea rd a bea tin f o r in H , K g g tervals as la rge as 8 : 15 . e sa ys no thing

f where thes bea ts r o e a ea ed to be l a e . x pp oc t d Cf . Qnelqsses E perience ! A t o ne

'

s n e . 0 t q sse, o t , p 13 . t I 6 J E H ETE N OS P P RSO . int x t the a u nt t th i ervals do e is , whereas above rg me posi s e r - B it to a o ne. ut no n existence . This o bjection is valid is be remarked that only those theories which a lrea dy admit the i th existence o f summation tones will urge th s o bject io n . Wi t s e h such theories we have no quarrel . All heorie , how ver, whic deny the existence of summation tones a lso deny tha t of dif “ ference tones of large intervals . The immediately foregoing sta tement is hardly nec essary f It x u e x . a fter th e cellent work o Krueger is e c sable , however, from the fact that Krueger studied summation tones on ly in ll m to he t nt t o ur in cidenta y. But no wwe co e t impor a par of i a re these su o sed summa tio n to nes to be ex vest g tio n . A pp pla ined a s difference to nes of higher o rders This q uestion u tt it m to in a ll s mm Kr eger has also se led , see s me, case of u a n t r s tion tones which are very weak . His ge era o s were pr c m t n et o ent ca d tically free fro over o es , y his bservers frequ ly lle h t s m t n t n . But t in n att ention to weak su ma io o es wi cer a i terval , h fi d the t the t n ton re no t e . a u a . t e t n g , f h four h, s mma io es hard to hear when the interval is no t pitched high . Koenig admitted that the summation to nes of these intervals ha d been t to n heard direc ly in case of siren tones . Since these es clea rly po ssessed overtones he suppo sed that the summation tones gen era ted t t-t n a t al by hem were really bea o es of upper p r i s. cker a d ser t t e n a - Rii n Ed showed tha hey a r o t such be t tones. ’ In t n nt to l t w la er years, eve such a close adhere He mhol z s vie S o t t t on t n s as K . L . chaefer has supp sed ha summa i o e which are audible under ordinary conditions may be due to the

n o f t i . e. t t in t are prese ce upper par ials , , ha cases where hey perceptible with the unaided c a r t hey are really difference " x M n in tones . M a eyer, as we have see , was of like m d. The results that follow refute such noti o ns . It will be recalled that two quite different possibilities have been suggested fo r explaining the summation tone as a mere difference tone :

’ N o ne o f these sta tements c a n o f c o urse a l to Me er , , pp y y s theo ry . He admits tha t o n the H elmho ltz i an pri nc iple o f tran sf o rma ti on due to asymmetry e drum n o ri in a te in the mid H in th these to es ma y g dle ear. e rightly ho lds them

v r win thi s c o nn ec tion to be e weak . On his vie see abo ve y , pp . 9! if .

S u r 1 1 . p a , p . 0 N A ND ELA TE HEN M ENA H COM BINA TI ON TO ES R D P O . 7

1 . fi t t x n ti n u t The rs of hese e pla a o s , s gges ed by Apunn, a d the t Preyer, n others , makes summa ion tone a di fference tone of the second order according to the formula — ah ( h

2 . t x n t n n e The o her e pla a io , give by Koenig, makes th sum ma tion tone a di fference tone of the fi rst o rder resulting from appropriate upper part ials according to the equation

n ( h

’ S h x n t n t o ne to o c aefer s e pla a io , where his requires high orders “ t t n s nt th nh : of par ial o es , may be repre e ed us mt h t. The fi rst of these views is closely allied to the o ne which derives the second dili erence tone from the first with on e of the ri a t p m ries , in his way

fl — t = 2 t — h fi ( ) .

nt n t ffi t as has t n n nt o ut The i e si y di cul y , of e bee poi ed , at once r t ex conf onts his view . The following periment is given simply l t t the ffi T o to lt . he tw Ut il us ra e di cu y forks , and M i, were fastened securely to the side of the table in such a position that t n n x o int in t i t heir reso a ce bo es p ed a ver ical d rec ion . They were then simultaneously actuated by dro pping o n their hori z o ntally projectin g prongs two small rubber stoppers each weighing gr . The stoppers fell o n the prongs near the end and immediately bounded 06 . To prevent any noise a piece of twine was a tt ached to each stopper a nd held so that the stop per could no t fall o n the floor . By this means it was possible to regulate fairly well the i ntensi ties of the primaries t the for purpo ses o f comparison . Bo h of forks used gave very clear tones wi thout any percept ible overtones fo r the ia M i o nt n n n n tensi ties used . , c i ued sou di g lo ger, however, when t the the two were actuated with abou same force . N owit n o n t ts t n t t the was fou d , various es of his ki d , ha t n So l t sec o nd difference o e , a was easily percep ible when the st oppers dropped upon the prongs of the forks fro m a posi tion

t . i t n in u s cm . above hem The pr mary o es s ch ca es were

” n an d m sund for whole numbem b and t o f eo urse f o r u er and lower , , p p

c vel p rimary to nes respe ti y . I 1 8 J E T OS PH PE ERSON .

t r . seco n ff n to n it was hemselves ve y feeble The d di ere ce e ,

n the use to t h ntin to a for fou d by of a s p wa c , co ued be udible to 1 se re from 5 6 conds after the forks were actuated . The

et the . was y absolutely no trace of first difference tone Ut, The distance through which the sto ppers ha d to fa ll was now

n unt th t n a to o . i creased gradually il is o e bec me audible, This e occurred when the falling distance became so to 5 6 cm . Thes figures were confirmed by tests in which the drop-dista n ces o f the weights were gradually decreased from a positio n a t which both difference tones were audible to po sitions where t hey be ‘ n fi t the rst th n the seco nd ff n t n came i audible ; rs fi , e di ere ce o e h n ff n t n o t disappeared . While t e seco d di ere ce o e was smo h a nd clear and continued relat ively long [ nearly as long as the

the fi t ff n t n o u and lower primary] , lower rs di ere ce o e was r gh

. h e. e new s n . t t oo disappeared With sui able forks , such , g , as

E l nn se t x m nt fit o ut de ma t, his e peri e could pro ably be carried nt m t t t in to considerable length . Various i ervals igh be es ed m n . ts differe t parts of the scale The above fac , however, see conclusive against the view that the second difference tone always arises from the first difference ton e and the higher pri

r t n . i nn t to t ma y o e Th s view , therefore , ca o be used refu e by analogy as suggested the existence o f real summation to nes in the m tz n n Hel hol ia se se . It is this same interval that Hermann and others used as illustrative of the intensity relations which they considered co n ’ tradicto ry to Helmholtz s theory of different orders of co m n t n t bi a io ones . The object ion has less force when urged n t m t m t t r agai s his a he a ical heo y . According to this theory, n t the m n t as has bee sugges ed , all co bi a ion to nes arise directly m the m fro pri aries . The intensity relations for this same interval ( the major third ) differ with different pitches of the

m . E . . n Ut and M i pri aries g , whe a 3 are used as the primary t n the fi t ff n to n o es , rs di ere ce e appears , for me, before the sec o nd the nt n t o f the m as i e si y pri aries i ncreases . When the m Ut and M £ the t wo ff pri aries are ts 5 di erence tones usually ap and t t t n but h t pear disappear oge her, las i g a very s or time . In ’ t i tt the m t o n un h s la er case pri aries hemselves so r o ut . Fo r t 20 TE N J OSEPH PE RSO .

h at o n num er i . e. t e the vibrations of these forks is 896 , , vibr i b th mat o n to n in t case was inaud of e fork 7 . The sum i e his ible,

" but beat very plainly with the wea k tone of 896 vibra tions . h a o v The same result was obtained in this case as in t o se b e, Ut Ut I : 8 at ver z . e ff he s vi , th di erence tones of t fork 2 5 ( ) be y

feebly with the auxilia ry tone . In ll e n r the r s ta en to a three of th ca ses just co side ed, fo k k represent the interval ( h t ) a h were so unded with co nsid cra nt n n th e t n s but cer ble i e sity. Whe ey give mor feeble o e , ta inly louder than the difference tone and the first upper pa rtial fo r t urel ne a tive results were which hey are substituted , p y g o bta ined : the auxiliary tone revealed no bea ts whatever. These few experiments seem conclusive aga inst such an acco unt of summation tones as that given by Preyer and o thers . The x n t n it t nt n to nt o n mma tion e pla a io , is rue , is i e ded accou ly for su t n h t tta n d a o es whic are ac ually audible . We may a ch co si er ble t ne h x n e sut importa nce o the o suc case here e ami ed. Th re l of that o ne case seems absolutely decisive against the exp lana t n n io i question . no w n the se n — e. We shall co sider co d type of view i . that — , of Koenig which regards the summation tone as a fi rst differ en t n t-t n t the a Is ce o e ( bea o e ) of upper par ials of s me order . an x n t n n ss in a n a se ? such e pla a io probable , or eve po ible , y c In the first place tests with auxilia ry forks for the existence t h nt t t e . n of upper par ials co radic view The seco d pa rtia l, he i . e. t fi t t Ut to h , rs upper par ial , of a was audible t e una ided ear . The auxiliary fork beat with part ials as high as the fifth

n . n the t i clusive Whe fork U a was damped by holdin g one n n the t t t and fin the fi t a pro g ear s em wi h humb a ger, f h p rtial to o ut n n o s nt . seemed drop , leavi g o ly f ur pre e Of 8013 the n seco d partial was audible . The auxiliary tone beat with p artials as high as the fourth . No fork was available to test the fi but the o for fth , fourth was s weak that the fi fth pro bably m did no t exist . Da ping does no t seem to affect the upper t m it par ials as uch as was at first supposed . I am no t sure that the t t in t m n t n four h par ial his case was eli i a ed by dampi g . Of i th M 3 e partials as high as the fourth beat plainly with the x n t auxiliary tone . I did no t e te d his test to the other forks M BI A TI T ES AND ELA TE HE M E A i “ CO N ON ON R D P NO N .

t t the r o t used below , bu i may be safe from above esults t say tha if t n the fi t nt a t all t ext r par ials beyo d f h are prese , hey are emely ” a we k . I f upper partial tones are efi ective in the production of com b nati n t n n a u t t i o o es, as some perso s h ve s pposed , his fac should c n the he s n d e easily be dis overable i case of t eco d ifi erenc tone . h r As has already been sta ted t e second difi e ence tone (Sol, ) of t i the interval U , : M , is very prominent even when the primaries n i n i are sou ded very feebly. Th s difference to e s clear and co n “ i e t nus nearly as long as the lower of the primary tones . When t st erence the primaries are made s ronger the fir difi tone appears . he Ut It and t but t ti . N o ww n is sub is rough las s a shor me , a stituted o r Ut in this interva l the di erence to ne So l sho uld f , , ff 3 , as a rule be even m re ro minent i it b de endent u o n the , o p , f p p u er a tia l o the lo wer to ne o the interva l Ut : M i pp p r f f 4 , , as has t h the been supposed from he pi tch 2 t . But when substi tut io n t ff n t n at n its n is made , his di ere ce o e o ce loses clear ess and takes o n the characteristics o f the first difference tone just he m t no o n co . t w described M oreover, pri aries mus be s u ded n o c he siderably l ud r for t appearance of this tone . This seems o the u e a rtia l o Ut n t n erv t show conclusively that pp r p f , i he i t a l Ut : M i has ctica ll o thin a t l to with he ener o n , , pra y n g a l do t g a ti o o n i erence o e 801 nd the s t n . a f ec d d ff 3 M eyer Krueger by ano ther method have i ndeed already proved more or less co n elusively that co mbi na tion to nes are pract ically no t a t all de n n n t o f the t t t pe de t upo upper par ials primaries , so fur her es s alo ng the li ne indicated here would be needless even i f the necessary fo rks were at hand . What holds fo r diflerence tones with respect to upper — pa rtials ought also to hold for summation tones espec ially i f re d er n e t n ! But n n the latt er a ifi e c o es si ce Koe ig and others ,

nt an x nt . . S and even as rece e perime er as K L chaefer, have t t n t n to t i attribued audible summa io o es upper par ials , t seems necessa ry to in vestigate the matter furt her . “ It is of co urse true tha t o cc a si o n all a ver hi h lo u ar . , y y g d p tial i s hea rd. but such cases are us ually irregular and the pa rti al may rea dily be damped w bi o ut itho ut a ff ec ti n the com na tion tones . The ex eri ment fo llo win an wa g p g. y y. rules o ut the o ssibi li t o f the eff ec ti vene ss f o r the r p y , p o duc ti on o f summa ti o n s f l h ue . o up per parti a s . “ Al it will b remem e con l“ e ber d. ti nues longer than Uls t 22 J E H ETE OS P P RS ON .

If the fi fth part ia ls of the fi fth produce a n a udible difierence to ne ( 5 X 3 5 X 2 5 ) it is rea so nable to sup pose that in most ca ses partials belowthese p ro duce ev en lo uder ” se n t s o u v diflerence tones . The co d par ials h ld gi e rise to the — h n . 2 t h to the t the t n 2 i . e 2 2 e t rt o e ( , X3 x ) ird o e 3 ; fou h s t n all s o n to the i r e t o so m to 4 . The e o es corre p d pr ma i s or e di eren th of their upper part ials . The ca se is fl t wi the major third ( 4 : If we loo k a t the upper partials

1 2 I 6 2 0 2 2 8 2 6 4 , 8, , , , 4 , , 3 , 3 , 1 0 1 2 0 2 0 0 5 , , 5 1 : 5 , 3 , 35 , 4 » 4 5 :

it is i ent tha t es s the t n i . e . 6 t ev d b ide o e 9 ( , 45 3 ) we ough to hear ( other than upper partials and primaries ) tones corre ’ s o ndin to 2 6 and n to n ex nat on p g I , , 3 , 7 , accordi g Koe ig s pla i t ne 1 and a re so di e o f summation tones . The o s 3 al fi rence

s he r r s . can test t en o n tone of t p ima ie , however We , h , ly for

2 and . , 6 7 t a l i e When the forks U 3 nd A , ar so unded loudly toget her ti a re but r ro u evi their first upper par als audible ve y gh, due dently to the summation tone lying between them This 13 much less noticeable when the interval is pitched an octa ve i b t r xi to n t h n e to h gher, u a ve y weak au liary e of pi c early qua l t t the mm t n t n ts nl t a ha o f su a io o e bea plai y . Now is here ny evidence o f the existence of those other tones which we o ught to hea r if Koenig is right ?

U n the im s Ut : 1lI i : si g pr arie , 3 ( 4 5 ) we may test with the x r o l the t n o n auilia y fo rk S ; for o e corresp n di g to 6 . When the p rirn a ries a re so unding very loudly a nd so undin g very to the ca r the t n t ll a feebly , is held o e is ac ua y he rd to beat In se the tin n . t no plai ly his ca , however, bea g is proof of the existence of the tone 6 . It is doubtles s due here to the inter ference o f two fi rst diflerence tones as is evident fro m these numbers :

‘ “ 6 — = 1 and — = 1 5 5 4 .

It will be reca lled that in ca ses of this kind the bea ting is very

See the no te ust recedi ho wv j p ng this. e er. ” ‘ f r C . ueger s remarks on the interto ne wi schenton in Phil K ( Z ) . Studien,

XVII” t oo t .

1 3 T 4 Josem PE ERSON. to the ca r has r nc o f‘six er se on nt rom , a f eque y p c d as is evide f the equatio ns : 896 5 I O 3 86 — = 2 x 5 1 0 64o 3 8o .

n the t n M i 6 0 n t Ur Whe , however, o e , ( 4 ) i s ead of , is de pressed two beats per second the auxiliary fork beats only t wice

n the x n t n t n nt re i . per seco d , as e pla a io jus give evide ly qu res hea tin the x t n t c h d The g of au iliary o es , herefore , whi h was ear in t t n the t n 2 and nd n to at-t n es i g for o es 7 , correspo i g be o es of upper partials of the major third ( 4 : 5 ) does no t prove the ’ x t n t t-t n but must x a n e is e ce of hese alleged bea o es , be e pl i ed ’ - t . t n n t a o o herwise That such beat o es , arisi g from par i ls l wer than tho se which give rise ( according to Koenig) to the sum t n t n no t x t th n I the t t t ma io o e , do e is is proved , e , ( ) by fac ha in t the t n nt n to nt several cases tes ed auxiliary o e , i e ded i erfere th t no t t a t 2 the t t t n wi hem , does bea all ; ( ) by fac ha whe ever x t the heatin tt t to lo w the au iliary tone does bea , g is a ribu able diflere c t n a nd do es no t have the re uenc re uired b n e o es , f q y q y ’ eni s o s i n I he o s u it o . n t t t n the K g pp illus ra io above , where beating o f the auxiliary fork had a frequency of six per second n Ur two t n i n whe , was depressed vibra io s , t is evide t that if the ton e 7 existed at all and was produced by a beat-to ne of the nt t the the re uenc o the bea ts seve h par ials of primaries , f q y f sho uld ha ve been sev en times two er seco nd since the seventh p , pa rtia ls wo uld be depressed seven times as ma ny bea rs as the m n t r r t n n . fi s . On this poi t we shall dwell ore i de ail ve y soo t n o n r o o un t t We have assumed , he , ve y g od gr d, ha if the mm t n to n t-t n t a su a io e be a bea o e of upper par i ls , as n t t t-t n rt Koe ig sugges ed , several o her bea o es of lower pa ials should be even more plainly audible ; and it has no wbeen n t n t t t-t n o proved co clusively , I hi k, hat such o her bea o es d n t the i o t exis . Aside from proof here given t should be noted that these lo wer bea t-to nes o f upper pa rtia ls are never 27 a udible to the una ided ea r the m t n t n f t c , whereas , sum a io o e quently is audible .

A t any rate I ha ve never been able to hea r them and no one else seems to

H m tho uh he heard a — do ne so . iillstrii t to n alt ar have g e , as has been seen,

obabl i ed. f co urse thi s but was pr y dece v O , to ne may exi st when the second p M BINA TI N T NES A ND ELA TE HEN M ENA 1 2 CO O O R D P O . 5

We have yet o ne more method of att ack which meets even t the t and t h l more direc ly heories here opposed , which mus o d qui te i ndependently of the va lidi ty of the use o f auxilia ry tones in i h n establish ng t e existence o f summation tones . The pri ci le t i p of h s method has already be en suggested . After having con vi nced myself t hat the summation tones of

the Ra ma h Fa zLa primaries a 3 a . a nd Ut zLa to the n s a are audible u aided ear, I experimented with other students o f the laboratory to assure t t the no a myself ha summation tones were t imagin ry . The t nt s ude s selected were all trained in experimental psychology .

Those who served in this connection were M iss Elizabeth R. S F and M R n M M . haw , iss lorence E. ichardso , iss Grace al Fem d. Each subject was asked t o select from a number o f high pi tched forks the tones that were audible in addi tion to h t e t n nt n . n o ne primaries , when cer ai i ervals were sou ded O ly the u t n t a t t and in no s of s bjec s was prese a ime , ca e did she h h h know whic fork of t e group represented t e summation tone . The primaries were soun ded a little above medium i ntensity t nt t t wi h a rubber hammer . The i ervals used were hose jus

named . A t least o ne of the second partials of the primaries a t ws always heard . The fork representi ng the summation one wa s i n every case selected . Some of the subjects found it more

t n t . t n t t an the m t n t n easily ha o hers No o e , o her h su ma io o e the fir or st upper partial tones , was ever selected as a final s he x m n judgment, ev en though in some ca es t e peri e ter called

specia l att ention to other tones to try the force of suggestion . In some cases the subject readily sang the note representi ng the It t t t t t t the summation tone . seems eviden from hese es s ha

summation to nes of these intervals are actually audible . Now these summation tones ought to beat with auxiliary t t and o n t t tones of nearly heir pi ch , his basis i should be possible to as certain some facts concerning the mode of origin of these the t n t n dillerence t n ton es . I f summa io o e is a o e , or a bent ’ t n e ti t i u the o e , of upp r par als h s sho ld be revealed by frequency

b smee wfll no t aflec t the sbo n n M ala m strong nt i ts exi gumen t. In my

' c ase i t ought to be very muc h stron ger than the snmma tion tone if Koenig s view

ho eorrect. t 26 ETE N JOSEPH P RSO .

its t t a n n s of bea s wi h auxiliary tone . From these umber , rep r n tin the fifth t its t ese g wi h upper par ials,

6 , 89 1 0’ s . 6 1 2 1 3 , , 9, , 5 ,

it is evident that if the tone represented by 2 is dep ressed one t n the n t two vibw vibra io , seco d par ial , 4 , will be depressed t n the t t e t n the t io s ; hird , 6 , hr e vibra io s ; four h , 8, four vibra t n the t 1 0 i t n a d o n io s ; fif h , , f ve vibra io s ; n so . This is easily

’ t : t t proved hus if U a beats once per second wi th U 8 it is found

to t t t Ut t t t o l and o n . bea wice wi h , , hree imes wi h S , , so Now ’ ‘ i f Koenig s explanation of summation tones as beat-tones o f t t t it nt t t a de appropria e upper par ials , be correc , is evide ha pressio n o f o ne vibra tio n o f the primari es o f the fifth will de r s h n o this t we p e s t e summa ti o n to ne fio e vibra tio s . Fr m fa c e i e n ca n ma ke a d c sive t st o f the va lidity o f this expla na tio . In the attempt to test this explanation I soon enco untered

ffi lt o ne no t n . n a di cu y , , however, which was i superable Whe o ne o f the primaries is depressed the i nterval becomes imperfect and the dissonant beats interfere with a careful study of the n t n b t the t to the t n summatio o e , u bea s due imperfec io of the i nterval are disti nguishable from those of the summation t t et o ne wi h the auxiliary of nearly the same pi tch. To g rid of the beats of the imperfect interval the following procedure

. o n n d n was adopted Fi rst , after e of the primary to es ha bee s o ne i t n n the x depres ed v bra io per seco d , au iliary fork was gradually depressed until the only bea ts remaini ng we re t hose nt I the the of the imperfect i erval . n case of fifth given above these beats had a frequency of three per second when the lo wer

primary tone was depressed o ne vibra tion . This is evident from the fo llo wing numbers

h ti the 3 84 is t e vibra on number of higher ton e . 2 5 5 is the vibration number of the depressed lower p rimuy

t n o e .

- 1 2 9 is the first diflerence tone . e he e 1 2 6 i . . 2 2 8 t n difi rence t n . ( , 5 5 X 3 4 ) is seco d o e

3 is the number of beats per second due to the i nterference e n of these difi rence to es .

1 2 8 J SE H ETE O P P RSO N.

’ Our conclusio n which is in harmony wi th Helmholtz s t r n t n t n e nt t the t heo y of combi a io o es , agr es e irely wi h resul s ’ of Krueger s recent experiments in which he fo und tha t sub jective summation tones are audible even when upper partials h er men t e a re n . e ex i tal of primaries elimi ated On th whole, p evidence co ntra dicts a ny theo ry whic h regards summa tio n to nes a s dependent upo n upper pa rtia l to nes or difference to nes o f the ri s p mari e .

' SUMM ARY o r DATA B EA RI N G UPON r m: O HM s LAW

Quest i o n .

Obj ectio ns to the Ohm-Helm Replies a nd Po ints Favo ring t he

holtz ia n View.

. e i u 1 ntil oocasio n is fi or to 1 Upp r part al to nes act ally . U a ded p rwent are heard either rela tively analyze the tonm o f a clang we e wak or not at all ( Seebeck) . t raining we can perceive di e up per partials relatively lo uder than usul elmho ltz a ( H ) . No 2 . bea hus 2 . The vibratio n numbers o f ts t co rra po o d to ff the umma io ne e di erence t ones equal the f re s t n to ( H lmholtz ) . q uency o f the bea ts o f the p ri

maries i ( Ko en g , et

. uch bea whe 3 . Large intervals may beat even 3 S ts, n there are wh th no overto nes to ro uce them en ere are no overtones p res p d , ent Koeni Wun t et due to the resence o f difi erence ( g , d , p to nes which may be too weak to be irectl hear elmho ltz d y d ( H , Bosan

quct Kmeeer) . ‘ ' nterru . These to nes can be 4. I ption to nes are 4 reso nated he r hen a iven to ne is erio with h si al reson ators ro vin a d w g p d p y c , p g

' ically intermitted o r partially so that they co rrespond to physical intermitte o eni ennert et en ular vibr tio d ( K g , D , p d a ns ( Schaefer and

h e u 5 . T es s pposed phase chang ing vibratio ns have been resonated

that they are actually sinus-fo rm E A TE H N M ENA 1 2 COM BINA TI ON TONES AND R L D P E O . 9

r unploys them in lo caliz atio n o f ( Schaefer and Ab aham) .

is r ba le that the instru 6 . Shifts o f phase in upper pnr 6 . It p o b tial to nes of a clang affect the men t used ( the wa ve-siren ) did quality o r timbre o f the clang no t rep ro duce with accuracy these

i bre hif o f hase ermann an ( Koenig ) . The chan ge in t m s ts p ( H ) ; y is due no t to shift of phase but to way the cha nge in t imbre is but

e e e o i The seco n chang in obj ctiv fo rm f the sl ght. d sta tement hm ud wave H never ee ve nt o o . co p n ( ermann ) . b n pr d

f m ~ N f n bin . o t satis ril n re 7 . The i t ensities o co a 7 ( acto y a swe d b an r y y theo y . ) the pro po rtio ns required acco rding

to the Helmho ltz ian theo ry ( Vo igt ,

tum f e m nn M e er et al . S p , H r a , y , A to ne ma o bli ra a 8. low y te te higher o ne but the reverse is no t

e c nno 9 . W a t be quite sure tha t rarel one to each ear can be heard b me ns o f bo ne con uc tion o r y y a d , to t o i b so me h r mo o bea Cross and o wn . o t e e f co n utio n ( G d ) y d d c , bo th tones did no t operate in the

same ca r.

’ t ~ o . Helmholtz s theory o f oh jective co mbina tio n tones is sup sarily contra dicto ry to the o pposi ported by the fact tha t such tones have been reso nated with va rio us

h sical reso nato rs lmho l p y ( H e tz ,

t t e e e . Th int rton ( Zwischen ton ) is co ntradicto ry to a strict

acco unts ; it is mo re nearly a ‘ ’ bea ting co mplex than a ton e rue er ( K g , Peterso n ) . 1 2 . mch 1 2 Only stunmation tones . Very weak summatio n tones as are att ributable ( as dif can in most cases be heard o n m oc to nes) to upper partials 1 0 E H T 3 JOS P PE ERS ON .

either wholl oeni o r in art the ear and en o y (K g ) p , att ti n ( Krueger)

A unn re er et o b ective summ tio n to n are ( pp , P y , j a es no t explicable as difi erence to nes

’ Rticker and Edser) subjective summatio n to nes are no t explicable r as diffe ence to nes ( Peterso n ) .

I 3 . Bo th o bj ective ( Riicker and Edser) and subj ective ( Krueger) intermediate ( z wisch

enlie e e g nd n ) difference to nes exist.

Ta s man Srarz uanr or me Ern crmcv o r Tn soxues Drscussan i n ra ts

Tn r as s.

A c ro ss ( X) mean s that the theo ry can explain the f ac t o pposi te which it

occurs.

i s A question ma rk signifies either tha t the efi ciency o f the theory lemen do ubtf ul o r tha t the theo ry needs modifica tio n o r supp t.

’ ’ Note the inconsistencies in Wundt s and Ebbinghaus s t heories pointed o ut in the text . t a T 3 Jo sEm PE ERS ON .

o o a n f o r ti n 2 t t n f re , f ur phase ch ges each revolu o ; ( ) a s a io ary fo rk which actuated a telephonic diaphragm under a curren t which was periodically reve rsed by means of a rota ti ng co m t t in two n at t o n muta or, hus caus g cha ges of phase each revolu i : ( 3 ) a rotati ng sto pcock which brought alternately to the c a r the waves from the side and from the face of a continuous ly t t t nfi sounding fork . The resul s reached by hese me hods co rm ’ he t t n u t n n The t au hors hypo hesis regardi g a di ory reso a ce . y n t t the n n t wit fou d ha sudde reversal of phase , whe i co mes h

u ient n t the t n . t a t s fiic freque cy , des roys o e A cri ical r e of phase

n v e. . 1 0 t n tw n a two cha ge was disco ered [ g , vibra io s be ee e ch t successive phase changes for a tone of 2 40 vibra tion] . A this rate (which was fairly co nstant under the given conditions ) the n the t n n and n sou d of u i g fork disappeared , reappeared o ly f ”2 when the rate o revolution was diminished . Bentley a nd Sabi ne repea ted the experiment under so me» 3 what similar condi tions . t t Their results did no t agree entirely wi h hose given above. h n t t no t et T ey promise a o her ar icle which , I believe , has y appeared .

Dr. eo E. S nt re u G . hambaugh has rece ly made a very ca f l ' ‘ m n i the n study of the tectorial embra e n pig s ear . He fou d t t n t its un t n n n m in hat his orga , wha ever f c io , i creases e or ously the the ch to w the x a nd su ts size from base of co lea ard ape , gm that this structure may serve as the resonant a nalytic organ in I t t n . t t n s t heari g is doub ful , however, whe her his orga , de pi e its fibrilla r st t ca n act n t N ruc ure , as a reso a or . O physical

n t o f t n n n b ist . reso a ors his ki d are k ow , or descri ed by physic s fin t t at the f Dr. Shambaugh ds ha base o t he cochlea the basilar n t un membra e is comple ely derlaid by a bony bridge . “ o n the o t n fin t t the K . Kishi , her ha d , ds ha basilar membra ne not n in its t t t is so u favorable , s ruc ure , o a view like that of z Helmholt .

‘ m V I 1 - A ur P ch l X . 0 . . J o . a] sy o a , 9 4, pp 489 90. ’ A f T n e A nal si s i id — Stud o b . y o y . ., pp 484 98.

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