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DOCUNINT MORI I. 00.175 695 SE 020 603 *OTROS Schaaf, Willias L., Ed. TITLE Reprint Series: Mathematics and Music. RS-8. INSTITUTION Stanford Univ., Calif. School Mathematics Study Group. SKINS AGENCY National Science Foundation, Washington, D.C. DATE 67 28p.: For related documents, see SE 028 676-640

EDRS PRICE RF01/PCO2 Plus Postage. DESCRIPTORS Curriculum: Enrichment: *Fine Arts: *Instruction: Mathesatics Education: *Music: *Rustier Concepts: Secondary Education: *Secondary School Mathematics: Supplementary Reading Materials IDENTIFIERS *$chool Mathematics Study Group ABSTRACT This is one in a series of SBSG supplesentary and enrichment pamphlets for high school students. This series makes available expository articles which appeared in a variety of athematical periodicals. Topics covered include: (1) the two most original creations of the human spirit: (2) mathematics of music: (3) numbers and the music of the east and west: and (4) Sebastian and the Wolf. (BP)

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THISLX)( 'API NT HAS141 IN WI P410 Oti(10 4 MA, Yv AS WI t1,4- 0 4 tiONI TH4 Pi 40SON 4)44 fwftAN,IA t.(1N 041.G114. AT,sp, .1 po,sos 1)1 1 ev MW OP4NIONS STA 'Fp LX) NE.IT NI SSI144,t Y RIPWI. TO THE EDUCATIONAL RESOURCES SI NT (7$$ M A NAIhj N',1,11)11E 01 INFORMATION CENTER (ERIC)." E.),,f T P(1,1140N ,1 V 0 1967 by The Board cf Trustees of theLeland Stanford AU rights reeerved Junior University Prieted ia the UnitedStates of Anverka

Financial support for tbe SchoolMatbernatks Study provided by the Group has been Nasional ScienceFoundation.

3 Mathematics is such a vast andrapidly expanding field of studythat there are which do not find inevitably many important andfascinating aspects of the subject well a place in thecurriculum simply because of lackof time, even though they are within the grasp of secondaryschool students. find time to pursue Some classes and manyindividual students, however, may mathematical topics of specialinterest to them. The SchoolMathematics Study Group is preparing pamphletsdesigned to make material for suchstudy readily accessible. Some of the pamphletsdeal with material found in theregular curriculum but in a more extended manner orfrom a novel point of view.Others deal with topics not usually found atall in the standard curriculum. This particular series ofpamphlets, the Reprint Series, makesavailable ex- :ository articles which appeared in avariety of mathematical periodicals.Even if the periodicals were available toall schools, there is convenience inhaving articles on one topiccollected and reprinted as is donehere. This series was prepared forthe Panel on SupplementaryPublications by Professor William L. Schaaf. Hisjudgment, background, bibliographicskills, and editorial efficiency weremajor factors in the design andsuccessful completion of the pamphlets. Panel on SupplementaryPublications Rouge R. I). Anderson (1962-66) Louisiana State University, Baton M. Philbrick Bridgess (1962-64) Roxbury Latin School, Westwood,Mass. Michigan Jean M. Calloway (1962-64) Kalamazoo College, Kalamazoo, Ronald J. Clark (1962-66) St. Paul's School, Concord, N.H. Roy Dubisch (1962-64) University of Washington, Seattle Mass. W. Engene Ferguson (1964-67) Newton High School, Newtonville, Montclair. N. J. Thomas J. Hill (1962-65) Montclair State College. Upper L. Edwin Hirschi (1965-68) University of Utah, Salt Lake City Karl S. Kalman (1962-65) School District of Philadelphia Richmond, Va. Isabelle P Rut ker (1965-68) State Board of Education, Augusta Schurrer (1962-65) State College of Iowa, CedarFalls Merrill E. Shanks (1965-68) Purdue University, Lafayette,Indiana Henry W Syer (1962.66) Kent School, Kent, Conn. Frank L. Wolf (1964-67) Carleton College, Northfield, Minn. John E. Yarnelle (1964-67) Hanover College, Hanover,Indiana PREFACE

things to differentpeople and exhibits manyfaces: Music means different harmony and musical sounds and tones;scales and modes;musical notation; musical compositionand forms; dissonance; rhythm,melody and counterpoinr, orchestral and symphonicmusic; acoustics the human voiceand choral music; of music by phonograph,radio, T-V, sound motionpic- and the reproduction facets of music related tomathe- tures. In what ways,if any, are these various mathematics contributed tomusical notation? tothe theory matics? What has instruments? to thehigh-fidelity of composition? tothe design of musical the composerawire of mathematicalrelations in- reproduction of musk? Is mathe- volved in music andmusical composition?Can the mathematician, as These are questions moreeasily matician, enrich thedomain of the musician? have been given are,for the asked than answered.Moreover, such answers as periodicals, often inaccessible.That is why most part,scattered through various enjoyment. It is hopedthat brought these essaystogether for your we have if they do not answer your they will at least open newhorizons for you, even You may then agreewith Morris Klinewhen he says questions completely. abstract of the of the arts can betranscribed into the most "the most abstract clearly recognized to beakin to sciences, and the moureasoned of the arts is the music of reason." William L. Schaaf

111 Contents

ACKNOWLEDGMENTS

FOREWORD .

HUMAN SPIRIT THE Two MOSTORIGINAL CREATIONS OF THE ELMER B. MODE

11 MATHEMATICS OF MUSIC Au R. AMIR-Maa

17 WEST NUMBERS AND THEMUSIC OF THE EAST AND ALI R. AMIR-MOEZ 21 SEBASTIAN AND THEWOLF . THEODORE C. RIDOUT 25 FOR FURTHERREADING AND STUDY ACKNOWLEDGMENTS STUDY GROUP takes thisopportunity to The SCHOOL MATHEMATICS in exptess itsgratitude zo the authors of thesearticles for their generosity allowing their material to bereproduced in this manner: Ali R.Amir-Moez, who, at the time his articles werefirst published, was associatedwith Queens College of the City Universityof New York; Elmer B.Mode, who was associated with Boston Universitywhen his paper first appeared;and Theodore C. Ridout, formerly master atBrowne and Nichols School,Cambridge, Massa- chusetts, who was a member ofthe Editorial staff of Ginnand Company at the time his article was written. its The SCHOOL MATHEMATICS STUDYGROUP is also pleased to express sincere appreciation to the severaleditors and publishers who havebeen kind enough to allow these articles tobe reprinted, namely: MATHEMATICS MAGAZINE ELMER B. MODE, "The Two MostOriginal Creations of the Human Spirit", vol. 35 (1962), pp. 13-20. RECREATIONAL MATHEMATICSMAGAZINE ALI R. AMIR-MoEz, "Mathematicsof Music", vol. 3 (1961), pp. 31-36. SCRIPTA MATHEMATICA Au R. ANTIR-M0Ez, "Numbersand the Music of East and West", vol. 22 ( 1956), pp. 268-270. THE MATHEMATICS TEACHER THEODORE C. RIDOUT, "Sebastian and the'Wolf' ", vol. 48 (1955), pp. 84-86.

vii FOREWORD Leibniz, the philosopherand a coinventor Nearly three hundred years ago, soul had this to say:"Music is the pleasurethat the human of the calculus, being aware that it iscounting". In more experiences fromcounting without architect, theosophistand philosopherClaude recent times,the renowned architecture is Bragdon once observedthat "music isnumber made audible, number made visible". the conviction thatmusic, These two observationswould seem to justify intimately and inextricablyassociated at least in someof its aspects, is somehow mathematicians of the with numbers andtheir properties. Theearly Greek firmly convinced of this.Since ancient times, men Pythagorean School were depends upon its have known that thepitch of a soundfrom a plucked string the ratios of thelengths of the strings aresimple whole length, and chat if Specifically, Pythagoras numbers, the resultingsounds will be harmonious. sounded a note, its fifthand its octave werein was awarethat lengths which the dis- the ratio 2:3:4. Infact, Pythagoras and hisdisciples believed that astronomical planets fromthe earth were alsoin a musical pro- tances of the through space, gression, and thattherefore the heavenlybodies, as they moved harmonious sounds: whence arosethe phrase "the harmonyof the gave forth only explanation ofthe spheres". The Pythagoreansfully believed that the Universe was to befound in the order and harmonyand perfection in the science of numbers, orarithmetike. convictioh was so deep-rootedthat for 1500 years, fromthe !ndeed, this classified knowledge as theSeven time of Pythagoras tothe Middle Ages, men (grammar, rhetoric,and logic) and thequadrivium Liberal Arts: the trivium the mathematical (arithmetic, astronomy, geometry,and music). Furthermore, of as follows: numbersabsolute, or arithmetic:numbers sciences were thought magnitudes in motion, or applied, or music;magnitudes at rest, or geometry; astronomy. observers said about musicand mathematics? Listen to What have other mathematician of J. J. Sylvester,the brilliant, poetic,temperamental British who contributed somuch to the theory ofinvariants the mid-nineteenth century The musician feelsMathe- and matrices: "Mathematicsis the music of Reason. thinks Music". Or again,the opinion ofHelmholz, matics, the mathematician and Music, the most physicist than themathematician: "Mathematics more the so related as toreveal the sharply contrastedfields of scientific activity, are yet binding together all theactivities of ourmind". Finally, from secret connection author of the "Danceof Life" and the pen of HavelockEllis, the celebrated culture: "It is not surprisingthat the perceptive interpreterof civilization and and again appealed tothe arts in order to greatestmathematicians have again their own work. Theyhave indeed found it inthe most find some analogy to although it would certainly varied arts, in poetry, inpainting, and in sculpture, abstract of all the arts,the art of number seem thatit is in music, the most and of time, that wefind the closest analogy." The Two MostOriginal Creations of the HumanSpirit ELMER B. MODE claim to "The science of Pure Mathemarks,in its modern developments, may claimant for this psi- be the most original creationof the human spirit. Another don is music." World. A. N. WHr1111113AD,Sanwa grad the Modem the luotation given above a greatAnglo-American I. Introduction. In interest, one a philosopher (I) characterized twodistinct fields of human science, the other an art. The artsand the sciences, however, are notmumaly to solv;s exclusive. Art has oftenborrowed from science in its attempts higher forms has many problems and to perfect itsachievements. Science in its Vivid aesthetic feelings are not atall foreign in the of the attributes of an art. in fact, wrote as work of the scientist. The lateProfessor George Birkhoff, follows: be beautiful, or amathematical proof maybe "A system of laws may either elegant, although no auditory orvisual experience isdirectly involved in desirability which is morethan case. It would seemindeed that all feeling of aesthetic feeling." (2) mere appetitehas some claim to be regarded as "there exists a Serge Koussevitzky, notedconductor, has stated also that profound unity between scienceand art." (3) It is not, however, the purposeof this paper, to discussthe relationships but rather to enumerate someof the lesser between the sciences and the arts, There is no known attributes which musicand mathematics have in common. attempt to establish athesis. The study of mathematicsusually begins with 2. Number and Pitch. has the natural numbers orpositive integers. Theirsymbolic representation by means of a radix orscale of ten, the principle been effectively accomplished indicates the power of ten tobe of place-value wherethe position of a digit multiplied by it, and a zero.The concept of numberis most basic in mathe- cardinal number, such asfive, is matics. We cannotdirectly sense number. A which comes to us from many concreteinstances each of which an abstraction which attributes not even remotelyconnected with the one upon possesses other of the hand, the is fixed. Such widelydiffering groups as the fingers our interest Dionne quintuplets, are sides of the pentagon, the armsof a starfish, and the each group to be matched all instances of "fiveness,"the property which enables correspondence with the other.The establishmentof or placed into one-to-one only good eyesight. such equivalence requires noknowledge of mathematics, mind we may state adefinition familiar tomathematicians. With these facts in of the The (cardinal) numberof it group of objects isthe invariant property group and all other groupswhich can be matchedwith it. constitute, however, but asmall portion of the The positive integers intervals in the con- numbers of mathematics. Theformer mark off natural

3 tinuum of teal numbers.The difference is readily sensed; between two smallgroups of objects man finds no difficultyin distinguishing between three and four, visually, atonce, objects, but thedistinction between, thirty-three objects say, thiny-two and calls for somethingmore than good vision. In music, study beginswith notesor tones. In western music representation is accomplished their symbolic by means ofa scale of seven, a principle tion, and therest, which denotes cessation of posi- of tone. Thereis something nent and unchangeableabout a given perma- may emit it, the clarinet note. You may sing it, theviolin string may sound it, and thetrumpet may fill the with it. The qualityor timbre, the loudness room soundmay be markedly different or intensity, and durationof one from another;yet among these differencesof sound thereremainsone unchanging attribute, a single such its pitch. This isthe same for now or any combinationof them. The be definedas the invariant pitch ofa note may then property of :henote and all other be matched withit. Notes which notes which way can be matchedare said to be in unison. Pitch, also, as an abstraction,derived from many auditory experiences.The establish- ment of pitch equivalencedoes not require ear. a knowledge of music,only a keen The notes of the diatonic scale markoff convenient tinuum of pitches.Within a given intervals ina con- range, the interval betweentwo tones of the scale is, in general,readily sensed, but outside of sucha range the humanear may fail to distinguishbetweenor even to hear ter of fact, "tones" removed two differing tones. Asa mat- from therange of audibilitycease to be such. As psychological entitiesthey disappear and in a physical medium. may be identified onlyas vibrations Invariance of pitchis an importantmusical musician not playinga kied instrument property and the ability ofa to maintain thisproperty for a given note is a necessary, butnot a sufficient condition for his artistry.This recalls the story of the distractedsinging teacher who, pupil, sprang suddenly after accompanyinghis none-too-apt from the piano,thrust his fingers hair, and shouted:"I play the white wildly throughhis sing in the cracks." notes, and I play the blacknotes, but you 3. Symbols. Mathematics ischaracterized by They are indispensable an extensive use ofsymbols. tools in the work;they constitutethe principal for the preciseexpression of ideas; vehicle be non-existent. without them modernmathematics would The mostimportant mathematical exceptions, in universal symbolsare, with few use among the civilizedcountries of the world. Music also isdistinguished bya universal symbolism. anything but thesimplest musical The creation of compositionor the transmission ofsig- nificant musical ideasis difficult ifnot impossible without the symbols ofmusic. Incidentally itmay be remarked that the page of a book in calculus page of a musicalscore and the are equally unintelligibleto the uninitiated. are very few fields of activityoutside of mathematics There (including logic)and music which havedevelopedso extensively their Chemistry and phonetics own symbolic language. are nearest in thisrespect.

4 mathematics preliminarytraining involves the acquir- In both music and symbols ing of technique.Mathematics demands suchfacile manipulation of become mechanical. We areencouraged to elimi- that the detailed operations the funda- nate the necessityfor elementary thinking asmuch as possible, once mental logic is made plain.This clears the way for morecomplicated processes of reasoning. copy-books and by "It is a profoundly erroneoustruism, repeated by all eminent people when they aremaking speeches, that weshould cultivatr the habit of thinking of what we aredoing. The precise oppositeis the case. Civilization advances by extendingthe number of importantoperations which we canperform without thinkingabout them." [4] In music also, the preliminarytraining involves a learningof technique. The aim here is to be able toread, or to write, or totranslate into the appro . priate physical actions, notesand combinations ofthem with such mechanical perfection that the mind is freefor the creation ;Ind theinterpretation of more profound musical ideas. 4. Logical Structure. Theframework of a mathematicalscience is well known. We select a classof objects and a set ofrelations concerning them. Some of these relations areassumed and others arededuced. In other words, from our axioms andpostulates we deduce theoremsembracing important properties of the objectsinvolved. Music likewise has its logical structure.The class of objects consistsof such musical elements as tones,intervals, progressions, and rests,and various relations among these elements.In fact, the structure of musichas been formally procedure of mathe- described as a set of postulatesaccording to the customary matical logic. [5] In mat. ..matics a developmentis carried forwardaccording to the axioms If these are obeyed the results are correct,in the mathematical or postulates. law sense, althoughthey may not be interesting oruseful. Mere obedience to does not create an original pieceof mathematical work. Thisrequires technical skill, imagination, and usually adefinite objective. Music also has its axioms orlaws. These may be assimple as the most of all obvious things in elementarymathematics the whole equals the sum they may be less obvious to its parts if we are counting beats in a measure; the layman, such as the canonsof harmony or the structurallaws of a classical symphony. Here again we mayfollow the laws of musicscrupulously without Technical skill, imagination, cvcr creating aworth-while bit of original music. the fortunate mood, andusually a definite objective arerequh.ites for the creation of a composition which notonly exhibits obedience tomusical lay s but and expresses significantideas also. Occasionally themusician becomes bold violates the traditional musical axioms sothat the resulting effects may atfirst sound strange or unpleasant.These may become as useful,provoking, and en- joyable, as a non-Euclidean geometry or anon-Aristotelian logic. In such manner did Wagner,Debussy, Stravinsky, and othersextend the bounds of musical thought. In mathematics as well asin music one may have tobecome accus- tomed to novel developmentsbefore one learns to like them.

14 Benjamin Peircedefined mathematics sary conclusions." The as "the science whichdrawsneces- operations fromhypothesis order withoutlogical hesitation to theorem proceedin logical tions flows swiftly or error. When theseries of deductive and naturallyto its inevitable opera- structure gives a conclusion, themathematical sense of satisfaction,beauty, and acterizes the completeness. Sullivanchar- opening theme ofBeethoven's Fifth mediately, in itsominous and Symphonyas one which "im- state of expectance, arresting quality,throws the mindinto a certain a st.-te wherea large number of a certain class,can logically follow." happenings belongingto phrase of the [6) Thesame is true of the prelude to Tristtanand Isolde, opening masterpiece. or any reallygreat enduring An interesting departure fromthe usual logical compositionoccurs in the Symphonic structure of a musical Instead of the Variations, "Istar"by Vincent initialannouncement of the d'Indy. variations, "the musical themewith itssubsequent seven variations proceedfrom the point tation to the finalstage of bare thematic of complexornamen- Boston musical simplicity." Philip critic relatedthe following Hale, the eminent Bulletin of April23, 1937. anecdote in theBoston Symphony "M. Lambinet, a professor ata Bordeaux public text 'Pro Musica' forhis prize-day school, chose in1905 the speech. He toldthe boys that the the study ofmusic wouldteach them first thing ment logic plays would be logic.In symphonic as great a partas sentiment. The deveiop- full of musicaltruth, whence theme isa species of axiom, sounds proceed deductions.The musician as the geometricianwith lines and deals with master wenton to remark: 'A the dialectitianwitharguments. great moderncomposer, M. Vincent has reversedthecustomary process &Indy, degrees unfolds in his symphonic from initialcomplexity the poem "Istar." He by therein and simple idea whichwas wrapped appears only at theclose, like Isis up discovered andformulated.' The unveiled, likea scientific law a musical work speaker foundthis happy definition 'an inductivesymphony.' " for such 5. Meaning. Amathematical formula accurate representation represents a peculiarly of meaningwhich succinct and means. It is concerned cannot be duplicatedby any other with thephenomenon of functionconcept. "A mathematical variability; itinvolves the formulacan never tell but only howit behaves." [7] us what a thing is, How true this isof music! Atheme of small interval of great musiccompresses into space or time,inimitably and a of meaning.Music is accurately,a remarkable wealth not fundamentallyconcerned with static physical objects,but with the the descriptionof impressions theyleave undervarying Debussy's "La Met"is a fine exampleof this aspects. is oftennot in the physical type of description.Music's interest One of the man but in his changingmoods, in his sources of thegreatness of "Die emotions. portraying vividly Walkiire" isWagner's genius the conflictingaspects of Wotan's for man. nature as god and as The meaning of musicalmotive grows with or developed and from study. It isusually exploited it are derivednew figures of musical expression. Agood 6 I .? than the casual hearingbefore its deepsignificance is theme demands more insignificant completely appreciated. Itis often worked upfrom an entirely Fifth Symphony or inMozart's G minorsymphony. motive as in Beethoven's which can In mathematics a basicformula or equation mayhave implications after much study. It may appear tobe almost trivial asin be understood only elegant as in the case of a + b =6 + a or it may beless obvious and more the case of Laplace'sequation, Zati+viu 0 .

gives expression to concepts Music consists ofabstractions, and at its best Beethoven's music expresses which represent the mostuniversal features of life. struggles, joys, andtragedies of human exist- powerfully the great aspirations, Napoleon in mind ence. TheEroica symphony mayhave been composed with than the career of asingle man. It is aportrayal of but it portrays far more known the heroic in man and assuch is universal in itsapplication. It is well composition may producedifferent responses among that a musical passage or constitues one of the sources people. The possibility ofvarying interpretation It is an evidenceof its uni- of music's uniquenessand a reason for its power. fundamental differencebetween music and painting or versality. Herein lies a emotional sculpture. The effect of amusical episode is due toits wide potential painting or piece of sculptureis due to its con- applicability; the effect of a been gen- creteness. Attempts atabstract representationsby painters have not erally successful; attempts atstark realism in musichave likewise failed. Music form, as for example,Bach's or Mozart's, oftendefies ap- in its most abstract in the realm of plication to the concrete. It seems tobe above mundane things, pure spirit. abstract, and universal So it is with mathematics.Our conclusions are always originated from a specialprob- in their application,although they may have of interpretation andapplication of a giventheorem or lem. The possibilities said that even the same formula are unlimited.Poincar6 is reported to have mathematical theorem has notthe same meaning for rwodifferent mathema- Laplace's equation is set up ticians. What differingreactions may ensue when degrees of abstractness before an audience ofmathematicians! What differing are suggestedby the two equationspreviously written! 6. The Creative Process."It is worth noting . ..that it is only in mathe- .the boy mathematics and music that wehave the creative infantprodigy;. utilizing a store of impressions, matician or musician,unlike other artists, is not emotional or other, drawnfrom experience or learning;he is utilizing inner resources. ..."(8) Statements of this typehave led many to believe thatmathematical talent and musical talent have morethan an accidental relation.Some feel that mathematicians are morenaturally drawn to musicthan musicians are to mathematics. As far as the writerhas been able to ascertain, noserious investi- gations on the relationbetween the two talents havebeen published. A brief the child prodigy study of exceptionally giftedchildren yields no testimony that

7 in music hasmore than the prodigy in average mathematicalsense, or that the child mathematics hasexceptional ability In a in music. recent article, Mind andMusic, [9] the critic, ErnestNewman, discusses inimitable Englishmusic play in the the role thatthe subconscious creativeprocesses of music. Hampered mind might mony on this subject, by a dearth ofreliable testi- he attempts,nevertheless, and Wagnerhad written of to estimate this role.Berlioz their creativeexperiences without self-analysis. So alsohad Mozart attempting any Newman feels that although Newmandoes not referto him. the Memoirs ofHector Berliozare not too reliable in respect Wagner's letters,however, this ideas seem to indicate thatmany of his musical were the..,sult ofan upsurge from the ideas long hidden unconscious depths ofhis mind of but suddenlycrystallizing. The was often displaced by activity of hisconscious mind the upward thrustof these latent The interpretation creative forces. thus suggestedis strikinglysimilar in to that described byJacques Hadamard many respects in his Essayon the Psychology of vention in theMathematical Field. In- the related (10] This notedmathematician draws experiences ofPoincaré, Helmholtz, on the origin of Gauss, and othersto discuss the inspirationor sudden insight that or initiates an original contributesto, completes, work. The roleof thoughtsthat lie cernible in thesubcor cious,only to become, vague and undis- after a period of of a sudden,clear and discernible unsuipectedincubation, is described cannot affirm, of in undogmaticterms. One course, that theseopinions concerning are confined solelyto music and the creativeprocess are voiced by mathematics, butit is interestingthat they two eminent scholars,one from each field. The greatest worksof music as well as by their are distinguished bytheir intellectualcontent emotional appeal.The sacred music of Beethoven, of Bach, thesymphonies or the operas ofWagner, offer subjects sion, as wellas opportunities for for analysis anddiscus- ideas to "work emotional experience.Each out," ideasto be developed and composer had artifices of music, clarified by theforms and the object beingto make their full appreciative listener. significance feltby the Mathemsticalcreativity involves ment. Concepts very much thesame general develop- must be clarified,operations carried vealed. If theseare significant and out, latent meaningsre- and asense of completeness logically developedthe result hasa unity which bringsintellectual and both author andreader. aesthetic satisfactionto 7. Aesthetic Considerations. Tomany, mathematics bidding subject.Its formseems to be more like seems to be a for- of a living, that ofa skeleton than that breathing, humanbody. This idea, abstract character is, ofcourse, derived from and from thedemands which its concepts, terse methods it makes forsharply defined of expression,and precise sense, mathematics lacks rules ofoperation. Ina richness, if by richnesswe mean the impurities whichimpart savor and presence of those of concrete color. Theseimpuritiesmay be in the examples, illustrationsfrom, or nature mathematics. They applicationsto fields other than may represent departuresfrom the normal development, andmay make no contributions abstract logical whatsoeverto the formalstruc- 8 from the richness of turc whichconstitutes mathematics.But if we subtract its purity, for inmathematics the structure orform mathematics, we add also to mathematics to than its applications.We may apply the is more important these applications are manyproblems associated withhuman existence, but they lie apart from it. not essential partsof pure mathematics, but because it is purestit is also "In music the flavorof beauty is purest, form. Its content is itsform and its form is least rich. ...A melody is a pure in one means a changein other. We can,of course, force its content. A change do so upon it, readinto it stories orpictures. But when we an external content music." [I I] we knowthat they are extraneousand not inherent in the for its fields areunlimited in In a different sensemathematics is over-rich extent andfertility. multiple fields of modern "But no one can traversethe realm of the realize that it deals with aworld of its own aeation,in Mathematics and not anything to be found inthe which there are strangelybeautiful flowers, unlike life of their own, different world of external entities,intricate structures with a new andfascinating laws from anything in therealm of natural science, even drawing conclusions morepowerful than those wedepend of logic, methods of different from those wecherish upon, andidealcategories very widely most." f 12) unique One needs here but tochange a few words inorder to describe the and harmonies of music areits own and lovely creations ofmusic. The melodies mysteriously beautiful,incapable of descriptionby inventions. They are often about us. A other means andwithout counterpartelsewhere in the world be of the utmostsimplicity or of the mostintricate musical composition may inexpressible." It is exactlythis character, yet it may"well-nigh express the "inexpressible" ideas that givemathematics and music ability to convey the meaning or much in common.The mathematics studentwho seeks always a appreciate the powerof that picture of each newproposition often fails to which defies representation. There is much of interest tothose who love both music 8. Conclusion. mathematicians on the bear- and mathematics, andmuch has been written by other. Archibald haswritten delightfully of someof ings of one field on the evalua- their human aspects aswell as the scientific.Birkhoff has attempted the aesthetics by quantitativemethods. Miller and othershave tion of musical problems of musical tone brought the instruments ofphysics to bear upon the and acoustics. depend upon very muchthe Success in musicarid in mathematics also fine technical equipment,unerring precision, andabundant same things beauty. imagination, a keen senseof values, and, aboveall, a love for truth and REFERENCES 1. WHITEHEAD, A.N., Science and the Modern W orld,NewYork, 1925,p. 29. 2. ButscHoFF,G. D., Anthrtic Maws, Cambridge,1933, p. 209. 3. At a symphonyconcert given in honor of the AmericanAssociation for the Advance- ment of Science, Boston,Mass., Dec. 27, 1933. 4. WHITEHEAD,A. N., As hstrodartiott to Mathematics, NewYork, 1911, 5. LANGER, S. K., p. 61. A Set of Postulatesfor the Logical Structure 1929, pp. 561-570. of Music, Monist,Vol. 39, 6. SULLIVAN, J.W. N., &peas of Science, london,1923, p. 167. 7. JEANS, J. H., The Mysterious Universs,Cambridge,1930, pp. 151, 152. 8. See 6, p. 168. 9. NEWMAN, E,Mind mod Music, SundayTimes ( London ),July 8, 1956. 10, HADAMARD, J.,Ars Essayon the Psychology of Princeton, 1945. Itivention us theMathematical Field. 11. SCHOEN, M., Ars and Beauty, NewYork, 1932,pp. 190-192. 12. SHAW, J. B.,Cause, Purpose, Creativity, Monist, VoL33, 1923, p. 355.

10 Mathematics of Music ALI IL AMIR-MOEZ Though music may seem far removedfrom what many think are thecold logical aspects of mathematics,nevertheless, music, with its emotionalappeal, has a mathematical foundation.The following article will showhow highly mathematical are the sounds, the scalesand the keys (the parts, so tospeak) of music. 1. Harmonics of a Sound:When a sound is made, for example,by striking a string of a musical instrument,each particle of air next to the source of the sound vibrates. Weshall call the number of vibrationsof that particle of air in one second the numberof vibrations of the sound. Thelarger this number is, the higher the pitch ofthe sound becomes. Suppose a sound is called C,and its number of vibrations is c. Thatis, if, for example, the sound C, andits number of vibrations is C. Thatis, for example, the sound C makes a particleof air vibrate five hundred times in one second, we say c=500. It wasdiscovered by Greek mathematicians that ifafter the sound C is heard wemake another sound S whose number ofvibrations is twice the number of vibrationsof C, i.e., 2c, then S will be pleasant toheat. As far as the history of mathematicsshows, this idea is due to Pythagoras.The sound T with three times as manyvibrations, i.e., with 3c vibrations, is also pleasant to hear after C. This fact is truefor sounds with vibrations c, 2c, 3c,4c, 5c, etc. Usually, if we play thesesounds successively in some order with acertain rhythm, we call it a melody. If we play afew of these sounds together, wecall it harmony. We shall call the sounds withvibrations 2c, 3c, 4c, etc. harmonics of C. 2. A Primitive Scale: In thework of Omar Khayyani*, it ismentioned that the study of the ratios of integers isessential for the science of music.That was the onlymathematics used in the Greek theoryof music. To explain the idea, we start with the sound C, and we supposethat C, is the name of the sound with 2c vibrations. Let us call G, thesound whose number of vibrations is 3c. ( We shall explain why wehave chosen these names. )If a sound with twice as many vibrations is aharmonic of a given sound, it isreasonable to believe that the sound G with one-half as manyvibrations as G, is a harmonic of C. Thus we can say thatthe sounds C, G, and C, are harmonicof one another, and their vibrations are, respectively, c, " and 2c.We can compare these sounds and their vibrations by constructing thefollowing table.

Sound C G C, 3 c I -,..--, 2 .4

°Omar Khayyam. "Discussion of Difficuhies in Euclid,"Scripm Aftobtaramica V. 24, pp. 275 303 (1959). I I The first line of thetable shows the line shows the name of each sound, and the second corresponding number ofvibrations. For example,under G we see 3/2, which means that Ghas 2(.12 vibrations ina second. The names chosen hereare actually those chosen in the natural C of the scale, scale. If C is the then G has 3/2as many vibrations as C. C, is thenext so-called C, which isusually called theoctave of C. In this scale we have only three sounds. Ifwe play C, G, and octave of C on the piano, we can almost see how they sound. Ofcourse, we cannot make much music with threesounds. 3. Oriental Scale: Letus extend the idea of section 2 the fifth and seventh further. We rake harmonics of C, i.e., thesounds whose numbers tions are 5c and 7c. of vibra- We call these souads,respectively, E, and K. explain the choice of We shall the subscripts shortly.Let us compare these sounds C the octave of C, and with with C2, theoctave of C. Note that 2c is thenumber of vibrations of Cand 4c is the number ofvibrations of C,. Thus, if sound with half E, is a as many vibrations as E thenwe see that 'el: is the number of vibrations of E. Similarly,we can choose a sound K, whose tions is If we number of vibra- compare these sounds accordingto their pitch, we get them in the order C E K,,C. This is clear because 5 7 2 << < 4. 2 2 Since these soundsare all harmonics of C, the sounds half as many vibrations E and K, which have as E, and K respectively, i.e., '".14and 'ci are also harmonics of C. As insection 2, we can makea table as follows: Sound CEGK C1 c 7 -4- 2 . These five sounds together approximately constitute theoriental scale. 4. Middle-East Scale: If we proceed withwhat was done in section3, we get more sounds in the scale.Since the sounds with etc. do not contribute vibrations 2c, 4c, 8c, 16c, to the scale, we choose the soundsbetween them. In particular, let us call D, thesound with 9c vibrations. and B, with vibrations, We also choose P H respectively, 1 lc, 13c, and15c. As before,we may choose D2, P, H and B,with vibrations "" "`/2, ""and "" respectively. Then we choose D, H, and B with vibrations ""/, ", fr/, and '"/,respectively. We shallconstruct a table as before. Sound -CD.E P GH K B C, c 9 5 11 3 13 7 15 1 -11- -4.1 T 2 i _ I T A scale may be madeout of these sounds with eight instead of names in the scale seven. Before we discuss thisset of sounds, we makea table using the theoretical (physical) sounds of the scale.

12 13 , - Sound C D E G ABC1 5 15 4 3 2 c 1 jj _ If we compare P and F, we seethat that the ratio of thenumber of vibrations of P to the number of vibrationsof F denoted by P 11 4=33 F 8 3 32 This shows that P is sharperthan F. This is where themiddle-east music is different from the physical scale. Thesound H with "" vibrations is notused in the middle-east music. Thus,C, D, E, P. G, K, B, C,approximately constitute the sounds of themiddle-east scale. We see that K 7 5 21 A 4 3 20 Therefore, K is also sharper than A. 5. Tones and half-tones:If we study the physical scale, weobserve that 9 _E 10 F 16 ("i _9 A10 B_9 C16 p 8 'B 15 C 8 ' D 9 *E 15 ' F 8 9 A This suggests the idea of smalland large intervals or tonesand half-tones. We shall write this as follows:

SoundC D E F A B c 1 I i ; 1 : I A I 1 i i I 1 I 1 i Tone I 1 I l'-: : 1 I, i :l z- L I , I . 1. . ,

The above table indicates whichinterval is a tone and whichis a half-tone. For example, between E and Fis a half-tone. But, we reallyshould say large and small intervals. 6. Geometric Progression:An ordered set of numbersis called a geometric progression when theratio of each one to itspredecessor is always the same. For example, the set 5,10,20,40,80, ... of each number to is a geometric progression.The ratio is 2, that is, the ratio the one before it is two.Indeed, we can produce as manymembers of this set as we desire. If one member of a set and theratio are given, we can alwaysproduce as many members asneeded. For example, if 1/2 is amember of the progression and the ratio is VI, then we canwrite some of the membersof this progression, such as 1 Ir2) ( V2) = 2 (VD) -1 2 '(2 2 2 of them 7. Geometric Means:For two numbers, the geometric mean is the square root of theproduct of them. This is a sortof average, similar to one-half of the sum, which iscalled the arithmetic mean. Asfor the arithmetic of average of a fewnumbers, we add them anddivide the sum by the number

13 them; for thegeometricmean of a few numbers the root of orderequal to the number we multiply them and take mean of of them. Forexample, the geometric 5,7,2,6 is 'V(5)(7) (2) (6)= 8. Modern Scale: Sincetwo sounds are compared of their numberof vibrations in terms of theratio rather than thedifference of the brations, in orderto make all intervals number of vi- we need to take the equal and calleach onea "half tone," geometricmean of twelve half-tones of the scale. Thusthe number of vibrationsof the soundsin the modern progression whichhas 1 as scale forma geometric a member and "VIas its ratio. Thus the modern scale can beshown in thefollowing table: 'Sound IC D E F G- A B c 1 _(l V-2)2 J (uNri)'r VIY("11/)'(" VI)'(I' V2)",2 As we observe,the power of "V2 and it increases increases by 2whenever we havea tone: by one wheneverwe have a half-tone. The modernscale is not really but with slight as natural to theear as the old Greek scale; training, theear gets used to it. The modulation from important fact is that one key to anotherbecomes extremely There is easy. one disadvantage inthe modern scale, monic of C, i.e.,G, becomes namely, the thirdhar- slightly flat. Thesound G is called of the scaleand, being flat, the dominant makes the muskdull. We shall mathematically. Inthe modern scale show this fact G= = 1.498

But, in the naturalscale

1.5 C 2 This mistake isalways correctedin the violin. This an orchestra with string is one of thereasons that instruments soundsmuch better thana piano solo. 9. Major Keys:A sample of thescale of section 8. This is a major key is theone in called "C major"since it starts with tonic of the scale. C. C is also calledthe In any major key,the sound (notes) same relation to of the scalehave the one another as theones in C major. That is, tween the third and fourth the interval be- elements isone half-tone; also the the seventh andeighth elements is interval between one tone. a half-tone, and theother intervalsare all The next majorkey is G major.This has been One is that thenote G is the third chosen fortwo reasons. has a higher harmonic of C; theother is that this pitch. Note thatgoing from C key change the scale. to its second harmonicdoes not The table of thescale of this keyis as follows:

14 - - Es Fi* Gi fa A B I Cs Di 1p2)" 2 es V2)" (a VZ° ea ("WZ"-' C e vu:ev'ir'(-viz" between the seventhand We observe that inorder to have the interval have to use F (F, sharp) eighth, i.e., subtonicand tonic, a half tone, we "N(7) " instead of F, with ( with vibrations ( the first of a newscale, we get If we choose thefifth note of this scale as the key of D major.This key needs two sharps. tables for several majorkeys The reader may trythis idea and work out major. which come after G pitch, it is also possible As it was possible to getmajor keys with higher to get majorkeys with lower pitch. for which Suppose we look atthe table in section 8and consider a scale the fifth note is C.This will have thefollowin,g table: , E F. G. A. B.' IC D SoundI (13 1 (1'1,11? C3Nr2Y V2). c C" V12)-* (2 V2)-* (1'V-2)4 between the Here we have to useBb, i.e., B flat, in order tohave the interval be a half-tone. third and fourth notes flat. We leave it If we proceed in this way,each lower key has an extra and write tables for thecorresponding to the reader toproduce many major keys scales. To imitate the cryingsound of middle-eastmusic, 1 a Minor Keys: written in minor keysavoid minor keys seem to be proper.Most older pieces but we find thiscombination the very large intervalfollowed by a half-tone, of sounds in recent pieces. only Many forms of minorkeys have been considered.We shall describe one of the most recentpieces. instead of going to thethird harmonic of C, we To obtain a new scale, key. the fifth harmonic of C.But, this key is notthe simplest minor may go to The table Thus we move down toA, whose fifth harmonicis approximately C. of the scale for A minoris the following: 0-# A B. C D E F Sound es Iry (.1v2) c vly. (i, v2). (u v.2).s (s1.(2), I (' v-2),

shows, it is desirable tohave a half-tone intervalbetween As the physical scale As we see, the subtonic and thetonic of a scale. Thisbrings Qt into the scale. and G# is one and a half tones. the interval between F similar to that by Other minor keys areobtained from this in a manner leave it to the reader to which the major keys areobtained from C major. We obtain them. his knowledge of music It would be veryinteresting for one to compare with what has beensaid here.

1 5 Numbers and the Musicof the East andWest ALI R. AMIR-MOEZ Let C be a sound whose numberof vibrations is c. Having heardC, any sound with number of vibrationsequal to kc, k 1, 2,... ,is pleasant to hear. These sounds are called the harmonicsof C. If we play a few of thesesounds successively with some rhythm, apleasant melody is made. If weplay a few of these sounds together, a richsound comes our and it is calledharmony. A sound C with vibrations 2c is calledthe octave of C. Let us consider two octavesof C, say e and C", and constructthe following table. Sound C G Ci G' C"

c l 1 2 3 4 2

As shown in the table letG' be the third harmonic of C. Sincethe sound G' with 3c vibrations is pleasant to beheard with C, it is reasonable tothink that G with half as manyvibrations, i. e., c, would alsomake harmony with C. In fact this sound is calledthe dominant of the scale. This way weobtain only three sounds in the scale.Now in order to get more soundsin the scale let us consider three octaves of C, sayC' C", and CH. Let us construct thefollowing table. K" SoundCE G KCEiG'leC"E"G" 0" 5 7 5 3 7 2 i 3 4 5 6 7 8

As shown in the table the5th harmonic of C is calledE". Again it is conceivable that since E" is aharmonic of C, also E' with vibrations c,and E

with vibrations c will be harmonicwith C. A similar process can beused for the 7th harmonic of C. calledK", and K', K could be foundaccordingly. This sound K is missing in the physicalscale. If we consider only thesefive sounds. the scaie will be as follows: ScundODD= MEI111 An approximation ofthese sounds appears frequentlyin the music of the Far East. Now let us consider four octavesof C, say C, C", C8", andC".. In order to get moresounds in the scale, we constructthe following table by considering the harmonics of orders 9,10, 11, 12, 13, 14, and 15of C, and putting tones withof these frequencies in thescale.

17 Sound3 D DOG Dm EICKEiltacin EwGISICSIMIGIS c IIDOEMBEINIIIIIIIII10ell131111116 Now theoretically speaking the soundsof the scale standas follows: SoundCD mac am 1 IffilliKill Comparing this tablewith the previousone we see that 1:F.= if. 4 33. 8 3 1. e., P is sharperthan F. Some people half tone exists in are of the opinion that halfof a the music of themiddle east, but thisis not true. The difference is actually theuse of P instead of F. As we shall show in the modernor practical scale, F isslightly sharper in be equal to P. theoretical scale, but itis still too flatto We see also that Hand K are replacedby A. An experiment of Science, University at the School of Teheran, madeit certain that H,i. e., the 13th har- tnonic of C doesnot exist in the music of the instead of A. The ratio middle east. ButK is also used K:A po

shows that K is sharperthan A. Now if we look at the intervals in the theoreticalscale we see that

D:C=:,E:D=142,F:E=1,6:F=:.A:G=8:A=2,andC:B=9 8 116.5 This has suggested the idea of tones and halftones for today's music. Looking at the preceding intervalswe have approximately 12 half C.In practical scale all tones between C and half tonesare at equal intervals, i.e.,we have to have the geometricmean of these half tone intervals. Clearly the productof intervals, i.e.,9 10 16 9 10 916 2. 8 9 15 8 9 8 Therefore the geometric mean, considering 12 halftones, will be "Na. Wecan really describe each halftone as a term in a geometric term is 1 and its ratio is "1/2. progression whose first Let us compute theratio of the dominant Clearly G is the 8th and the tonique,i. e., term of the progression andits frequency willbe

18 23 a 1.498 which is a little moreflat than -; = 1.5. This is the place that in violin is always corrected. Theadvantage of the modern scale is that modulation from one key to another is very convenient.In fact musicians say: "If the sensitive ear of a musician does not distinguish theseslight differences of sounds, the mathematiciins ear of course wouldn'tdistinguish them either." But the musician's ear has been trained toappreciate the sounds of practical scale. However any simple melody of this sortsounds harsh to a Persian tribesman.

1 9

2 Sebastian and the Wolf THEODORE C. RIDOUT This is not another parody onLittle Red Ridinghood. It israther a tale of how a musician battled a "wolf,"and how a schoolteacherfought the same fight in the classroom. The "wolf" we are interested inis a beast that plaguedthe makers of musical instrumentz for many centuries.Although many attacked himwith vigor, and Johann Sebastian Bachlaid him fairly low, the ghostof the critter still haunts our concert hallsand keeps turning up in thephysics laboratory. Since he has a mathematical origin,it seems fitting that weshould discuss him in these pages. The problem is something likethis. You tune your violinby fifths, ad- justing string tension until youhear a perfect fifth when twostrings are sounded together. The fifth is a basicunit in musical tuning. But so, also, is the octave.Take a very simple one-stringedinstrument, Pythagoras' monochord. Every time wequadruple the tension on the string we double the rate of vibration, andthe tone goes up an octave. Starting at a single vibration per second, the processof doubling the frequency ofvibration takes us up in geometric progression tofrequencies of 2, 4, 8, 16, . .. until we reach 256, where we pause forbreath and call the tone "middle C."If we double again we get "upper C," at afrequency of 512 per second, and so on, until the pitch is too highfor even your dog to hear. According to the laws of physics,if C has a frequency of 256, theperfect fifth above it, G, will have afrequency of 256 X 1.5, which is 384.Here C:G 1:1.15, or 4:6. Moreover thetriad (or chord) C-E-G sounds bestif the frequencies of these notes havethe exact ratio 4:5:6. Thesecombinations are pleasing to the ear because theharmonics, or overtones, as well as thefunda- mental topes, combine with aminimum of conflicts or undesirablebeats. Filling in, we have the following notesand their relative frequencies: 9k) Ei 5 F ( k )- G( k )- A(5- k B k )-- C.,(2k) C (frequency k)-- D 4 1 2 3 8

This is known as a purescale, and its tones have exact harmonicrelations to the keynote. Music played on aninstrument so runed sounds richand ethereally beautiful. So much for theory. I will nowretire to my workshop and construct a piano. Starting at a verylow C (about 32 vibrations persecond) I go up bv perfect fifths along the musicalalphabet. The nores will be C, G, D, A, E, B, . G$, Eb, Bj, F, and C. Frombottom C to top C is just seven octaves.This looks like the beginning and theend of a complete and perfectkeyboard, and I flatter myself I can fill inthe other notes in proper ratios tomake a shining row of ivories. Butfirst I had better check myfifths. Going up from C to G, 1increased my frequency by the correctfactor, 1.5. From G up to D. I againmultiplied by 1.5. In all, I multipliedtwelve

21 times bthis factor, so that rny top C vibratesat a frequency that is (1.5)" times that of the bottomC, or 129.75times the original. But suppose I go up by octaves? Startingat the same C as before and doubling frequencyuntil I have that is 21, gone up seven octaves, Iarrive at a frequency or 128, times the original.I now have difference of about two top C's, and there isa a quarter of a semitonebetween them. Whatto do? VariOLIS solutionshave been offered. known as the For many centuric4a compromise mean-tone scale was used. Thiscan best be illustrated by the twelve fifths arranging around the dial ofa clock, with C at twelve o'clock, and o'clock, G atone so on. Instead of the fifths beingtuned in the are tuned in the compromise correct ratio 1.5, they ratio Y5, which equals1.49535, introducinga very slight error. Thiserror multiplied twelve times Suppose I have the leaves a gap somewhere. gap come after 11on the clock, between F and my keyboard. Instead of top C on a normal interval of 7semitones here, I have interval of 7.4semitones. an A musical compositioninvolving this overgrown interval would beany- thing but harmonious,and mighteven howl like a wolf in interval cameto be known as the quinte-de-loup, the forest; hence the or wolf fifth. One couldplay only in certain keysthat did notrun afoul of this beast. Mathematicians ofcourse came to therescue, but their aid appreciated. As far back was not always as 1482 a Spaniard namedBartolo Rames proposed an equalized tuning in whichall semitones ..hould or 1.05946 times the go up in the ratio of "VI, note below. This dividesthe error exactly cessive intervals, slightly between suc- reducing each fifth,so that a completeset of twelve fifths can begin andend on exactlythe sax_ With the intervening tones as a set ofseven octaves. notes filled in, we havea slightly imperfect but useful tuning knownas equal temperament. very This solution made littleheadway, in spite of appearance of a mighty figure various advocates, untilthe on the scene. Thiswas none other than Johann Sebastian Bach, who proposedequal temperamentfor all keyboard and proceeded instruments, to rune his clavichord andharpsichord accordingly. thus able to play inany one of the tweleve possible He was from one key keys, and couldmodulate to another without encounteringany wolves. To demonstrate thesystem he composeda series of twenty-four and fugues, making preludes use of all twelve keys. Thiswas in 1722. He later peated theprocess, giving us in all forty-eight re- pieces under thetitle "The Well- Tempered Clavichord."The term "well-tempered"of course refers temperament. It is generally agreed to equal that this workwas chiefly responsible for the present universaluse of equal temperament. It was many generations beiore themean-tone scalewas abandoned in the tuning of pipeorgans; and Bach was forced to compromise with thewolf at the car note. For thisreason his organ compositions simpler kc are written only in the My a :bra class was naturally interested in allthis. Plainly "Nr2 to the twelfth power is 2,so that twelve semi-tones raised will fit perfectlyinto an 22 octave, forming acontinuous chromatic scale. Wechecked the value of the radical on a slide rule. The physicsdepartment demonstrated a sounddisk for us, proving to allthe neighborhood that the frequenciesunder discussion produced a musical scale. The disk contained8 rows of holes, corresponding tothe notes of the scale, asfollows: C (24 holes), D (27 ), E (30), F(32 ), G (36 ), A (40), B (45 ), C (48 ). When spunby an electric motor and played upon with a jet of compressed air, such adisk gives off "musical" tonesapproach- ing the power of a steam calliope, andin the exact harmonic ratios ofthe pure scale. My students wrote papers on suchtopics as the clavichord, pipe organs, orchestration, electronic instruments, acoustics,and so on. One or two whose musical background was strongerthan the mathematical weresomewhat shocked to find figures encroaching onthe province of the Muse. Thesedis- senters wgre cheered,however, to know that Robert Smith in1759 characterized equal temperament as "extren-ely coarseand disagreeable," and that Helmholtz in 1852 considered that itmade every note on the piano sound "false and disagreeable," and that onthe organ it produced a "hellishrow." Helmholtz had used just intonation, ashe called it, or pure tuning, forhis experimental harmonium, and like many amusical expert, became so con- ditioned to perfect harmonies that hefound those of equal temperament very distasteful. Thus, thanks to the Queen ofSciences, the "wolf" has now become a thing of the past, though my youngdaughter tells me that one or two of hiscubs show up occasionally at thehigh-school band rehearsals. FOR FURTHER READING AND STUDY The following references are not all equally significant.Some of them are quite general, others are either marescholarly or more technical. All of them, however, are relevant Those that are probably mostilluminating and readily accessible have been indicated by. an astriisk. Not tohave been so labeled does not imply that the reference is any way unscholarly orwithout merit. °ARCHIBALD, P.. C Mathematics and music AmericasMathematical Monthly 31:1-25; 192t °BARBOUR...J. MURRAY. Musical logarithms. Scripta Mathernafica 7 :21-31;1940. Beabout, J. MURRAY. The persistence of Pythagorean tuning system.Scripta Maths- viatica 1:286-304; June 1933. HAMILTON, E. R. Music and mathematics. New Era 15: 33-36; January,193t Huu., A. E Music and mathematics. The Monthly Muria Record 46: 133+ ;1916. *HurrnNoToN, E. V. A simplified musical notation. Scientific Monthly11:276-283; 1920. °KLINE, Moms. Mathematics in Wtfterli Culture. New York: Oxford University Press, 1953. 'The sine of G major", pp. 287-303. LANGER, SUSANNE. A set of postulares for the logical structureof music. Monist 39: 561- 570; 1929. °LAND, FRANK. The Language of Mathematics. London: John Murray,1960. "Logs, Pianos and Spirals", pp. 117-132. MORE, ThENCHARD. sem ial expression for music The DuodecimalBulletin 5:25. BENSE, MAX. Kowtows einer Gektesgeschicbte der Matbemafik.Hamburg: Classen & Govers, 1949 ( 2 vol.). "Die Vereinbarung von Musik undMathernatik", voL 2, pp. 183-206. PETERS, I. Die math-mats:schen sod physikalischenGrundlagen der Musik. Leipzig: Teubner, 1924. PETERS, I. Die Grundlagen der Musik. Einfiirhung inihre mathematisch-physikalen und physiologisch-psychologischen Bedingungen. Leipzig: Teubner, 1927.

W L.S.