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) *********************************************************************** Reproductions snpplied by EDRS are the best that cam be made from the original document. *********************************************************************** "PERMISSION TO REPRODUCE THIS U S DE PE* /NE NT Of MATERIAL HAS BEEN GRANTED SY EOUCATION WELFAIE NATIONAL INSTITUTE OF IEDUCAT1ON 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 curric- ulum 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 as- sociated 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