Mathematics

a practice in the mathematics of tuning instruments and analyzing scales

 

© 2000-2008 Cristiano M.L. Forster
All rights reserved.

 

www.chrysalis-foundation.org

 

 

Dear Reader,             Since July 2002, more than 250,000 visitors have logged on to www.chrysalis-foundation.org.  Your interest has encouraged us to plan a self-publication of Cris Forster's manuscript Musical Mathematics: A Practice in the Mathematics of Tuning Instruments and Analyzing Scales.  To accomplish this task, we will be applying to organizations and individuals for grants.  Our goal is to publish a complete and unabridged first edition of 500 copies, with a retail cost of $90.00 per book.  If you would like more information on Mr. Forster's 1300-page manuscript, please visit our Musical Mathematics page, which shows the Table of Contents of this work.              If you would like to see Musical Mathematics in print, please write to us so that we may include your emails and letters in our grant applications.  Kindly let us know whether you are a musician, student, teacher, professor, instrument builder, etc., and indicate any institutional affiliations you may have.  Your name and email address will only appear in confidential grant applications.  We will honor anyone's wish to remain anonymous.   Please send your email to:   info@chrysalis-foundation.org   or send your letter to:   The Chrysalis Foundation 1459 18th Street, PMB #137 San Francisco, CA 94107             Thank you for your interest and support of the Chrysalis Foundation book publication project.   The Board of Directors

 

 

BIBLIOGRAPHY

 

Chapters 1–9

  

Askenfelt, A., Editor (1990).  Five Lectures on The Acoustics of the Piano.  Royal Swedish Academy of Music No. 64, Stockholm, Sweden.

 

Askill, J. (1979).  Physics of Musical Sound.  D. Van Nostrand Company, New York.

 

Baines, A. (1967).  Woodwind Instruments and Their History.  Dover Publications, Inc., New York, 1991.

 

Barbera, A., Translator (1991).  The Euclidean Division of the Canon: Greek and Latin Sources.  University of Nebraska Press, Lincoln, Nebraska.

 

Barker, A., Translator (1989).  Greek Musical Writings.  Two Volumes.  Cambridge University Press, Cambridge, Massachusetts.

 

Bell, A.J., and Firth, I.M.  (1986).  The physical properties of gut musical instrument strings. Acustica 60, No. 1, pp. 87–89.

 

Benade, A.H., and French, J.W. (1965).  Analysis of the flute head joint.  Journal of the Acoustical Society of America 37, No. 4, pp. 679–691.

 

Benade, A.H. (1967).  Measured end corrections for woodwind toneholes.  Journal of the Acoustical Society of America 41, No. 6, p. 1609.

 

Benade, A.H. (1976).  Fundamentals of Musical Acoustics.  Dover Publications, Inc., New York, 1990.

 

Berliner, P.F. (1978).  The Soul of Mbira.  University of California Press, Berkeley, California, 1981.

 

Blevins, R.D. (1979).  Formulas for Natural Frequency and Mode Shape, Reprint.  Krieger Publishing Company, Malabar, Florida, 1993.

 

Boehm, T. (1847).  On the Construction of Flutes, Über den Flötenbau.  Frits Knuf Buren, Amsterdam, Netherlands, 1982.

 

Boehm, T. (1871).  The Flute and Flute-Playing.  Dover Publications, Inc., New York, 1964.

 

Boyer, H.E., and Gall, T.L., Editors (1984).  Metals Handbook, Desk Edition.  American Society for Metals, Metals Park, Ohio, 1989.

 

Bray, A., Barbato, G., and Levi, R. (1990).  Theory and Practice of Force Measurement.  Academic Press, San Diego, California.

 

Burkert, W. (1962).  Lore and Science in Ancient Pythagoreanism.  Translated by E.L. Minar, Jr.  Harvard University Press, Cambridge, Massachusetts, 1972.

 

Cadillac Plastic Buyer’s Guide.  Cadillac Plastic and Chemical Company, Troy, Michigan, 1986.

 

Campbell, M., and Greated, C. (1987).  The Musician’s Guide to Acoustics.  Schirmer Books, New York, 1988.

 

Capstick, J.W. (1913).  Sound.  Cambridge University Press, London, England, 1932. 

 

Chapman, R.E., Translator (1957).  Harmonie universelle: The Books on Instruments, by Marin Mersenne.  Martinus Nijhoff, The Hague, Netherlands.

 

Cohen, H.F. (1984).  Quantifying Music.  D. Reidel Publishing Company, Dordrecht, Netherlands.

 

Coltman, J.W. (1979).  Acoustical analysis of the Boehm flute.  Journal of the Acoustical Society of America 65, No. 2, pp. 499–506.

 

Cremer, L., Heckl, M., and Ungar, E.E. (1973).  Structure-Borne Sound, 2nd ed.  Springer-Verlag, Berlin and New York, 1988.

 

Cremer, L. (1981).  The Physics of the Violin, 2nd ed.  The MIT Press, Cambridge, Massachusetts, 1984.

 

Crew, H., and De Salvio, A., Translators (1914).  Dialogues Concerning Two New Sciences, by Galileo Galilei.  Dover Publications, Inc., New York.

 

Den Hartog, J.P. (1934).  Mechanical Vibrations.  Dover Publications, Inc., New York, 1985.

 

Den Hartog, J.P. (1948).  Mechanics.  Dover Publications, Inc., New York, 1984.

 

D’Erlanger, R., Bakkouch, ‘A.‘A., and Al-Sanusi, M., Translators (Vol. 1, 1930;  Vol. 2, 1935;  Vol. 3, 1938;  Vol. 4, 1939;  Vol. 5, 1949;  Vol. 6, 1959).  La Musique Arabe.  Librairie Orientaliste Paul Geuthner, Paris, France.

 

Diels, H. (1903).  Die Fragmente der Vorsokratiker, Griechisch und Deutsch.  Three Volumes.  Weidmannsche Verlagsbuchhandlung, Berlin, Germany, 1951.

 

D’Ooge, M.L., Translator (1926).  Nicomachus of Gerasa: Introduction to Arithmetic.  The Macmillan Company, New York.

 

Dunlop, J.I. (1981).  Testing of poles by using acoustic pulse method.  Wood Science and Technology 15, pp. 301–310.

 

Du Pont Bulletin:  “Tynex 612 Nylon Filament.” Du Pont Company, Wilmington, Delaware.

 

Düring, I., Translator (1934).  Ptolemaios und Porphyrios über die Musik.  Georg Olms Verlag, Hildesheim, Germany, 1987.

 

Einarson, B., Translator (1967).  On Music, by Plutarch.  In Plutarch’s Moralia, Volume 14.  Harvard University Press, Cambridge, Massachusetts.

 

Elmore, W.C., and Heald, M.A. (1969).  Physics of Waves.  Dover Publications, Inc. New York, 1985.

 

Fenner, K., On the Calculation of the Tension of Wound Strings, 2nd ed.  Verlag Das Musikinstrument, Frankfurt, Germany, 1976.

 

Fishbane, P.M., Gasiorowicz, S., and Thornton, S.T. (1993).  Physics For Scientists and Engineers.  Prentice-Hall, Englewood Cliffs, New Jersey.

 

Fletcher, H., Blackham, E.D., and Stratton, R.S. (1962).  Quality of piano tones.  Journal of the Acoustical Society of America 34, No. 6, pp. 749–761.

 

Fletcher, H. (1964).  Normal vibration frequencies of stiff piano strings.  Journal of the Acoustical Society of America 36, No. 1, pp. 203–209. 

 

Fletcher, N.H., and Rossing, T.D. (1991).  The Physics of Musical Instruments, 2nd ed.  Springer-Verlag, Berlin and New York, 1998.

 

Fogiel, M., Editor (1980).  The Strength of Materials & Mechanics of Solids Problem Solver.  Research and Education Association, Piscataway, New Jersey, 1990.

 

Goodway, M., and Odell, J.S. (1987).  The Historical Harpsichord, Volume Two: The Metallurgy of  17th– and 18th– Century Music Wire.  Pendragon Press, Stuyvesant, New York.

 

Gray, D.E., Editor (1957).  American Institute of Physics Handbook, 3rd ed.  McGraw-Hill Book Company, New York, 1972.

 

Halliday, D., and Resnick, R. (1970).  Fundamentals of Physics, 2nd ed.  John Wiley & Sons, New York, 1981.

 

Hamilton, E., and Cairns, H., Editors (1963).  The Collected Dialogues of Plato.  Random House, Inc., New York, 1966.

 

Helmholtz, H.L.F., and Ellis A.J., Translator (1885).  On the Sensations of Tone.  Dover Publications, Inc., New York, 1954.

 

Hoadley, R.B. (1980).  Understanding Wood.  The Taunton Press, Newtown, Connecticut, 1981.

 

Hubbard, F. (1965).  Three Centuries of Harpsichord Making, 4th ed.  Harvard University Press, Cambridge, Massachusetts, 1972.

 

Ingard, U. (1953).  On the theory and design of acoustic resonators.  Journal of the Acoustical Society of America 25, No. 6, pp. 1037–1061.

 

Ingard, K.U. (1988).  Fundamentals of Waves and Oscillations.  Cambridge University Press, Cambridge, Massachusetts, 1990.

 

Jan, Karl von, Editor  (1895).  Musici Scriptores Graeci.  Lipsiae, in aedibus B.G. Teubneri.

 

Jerrard, H.G., and McNeill, D.B. (1963).  Dictionary of Scientific Units, 6th ed.  Chapman and Hall, London, England, 1992.

 

Jones, A.T. (1941).  End corrections of organ pipes.  Journal of the Acoustical Society of America 12, pp. 387–394.

 

Kinsler, L.E., and Frey, A.R. (1950).  Fundamentals of Acoustics, 2nd ed.  John Wiley & Sons, Inc., New York, 1962.

 

Klein, H.A. (1974).  The Science of Measurement.  Dover Publications, Inc., New York, 1988.

 

Land, F. (1960).  The Language of Mathematics.  Doubleday & Company, Inc., Garden City, New York.

 

Lemon, H.B., and Ference, M., Jr. (1943).  Analytical Experimental Physics.  The University of Chicago Press, Chicago, Illinois.

 

Levin, F.R., Translator (1994).  The Manual of Harmonics, of Nicomachus the Pythagorean.  Phanes Press, Grand Rapids, Michigan.

 

Liddell, H.G., and Scott, R. (1843).  A Greek-English Lexicon.  The Clarendon Press, Oxford, England, 1992.

 

Lide, D.R., Editor (1918).  CRC Handbook of Chemistry and Physics, 73rd ed.  CRC Press, Boca Raton, Florida, 1992.

 

Lindeburg, M.R. (1988).  Engineering Unit Conversions, 2nd ed.  Professional Publications, Inc., Belmont, California, 1990. 

 

Lindeburg, M.R. (1990).  Engineer-in-Training Reference Manual, 8th ed.  Professional Publications, Inc., Belmont, California, 1992.

 

Lindley, M. (1987).  “Stimmung und Temperatur.”  In Geschichte der Musiktheorie, Volume 6, F. Zaminer, Editor.  Wissenschaftliche Buchgesellschaft, Darmstadt, Germany.

 

McLeish, J. (1991).  Number.  Bloomsbury Publishing Limited, London, England.

 

Moore, J.L. (1971).  Acoustics of Bar Percussion Instruments.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Morse, P.M., and Ingard, K.U. (1968).  Theoretical Acoustics.  Princeton University Press, Princeton, New Jersey, 1986.

 

Nash, W.A. (1957).  Strength of Materials, 3rd. ed.  Schaum’s Outline Series, McGraw-Hill, Inc., New York, 1994.

 

Nederveen, C.J. (1969).  Acoustical Aspects of Woodwind Instruments.  Frits Knuf, Amsterdam, Netherlands.

 

Nederveen, C.J. (1973).  Blown, passive and calculated resonance frequencies of the flute. Acustica 28, pp. 12–23.

 

Newton, R.E.I. (1990).  Wave Physics.  Edward Arnold, A division of Hodder & Stoughton, London, England.

 

Norton, M.P. (1989).  Fundamentals of Noise and Vibration Analysis for Engineers.  Cambridge University Press, Cambridge, Massachusetts.

 

Oberg, E., Jones, F.D., Horton, H.L., and Ryffel, H.H. (1914).  Machinery’s Handbook, 24th ed.  Industrial Press Inc., New York, 1992.

 

Olson, H.F. (1952).  Music, Physics and Engineering, 2nd ed.  Dover Publications, Inc., New York, 1967.

 

Pierce, A.D. (1981).  Acoustics.  Acoustical Society of America, Woodbury, New York.  1991.

 

Pierce, J.R. (1983).  The Science of Musical Sound.  Scientific American Books, W.H. Freeman and Company, New York.

 

Pikler, A.G. (1966).  Logarithmic frequency systems.  Journal of the Acoustical Society of America 39, No. 6, pp. 1102–1110.

 

Rao, S.S. (1986).  Mechanical Vibrations, 2nd ed.  Addison–Wesley Publishing Company, Reading, Massachusetts, 1990.

 

Richardson, E.G. (1929).  The Acoustics of Orchestral Instruments and of the Organ.  Edward Arnold & Co., London, England.

 

Rossing, T.D. (1989).  The Science of Sound, 2nd ed.  Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1990.

 

Sadie, S., Editor (1984).  The New Grove Dictionary of Musical Instruments.  Macmillan Press Limited, London, England.

 

Schlesinger, K. (1939).  The Greek Aulos.  Methuen & Co. Ltd., London, England.

 

Schuck, O.H., and Young, R.W. (1943).  Observations on the vibrations of piano strings.  Journal of the Acoustical Society of America 15, No. 1, pp. 1–11. 

 

Sears, F.W., Zemansky, M.W., and Young, H.D., University Physics, 7th ed.  Addison-Wesley Publishing Company, Reading, Massachusetts, 1988.

 

Skudrzyk, E. (1968).  Simple and Complex Vibratory Systems.  Pennsylvania State University Press, University Park, Pennsylvania, 1981.

 

Smith, D.E. (1925).  History of Mathematics.  Two Volumes.  Dover Publications, Inc., New York, 1958.

 

Standards Handbook, Part 2 — Alloy Data, Wrought Copper and Copper Alloy Mill Products, Eighth Edition, Copper Development Association, Inc., Greenwich, Connecticut, 1985.

 

Stauss, H.E., Martin, F.E., and Billington, D.S. (1951).  A piezoelectric method for determining Young’s modulus and its temperature dependence.  Journal of the Acoustical Society of America 23, No. 6, pp. 695–696. 

 

Steinkopf, O. (1983).  Zur Akustik der Blasinstrumente.  Moeck Verlag, Celle, Germany.

 

Stiller, A. (1985).  Handbook of Instrumentation.  University of California Press, Berkeley, California.

 

Suzuki, H. (1986).  Vibration and sound radiation of a piano soundboard.  Journal of the Acoustical Society of America 80, No. 6, pp. 1573–1582.

 

Thompson, S.P. (1910).  Calculus Made Easy, 3rd. ed.  St. Martin’s Press, New York, 1984.

 

Thomson, W.T. (1972).  Theory of Vibration with Application, 4th ed.  Prentice Hall, Englewood Cliffs, New Jersey, 1993.

 

Timoshenko, S., and Woinowsky-Krieger, S. (1940).  Theory of Plates and Shells, 2nd ed., McGraw-Hill Book Company, New York, 1959.

 

Timoshenko, S.P. (1953).  History of Strength of Materials.  Dover Publications, Inc., New York, 1983.

 

Towne, D.H. (1967).  Wave Phenomena.  Dover Publications, Inc., New York, 1988.

 

Tropfke, J. (1921).  Geschichte der Elementar-Mathematik.  Seven Volumes.  Vereinigung Wissenschaftlicher Verleger, Walter de Gruyter & Co., Berlin and Leipzig, Germany.

 

U.S. Business and Defense Services Administration (1956).  Materials Survey: Aluminum.  Department of Commerce, Washington, D.C.

 

Weaver, W., Jr., Timoshenko, S.P., and Young, D.H., Vibration Problems in Engineering, 5th ed.  John Wiley and Sons, New York, 1990.

 

White, W.B. (1917).  Piano Tuning and Allied Arts, 5th ed.  Tuners Supply Company, Boston, Massachusetts, 1972.

 

Wogram, K. (1981).  Akustische Untersuchungen an Klavieren.  Teil I: Schwingungseigenschaften des Resonanzbodens.  Das Musikinstrument 24, pp. 694–702, 776–782, 872–879.  English translation:  Acoustical research on pianos.  Part I: Vibrational characteristics of the soundboard.  In Musical Acoustics: Selected Reprints, T.D. Rossing, Editor, pp. 85–98.  American Association of Physics Teachers, College Park, Maryland, 1988.

 

Wolfenden, S. (1916).  A Treatise on the Art of Pianoforte Construction.  The British Piano Museum Charitable Trust, Brentford, Middlesex, England, 1975.

 

Wood, A.B. (1930).  A Textbook of Sound.  The MacMillan Company, New York, 1937. 

 

Wood, A. (1940).  Acoustics.  Dover Publications, Inc., New York, 1966.

 

Young, R.W. (1952).  Inharmonicity of plain wire piano strings.  Journal of the Acoustical Society of America 24, No. 3, pp. 267–272.

 

Zanoncelli, L., Translator (1990).  La Manualistica Musicale Greca.  Angelo Guerini e Associati, Milan, Italy.

 

Zebrowski, Jr., E. (1979).  Fundamentals of Physical Measurement.  Duxbury Press, Belmont, California. 

 

 

 

Chapter 10

 

 

Adkins, C.D. (1963).  The Theory and Practice of the Monochord.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Al-Faruqi, L.I. (1974).  The Nature of the Musical Art of Islamic Culture: A Theoretical and Empirical Study of Arabian Music.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Asselin, P. (1985).  Musique et Tempérament.  Éditions Costallat, Paris, France.

 

Barbera, C.A. (1977).  Arithmetic and geometric divisions of the tetrachord.  Journal of Music Theory 21, No. 2, pp. 294–323.

 

Barbera, A., Translator (1991).  The Euclidean Division of the Canon: Greek and Latin Sources.  University of Nebraska Press, Lincoln, Nebraska.

 

Barbour, J.M. (1933).  The persistence of the Pythagorean tuning system.  Scripta Mathematica, Vol. 1, pp. 286–304.

 

Barbour, J.M. (1951).  Tuning and Temperament.  Da Capo Press, New York, 1972.

 

Barker, A., Translator (1989).  Greek Musical Writings.  Two Volumes.  Cambridge University Press, Cambridge, England.

 

Barnes, J. (1979).  Bach’s keyboard temperament.  Early Music 7, No. 2, pp. 236–249.

 

Beck, C., Translator (1868).  Flores musice omnis cantus Gregoriani, by Hugo Spechtshart [von Reutlingen].  Bibliothek des Litterarischen Vereins, Stuttgart, Germany.

 

Bower, C.M., Translator (1989).  Fundamentals of Music, by A.M.S. Boethius.  Yale University Press, New Haven, Connecticut.

 

Briscoe, R.L., Translator (1975).  Rameau’s “Démonstration du principe de l’harmonie” and “Nouvelles reflections de M. Rameau sur sa démonstration du principe de l’harmonie:” An Annotated Translation of Two Treatises by Jean-Philippe Rameau.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Brun, V. (1964).  Euclidean algorithms and musical theory.  L’Enseignement Mathématique X, pp. 125–137.

 

Burkert, W. (1962).  Lore and Science in Ancient Pythagoreanism.  Translated by E.L. Minar, Jr.  Harvard University Press, Cambridge, Massachusetts, 1972.

 

Chalmers, J.H., Jr. (1993).  Divisions of the Tetrachord.  Frog Peak Music, Hanover, New Hampshire.

 

Chandler, B.G., Translator (1975).  Rameau’s “Nouveau système de musique théorique:” An Annotated Translation With Commentary.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Chapman, R.E., Translator (1957).  Harmonie universelle: The Books on Instruments, by Marin Mersenne.  Martinus Nijhoff, The Hague, Netherlands.

 

Coelho, V., Editor (1992).  Music and Science in the Age of Galileo.  Kluwer Academic Publishers, Dordrecht, Netherlands. 

 

Cohen, H.F. (1984).  Quantifying Music.  D. Reidel Publishing Company, Dordrecht, Netherlands.

 

Compact Edition of the Oxford English Dictionary.  Oxford University Press, Oxford, England, 1974.

 

Crew, H., and De Salvio, A., Translators (1914).  Dialogues Concerning Two New Sciences, by Galileo Galilei.  Dover Publications, Inc., New York.

 

Crocker, R.L. (1963).  Pythagorean mathematics and music.  The Journal of Aesthetics and Art Criticism XXII, No. 2, Part I:  pp. 189–198, and No. 3, Part II:  pp. 325–335.

 

Crocker, R.L. (1966).  “Aristoxenus and Greek Mathematics.”  In Aspects of Medieval and Renaissance Music: A Birthday Offering to Gustave Reese, J. LaRue, Editor.  Pendragon Press, New York.

 

Crone, E., Editor;  Fokker, A.D., Music Editor;  Dikshoorn, C., Translator (1966).  The Principal Works of Simon Stevin.  Five Volumes.  C.V. Swets & Zeitlinger, Amsterdam.

 

Crookes, D.Z., Translator (1986).  Syntagma musicum II – De organographia: Parts I and II, by Michael Praetorius.  The Clarendon Press, Oxford, England.

 

Daniels, A.M. (1962).  The De musica libri VII of Francisco de Salinas.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

De Haan, D.B., Publisher (1884).  Vande Spiegeling der Singconst, by Simon Stevin.  Amsterdam.

 

D’Erlanger, R., Bakkouch, ‘A.‘A., and Al-Sanusi, M., Translators (Vol. 1, 1930; Vol. 2, 1935;  Vol. 3, 1938;  Vol. 4, 1939;  Vol. 5, 1949;  Vol. 6, 1959).  La Musique Arabe, Librairie Orientaliste Paul Geuthner, Paris, France.

 

Diels, H. (1903).  Die Fragmente der Vorsokratiker, Griechisch und Deutsch.  Three Volumes. Weidmannsche Verlagsbuchhandlung, Berlin, Germany, 1951.

 

D’Ooge, M.L., Translator (1926).  Nicomachus of Gerasa: Introduction to Arithmetic.  The Macmillan Company, New York.

 

Dupont, W. (1935).  Geschichte der musikalischen Temperatur.  C.H. Beck’sche Buchdruckerei, Nördlingen, Germany.

 

Düring, I., Editor (1930).  Die Harmonielehre des Klaudios Ptolemaios.  Original Greek text of Ptolemy’s Harmonics.  Wettergren & Kerbers Förlag, Göteborg, Sweden.

 

Düring, I., Translator (1934).  Ptolemaios und Porphyrios über die Musik.  Georg Olms Verlag, Hildesheim, Germany, 1987.

 

Farmer, H.G. (1965).  The Sources of Arabian Music.  E.J. Brill, Leiden, Netherlands.

 

Farmer, H.G., Translator (1965).  Al-Farabi’s Arabic-Latin Writings on Music.  Hinrichsen Edition Ltd., New York.

 

Fend, M., Translator (1989).  Theorie des Tonsystems: Das erste und zweite Buch der Istitutioni harmoniche (1573), von Gioseffo Zarlino.  Peter Lang, Frankfurt am Main, Germany.

 

Fernandez de la Cuesta, I., Translator (1983).  Siete libros sobre la musica, by Francisco Salinas.  Editorial Alpuerto, Madrid, Spain.

  

Flegg, G., Hay, C., and Moss, B., Translators (1985).  Nicolas Chuquet, Renaissance Mathematician.  D. Reidel Publishing Company, Dordrecht, Holland.

 

Gossett, P., Translator (1971).  Traité de l’harmonie [Treatise on Harmony], by Jean-Philippe Rameau.  Dover Publications, Inc., New York.

 

Green, B.L. (1969).  The Harmonic Series From Mersenne to Rameau: An Historical Study of Circumstances Leading to Its Recognition and Application to Music.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Guthrie, K.S., Translator (1987).  The Pythagorean Sourcebook and Library.  Phanes Press, Grand Rapids, Michigan.

 

Hamilton, E., and Cairns, H., Editors (1966).  The Collected Dialogues of Plato.  Random House, Inc., New York.

 

Hawkins, J. (1853).  A General History of the Science and Practice of Music.  Dover Publications, Inc., New York, 1963.

 

Hayes, D., Translator (1968).  Rameau’s Theory of Harmonic Generation;  An Annotated Translation and Commentary of “Génération harmonique” by Jean-Philippe Rameau.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Heath, T.L., Translator (1908).  Euclid’s Elements.  Dover Publications, Inc., New York, 1956.

 

Heath, T. (1921).  A History of Greek Mathematics.  Dover Publications, Inc., New York, 1981.

 

Hitti, P.K. (1937).  History of the Arabs.  MacMillan and Co. Ltd., London, England, 1956.

 

Hubbard, F. (1965).  Three Centuries of Harpsichord Making, 4th ed.  Harvard University Press, Cambridge, Massachusetts, 1972.

 

Hyde, F.B. (1954).  The Position of Marin Mersenne in the History of Music.  Two Volumes.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Ibn Sina (Avicenna):  Auicene perhypatetici philosophi: ac medicorum facile primi opera in luce redacta...  This Latin translation was published in 1508.  Facsimile Edition:  Minerva, Frankfurt am Main, Germany, 1961.

 

Jacobi, E.R., Editor (1968).  Jean-Philippe Rameau (1683–1764): Complete Theoretical Writings.   American Institute of Musicology, [Rome, Italy].

 

James, G., and James, R.C. (1976).  Mathematics Dictionary, 4th ed.  Van Nostrand Reinhold, New York.

 

Jorgensen, O. (1977).  Tuning the Historical Temperaments by Ear.  The Northern Michigan University Press, Marquette, Michigan.

 

Jorgenson, D.A. (1957).  A History of Theories of the Minor Triad.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Jorgenson, D.A. (1963).  A résumé of harmonic dualism.  Music and Letters XLIV, No. 1, pp. 31–42.

 

Kastner, M.S., Editor (1958).  De musica libri VII, by Francisco Salinas.  Facsimile Edition.  Bärenreiter-Verlag, Kassel, Germany.

 

Kelleher, J.E. (1993).  Zarlino’s “Dimostrationi harmoniche” and Demonstrative Methodologies in the Sixteenth Century.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Lawlor, R. and D., Translators (1978).  Mathematics Useful for Understanding Plato, by Theon of Smyrna.  Wizards Bookshelf, San Diego, California, 1979.

 

Levin, F.R., Translator (1994).  The Manual of Harmonics, of Nicomachus the Pythagorean.  Phanes Press, Grand Rapids, Michigan.

 

Lindley, M. (1984).  Lutes, Viols and Temperaments.  Cambridge University Press, Cambridge, England.

 

Litchfield, M. (1988).  Aristoxenus and empiricism: A reevaluation based on his theories.  Journal of Music Theory 32, No. 1, pp. 51–73.

 

Mackenzie, D.C., Translator (1950).  Harmonic Introduction, by Cleonides.  In Source Readings in Music History, O. Strunk, Editor.  W. W. Norton & Company, Inc., New York.

 

Macran, H.S., Translator (1902).  The Harmonics of Aristoxenus.  Georg Olms Verlag, Hildesheim, Germany, 1990.

 

Marcuse, S. (1964).  Musical Instruments: A Comprehensive Dictionary.  W. W. Norton & Company, Inc., New York, 1975.

 

Maxham, R.E., Translator (1976).  The Contributions of Joseph Sauveur to Acoustics.  Two Volumes.  Ph.D. dissertation printed and distributed by University Microfilms, Inc., Ann Arbor, Michigan.

 

Mersenne, M. (1636–37).  Harmonie universelle contenant la théorie et la pratique de la musique.  Three Volumes.  Facsimile Edition.  Éditions Du Centre National De La Recherche Scientifique, Paris, France, 1963.

 

Meyer, M.F. (1929).  The Musician’s Arithmetic.  Oliver Ditson Company, Boston, Massachusetts.

 

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