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Music as a Branch of Mathematics

Music as a Branch of Mathematics. A presentation prepared for the Mathematics Across the Community College Curriculum MAC^3 2007 Winter Institute by Ed Calle January 12, 2007. Music.

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Music as a Branch of Mathematics

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  1. Music as a Branch of Mathematics A presentation prepared for the Mathematics Across the Community College CurriculumMAC^32007 Winter Instituteby Ed CalleJanuary 12, 2007

  2. Music • Nurtured by earth’s atmosphere, supported by mathematical pillars anchored in study, fueled by imagination, necessitated by the need of a creative soul to share unique visions, and realized by a tireless dedication to the celebration of talented passion, music fills the air with sound, minds with wonder, and hearts with joy.

  3. Raymond Clare Archibald Mathematical Association of America presidential address (1923) • In his presidential address on September 6th, 1923 regarding mathematics and music to the Mathematical Association of America, Brown University’s Raymond Clare Archibald celebrated the ties binding mathematics and music from a historical perspective (http://www-history.mcs.st-andrews.ac.uk/Extras/Archibald_music_1.html, 2006).

  4. Archibald (cont.) • From Helmholtz’s suggestion that math and music share a “hidden bond” visible through the study of acoustics by Fourier, to the proclamation by Leibniz that “Music is a hidden exercise in arithmetic, of a mind unconscious of dealing with numbers” (http://www-history.mcs.st-andrews.ac.uk/Extras/Archibald_music_1.html, 2006), the history of mathematics is replete with great spirits fascinated by music.

  5. Musician mathematicians • Pythagoras - The Pythagorean theorem, musical ratios, and the Greek music modes. • Pierre-Louis Moreau de Maupertuis - Formulated the principle of least action and accomplished German guitarist. • William Herschel - Astronomer who discovered Uranus, first described the actual form of the Milky Way, played violin, conducted, and published a symphony.

  6. Musician mathematicians (cont.) • János Bolyia - Violinist who, “Prepared a treatise on a complete system of non-Euclidean geometry” (http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Bolyai.html, 2004). • Augustus De Morgan - Flutist who added rigor to science through the process of mathematical induction. • Joseph Lagrange - Used music as an environmental workspace while contemplating “analysis and number theory and analytical and celestial mechanics” (http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lagrange.html, 1999).

  7. Albert Einstein • “If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music… I get most joy in life out of music” (http://www.if.ufrgs.br/einstein/frases.html, 2003).

  8. Pythagorean musical ratios • The Greek philosopher, mathematician, and musician Pythagoras defined the octave as a ratio of 1:2 (Archibald, 1923) by discovering that two tones produced on either side of a string bridged in a manner dividing the string into two sections measuring a single unit on one side of the bridge and two units on the other differed in pitch or frequency by one octave.

  9. The Monochord

  10. Illustration of the octave The octave

  11. Illustration of the fifth • The fifth

  12. Illustration of the fourth • The fourth

  13. Doriana (Pythagorean tuning) • This recording features an original composition by Ed Calle based on the Dorian mode. The soprano saxophone was tuned applying Pythagorean tuning using Antares Corporation Auto-Tune 5 software.

  14. Doriana (chromatic tuning) • This recording features an original composition by Ed Calle based on the Dorian mode. The soprano saxophone was tuned applying chromatic tuning using Antares Corporation Auto-Tune 5 software.

  15. Auto-Tune 5 Pythagorean tuning window

  16. Auto-Tune 5 chromatic tuning window

  17. References • Antares Audio Technologies. (2007). Auto-Tune 5 software. Scotts Valley, CA • Armagh Observatory. (2005). Portrait of Sir William Herschel (1738-1822). Retrieved January 11, 2007 from the web site: http://www.arm.ac.uk/history/herschel.html • O’Connor, J. J., and Robertson, E. F. (2006). Augustus De Morgan [Electronic version]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.standrews.ac.uk/Mathematicians/De_Morgan.html • O’Connor, J. J., and Robertson, E. F. (2004). JánosBolyia[Electronic version]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.st andrews.ac.uk/Mathematicians/Bolyai.html • O’Connor, J. J., and Robertson, E. F. (1999). Joseph-Louis Lagrange [Electronic version]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.st andrews.ac.uk/Mathematicians/Lagrange.html • O’Connor, J. J., and Robertson, E. F. (2004). Pierre Louis Moreau de Maupertuis[Electronic version]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.st- andrews.ac.uk/Mathematicians/Maupertuis.html

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