Music as a Branch of Mathematics

A presentation prepared for the Mathematics Across the Community College Curriculum

MAC^32007 Winter Institute

by Ed CalleJanuary 12, 2007

MusicMusic

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.

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.

Raymond Clare Archibald Mathematical Association of

America presidential address

(1923)

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).

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).

Archibald (cont.)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.

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.

Musician mathematicians

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.

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.

Musician mathematicians (cont.)

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).

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).

Albert EinsteinAlbert 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).

“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).

Pythagorean musical ratios

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.

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.

The MonochordThe Monochord

Illustration of the octave

Illustration of the octave

The octaveThe octave

Document

Illustration of the fifth

Illustration of the fifth

The fifthThe fifth

Document

Illustration of the fourth

Illustration of the fourth

The fourthThe fourth

Document

Doriana (Pythagorean tuning)

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.

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.

Doriana (chromatic tuning)

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.

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.

Auto-Tune 5 Pythagorean tuning

window

Auto-Tune 5 Pythagorean tuning

window

Auto-Tune 5 chromatic tuning window

Auto-Tune 5 chromatic tuning window

ReferencesReferences

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). János Bolyia [Electronicversion]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.standrews.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.standrews.ac.uk/Mathematicians/Lagrange.html

O’Connor, J. J., and Robertson, E. F. (2004). Pierre Louis Moreau deMaupertuis [Electronic version]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Maupertuis.html

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). János Bolyia [Electronicversion]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.standrews.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.standrews.ac.uk/Mathematicians/Lagrange.html

O’Connor, J. J., and Robertson, E. F. (2004). Pierre Louis Moreau deMaupertuis [Electronic version]. Retrieved January 11, 2007 from the web site: http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Maupertuis.html