**Some interesting quotations:**

“Music is the pleasure the human soul experiences from counting without being aware that it is counting.” — German mathematician and philosopher Gottfried Wilhelm von Leibniz (1646-1716) who co-discovered calculus

“Mathematics and music, the most sharply contrasted fields of scientific activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind, and which leads us to surmise that the manifestations of the artist’s genius are but the unconscious expressions of a mysteriously acting rationality.”** –**19^{th} century** **German physician and physicist** Hermann von Helmholtz***, Vorträgeund Reden, Bd. 1 (Braunschweig, 1884), p. 82*

“May not music be described as mathematics of the sense, mathematics as music of the reason?”** –**19^{th} century English mathematician** James Joseph Sylvester,** *On Newton’s Rule for the Discovery of Imaginary Roots; Collected Mathematical Papers, Vol. 2, p. 419*

“If all art aspires to the condition of music, all the sciences aspire to the condition of mathematics.” – Spanish philosopher/writer **George Santayana **(1863-1952)

“Musical training is a more potent instrument than any other, because rhythm and harmony find their way into the inward places of soul, on which they mightily fasten, imparting grace, and making the soul of him who is rightly educated graceful…” **— **classical Greek philosopher** Plato **(428-348 BCE), *The Republic, III*

“Educators have always known that learning and life are maximal where play and work coincide.”** — L. W. Gibbs**

“In the future, we can expect that not much difference will exist between education and entertainment. We just have to put intelligence behind the entertainment.” **— North Carolina State University’s James Lester, ***quoted at the 12th International Conference on College Teaching and Learning*

“It is harmony which restores unity to the contrasting parts and which moulds them into a cosmos. Harmony is divine, it consists of numerical ratios. Whosoever acquires full understanding of this number harmony, he becomes himself divine and immortal.” –20^{th} century Dutch mathematician** B. L. van der Waerden***, *describing the beliefs of the followers of Pythagoras

“We do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist;

but if a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic music.” **—**19^{th} century American philosopher/writer** Ralph Waldo Emerson, ***Society and Solitude, Ch. 7, Works and Days*

“Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders, designs and harmonies.” **— William L. Schaaff**, author and mathematics education professor

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas. His patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way.”** — **English mathematician** G. H. Hardy** (1877-1947)

“Music and math together satisfied a sort of abstract ‘appetite’, a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical.” –music critic/composer **Edward Rothstein** (p. xv of his 1995 book *Emblems of Mind: The Inner Life of Music and Mathematics*)

“We can no more come to understand mathematics by examining its final product than we can understand the experience of music through simply looking at a score or an analysis of one; there is an experience that lies underneath and behind the systematic organization of the material.” –music critic/composer **Edward Rothstein** (p. 38 of his 2006 book *Emblems of Mind: The Inner Life of Music and Mathematics*)

“Each time I’ve learned a little bit more about the inner structures of music, the math of it and the shape of it, my joy in it has increased.” — recording artist **Peter Mulvey, **

*in the Nov. 2000 Performing Songwriter*

“Music is *true*. An octave is a mathematical reality. So is a 5th. So is a major 7th chord. And I have the feeling that these have emotional meanings to us, not only because we’re *taught* that a major 7th is warm and fuzzy and a diminished is sort of threatening and dark, but also because they actually do have these meanings. It’s almost like it’s a language that’s not a matter of our choosing.

It’s a *truth*. The laws of physics apply to music, and music follows that. So it really lifts us out of this subjective, opinionated human position and drops us into the cosmic picture just like that.” — recording artist **James Taylor, ** in the May 2002* Performing Songwriter*

“The syntax and the grammar of the language of music are not capricious; they are dictated by the texture and organization of the deep levels of the mind, so with mathematics.”** – **mathematician** H. E. Huntley**

“The most distinct and beautiful statement of any truth (as of music) must take at last the mathematical form.”** –**19^{th} century American writer/philosopher** Henry David Thoreau, ***A Week on the Concord and Merrimac Rivers (Boston, 1893), p. 477*

“You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.” **— Alfred North Whitehead **(1861-1947), English mathematician

“Musical form is close to mathematics — not perhaps to mathematics itself, but certainly to something like mathematical thinking and relationship.” — 20^{th} century Russian composer** Igor Stravinsky**

**Some interesting books:**

- Garland & Kahn’s
*Math and Music: Harmonious Connections*(Dale Seymour, 1995) - Scott Beal’s
*Functional Melodies: Finding Mathematical Relationships in Music*(Key Curriculum Press, 2000) - Leon Harkleroad’s
*The Math Behind the Music*(Cambridge U. Press, 2006) - David Benson’s
*Music: A Mathematical Offering*(Cambridge University Press; 2006) - David Temperley’s
*Music and Probability*(MIT Press, 2007) - Gareth Loy’s
*Musimathics**: The Mathematical Foundations of Music*(MIT Press; 2006, 2007) - Fauvel, Flood & Wilson’s
*Music and Mathematics: From Pythagoras to Fractals*(Oxford University Press, 2006) - Jan Beran’s
*Statistics in Musicology*(Chapman & Hall/CRC, 2004). - David Wright’s
*Mathematics and Music*(American Mathematical Society; 2009)

Note: the above books vary greatly in their level of technical precision and in classroom applicability, so browse before buying! (classroom teachers may generally prefer the ones near the beginning of the above list)

**Some interesting articles…..**

- My 2000 paper Sum of Songs: Making Mathematics Less Monotone!
*Mathematics Teacher, 93*(5), 372-7. - My 2001 article Musical Means: Using Songs in Teaching Statistics,
*Teaching Statistics*,*23*(3), 81-85. *My July 2013 paper (with W. Robertson) “*Scientific skateboarding and mathematical music: Edutainment that actively engages middle school students”.*European Journal of Science and Mathematics Education*,*1*(2), 60-68. http://www.scimath.net/articles/12/123.pdf- My Sept. 2014 paper “Mathematical lyrics: A noteworthy endeavor in the education”,
*Journal of Mathematics and the Arts*,*8*(1-2), 46-53. http://www.tandfonline.com/doi/pdf/10.1080/17513472.2014.950833?needAccess=true - My Dec. 2015 paper “ ‘American Pi’: The Story of a Song about Pi” in
*Journal of Mathematics Education, 8*(2), 158-168. - Lesser, L. M., Pearl, D. K., & Weber, J. J. (July 2016). Assessing fun items’ effectiveness in increasing learning of college introductory statistics students: Results of a randomized experiment.
*Journal of Statistics Education*,*24*(2), 54-62. http://tandfonline.com/doi/pdf/10.1080/10691898.2016.1190190?needAccess=true - Lesser, L. (Autumn 2018). Modulating Misconceptions with Musical Means.
*Teaching Statistics, 40*(3), 79-82. https://onlinelibrary.wiley.com/doi/pdf/10.1111/test.12157 - Lesser, L. (May 24, 2018). Student-Created Songs in Statistics Class, competitively-selected contributed videoposter, 4
^{th}Electronic Conference on Teaching Statistics, archived at https://www.CAUSEweb.org/cause/ecots/ecots18/posters/4-04 - Lesser, L. (Sept. 27-28, 2017). A continuum of interactivity with (mathematics/statistics) songs. Invited reviewed presentation, VOICES: Virtual Ongoing Interdisciplinary Conferences on Educating with Song. https://www.CAUSEweb.org/voices/2017/panel/1-3
- Craig Johnson’s “Functions of Number Theory in Music” in the Nov. 2001
*Mathematics Teacher* - Barger & Haehl’s article in the April 2007
*Mathematics Teaching in the Middle School* - Robert Osserman’s chapter in
*Essays in Humanistic Mathematics*(*MAA Notes #32*), 1993 - Math and the
*Musical Offering* - recent NCTM journal articles include Craig Johnson’s “Introducing Group Theory through Music” (Sept. 2009
*Mathematics Teacher*), Brett Cooper & Rita Barger’s “Listening to Geometry” (Sept. 2009*Mathematics Teacher*), and Margaret Meyer’s “Teacher as Musician versus Teacher as Composer” (Sept. 2009*Mathematics Teaching in the Middle School,*Matson & Grigoriadou’s “Don’t Fret over Exponential Functions” in the Dec2012/Jan2013*Mathematics Teacher* - http://www.khanacademy.org/math/vi-hart/v/what-is-up-with-noises—the-science-and-mathematics-of-sound–frequency–and-pitch
- http://vihart.com/papers/binary/
- http://vihart.com/papers/symmetry/MusicalSymmetry.pdf

**Some more advanced articles:**

- Fall 2004
*Chance*(vol. 17, no. 4) has several articles about music and statistics *Journal of Mathematics and Music*- https://www.causeweb.org/ecots/ecots12/posters/12/ (Ethan Brown’s work on sonifications)
- The impressive work of Rachel W. Hall: http://www.sju.edu/~rhall/research.htm and http://www.sju.edu/~rhall/Mathofmusic/MathandMusicLinks.html
- J. Bamberger & A. Disessa (“Music as Embodied Mathematics: A Study of Mutually Informing Affinity”;
*International Journal of Computers for Mathematical**Learning*, 8, 123-160, 2003)

Some **bibliography** **for** **educational uses of song**:

http://www.songsforteaching.com/references.htm

http://singaboutscience.org/wp/educating/research/

https://www.tandfonline.com/doi/pdf/10.1080/17513472.2014.950833?needAccess=true,

**Searchable song databases:**

- math/stats/science songs: http://www.science-groove.org/MASSIVE/; http://singaboutscience.org/wp/findandaddsongs/
- statistics songs: https://www.causeweb.org/cause/resources/fun/all?combine=&type_1=song&field_tags_tid=All&items_per_page=100
- educational songs: http://www.songsforteaching.com or http://www.learningfromlyrics.org/songs.html

**Applets/Videos/Software:**

- http://illuminations.nctm.org/LessonDetail.aspx?id=U163
- http://www.sju.edu/~rhall/Multi/drums.html
- https://thefutureschannel.com/videogallery/the-rhythm-track/
- http://www.5min.com/Video/Physics-in-Action-Musical-Instruments-and-Waveforms-174071026
- https://dotpiano.com
- http://chenalexander.com
- https://musiclab.chromeexperiments.com/

**Mathematically-generated compositions**: