Thursday, October 11, 2012

Active Engagement in Science and Mathematics through Music- Pattern Recognition leading to Higher Mathematics needed for Deeper Study in the Sciences

Here is a peaceful video to watch on the interconnections of mathematics and music.  Watch it and see what ideas come to your mind for connections of math and music to science.  Think about what questions come to your mind, if any, when you watch the video?  Jot them down.  

Since in this blog, the focus is on science literacy and science education, how can the video enhance them?  For example, did you hear new vocabulary?  A video can bring new vocabulary to life because we hear the words pronounced and also see the usage of the word, the context.  Mathematics is important to science literacy.  That is why the STEM- science, technology, engineering and math- education focus prominent today includes the, "M."Another valuable M, however, is music.  Music can engage students in science (That is why I once submitted a grant proposal to FIPSE called, Humanities at the Heart of Science.  Music is one of the humanities that engages students).  Vocabulary development is one way.  Another is the ability of music in the video to engage students in science and mathematical studies.

Many skill sets of musicians and scientists overlap.  Can you think of some?  List as many as you can.  Now circle on the list skills you think you have or could develop.  Write a reflection on what you discover by this exercise.

How can you use the video to further enhance science reading?  Do you know Harlan Brothers?  Have you seen his journal articles?  Do you know how to find them?  Remember that in journal articles, you might find new and difficult vocabulary.  Remember as Joan Beinetti says, "No one knows all the words," (personal communication, 1989).  After you have slowly read one or two articles on a topic, you will start to develop more vocabulary in the field of the article.  The more vocabulary you develop, the easier other articles on the same topic are to read.  By the time you have read 5 or 6 articles, you are likely to be looked at by others as an expert...or, at least, quite knowledgeable, in that area.

Let's go about finding some articles.  Just searching using his name, we find a number of links.  Let's look at one:

H. J. Brothers, "The Nature of Fractal Music," in Benoit Mandelbrot - A Life in Many Dimensions, edited by Michael Frame, World Scientific Publishing (Fall, 2012).
H. J. Brothers, "Pascal's triangle: The hidden stor-e ." The Mathematical Gazette, Vol. 96, No. 535, 2012; pages 145-148.  [Try this, too:]
H. J. Brothers, "Pascal's prism." The Mathematical Gazette, Accepted for publication, July 2012.  (See here also.[and here:]

Pick one of the articles or books above as a starting point, or, try to find an article on your own to start with.  

You might be interested in the following biographical information quoted from Wikipedia:

In 1997, while examining the sequence of counting numbers raised to their own power ( {an}=nn ), Brothers discovered some simple algebraic formulas [1] that yielded the number 2.71828..., the universal constant e, also known as the base of the natural logarithm. Like its more famous cousin πe is a transcendental number that appears in a wide range of formulas in mathematics and physics.

Having no formal college-level mathematics education, he sent brief descriptions of his findings to the host of the National Public Radio show “Science Friday” and also to a well-known mathematician at Scientific American.

His communication with “Science Friday” led to a fruitful collaboration with meteorologist John Knox. Together they discovered over two dozen new formulas and published two papers on their methods. These methods subsequently found their way into the standard college calculus curriculum by way of a popular textbooks on the subject.[2] [3]
Brothers went back to school to study calculus and differential equations. He went on to publish methods for deriving infinite series that include the fastest known formulas for approximating e.[4]  (Source:  Accessed 11 October 2012.    See also:
So, there is a fine example of a student communicating on an interest leading to a fascinating career in an subject that blends his love of math and music.  Before digressing too much, let's return to fractals and music, remembering that reading slowly and carefully, you will develop the needed vocabulary to understand even very difficult topics. And, you can contact people, as did, Harlan Brothers, and meet the people you need to grow your knowledge.

Let's look at the abstract of another article on fractals and music (, accessed 11 Oct 2012):

ABSTRACT The objective of this paper is to identify  some distinctive features of fractal music – offering a possible answer to the question: “What does fractal music mean?”.  Following an introduction to the general concept of fractals, it discusses their fundamental characteristics, that is the scale invariance and self-similarity derived from a power law. The understanding of the fractal nature of music requires a clear grasp of the fundamental physical characteristics of sound, such as pitch, duration and timbre. The perception of music, however, is  a psychological experience, so the paper briefly explores some amazing but widely known examples of aural illusions, deriving from our logarithmic sensitivity. Following a brief  outline of the main areas of current research in this sector, the paper proposes a formal definition of fractal music, based on its physical, mathematical and psychological characteristics.  Finally  a musical composition is analyzed, showing that it is indeed real fractal music according to the proposed definition. The paper concludes by suggesting possible areas for further exploration."
It looks like this article might be readable, but, even here, students will have to develop some vocabulary as the authors definition may be difficult for many people (children and adults, alike).  Also, the author is new to this research area and the understanding is limited as a result.  Check out the definition of fractals in the article here:
 and compare it to what you find here:

or here:

Have students note that not every article presents the rigorous academic nature needed for serious study fractal music.  Have the students check for these concepts and other possible errors:

Let students have fun finding articles and slowly reading them, using the dictionary at hand or an on-line dictionary as needed.  For example, fractals are defined here:

A deeper study of fractals and related material can be found here,  "a collection ... meant to support a first course in fractal geometry for students without ... strong mathematical preparation, or any particular interest in science:"

Have students do their own searches for definitions and journal. (Be sure your virus protection is on.) Have them check the academic rigor of any definitions they find.  Similarly, are the journal articles they look at refereed (peer-reviewed by those knowledgeable and recognized in the field of study)?  Have the students each find an article abstract by a leading researcher in this field.

After the students read the article abstract(s), they might wish to get the article(s) by inter-library loan from the local library and then read the full article and even e-mail the author with a question.  They might even pick up a musical instrument and try to make some musical fractals.  Enjoy!

Note that Michael Frame and Harlan J Brothers have worked hard to establish a rigorous framework for the study and discussion of fractal music. Comments or questions can be posted below in the comment section and also submitted to:

"Dr." J.  and to Harlan Brothers

(c)2012 J S Shipman

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