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Functional Melodies
Finding Mathematical Relationships In Music

Author Advise and Comments

My intention on this page is to provide a forum where I can share my experiences and candid advise for teaching Functional Melodies activities. Each chapter is represented so scroll down until you find what you need. The integration of mathematics and music is a work in progress with enormous potential to enhance student learning in mathematics and music. These activities are launching points for further work and connections, and in this forum I can share insights as they evolve and elaborate on those presented in the book. This page will continually grow and change, so check back periodically!

Sound Shapes
Hearing Geometry As Function and Metaphor
This activity is very abstract for many students and the least exacting from a mathematical point of view from any other in the book. I have found it especially useful to bring marginal students into the mainstream of activity in my math class. Some of my most adept and mathematically strong students have had difficulty with it, probably because of the inherent ambiguity and subjectivity in some of the connections that are sought by the activity. For the same reasons, my under performers in math class have come to life in this activity, feeling a degree of safety in the fact that for all of the exercises there is room for interpretation as to what geometric concept is represented by the music. The fear of being unequivocally wrong is diminished, hence the class is safer to take risks. Also, in a curious way, many of my underachieving math students could actually think in this abstract way more readily. It can be valuable for subsequent activities in class for students to have this experience; the shift in self esteem bleeds over to other classroom activities.

Measures of Time, Part 1
Hearing,Writing, and Computing Fractions of Time
Measures of Time fuses time quantities with geometric and symbolic representations. It is actually an excellent exercise for teaching musical rhythmic notation and dictation. While this may be of a lesser concern for a mathematics teacher in a mathematics class, Measures of Time reinforces the conceptualization of fractional quantities of time the computation of those fractions.

Measures of Time, Part II
Tempo and Rate Problems Facing Musicians
This activity is essentially a study in dimensional analysis. Used with Measures of Time Part I it provides a rich mathematical counterpart to the very musical exercises of that activity. Students compute authentic problems of time encountered by composers, musicians and conductors. It is important to note that In practice these computations are often performed mentally by musicians and/or are estimated and a musician will rarely need to calculate these problems to tenths of notes. This fact can be ground for discussion with your students as to the importance of relative degrees of accuracy in mathematics and when when estimation is valuable. Computing the problems with dimensional analysis does present the line of thinking required for their solution and provides useful computation practice and skill applying dimensional analysis. Interest is enhanced by the musical context. Indeed, students learn more than just mathematics in this activity.

The Multiples of Drummers
The Mathematics of Polyrhythms
This is a very accessibly activity, and always a lot of fun. There are many ways it can be adapted. A kinesthetic version comes to mind. I observed a choreography workshop at the Julliard School in New York last fall and students pairs were instructed to dance the same dance at different rates, so that student A would perform the dance twice in the same time that student B performed the sequence once. This is essentially the same notion of polyrhythms as presented in Multiples of Drummers(MD). The quantitative relationships in MD are present in many phenomena, and can be used to demonstrate the idea of resonance of vibrating systems in nature.
Teaching Tip: When it comes time to create a rule for the LCM, you may want to extend the tables #1 and #2 to include more examples. I have conducted this activity with elementary students and the inductive process requires more examples at that level. If you take the time and have them factor the examples, they can usually observe the patterns and come up with their own rule.

Record-Producer Algebra
Using Algebra To Perform Rap Music
This activity is a great way to get all kids involved. Its an opportunity for the performers to shine in math class, where the "bookish" types may feel more intimidated. As in Sound Shapes, this can be a useful event to shift the perceived status relationships in your class culture.

Teaching Tip: First and foremost, be patient. Allow students time to determine the solutions on their own. Have students present their solutions (there will be a wide variety) and use the solutions as a basis for discussion. It is also a great opportunity to have students debate the relative usefulness of the developed formulas. Appeal to musically oriented students to lend insights into where and when a formula might be useful, and another relatively meaningless.

Functional Composer, First Movement
A Mathematical Solution to Writer's Block
Click here for a graph and audio demo of this activity.
This activity is rich in music and math content. Pedagogically it is more teacher directed than man of the other, but extent to which students can generate the material themselves can be adjusted by you.
Teaching Tip: Consider having students keep notes after the transformations, of performing several of each family so that conjectures can be made that connect families of operations with graphic/musical characteristics. If you choose to do this have a student bring in a musical instrument to play the extra transformations. I have actually applied the transformations to a simple quadratic function simultaneously along with the musical application as the activity was conducted. This can be very interesting and help to make the connection to pure mathematics. The transformation techniques of Functional Composer are excellent tools to enrich the tone rows generated in Inside Out. Bringing these ideas back in that activity not only enriches that activity but serves as another way to spiral the concepts through the curriculum, resulting in deeper understanding and retention.

Functional Composer, Second Movement
The Relentless Composer
This activity applies transformations on the input value of functions.
It is especially rich in music and math content. Actually, in many respects it is my favorite activity in the extent to which substantive mathematics and music concepts alike are seamlessly integrated in the process of the activity. I personally made several discoveries while exploring this work--the mathematical context to view counterpoint can be a great way to teach the topic to music students. I stumbled upon a mathematical notion while experimenting with various phase shifts of a periodic melody. I was recording a version that I chose at random to hear how it would sound with the original motif. Oddly on playback could not hear my transformation, as though it did not record. I soon discovered that it in fact had recorded, but the transformation that I chose was the period of the original melody! They were playing back in perfect unison. It was an intriguing application (and a very direct experience) of the notion that a phase shift of the amount of the period will yield an identical function.

Teaching Tip: The audio examples that ask students "which is more harmonious" may be difficult for many to judge. This is a very subjective area as well as the fact that the transformations are somewhat short in their strict mathematical generations.

Name That Function
Determining Function Transformations By Listening
This activity might have been called Functional Composer--The Sequel. It can be difficult to use this activity without having first done Functional Composer. In any event, it is a fabulous warmup for the beginning of any class session throughout the year. You might consider making up more of the exercises if you are musically inclined, or have students generate them. Another great extension is to play Name That Graph. In my math and music units for the elementary level I use this activity daily. One version would be to place three graphs on the overhead and play a melody represented by one of them.

Inside Out
Hearing Pictures As Music Through Polar Coordinates
Click here to see and hear a demo of this activity.

Students love this activity. I have conducted it with elementary students and they have no trouble with the polar coordinate system, while gaining a valuable background in angle measurement and the notion of angular velocity. I strongly urge you to collaborate with a music teacher to incorporate this activity into a music composition project. At the Center for Educational Enrichment in Beacon, New York, I have had students explore and discover many essential aspects of music composition as an outgrowth of what was initially a purely visual arts/mathematics lesson.
Teaching Tip: The degree to which you will be able to have students match graphs to melodies will vary markedly between classes. It can be a cumbersome process if you don't have a fair number of musically proficient students to play the music accurately and consistently. The keyword here is be adaptable. The activity is very valuable if only a few of the graphs are demonstrated at the end of class.

Scaling the Scale, Part I
Natural Vibrations and Pythagorean Tuning
Teaching Tip: Use your own discretion as to how much of the audio CD you use for this activity. As you will see, it is very simple. A valuable resource is to have a student bring in a guitar for this lesson.

Scaling the Scale, Part II
A Solution to the Limitations of Pythagorean Tuning
Teaching Tip: Do not be discouraged if your students cannot reach the final solution independently working from the prompts. The ideas are potentially confusing, especially for students with no familiarity with the piano keyboard or the notion of musical intervals. The message of the activity will still be strong if you need to help them through some of the steps, but do your best to do this individually so that all students have the opportunity for their own discovery.

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