Mathematical Relationships In Music
Advise and Comments
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!
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.
of Time, Part 1
and Computing Fractions of Time
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.
of Time, Part II
Tempo and Rate Problems Facing Musicians
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.
Multiples of Drummers
The Mathematics of Polyrhythms
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.
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
Composer, First Movement
A Mathematical Solution to Writer's Block
Click here for a graph and audio demo of
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.
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.
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.
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.
the Scale, Part I
Natural Vibrations and Pythagorean Tuning
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
the Scale, Part II
A Solution to the Limitations of Pythagorean
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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