DRAFT
Teacher Notes
Geometric Music
Teacher Notes
Geometric Music can be a stand-alone activity. Since geometric transformations are applied to a
musical setting, the concepts can be understood by anyone with a basic knowledge of mathematical
transformations.
In the pilot, this activity was used as the introductory activity to the complete music unit. It could
have been used at the end of the music unit just prior to the geometry unit or it could have been
taught as one of the applications of border patterns within the geometry unit. It is well-suited to
insertion in a variety of places within the curriculum.
Instructional Time Required for this Section
2-3 hours
Required Materials
•
•
•
•
•
•
student activity sheets, one per student
Geometric Music Grid, one per student
1/4” graph paper or Geometric Music Grid, one per student
3”x3” (approximately) square of transparency film, one per student
overhead pens, one per group
Transparencies
o Blank grid
o Example Composition
o Musical Names for Transformations
•
Evaluation forms
Optional Materials
•
keyboard with the keys numbered –24 to 24 (middle C is zero) with removable labels
Vocabulary
translation - a translation is a shift or a slide of a figure or design.
reflection - a reflection is a mirror image of a figure or design across a line of reflection
rotation - a rotation is a turn of a figure or design around a specific point and specific
angle.
time translation – a translation relative to time
tone row – a collection of tones (for this activity, a collection of six tones) depicted on a
graph
matrix for a tone row-list of numbers that represent the heights of the tones
repeat – a horizontal translation (also considered a time translation) of the original tone row
retrograde – a reflection through a vertical line of the original tone row, a time translation
(a horizontal translation) is then performed on the reflection if the retrograde is not the
second component of the composition
transposition – a vertical translation of the original tone row followed by a time translation
inversion – a reflection about a horizontal line followed by a time translation
Mathematical Models with Applications, Fine Arts Module
Music Unit, Geometric Music Charles A. Dana Center
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Teacher Notes
retrograde inversion – a reflection about both a horizontal and a vertical line (a rotation
about a point) followed by a time translation if the retrograde inversion is not the second
component of the composition
Procedures
If this is the first section of the music unit that you cover, you may want to provide an overview of
the music unit such as the one provided below.
Overview: From simple rhythms to the complex production of sound, mathematics has
helped defined the underlying foundation of music. Even early mathematicians such as
Pythagoras were aware of the similarity between the patterns and the structure of the two
disciplines. Both disciplines have helped to form and define our culture. In this unit, we
will investigate some of these connections by looking at rhythm, pitch, and composition.
Assign the students to predetermined groups which have at least one student with a musical
background in the group (if possible). Assure the students that they will not have to learn to
“read” music.
Describe how this unit will be assessed at the beginning of the unit, please see the Assessment
section of these Teacher Notes.
Ask the students to refer to the first section on their Notes . 1. Basic Tone Row
• Model the creation of a tone row on a transparency of a grid.
• Write the intervals for the tone row you have created.
• Make a list of the numbers for your tone row ; call it the matrix for the tone row.
Allow time for students to create their own ton row on graph paper.
Make sure that they mark the x-axis (zero-note line) on the transparency and that they label top
(and front) of the transparency. Stress that they will be expected to include this piece of
transparency film when they turn in their final composition for use as a grading aid.
Model Steps 2 and 3 of the activity sheet using your tone row.
Review the geometric transformations with the students. The sample transparency from the
Geometry Unit has been repeated in this section.
Connect the musical names for the transformations to the mathematical names. Discuss the
examples in section 4: Transformations and on the transparency, Musical Names for
Transformations. : repeat = horizontal translation, transposition = vertical translation, retrograde =
reflection about a vertical line, inversion = reflection about a horizontal line, retrograde inversion =
rotation (both horizontal and vertical reflection). Model for the students how to create a
composition by using the example tone row and a piece of transparency film. Demonstrate how the
transparency is manipulated to generate each transposition.
The students will use the transparency film square to help them complete the composition.
The students may use the transparency as a manipulative to physically perform the transpositions
of the original tone row as previously demonstrated by the teacher. Once they see where the
transformed tone row is positioned, they can then copy it onto their graph.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Geometric Music Charles A. Dana Center
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Teacher Notes
Stress to the students that all transformations must be relative to the zero-note line (the x-axis).
While this is not necessary for composing, it does make grading the composition easier.
Emphasize that the line of reflection in the retrograde is located at the end of the original tone row
and the point of rotation for the retrograde inversion is at the end of the tone row. All the
transformations are transformations of the original tone row, not the previous six notes.
If the students use their own graph paper, they need to mark a horizontal center line (the zero-note
line) and mark vertical units on the left-hand side of the paper. Marked graph paper is included in
the Resources section of these Teacher Notes.
After the students have completed their compositions they may perform them on the keyboard,
They should turn in graph paper with their composition drawn, the piece of transparency film with
their original tone row, and the matrix list with transformations included.
Keyboard Option:
A keyboard can be used to perform the compositions. Place a removable label on the keys so that
middle C is Key 0, C-sharp is Key 1, D is Key 2, etc. going right on the keyboard. Label the B
below middle C as Key –1, B-flat as Key –2, A as Key –3, etc. going left from middle C on the
keyboard.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Geometric Music Charles A. Dana Center
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Teacher Notes
Geometric Music Resources
In addition to the installation instructions for the computer program, Geometric Music grid, and the
grading form that follow, the following resources may be useful or informative:
Beall, Scott. 2000. Functional Melodies: Finding Mathematical Relationships in Music. Key
Curriculum Press: Emeryville, California.
COMAP. 2002. Mathematical Models with Applications, Chapter 9. W.H. Freeman: New York.
Garland, Trudi Hammel, and Charity Vaughan Kahn. 1995. Math and Music: Harmonious
Conncetions. Dale Seymour Publications: Palo Alto, California.
O’Shea, Thomas. 1991. Geometric Transformations and Musical Composition in Applications of
Secondary School Mathematics, pp. 97-102. NCTM: Reston, Virginia. (Reprinted from the
October 1979 Mathematics Teacher.)
Mathematical Models with Applications, Fine Arts Module
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Scoring Guide
Name_________________________________Score_______________
Geometric Music
1. Creation of a basic tone row that is geometrically pleasing.
10
points _________
5
points _________
3. Use and identify an inversion.
10
points _________
4. Use and identify a retrograde.
10
points _________
5. Use and identify a retrograde inversion.
10
points _________
6. Use and identify a vertical translation.
10
points _________
7. Use and identify a fifth transformation.
10
points _________
8. Create a neat graphical design.
10
points _________
9. Use a matrix to list of the note numbers.
10
points ________
10.Enter composition into the Geometric Music program.
10
points _________
5
points _________
up to 10
points _________
2. Include transparency copy of the basic tone row.
11. Perform composition for the class.
12. Extra quality points.
Total _____________
Mathematical Models with Applications, Fine Arts Module
Music Unit, Geometric Music Charles A. Dana Center
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Student Activity
Geometric Music
Throughout history, music has evolved with all of the other aspects of
civilization. It has progressed from short simple songs with no structure to
today’s complex forms with a set of clearly defined rules governing the
manner in which the music is written. Students study music theory in order
to learn how to compose music that follows the rules that make the music
acceptable to the people who will listen. The structure controlling musical
compositions is very similar to the controlling structure of mathematics. As
the student of mathematics must follow the order of operation to
successfully solve a problem, a student of musical composition follows the
rules of harmony in order to write a composition that will be appreciated
by their audiences.
In the late 19th and early 20th century, many musicians, like the artists of
the time, began to question structure. They argued that the world would
run out of new music if the mold were not broken. Many began producing
compositions that did not follow any rules while others tried to create new
sets of rules. Arnold Schoenberg (1874 - 1951) experimented with a
system with a strong geometric form. He defined a 12-tone system for
music. We are going to compose music based on a 6-tone simplification
of Schoenberg’s system.
We will use a sheet of graph paper to represent our musical staff. The
vertical axis is scaled in half-tones with middle C located at 0. This means
that 1 is C-sharp, 2 is D, 3 is D-sharp. Going down the axis, –1 is B, –2 is
A-sharp, etc. Time is represented on the horizontal axis with each unit
representing one beat. Variation of rhythm will not be permitted; every
note will get one beat. The entire composition is limited to 36 beats, and
the tones must stay within the vertical interval [− 24,24].
You will compose music following the rules of transformational geometry.
Once the composition is constructed you are to transfer the design into
the computer program, “GeometricMusic”, so that your composition can be
performed. Your grade will be determined using the following grading
standard. You will turn in your composition on graph paper, the piece of
transparency film with the original tone row, and the matrix list with
transformations included.
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Music Unit, Geometric Music
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Draft
Student Activity
Grading Standard
1.
Creation of a basic tone row that is geometrically pleasing.
10 points
2.
Include transparency copy of the basic tone row.
5 points
3.
Use and identify an inversion.
10 points
4.
Use and identify a retrograde.
10points
5.
Use and identify a retrograde inversion.
10points
6.
Use and identify a vertical translation.
10 points
7.
Use and identify a fifth transformation.
10 points
8.
Create a neat graphical design.
10 points
9.
Use a matrix to list of the note numbers.
10 points
10.
Enter composition into the Geometry In Music program.
10 points
11.
Perform composition for the class.
5points
12.
Extra quality points.
up to 10 points
Mathematical Models with Applications, Fine Arts Module
Music Unit, Geometric Music
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Draft
Student Activity
The Rules
1.
The Basic Tone Row
On graph paper, draw a geometric pattern consisting of 6 horizontal
line segments connected with vertical line segments. This is your
original tone row, we will call it this because the height of each
segment on the graph paper represents a musical tone.
No two tones in the pattern can be at the same height and all of the
heights must be within an interval of no more than twelve units, [0,11],
[− 1,10], [2,13], etc.
Once the tone row is set, the pattern order may
not be altered.
The pattern at the right is an example of a basic 6tone row. If you make a list of the numbers for
your tone row, it is called a matrix; the matrix for
this tone row is {2,5,9,7,4,0}.
2.
Duplicate on Transparency
Place a small piece of transparency film over your basic tone row
then copy the zero-line and the basic tone row using an overhead
projector pen.
You will use this transparent copy to help place the transformation in
the correct position on your final composition.
3.
Structure of the Composition
The composition will consist of six versions, including the original, of
the 6-tone row.
The five copies must be modified using the transformations listed in
rule four.
The order in which the transformations are used does not matter;
however, the original tone row must occur first.
All transformations are made with respect to the zero-note
line (the y = 0 line).
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Student Activity
The composition must include an inversion, a transposition, a
retrograde, and a retrograde inversion. The other two, a repeat and a
mixture, are optional; however, the composition must consist of 36
notes.
All of the notes must stay in the interval [− 24,24 ].
4.
Musical Names for Transformations:
Since a horizontal translation - a time translation - is required in order
to make a copy, the time translation is not mentioned in the
descriptions of the transformations.
Row 1 of the matrices in the examples below is the original tone row;
row 2 represents the transformation.
Note: Required means you must include this transformation in your
composition.
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a. Repeat (horizontal or time
translation only)
Not Required
Make an exact copy of the
original.
Repeat the note values.
2 5 9 7 4 0


2 5 9 7 4 0
Student Activity
b. Retrograde (reflection about a
vertical line) Required
Reflect about the line y = 6.
Reverse the order of the note
values.
2 5 9 7 4 0


0 4 7 9 5 2
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c. Transposition (vertical
translation) Required
Student Activity
d. Inversion (reflect about a
horizontal line) Required
Subtract the amount of the
translation from each of the note
values. (Down three in this
example.)
2 5 9 7 4 0 


-1 2 6 4 1 -3 
Mathematical Models with Applications, Fine Arts Module
Music Unit, Geometric Music
Reflect about the line y = 0 .
Multiply the note value by -1.
 2 5 9 7 4 0


-2 -5 -9 -7 -4 0
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Student Activity
e. Retrograde Inversion
(Reflect about both a
horizontal and a vertical
line. This is also called a
rotation of 180° about a
point.) Required
f.
Mixture (combination of
several types of
transformations) Not
Required
This transformation combines a
retrograde inversion with a
transposition down 1 unit.
Reflection about x = 6 and then
about y = 0 OR rotate 180° about
the point (6,0 ).
Reverse the order of the note
values, multiply by -1, then
subtract 1 from the note
values.
Reverse the order of the note
values then multiply each note
value by -1.
7
4 0
2 5 9


-1 -5 -8 -10 -6 -6
2 5 9 7 4 0 


0 -4 -7 -9 -5 -2
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Student Activity
Example Composition
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
-18
basic tone row
2 5 9 7 4 0  

 0 -4 -7 -9 -5 -2  

retrograde
inversion

 

 3 7 10 8 5 1  

transposition up 1
=

 

retrograde
0 4 7 9 5 2  

-2 -5 -9 -7 -4 0  

inversion

 

 2 -2 -5 -7 -3 0  retrograde inversion with transpositon 2 units up 
Mathematical Models with Applications, Fine Arts Module
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Student Activity
Geometric Music
For Advanced Composers
Replacement rule: Any number can be replaced with an equivalent
number modulo 12. This means that 0, 12, 24, 36, –12, –24, etc. represent
higher or lower versions of the same note. 1, 13, 25, 37, –13, –25, etc.
also represent different notes with the same musical name. In mathematics
we call this modular arithmetic. The musical scale is modulo 12. Our
typical wall clock is also mod 12. Any numbers that have the same
remainders when divided by 12 are equivalent mod 12. The numbers 16,
4 and 28 are the same mod 12 because the remainder is 4 when each is
divided by 12.
Repeated note rule: On the graph, draw the horizontal lines two units
long with a short dividing vertical segment in the middle; this notation
indicates a repeated note. This repeated note does not alter the pattern
since a new note is not used.
Conversions to real notes: On a piano the basic names of the keys
beginning with middle C are C, C-sharp, D, D-sharp, E, F, F-sharp, G, Gsharp, A, A-sharp, B, and back to C. Notice that there is not a note named
E-sharp nor a note named B-sharp since there is only one half-step
between E and F and between B and C. The computer program converts
the numbers to note names. Middle C is 0, C-sharp is 1, D is 2 and so
forth on up the keyboard. The connection continues downward by
assigning –1 to B, –2 to B-flat or A-sharp, etc.
Mathematical Models with Applications, Fine Arts Module
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Teacher Notes
Rhythm
Teacher Notes
This section introduces the most basic connection between mathematics and music, that of rhythm.
The two activities in this section may be used together as the only music activities in the unit;
however, they are designed to be the beginning of a more comprehensive look at the connections
between mathematics and music.
The primary objective of the first rhythm activity is for students to understand the types of notes
and the organization of a time signature. Underlying objectives provide for a review of addition of
fractions and a review of the concept of a least common multiple. This first section is the most
musically demanding for the teacher, so, if the teacher is unfamiliar with musical notation, they may
want to ask for help from someone in the music department. The objective of the second rhythm
activity is to use fractions and a number line to see how complex rhythmic patterns are arranged
mathematically.
Instructional Time Required for this Section
Two-three hours
Required Materials
•
student activity sheets, one per student
Suggested Materials
•
•
video clip of Stomp, Riverdance, or Lord of the Dance (the library may be able to order
these videos; see the Resources section of these Teacher Notes for more information)
audio recording of rhythmic music (such as African music)
Vocabulary
rhythm – indicates the pattern of note values
measure – a group of notes or rests found between two bar lines on a musical staff
beat – a unit of musical time
time signature – two numbers written at the beginning of a line of music that resembles a
fraction without the fraction bar; the “numerator” indicates how many beats of equal length
occur in a measure, the “denominator” indicates the kind of note that is assigned one beat
Procedures
If this is the first part of the music unit (Geometric Music was not covered prior to this activity),
present an overview of the unit such as the one provided below.
Overview: From simple rhythms to the complex production of sound, mathematics has
helped to define the underlying foundation of music. Even early mathematicians such as
Pythagoras were aware of the similarity between the patterns and the structure of the two
disciplines. Both disciplines have helped to form and define our culture. In this unit, we
will investigate some of these connections by looking at rhythm, pitch, and composition.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
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DRAFT
Teacher Notes
Assign the students to predetermined groups that have at least one student with a musical
background in the group (if possible). Assure the students that they will not have to learn to
“read” music.
Mention how the assessment will work in this unit at the beginning of the unit, please see the
assessment section of these notes.
First hour:
Discuss with the students that rhythm pre-dates musical tone and that it creates the basic form for
music. Explain that music, like mathematics, uses a set of symbols. These musical symbols, like
the mathematical symbols, do not depend on a particular spoken language. They have the same
meaning in all languages. Musical notation consists of a set of notes that inform the musician
about the amount of time allotted to a particular sound. The location of a note on the staff (the
musical graph) provides information to the musician about the pitch that is to be produced. The
time signature on the musical staff sets the type of rhythmic pattern for the music and tells the
musician how the notes will relate to this rhythmic pattern. To understand how notes define the
rhythm, the students are to review (from their elementary school music classes) how this set of
symbols indicates time the time value of note.
Show a clip from a rhythmic performing group (such as Stomp, Riverdance, or Lord of the Dance)
and ask the students to focus on the rhythm. If Stomp is used the individual routines can be used
as class openers throughout the entire unit.
Discuss the types of notes with emphasis on the analogy to fractions. A time signature of 4/4 is the
foundation for most music. In 4/4 time, a quarter note gets one beat (or count) and there are four
beats in a measure. The time signature 3/4 still uses the quarter note as the “basic” note, the note
that gets one beat; however, there will only be three beats in the measure. The “numerator” tells
the musician how many beats will be in a measure, while the “denominator” tells the musician the
type of note that is to receive one beat.
A whole note is equivalent to four quarter notes and takes up an entire measure in 4/4 time.
A half note is equivalent to two quarter notes and takes up half the measure in 4/4 time.
A quarter note takes up one fourth of the measure in 4/4 time.
Working with fractions instead of notes, show that the “measures” in the examples on the second
and third pages of the student activity work or do not work.
Assign Rhythm Worksheet 1.
Second hour:
Lead the students in a discussion of how different cultures use rhythm. Tell them that many
cultures often have “music” in which the rhythm is much more important than the melody. If this
type of music is available, have the students listen to some African music; this musical type is
identified as polyrhythmic music in that each “musician” establishes a unique rhythm that must be
performed with others who are also playing their unique rhythm. The resulting rhythmic patterns
are extremely complex.
Play more from the performances Stomp, Riverdance, or Lord of the Dance. Ask the students to
pay attention to the interaction of the rhythms produced by the different performers.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
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Teacher Notes
Review the concept of least common multiple as necessary. Discuss the example of 2 against 3;
this is the idea of 2 beats “played” in a six-count measure against 3 beats “played” in a six-count
measure.
Ask a student to “direct” then divide the room into the “2-beat” side and the “3-beat” side. Have
each group "practice" their rhythmic patterns using a 6-count measure. First, have the “director”
clap while counting 1 2 3 4 5 6 1 2 3 4 5 6… then have the “2-beat” side clap on beats 1 and 4
until they feel the pattern. Next ask the “3-beat” side to clap on counts 1, 3, and 5. When both
groups have the basic idea have them clap together using their respective patterns. This generally
takes several tries and lots of laughter.
Ask the students to work on Rhythm Worksheet 2, then "perform" more of the rhythmic patterns.
Sometimes students have difficulty with the patterns in which the least common multiple is not just
the product of the two numbers. With 6 against 8, the least common multiple is 24 rather than 48,
which means there will be a 24-count measure. Thus, there will be a beat every 4 counts for the 6
pattern and a beat every 3 counts for the 8 pattern.
If there is a drummer in the class, be sure to have him or her demonstrate some of the patterns.
Invite any band members to bring percussion instruments to class and to explain how percussion
music is written.
Assessment
There will be a test on the last day of the unit; however, some of the test questions will be given
throughout the unit. The total of all the points from the end-of-section questions as well as the
questions on the test will be combined to calculate the exam grade.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
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Teacher Notes
Rhythm Resources
The Stomp Out Loud video is available to order on the web at www.stomponline.com or at
www.broadwaynewyork.com. Both offer the video for $29.95 (August 2001 price) plus shipping.
Also consider Riverdance (available to order on the web at www.riverdance.com, $35.00) and The
Lord of the Dance (available to order on the web at www.lordofthedance.com, $15.62) as
alternatives to Stomp.
In addition, you may find the following resources useful or informative:
Beall, Scott. 2000. Functional Melodies: Finding Mathematical Relationships in Music. Key
Curriculum Press: Emeryville, California.
COMAP. 2002. Mathematical Models with Applications, Chapter 9. W.H. Freeman: New York.
Garland, Trudi Hammel, and Charity Vaughan Kahn. 1995. Math and Music: Harmonious
Connections. Dale Seymour: Palo Alto, California.
Maor, Eli. 1991. What Is There So Mathematical About Music? in Applications of Secondary
School Mathematics, pp. 88-96. NCTM: Reston, Virginia. (Reprinted from the September 1979
Mathematics Teacher.)
Mathematical Models with Applications, Fine Arts Module
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Student Activity
Rhythm Activity 1
Notes
Mathematics and Music have much in common; however, the most
obvious connection between them is that of rhythm. To understand this
relationship, we first need knowledge of some of the symbols and the
vocabulary of music.
To begin you will need to recognize the types of notes and their value
relative to a quarter note.
whole note (4 quarter notes)
half note (2 quarter notes), add a stem
quarter note - fill in the note
note), add a flag
eighth note (half of a quarter
This represents two eighth notes. Connecting the flags together this
way is similar to using a set of parenthesis in mathematics.
Each additional modification on the note symbol cuts the note name in half.
For example, another flag attached to an eighth note creates a sixteenth
note.
Rhythm measures time. A piece of music is divided into equal
measures, each of which represents the same amount of time. Bar lines
divide the music into measures. When we listen to music, we often “feel”
the rhythmic organization of the music into measures.
time signature
bar lines
measure
In the music drawn above, notice the two numbers written at the beginning
of the line that resemble a fraction without a fraction line. These two
numbers indicate the time signature for the music.
The top number, in the “numerator” position, tells how many
beats of equal length will occur in a measure, and the bottom
number, in the “denominator” position, indicates the kind of note
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
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Student Activity
that is assigned one beat. In the music line shown above, the quarter
note receives one beat.
4
The time signature 4 (read “four four”) tells the musician that in one
measure there will be four counts and one quarter note receives one
count. The names of the notes are named relative to four-four time; these
names for notes are used in any time signature but are named with
4
3
respect to 4 time. For example, if the time signature is 4 then there will be
three counts in one measure and a quarter note will receive one count. If
3
the time signature is 8 there are three counts in the measure and an
eighth note receives one count.
4
If the time signature is 4 , the whole note receives all four counts in the
measure. In other time signatures the whole note does not receive all of
the beats in a measure. A whole note uses all of the time in a “four-four”
measure because there are four counts in the measure and one quarter
note (one-fourth of a whole note) receives one count. A half note is held
for two counts in a four-four measure; it uses half of the time in the
1
measure. An eighth note receives
of the time in a four-four measure or
8
half the time of a quarter note. Rests (absence of sound) are defined in
the same manner.
Adding a dot after any note increases its time-value by 50%. The
examples below shows that a “dotted half note” is equivalent to a half note
+ a quarter note and that a “dotted quarter note” is equivalent to a quarter
note + an eighth note.
This is where music makes a connection to fractions.
3 1 1
= +
4 2 4
3 1 1
= +
8 4 8
Musicians combine the different types of notes and rests (absence of
sound) to create interesting rhythmic patterns while keeping the correct
total of beats (counts) within each measure. Music with rhythmic mistakes
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
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Student Activity
sounds and feels wrong; therefore, musicians are very careful that the
measures contain the correct time value. The only exception to this
rule is the “pick-up measure” that is sometimes used at the
beginning of a song. The musical line example on the previous page
uses a pickup measure with two eighth notes.
Look at the time signature as a fraction. For instance think of “three four”
3
6
time as the fraction . “Six-eight” time becomes and so forth. Using the
4
8
note names, write the value of each note as a fraction. The sum of the
notes in each measure must equal the value of the time signature as a
fraction.
In
4
4
time the sum of the notes must be
4
. In
4
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
3
4
the sum must be
3
.
4
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DRAFT
Student Activity
EXAMPLES
I.
II.
The following examples show correctly written measures with a
variety of time signatures.
1.
1 1 1 4
+ + =
2 4 4 4
2.
1 1 1 1 4
+ + + =
4 4 4 4 4
3.
1 1 1 1 1 8 4
+ ( + )+ + = =
4 4 8 8 4 8 4
4.
1 1 1 1 3
+ + + =
4 8 8 4 4
5.
1 1 1 1 3
+
+ +
=
8 16 8 16 8
Each of the following examples illustrate incorrectly written measures.
Remember that the sum of the note values must equal the fractional
value of the time signature.
1.
1 1 1 7 4
+ + = ≠
8 2 4 8 4
2.
1 1 1 1 5 6
( + )+ + = ≠
4 8 8 8 8 8
3.
1 1 1 1 9 4
+( + )+ = ≠
2 4 8 4 8 4
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
25
DRAFT
III.
Student Activity
Determine which of the following measures are correct and which are
not. Justify your answers by writing the fractional expression for each
measure.
1.
2.
3.
4.
5.
6.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
26
DRAFT
Student Activity
Rhythm Worksheet 1
I.
For each of the following identify the type of note and tell how many
beats the note receives if the quarter note gets one beat. (Do not use
the decimal form of the answer.)
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
27
DRAFT
II.
Student Activity
In each of the following time signatures tell which type of note gets
one beat and how many beats there will be per measure.
Time
Signature
Note that gets one beat
(Name and symbol)
Beats per
Measure
3
4
3
8
6
8
2
2
3
2
4
4
4
2
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
28
DRAFT
III.
Student Activity
Some of the following measures have the correct number of beats
and some do not. Decide which measures are correct and which are
not then, using fractions, show the mathematics that will confirm your
answers.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
29
DRAFT
Student Activity
Rhythm Activity 2
Notes
Musical rhythm is not always a simple concept. For many musicians,
correctly performing the rhythm is far more difficult than producing the
correct notes. No matter how complicated a musical rhythm gets,
mathematics can be used to clarify the pattern. Modern music often
contains very complex rhythmic patterns which require the performers to
maintain several patterns at one time. For example, a drummer might play
one drum that maintains two beats per measure while playing a second
drum that will maintain three beats per measure.
This type of rhythmic pattern is called 2-against-3. One of the rhythmic
cycles divides the given time period up into two equal units, and the other
one divides it into three equal units. The problem of figuring out where the
notes will fall in relation to one another becomes one of finding a least
common multiple (as in fractions) for the two types of patterns.
The LCM (least common multiple) between two numbers is the smallest
number that can be divided by both of the original numbers. For example,
the LCM of 2 and 3 is 6, and the LCM of 4 and 6 is 12.
Example: Rhythm of 2 beats against 3 beats.
Find the least common multiple for 2 and 3. (The least common multiple is
the smallest whole number divisible by both 2 and 3.)
6 is the smallest number divisible by 2 and 3.
a.
Draw a number line using equal segments starting at 1. Number from
1 to 6. Instead of going on to 7 start over at 1 to show the counts
needed for two measures. Place a dot • to represent the locations
where the two beats will occur. (On count one and count four of
each measure so place the dots on 1, 4, 1, 4, ...)
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm
30
DRAFT
b.
Student Activity
Now place a n to represent the locations where the three beats will
occur. (On counts 1, 3, and 5 of each measure, so place the squares
on 1, 3, 5, 1, 3, 5, ...)
2 beats per measure in a six-count measure
•
•
•
1
n
2
3
n
4
5
n
6
1
n
2
•
3
n
4
•
5
n
6
1
n
3 beats per measure in a six-count measure
Notice that both sets of rhythms will always have a beat at the beginning of
each measure.
(The 6 counts of these measures must take the same amount of total time
as any other measure in the music.)
Complete the rhythmic patterns on the activity page.
Divide the class into two groups and perform the rhythmic patterns by
clapping. Good luck!
Mathematical Models with Applications, Fine Arts Module
Music Unit, Rhythm – Teacher Edition
31