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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 1 DRAFT 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 2 DRAFT 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 3 DRAFT 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 Music Unit, Geometric Music Charles A. Dana Center 4 DRAFT 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 5 Draft 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. Mathematical Models with Applications, Fine Arts Module Music Unit, Geometric Music 6 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 7 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). Mathematical Models with Applications, Fine Arts Module Music Unit, Geometric Music 8 Draft 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. Mathematical Models with Applications, Fine Arts Module Music Unit, Geometric Music 9 DRAFT 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 Mathematical Models with Applications, Fine Arts Module Music Unit, Geometric Music 10 Draft 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 14 DRAFT 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 Mathematical Models with Applications, Fine Arts Module Music Unit, Geometric Music 15 DRAFT 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 Music Unit, Geometric Music 16 DRAFT 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 Music Unit, Geometric Music 17 DRAFT 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 18 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 19 DRAFT 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 20 DRAFT 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 Music Unit, Rhythm 21 DRAFT 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 22 DRAFT 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 23 DRAFT 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 24 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