mathematical patterns and functions in musicmath.buffalostate.edu/~it/projects/stockings.pdf ·...

Mathematical Patterns and Functions in Music

Evaluations:Quiz

Homework Quiz

Alan M Stockings

December 1, 2004

Stockings MED 607 Final Project 2

Table of Contents

Day 1: Powers and Fractions in Music..........................….........…....Pg. 3Day 2: Reflections, Rotations, and Translations in Music........................Pg. 22Day 3: Sound Wave Sine Function (Technology Day)…....................... Pg. 42Day 4: Frequency Ratios and Cords in Music.......….......................…..Pg. 54Day 5: Did Pythagoras Discover Music?.………..........................…. Pg. 60Works Cited.............................……………............................. ..Pg. 64

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Day 1

Powers and Fractions in Music

Prerequisites: In this lesson students will review properties of powers/exponents and fractions inexamples like the following:

2° = 1 22 = 4

and use them to help understand written music.

Student Objectives: The students will:

a. Discover how the concepts mentioned above relate to writing musicb. Analyze phrases of music by using mathematics (reading the correct time signature)c. Write their own four to five bar phrases using correct time signatures

Equipment and Environment: Students will be in-groups of four (with at least one musically inclinedstudent) and have the appropriate handouts for the lesson with writing utensils (pencil and eraser).

NYS Core Curriculum Performance Standards:Operationsa. Use addition, subtraction, multiplication, division, and exponentiation with real

numbers and algebraic expressions

Modeling/Multiple Representationa. Choose appropriate representations to facilitate the solving of a problemb. Use multiple representations as tools to explain operations

Patterns/Functionsa. Represent, describe, extend, and create a wide variety of patternsb. Discover patterns in nature, art, music, and literaturec. Apply the concept of similarity in relevant situations

Opening Activity: This activity will remind students of the properties of powers/exponentsand fractions to help them understand how music is written. The following questions will beprovided in handout form (for students) as well as put on an overhead transparency (forteacher).

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2° = ? 22 = ?

The answers will be provided on another transparency.

Developmental Activity: The teacher will lead a discussion introducing time signature and note/restvalue to help the students with the information needed for the worksheets.

Students will be working in-groups of four with the following handouts, (Notes, Rests and Fractions,Bicentennial Beat, And the Beat Goes On) using mathematical concepts to analyze phrases of music.They will determine the value of the notes missing in musical phrases by looking at the timesignature and the portion written thus far.

Closing Activity: The students will take a brief quiz to see if they understand thematerial presented in class today.

Assignment: The students will use the music paper provided to "write" their own 5 bar phrases inthe following time signatures 4/4, _, 2/4 using combinations of notes and rests.

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Prerequisites: In this lesson students will review the three basic geometrictransformations (reflection, rotation, and translation).

Student Objectives: The students will:a. Find these geometrical transformations in various examples using written music

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b. Verify these transformations by analyzing them

Equipment and Environment: Students will work in the same groups as day 1 using the givenhandouts to complete the class work.

NYS curriculum Performance Standards:Operationsa. Develop an understanding of and use the composition of functions and

transformations b. Use transformations on figures in the coordinate plane

Modeling/Multiple Representationa. Investigate transformations in the coordinate planeb. Choose appropriate representations to facilitate the solving of a problem

Measurementa. Use geometrical relationships in relevant measurement problems involving geometric

concepts

Uncertaintya. Judge the reasonableness of results obtained from applications in algebra, geometry,

trigonometry, probability, and statistics

Patterns/Functionsa. Apply axiomatic structures to algebra and geometry

• Opening Activity: This activity will remind students of the properties of the three basicgeometrical transformations (reflections, rotations, and translations) by sketching them,

• a. Reflect triangle ABC over line segment m• b. Rotate triangle ABC about point D-180 degrees

• c. Translate quadrilateral ABCD 2 inches to the right. The answers will beprovided by a transparency before the next activity.

Developmental Activity: Students will work in the same groups as day 1 to completework sheets on reflections, rotations, and translations in musical phrases. The class willdiscuss the results and the teacher will explain how these transformations are used in 20th

century music.

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Closing Activity: Students will find these geometrical transformations in various etude exercisesform a music lesson study book.

Assignment: The students will complete the Finale: Variations worksheet and write their ownsix bar phrases (two or three phrases will suffice) using these transformations

Students' Worksheets: See attachments

Teacher's Notes: See attachments

Opening Activity: See attachments

Assignment: See attachments

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Answer Below:

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Day 3 Sound Wave Sine Function

(Technology Day)

Prerequisites: In this lesson students will review what they have learned about graphing the sinefunction, and the concepts of amplitude and frequency

Student Objectives: The students will:

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NYS Core Curriculum Performance Standards:Operationsa. Develop an understanding of and use the composition of functions and

transformationsb. Use transformations on figures and functions in the coordinate plane

Modeling/Multiple Representationa. Use graphing utilities to create and explore geometric and algebraic models

Patterns/Functionsa. Use function vocabulary and notationb. Represent and analyze functions using verbal descriptions, tables, equations,

and graphsc. Use computers and graphing calculators to analyze mathematical phenomenad. Translate among the verbal descriptions, tables, equations, and graphs

Developmental Activity: Students will be working in the same groups of four as the two previousdays using the TI-83 and the utilities provided (listed in the environment and equipment section) toanalyze the sound waves produced by the various musical instruments. Each instrument will playtheir tuning note (referred to as concert B flat). For example:Flutes will play B flat, clarinet, tenor saxes, and trumpets will play C, alto and baritone saxes will

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play G, etc....

Closing Activity: The students will discuss their results of the developmental activity if timeremains in the class period.

Assignment: The students will complete the Shapes of Sound work sheet

Students' Worksheets: See attached Lab

Teacher's Notes:

Opening activity: See attached sheet

Developmental Activity: Answers will vary slightly from group to group since there are differentmusical instruments being used.

Closing Activity: Class discussion Assignment:

See attached answer key

SOUND WAVE SINE FUNCTION Name __________________________

Materials1 TI-83 calculator with TUNE program installed1 Calculator Based Laboratory (CBL)1 CBL microphone1 link cable1 musical instrument (depending on the instrument of the musically inclined student inthat group. If there is no one who plays an instrument in that group then the teacher ormusic teacher could help out with this part)

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Collecting the sound data1. Connect the CBL, CBL microphone, and TI-83 calculator. Execute the TUNE program to

collect sound data produced by a wind instrument. TUNE collects data for 0.02475 seconds.

Creating the sine function2. Select the Y = menu. Enter Yl = A sin B (X+C)+D, a standard sine function. Store initial

values for 1 STO -> A; 1 STO •> B; 0 STO -> C; 0 STO -> Da. Record the current Yl using actual values rather than variables.b. Why is it reasonable to begin with the initial values given when exploring the creation

of a sine function representing the sound data?c. Why is the graph of Yl not on the same screen as sound data? Can you modify

domain and range of the WINDOW so that both are on the same screen? Explain.Explore WINDOW settings as needed.

Determining the vertical shift3. Under the ZOOM menu select ZoomStat to return to the original screen setting for

the sound data. Translate STO -> to the vertical position of the sound data byhanging the value stored in D.a. Determine the value of the line y = d on which the collected sound data are centeredvertically, that is, horizontal line midway between the oscillations. Explain how youdetermined this value. b. Enter the value into D using the STO -> key. Record the current Yl using actual

values rather than variables. c. Sketch the new graph screen, and explain the change in the graph of Yl.

Determining the period

4. Change the period of Yl by changing the value stored in B.a. Multiply the frequency in question 1 (b) by 2_.b. Enter this value into B using the STO -> key. Record the current Yl using actual

values rather than variables.c. Sketch the new graph screen, and explain the change in the graph of Yl. d. Explain why B is 27t times the frequency in cycles per second. HINT: Compare one

cycle of the sound wave with one cycle of y = Sin x.

5. Change the amplitude, the vertical height, of Yl by changing the value stored in A.a. Determine the height of the sound wave data from the line of vertical center found in

3 (a) to the maximum or minimum y value of the oscillations. Explain how theamplitude is related to the amplitude of y = Sin x.

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b. Enter the height into A using the STO -> key. Record the current Yl using actualvalues rather than variables,

c. Sketch the new graph screen, and explain the change in the graph of Yl.

Determining the phase shift6. Change the phase shift of Yl by changing the value of C.a. Determine the value that Yl needs to be shifted to the right or the left to coincide

with the collected sound data. Explain how you determined the value,b. Enter the appropriate, that is, positive or negative, into C using the STO -> key.

Record the current Yl using actual values rather than variables. c. Sketch the new graph screen, and explain the change in Yl.

Yl should be an algebraic model that represents the sound wave data. If the graph of Yl does notclosely approximate the graph of the sound wave data, check that Yl is entered correctly, as inquestion 2. Then modify the values stored in the parameters A, B, C, or D as needed to betterapproximate the sound data. Record your final Yl.

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Prerequisites: This lesson will review yesterday's activity

Student Objectives: The students will:a. Use frequencies of various music notes to find a mathematical relationship to musical

harmony

Equipment and Environment: This lesson will have students in the same groups of four as in theprevious three days with their graphing calculators and the work sheets provided

NYS Core Curriculum Performance Standards;Operationsa. Use addition, subtraction, multiplication, division, and exponentiation with real and

algebraic expressions

Patterns/Functionsa. Use graphing calculators to analyze and explore mathematical phenomenab. Describe and represent patterns and functional relationships using tables, charts andgraphs

Opening Activity: Students will complete the Can You Hit High C? worksheet, which will becorrected by transparency before the next activity

Developmental Activity: Students will work in their groups of four to find the frequencies andratios of the notes in the C, F, G, and A major triads as on the Are We On the Same Frequency?worksheet. There will be two groups assigned to each triad.

Closing Activity: The groups will put their findings on the board and discuss the patterns found ineach triad.

Assignment: The students will be asked if the ratios between the root and third, and root and fifthof the triads are the same for all major triads. They will also calculate these differences for the Cflat major triad and ask what other name this triad can go by (B major—hint use enharmonic tones)

Students' worksheets: See attachmentsTeacher's notes: See attachments for all activities

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Answer Key for Assignment :

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Day 5Did Pythagoras discover music?

Prerequsites: In this lesson, students have already learned the material on theprevious day's lab.

Student Objectives: The students will:a. Discover how the ratios they found in yesterday's assignment

relate to the ratio needed to make these notes possible on a string

b. Discover that this is how Pythagoras first discovered this concept.

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Equipment and Environment: The students will watch a video and then work onthe assignment individually. The teacher will need one TV one VCR and the movieDonald in Mathemagic Land by Walt Disney.

NYS Core Curriculum Performance Standards:

Uncertaintya. Judge the reasonableness of results obtained from applications in algebra, geometry,

trigonometry, probability, and statistics

Patterns/Functionsa. Recognize, describe, extent, and create a wide variety of patterns.

Opening Activity: Show part of the video Donald in Mathemagic Land dealing with howPythagoras discovered music.

Developmental Activity: Explain and work through the Online math Applications! Pythagoreanssheet, and answer any questions on the material.

Closing Activity: The students will do a homework quiz. The problems will include 1 5 barphrase written for day 1, 1 six bar phrase using the geometrical transformations and thefrequency tables for the C flat (B) major cord, and the C major cord.

Assignment: Find the sheet music for your favorite song online and analyze it for the threetransformations. Do you see any patterns? If so what are they?

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Works Cited

Principles and Standard for School Mathematics (2000). Reston, VA: The National Council ofTeachers of Mathematics Inc.

Resource Guide with Core Curriculum (2000). Albany NY. NYS Education Department

Learning Standards for Mathematics. Science, and Technology (1996). Albany NY, NYSEducation DepartmentMathematics and Science in Society (1977) University of Oregon

http://tqjunior.thinkquest.org/4116/music (April 2001)

http://thinkquest.org (April 2001)