Math in the Middle... of Music WIND INSTRUMENTS Key Terms aerophones scales octaves resin vibration waver fipple standing wave wavelength Math Concepts Integers Fractions Decimals Computation/Estimation Multiples/Factors Ratio/rates Proportion Patterns Harmonic series Variables/Solving Equations Measurement Video Fieldtrips See a middle school band perform Watch the Army band perform Objectives Discover how the concepts of proportion and ratio apply to instruments without strings Discover how instruments that don t have strings produce music Connect proportions and ratios to the way sounds are made Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 1
NCTM Standards Connections MATHEMATICS AS REASONING Using mathematics to determine the fundamental frequency of a pipe MATHEMATICAL CONNECTIONS Connection of ratios/proportions to the way sounds are made Relationship between the length of a string and the pitch Relationships between proportional length of reed-like instruments and a major triad Hearing a mathematical equation through music NUMBER AND NUMBER RELATIONSHIPS/NUMBER SYSTEMS AND NUMBER THEORY Frequencies expressed as a rate (cycles/second or hertz) The use of mathematical ratios and proportions in scales Octaves divided into sets of notes The relationship of elements of a harmonic series to a fundamental frequency Using mathematics to describe relationships of various frequencies Describing an octave in terms of the relationship of the frequencies of two notes PATTERNS AND FUNCTIONS The relationship of elements of a harmonic series to a fundamental frequency ALGEBRA Solving for the fundamental frequency of a pipe MEASUREMENT Measuring the length of a pipe to help determine its fundamental frequency Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 2
Getting Started - Discussion Topics It s common to speak of woodwinds and brasses as instrumental families. But medieval trumpets and other brass like instruments were made of wood, while such woodwinds as the modern flute and saxophone are constructed from metal. So the common way we have of classifying these instruments is really not very accurate. A better term groups all of the instruments into one category, that of aerophones. This literally means air-sound, which describes wind instruments. Making sounds with wind greatly differs from the string instruments we have looked at so far, or the rubber band instruments we have made to this point. There, a tight string was used to create the vibrations of sound. A box of some sort amplified this sound, and finally notes were changed by changing the length of the strings. With wind instruments, there is no apparent vibrating surface like that of the string instrument. Yet the principles of sound production remain the same; there must be a source for vibration, a way of making it more audible, and finally, a way to change notes in order to play music. Bearing those principles in mind, let s begin to look at how wind instruments work and what math can tell us about the application of these principles. Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 3
Discussion - Lesson Overview Using what has been learned about vibrations and pitch, this video will explore how wind instruments produce and manipulate sounds. Earlier lessons have shown various stringed instruments and how those strings vibrate to cause sound. Now we will look at the less obvious ways in which various wind instruments produce music. In watching wind instruments, we can see that the performer puts a part up to their mouths and wiggles their fingers. But where is the vibration coming from? And how exactly do they change notes? This lesson will begin with woodwind instruments and progress to the brass instrument family. Simple and complex wind instruments all operate on basically the same principle; a pipe is somehow altered in length in order to change pitches. Using this truth as a basis, reed wind instruments will be examined, starting with the simple tin whistle. Students will be shown how lifting a finger on each successive hole makes the pipe shorter and raises the pitch of the sound. But just as they become comfortable with that idea, fingers will be placed back over some holes and a yet higher sound will be produced. This manipulation of the air stream will lead to a discussion of how wind pressure can act with the overtone series of a pipe length to create a range of tones. More complex woodwind instruments will further reinforce this basic truth. The action of reeds will be investigated as sources of vibration for many woodwind instruments. The action of the fipple in the tin whistle and the blowing of air across the mouthpiece will include a discussion of air pressures that create different notes. This will provide experience with a number of different sound production methods. Brass instruments will be introduced and students asked to predict the source of the vibration that creates sounds on these instruments. It will be shown that it is the lips that do most of the work, both as vibrators and sources of note changes. It will also serve to demonstrate how the series of notes available on a brass instrument are exactly the same as the harmonic series that colors each individual note. The method for changing the length of the brass instrument will then be discussed. A trombone will serve to show how the instrument physically gets longer in order to play all the notes needed to make music. The bell acts as a megaphone, projecting the sound by organizing the sound waves. Students will be encouraged to make wind instruments out of found objects. Coke bottles can be filled with different amounts of water and played like a flute. A favorite exercise is to take a length of hose and attach a funnel to one end in order for it to act as a bell. Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 4
Activity - Understanding Woodwinds Purpose To use simple materials to understand how sound is produced by simple vibrations of reed like surfaces and how proportionally changing the length of the reed acts in the same manner as changing the length of a string. Background Most woodwinds use reeds to create vibrations for their sound. These reeds are made A B-Flat Clarinet. from a special sort of bamboo that is carefully shaped to create an efficient vibrating surface. This reed is then placed at the end of a megaphone and its vibration is refined to create the tone of a clarinet or saxophone. Materials Blades of grass (as flat as possible) Soda straws (paper is best, but plastic will work) Scissors Procedure First, let s see if we can make a sound using a reed like substance. Take a blade of grass and hold it tightly between two hands. There should be just enough tension to hold the grass tautly. With the mouth slightly open, place the blade of grass between your lips and blow. With a little experimentation you should be able to produce a sound. (Do this outside, as once you get the hang of it the grass makes a loud, shrill sound!). Experiment with different lengths of vibrating surface by holding the grass in different place. (Shorter lengths will give higher pitches). For a more refined experiment, give each student four or more soda straws and follow these directions: 1. Cut the first straw to a length of about 8 inches. 2. Flatten 1/2 inch of the straw at one end 3. Cut about 1/4 inch from each corner of the flattened end. It should look like this: 4. Test to see if it works by putting the cut end into your mouth and blowing. This cut part acts like a woodwind reed to produce a sound. 5. Repeat these steps with the other 3 straws, cutting the second straw to be 4/5 as long as the first, the third to be 2/3 as long as the first and the fourth to be 1/2 as long as the first. 6. These four straws can be banded together to form a pitch pipe. Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 5
Activity - Understanding Woodwinds cont'd 7. The pitches of these four straws form a major triad. To hear it, divide the class into four groups and have each group blow into a different length of straw. Violà, a chord! This exercise can be extended by creating more lengths of straws. Just as with the "strings" of our rubber band instrument, a major scale can be created by cutting straws to the following proportions: 1 8/9 4/5 3/4 3/2 3/5 8/15 1/2 Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 6
Activity - Oscillating Air Streams Purpose To use the same mathematical proportions given above to create a series of musical bottles. Background Another way to illustrate the mathematical proportions of scales with vibrating columns of air is to use a series of soda bottles filled with different amounts of water. These bottles can be sounded by blowing a column of air across the open ends, much in the manner of a flute. Materials At least 8 similarly shaped empty soda bottles (well cleaned!) Ruler Water Activity First, we need to learn to sound a bottle. Place your lower lip on the rim of the open end of the bottle closest to you. Now tilt the bottle until the opposite side of the opening is level with your upper lip. Blow air across the opening in the bottle. With some minor adjustments, you will hear a tone. To create a scale, first measure the total length of the bottle. Divide the length by half. The parameter of the scale will be from 0 (empty) to 1/2 the total length of the bottle. If the bottle is 8 inches high, the string of air will be a total of 4 inches long. Using the proportions given on page 6, divide this length into the eight notes of the scale. Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 7
Challenge Activity - Make a Brass Instrument Purpose To use a simple tube to create a brass instrument and sound its harmonics, or partials. Background As we saw in the video, playing a brass instrument is all in the lips. A brass player vibrates his/her lips at the end of a pipe of metal in order to produce its sound. Notes are changed in two manners; by tightening and loosening the lips, and by changing the length of the pipe. We can duplicate this by using a length of plastic pipe, such as surgical tubing or a hose. The difficulty in this exercise will be learning to produce the sound with the lips. Having a brass player available will greatly aid in this endeavor. Materials Around 9 feet of flexible plastic pipe, such as surgical tubing or an old hose. A funnel to fit in one end of the pipe (optional). A brass mouthpiece (optional). Procedure If a mouthpiece is available, put it into one end of the pipe. If a funnel is available, put it into the opposite end of the pipe. First hold the pipe out straight. Use several students to do this in order to get a sense of how long 9 feet of pipe is. Next, coil the flexible pipe in such a manner as to make it possible for one person to hold the instrument. Ask the students to name a brass instrument it looks like (usually it looks most like a French Horn). A Double French Horn. Now, try to play a note. Using the video as a guide, ask the students to make a buzz. This is done by tightening muscles at the corners of the mouth while leaving the center loose enough to vibrate. Listen to the on camera guest and try to imitate that sound. Another way to approach this is to purse your lips as if you had just eaten a lemon, and then blow air through the center of your mouth. Once you have a buzz, place the small end of the hose on your lips so that it is lightly touching. Now attempt to perform the same buzz; with practice you should begin to be able to hear sounds. By tightening and loosing the lips, you will begin to hear different notes. See if you can match some of the notes from the trombone on the video the 9 feet of hose you have is Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 8
Challenge Activity - Make a Brass Instrument cont'd about the same length as the trombone. These pitches are related mathematically as harmonics. The entire harmonic series of the pipe is a series of ratios: 1:1, 2:1, 3:1, 4:1, etc... Each doubling of the frequency is an octave higher (1:1, 2:1, 4:1, 8:1...). Take time to experiment with creating these doublings if you can sound them on the hose, you will be hearing (and feeling) math. Consider using different lengths of tubing to compare sounds that can be created. Math in the Middle... of Music WIND INSTRUMENTS LESSON 5 Page 9