Download Math/Musical Instruments Unit

The Mathematics of Instruments
and Sound Unit
8th Grade
Musical Standards Addressed:
2. Performing on instruments, alone and with others, a varied repertoire of music
4. Composes and arranges music with specified guidelines
5. Reading and notating music
8. Understanding relationships between music, the other arts, and disciplines outside the
arts
Core Curriculum Standards Addressed:
Mathematics:
 Solve real-world problems involving ratio, proportion, and percent
 Add, subtract, multiply, and divide intergers
 Select and use a computational technique (e.g., mental calculations, paper,-andpencil, technology) to solve problems
 Determine when an estimate is sufficient and an exact answer is needed in
problem situations
 Distinguish between experimental and theoretical probability; i.e., the results of
an experiment may not match the theoretical probability
Science:
 Design and conduct a scientific investigation (e.g., making systematic
observations, making accurate measurements, identifying and controlling
variables
 Explain the characteristic properties (e.g., wavelength, frequency) and behaviors
(e.g., reflection, refraction) of waves
Language Arts:
 Use a variety of primary and/or secondary sources to access information; i.e.,
computer catalogs, magazines, newspapers, and primary sources.
 Evaluate sources that present different perspectives; e.g., by identifying sources of
bias and distinguishing between primary and secondary sources.
 Use prior knowledge and experiences to aid text comprehension
 Use a variety of strategies to construct meaning from text; e.g., vocabulary
building strategies, skimming, paraphrasing, summarizing, brainstorming, and
discussing.
Purpose and Overview of Unit
The purpose of this unit is to understand how instruments are tuned and what causes the
pitches that are made. American bands tune to 440 mhz. This means that if each
instrument played a concert A in order to be perfectly in tune the vibration of sound
waves would vibrate at 440 waves per second.
Through this unit students will understand Pythagorean theorem in terms of length to
tone ratios. They will learn this through experimentations with and without their
instrument (see assignment examples). Students will learn how length of instrument
results in proper tuning and timbre. In order for proper understanding lesson and
assignments will need to be done in order.
Standards outside of music education are best taught by those specialized in those areas,
however the music teacher alone could still incorporate these standards effectively with
guidance and/or proper preparation.
Lesson Plan 1
Objective:
Students will give thoughtful ideas as to how sound is made with their instruments
Students will give thoughtful ideas as to what causes different pitches to occur in music
Students will demonstrate sound waves and resonance through experimentation
Materials:
Any scale
Cylindrical glass filled with water
Cylindrical tube
Procedure:
Begin by playing scale up and down on half notes.
How did we just make that scale happen? How did we make those sounds?
Anticipated answers: buzzing lips, air, reeds
We are making sound waves
Handout 1
Have students read “How is a sound wave made?”
Demonstrate a pencil dropping and blowing air over a bottle
Have students explain, based on reading, what is causing this sound
How does this relate to your instrument? (Air vibrates within the instrument)
Students will read “What is frequency”
Have students explain why a pencil being dropped is different from playing a
sound on an instrument
What do you notice about the difference in frequency between instruments?
Why?
Students will read “What is resonance”
Students participate in Assignment 1
Upon completion students will reconvene as a group to discuss what had happened
(resonance and sound waves) and how that translates into their instruments
Handout 1
How is a sound wave made?
Sound Wave = A sound wave is created as a result of a vibrating object.
Nearly all objects, when hit or struck or plucked or strummed or somehow disturbed, will
vibrate. If you drop a meter stick or pencil on the floor, it will begin to vibrate. If you
pluck a guitar string, it will begin to vibrate. If you blow over the top of a pop bottle, the
air inside will vibrate. When each of these objects vibrate, they tend to vibrate at a
particular frequency or a set of frequencies.
What is a frequency?
The particles of the medium through which the sound moves is vibrating in a back and
forth motion at a given frequency.
frequency = speed/wavelength
Some objects tend to vibrate at a single frequency and they are often said to produce a
pure tone. A flute tends to vibrate at a single frequency, producing a very pure tone.
Other objects vibrate and produce more complex waves with a set of frequencies which
have a whole number mathematical relationship between them; these are said to produce
a rich sound. A tuba tends to vibrate at a set of frequencies which are mathematically
related by whole number ratios; it produces a rich tone. Still other objects will vibrate at a
set of multiple frequencies which have no simple mathematical relationship between
them. These objects are not musical at all and the sounds which they create are best
described as noise. When a meter stick or pencil is dropped on the floor, a vibrates with a
number of frequencies, producing a complex sound wave which is clanky and noisy.
1 Hertz = 1 vibration/second
What is resonance?
The word resonance comes from Latin and means to "resound" - to sound out together
with a loud sound. Resonance is a common cause of sound production in musical
instruments.
In the absence of resonance, the sound of vibration is not loud enough to hear. Resonance
only occurs when the first object is vibrating at the natural frequency of the second
object.
http://www.physicsclassroom.com/Class/sound/u11l4a.html
Assignment 1
Materials:
Large cylinder filed ¾ with water
Small cylindrical tube
A tuning fork
(see picture)
Procedure
Step 1:
Fill large cylinder with water (3/4 full)
Step 2:
Have one person insert small cylinder into the water and hold above the surface of the
water
Step 3:
Another person within the group hits the tuning fork to create sound.
Step 4:
Hold the tuning fork over the small cylinder. Move the cylinder up and down to find the
most resonance (loudest sound).
Step 5:
Answer the following questions:
At what point did the cylinder produce the most resonance? How far above the
large cylinder?
Did anything happen to the water? If so, what was this?
Lesson Plan 2
Objective:
Students will demonstrate how the length of sounding device affects pitch
Students will discuss and demonstrate knowledge of Pythagoras and his musical ratios
and how it relates to pitch
Procedure:
What do you know about Pythagoras?
(Check with math teacher to know if they have studied Pythagorean theorem)
Discuss Pythagoras’ ratios for music
Using a guitar have one student measure from one tension rod to the next
Pluck string (G string)
Choose another student to find exactly halfway down the string and place one
finger – then pluck string
What happens? (Pitch is one octave higher)
Pythagoras discovered that a string at any given length produces x hertz (frequency), but
when the length of string is cut in half (2x hertz) the string sounds an octave higher.
Therefore x = fundamental, frequency x/2 = An octave higher
Give students handout 2
Demonstrate these ratios further using guitar
Are the ratios the same using the number 12?
Why?
Assignment 2
Handout 2
In dividing the length in this manner, Pythagoras exposed the first four overtones which
create the common intervals which have become the primary building blocks of musical
"harmony".
* An octave
* A Perfect 5th
* A Perfect 4th
* A Major 3rd
Pythagoras also acknowledged these intervals, not only as they relate to the fundamental
frequency, but to each other and found these ratios:
* 1:1 = Unison
* 2:1 = Octave
* 3:2 = Fifth
* 4:3 = Fourth
* 5:4 = Major Third
Pythagoras went a few steps further however and he acknowledged that twelve (12) was
the "most divisible" small number and that these basic ratios can be expressed in regards
to the number twelve (12).
* 12:12 (unison)
* 12:6 (octave)
* 12:8 (fifth)
* 12:9 (fourth)
He therefore surmised that twelve (12) was the most ideal musical number; thousands of
years of musical composition can attest to the notion that he was mostly right.
http://www.davesabine.com/Music/Articles/PythagorasMathematicalTheoruminMusic/ta
bid/169/Default.aspx
Assignment 2
Musical Glasses
Materials:
Several glasses of the same size
Water
Wooden spoon
Procedure:
Step 1:
Leave one glass empty and fill another glass with water to create an octave
Step 2:
What is the fundamental pitch of the glass? (Play on instrument or piano)
Step 3:
Fill glasses according to Pythagoras’ ratios (3:2, 4:3, etc.)
What were the pitches in each of the ratios?
Step 4:
Create a composition using various glasses and water heights (at least 3). Figure out
pitches and write out on staff paper. Be sure to give this piece a name, meter, and key
signature.
Lesson 3
Objective:
Students will briefly research to find length of their instrument (Percussion will
determine circumference of drums)
Students will be able to determine how pressing valves, keys, moving slides, and
tightening heads changes the pitch of the instrument
Procedure:
Break students up into individual sections throughout the room
Students will complete Assignment 3
If time remains students may share with the class their findings
Assignment 3
(For Brass and Woodwind)
Materials Needed:
Instrument
Staff Paper
Step 1:
On the first staff determine what notes you can play without pressing any keys, valves or
moving the slide?
Write these notes from lowest to highest
Step 2:
Woodwinds:
Press the key that adds the shortest amount of length to your instrument. Measure how
much longer that makes your instrument. Write the note you would play and the length of
the instrument at that point.
Continue doing this until you cover the length of the instrument (Use notebook paper)
Trumpet/Horn/Baritone/Tuba
Find the valve that has the shortest slide associated with it (follow the valve itself down)
Write all the notes you can play with that valve depressed. Continue this until you have
found all the combinations from shortest to longest.
*Remember valve combinations like 1 and 3 will add the lengths of both valve
tubing.
Trombone:
Move into 2nd position and write all the notes you can play in second. Continue until you
have written all notes in all positions in order.
Step 3:
Briefly research (using google search engine) how long your instrument is if it were
stretched out.
Brass – How long is your instruments with the various combinations? (complete on
separate sheet of notebook paper)
Woodwind – Using your measurements from Step 2 determine when your instrument is
nearly ½ the length.
When you have completed these steps hand in the staff paper stapled to the back of
the assignment sheets
Assignment 3
(Percussion only)
Materials needed:
Timpani
Staff paper
Pitch pipe
Procedure:
Step 1:
With a measuring tape carefully determine the circumference of each of the tympanis
Use the following chart
Largest
Circumference =
Lowest pitch =
Highest pitch =
Medium
Circumference =
Lowest pitch =
Highest pitch =
Smallest
Circumference =
Lowest pitch =
Highest pitch =
Step 2:
Briefly research how the peddle works to change pitches.
Lesson 4
Objectives:
Students will demonstrate proper tuning of instrument
Procedure:
Review length and how it changes sound
Have one student play a C on trumpet and another play a B.
Why does this hurt our ears? What is happening? (Students give suggestsions)
If we play exactly the same pitch our sound waves will look the same
Student A
Student B
Making a very strong sound
If we play a pitch that is slightly different the sound waves may actually collide which is
why we often flinch when something is close, but out of tune, those waves clash in our
ears.
How do we fix it??
Mechanically – lengthen or shorten instrument
If all notes are out of tune with the band you need to determine whether the pitch
is too high (sharp) or too low (flat)
Flat = push in
Sharp = pull out
All instruments should be pulled out slightly because that is how they’re made
Discuss where main tuning is for each instrument
Tuning on the fly
It is not always possible to fix the tuning mechanically, especially if it’s one note.
At home work on lip flexibility to be able to sharpen or flatten a pitch
Demonstrate
What am I doing to create a higher or lower pitch? (Altering the air
stream, which alters sound waves)
Break students into sections with tuner to determine if they are in tune and practice
tuning on the fly.
Percussion will practice tuning to various pitches on the timpani using a pitch pipe and
tuner to determine exact pitch. Work on speed.
Assessment for Unit
Musical Aspect:
Y
N
Demonstrates knowledge of tuning individual instrument
Y
N
Composed a piece using proper key and time signature, and correct notation
Y
N
Ability to hear pitches that are extremely flat or sharp
Mathematics Aspect:
Y
N
Demonstrates knowledge of ratios
Y
N
Able to problem solve
Science Aspect:
Y
N
Completed all experimental assignments with accuracy
Y
N
Can explain in own words how sound waves, frequency, and resonance
works
Language Arts Aspect:
Y
N
Researches correct material when necessary
Y
N
Demonstrates knowledge of vocabulary learned