DRAFT
Teacher Notes
Monochords
Teacher Notes
The objective of the activities in this section is to make connections between mathematics and music
through concretely measuring string lengths and hearing the changes in pitch that result. The first
activity in this section, Pythagorean Tuning, has students examining ratios in string lengths to hear
pitch changes. The second activity examines the modern scale and the ratio between string lengths
required to produce half-step tones. The third activity has the students applying what they have
learned in the first two activities to play simple musical pieces. Each group is responsible for one
or two notes in the composition and the “orchestra” plays the entire piece.
To accomplish this, a single-stringed musical instrument, a monochord, is used. The monochord
should have a movable fret so that students can easily change the string length. Monochords are
commercially available or can be made yourself; see the Resources section of these Teacher Notes
for information on both.
NOTE: The students will need the completed string length charts from the second activity for the
third activity. If there is only one monochord available, the third activity should be abbreviated.
Have the students do only one or two songs; have different students come forward to play each
note. Because the monochord will be shared, the songs will not be musical – it will be difficult to
create the correct rhythm for the songs even though the notes can be produced.
Instructional Time Required for this Section
Three hours
Required Materials
Pythagorean Tuning
•
student activity sheets, one per student
•
one monochord for each group of 3-5 students
•
transparency of the Monochord Introduction
The Well-Tempered Scale
•
student activity sheets, one per student
•
one monochord for each group of 3-5 students
•
graphing calculators, one per student
Playing a Tune Together
•
one monochord for each group of 3-5 students
•
transparencies of the music charts (Tunes 1-7)
•
copies of the string length charts from The Well-Tempered Scale activity
Suggested Materials
Pythagorean Tuning
•
video Donald Duck in Math Magic Land (optional)
The Well-Tempered Scale
•
keyboard (you may be able to borrow one from the music department; one option for
tuning monochords)
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
39
DRAFT
•
•
•
Teacher Notes
293.7 Hz D and 261.6 Hz C tuning forks (you should be able to borrow these from the
science department; one option for tuning monochords)
pitchpipe (one option for tuning monochords)
transparency of a keyboard (available in the Resources section of these Teacher Notes)
Preparation
Pythagorean Tuning
•
before the activity – order or obtain the video Donald Duck in Math Magic Land (your
librarian or media person should be able to order this video for you) if you plan to use it
•
before class begins – tune the monochords; for this activity it is not imperative that the
monochords be tuned precisely since they will not be played together (see the literature
included with commercially manufactured monochords for tuning instructions)
•
before class begins – cue the video Donald Duck in Math Magic Land to the section
about strings
•
copy one student activity sheet for each student
•
make the transparency of the Monochord Introduction
The Well-Tempered Scale
•
before class begins – tune the monochords; for this activity the monochords need to be
tuned to the same note since they will be played together (if possible tune them to middle
C), if you don’t have an ear for tuning instruments, you may want to get help from a
music teacher or musically inclined student; a tuning fork or a keyboard can be used to
match the pitch
•
copy one student activity sheet for each student
•
make a transparency of a keyboard
Playing a Tune Together
•
before class begins – tune the monochords; for this activity the monochords need to be
tuned to the same note since they will be played together (if possible tune them to middle
C; the D above middle C may make the commercially manufactured monochords sound
better but always consider that the note played on the open string is middle C when
playing the tunes)
•
copy one student activity sheet for each student
Vocabulary
chromatic scale – a sequence of the thirteen half-steps on a piano keyboard
major scale – a sequence of eight notes that follow the pattern whole-step, whole-step, halfstep, whole-step, whole-step, whole-step, half-step
Procedures
Pythagorean Tuning
Introduce this activity with the video clip from Donald Duck in Math Magic Land.
Before giving the students the monochords, review some safety issues as well as care and use of the
monochord. Use the Monochord Introduction transparency to direct this discussion. Ask the
students not to turn the tuners because the strings break easily if turned too tightly. Be prepared to
restring and retune the monochords if a string does break; safety glasses are recommended when
stringing or tuning.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
40
DRAFT
Teacher Notes
The monochord is not played like a guitar; the students should pluck rather than “twang” the
string. Explain and demonstrate how to press down on the fret and pluck the string. Placing the
monochord on the tabletop will amplify the sound by acting as a sound board.
Distribute the activity sheets and ask the students to read the introduction. Distribute the
monochords and allow the students to work through the activity.
The students should arrive at the conclusion that anytime the string is half the length of another one,
the sound produced is one octave higher.
This activity should take approximately an hour but can be condensed somewhat and used an
introduction to the second activity, The Well-Tempered Scale.
The Well-Tempered Scale
NOTE: Do not distribute part 2 of this activity until the students have completed part 1; part 2
supplies the answers to part 1.
Part 1 should take approximately 20 minutes.
In a well-tempered scale, the ratio between any two consecutive strings is the same; thus the
sequence of string lengths is a geometric sequence. The activity leads the students to draw this
conclusion. Emphasize that the relationship between lengths of strings is either an exponential
function or a geometric sequence. Give the students time to find this ratio experimentally.
Students with a stronger mastery of algebra may attempt or eventually realize that there is an
algebraic representation.
To find the constant ratio experimentally, you can use the recursion feature of a graphing calculator.
There are two versions of the second page of this activity in the teacher’s edition, one for TI
calculators and one for Casio calculators. The student edition has two complete activities.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
41
DRAFT
Teacher Notes
Directions for TI-82 or TI-83 calculators:
Set up the seed line: {1,60}, press enter.
(This is the note number (first, second, etc.) followed by the original string length; commercial
monochords have a 60cm string.)
Set up the procedure line: {Ans(1)+1,Ans(2)*number of your choice}, press enter
(This increases the note number and multiplies 60 by a guess of the constant ratio to find the length
of the string for the next half-step.)
Continue to press enter until you get to the 13th note and its corresponding string length; if you
have the correct multiplier (constant ratio), the string length will be 30.
To change the guess or begin again, recall (press 2nd ENTRY twice) or retype the seed line; press
enter to reset the seed. Recall (press 2nd ENTRY twice) or retype the procedure line. Change the
guess of constant ratio. Repeat this process until the final line is approximately {13,30}. (The note
number will be exactly 13, the string length should be very close to 30.)
Directions for Casio 9850 calculators:
Set up the seed line: {1,60} [EXE].
(This is the note number (first, second, etc.) followed by the original string length; commercial
monochords have a 60cm string.)
Set up the recursion procedure:
List Ans[1] à A [SHIFT][EXE]
List Ans[2] à B [SHIFT][EXE]
{A+1,B*number of your choice}
(This procedure increases the note number and multiplies 60 by a guess of the constant ratio to find
the length of the string for the next half-step.)
Press [EXE] until you get to the 13th note and its corresponding string length; if you have the
correct multiplier (constant ratio), the string length will be 30.
To change the guess or begin again, retype the seed line, press [EXE]. Retype the procedure lines
changing the guess of constant ratio. Repeat this process until the final line is approximately
{13,30}. (The note number will be exactly 13, the string length should be very close to 30.)
Move on to part 2 of this activity when most of the students are close to the correct ratio of
1
≈ 0.9438743127 ... .
12
2
Part 2 should take approximately 30 minutes.
In part 2, the students will use the exact ratio
1
to calculate string lengths to the nearest tenth of a
2
centimeter. Show the students, or have a student strong in algebraic skills show the class, how to
solve the equation 60 x 12 = 30 for x.
12
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
42
DRAFT
Teacher Notes
60 x 12 = 30
60 x 12 30
=
60
60
1
x 12 =
2
12
x 12 = 12
1
2
12
1
12
2
x=
x=
1
12
2
Discuss the formation of the major scale. Use a transparency of a keyboard or an actual keyboard
to help the students see how the whole-steps and half-steps fit on a keyboard and how the major
scale is built with 8 of the 13 notes in the chromatic scale.
A#
Bb
A B C
C# D#
Db Eb
D
E
F# G#
Gb Ab
F
G
For example, when counting both the white and black keys, from one note to the next note is a halfstep. On the keyboard above, from A to A# is a half-step or from E to F is a half-step. (The black
keys have two names; for example the third black key from the left is either A# , A-sharp, or Bb , Bflat.) The chromatic scale is thirteen notes with twelve half-steps between them. So the chromatic
scale beginning with A is A, A# , B, C, C# , D, D# , E, F, F# , G, G# , A.
From one note to the second note from it is a whole-step. A to B is a whole-step or E to F# is a
whole-step. A major scale is composed of seven steps (using eight notes); the first is a whole-step,
the second is a whole-step, the third is a half-step, the fourth is a whole-step, the fifth is a wholestep, the sixth is a whole-step, and the seventh is a half-step. So, beginning with C the major scale
would be C, D, E, F, G, A, B, C. Beginning with A, the major scale has the following notes: A, B,
C# , D, E, F# , G# , A.
Have the students complete the chart on the third page of the student activity. The chart is set-up
with the solfège names (do, re, me, …) for the notes of a major scale. Explain that the string
lengths for the scale will be set for the monochord; however, the actual pitches can be changed by
increasing or decreasing the tension in the string. (The solfège system provides relative, rather than
absolute, tuning.)
Make sure that the students have the string lengths chart completed correctly then have them play
the scale by adjusting the sliding fret.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
43
DRAFT
Teacher Notes
Tune all of the monochords to the same note. Assign “do” to group 1, “re” to group 2, “mi” to
group 3, etc., until all eight notes of the scale are assigned. (Do not give consecutive notes to the
same group.) Using a student director, play the notes in order up the scale and then back down the
scale. This is a good activity on which to take a group grade.
Remind the students that the musical scale is mathematically defined.
Depending on the time available, the students may be asked to play the chromatic scale, which
consists of all thirteen notes. This is a good opportunity for an informal assessment to determine
whether the students understand scales. Alternatively, the teacher may chose to move on to the third
activity, Playing a Tune Together, to play actual songs.
Playing a Tune Together
Assign the note names for the songs to each group, making sure that no group will have to play two
consecutive notes in the song. Choose a student to conduct the tune. Play tunes 1, 2, and 3 using
the solfège names of the notes. Either have a different student play each song or repeat each song
so that everyone performs. Students seem to prefer playing everything themselves, so you may not
use all of the songs.
Tune all of the monochords to middle C before proceeding to tunes 4, 5, 6, and 7 so that the pitches
played on the monochords will match the pitches indicated by the music. Assign the notes of a
song and choose a student director. Continue playing as many songs as time permits.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
44
DRAFT
Teacher Notes
Monochords Resources
Monochords are available from:
Foster Manufacturing Company
1504 Armstrong Drive
Plano, Texas 75074
972 424-3644 (phone & fax)
email: fmco@flash.net
website: www.flash.net/~fmco
$60 per monochord (model 115) or $330 for a set of six (model 116) (plus shipping)
Directions for making your own monochords can be found in both of the following:
COMAP. 2002. Mathematical Models with Applications, ancillary materials, Handout 9.2. W.H.
Freeman: New York.
Haak, Sheila. 1991. Using the Monochord: A Classroom Demonstration of the Mathematics of
Musical Scales in Applications of Secondary School Mathematics, pp. 143-149. NCTM: Reston,
Virginia. (Reprinted from the March 1982 Mathematics Teacher.)
In addition to the transparency masters that follow, you may find the following resources useful or
informative:
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.)
Acknowledgement
The activities and teacher notes in this section were used from and adapted from Monochord
Activities from Foster Manufacturing Company. We would like to thank FMCO for giving us
permission to use and adapt their materials.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
45
DRAFT
Student Notes
Monochord Introduction
A monochord is a very simple instrument whose invention is generally
attributed to Pythagoras. It consists of a single string with one fixed bridge
and a second sliding bridge. (The sliding bridge is referred to as a sliding
fret.) The one that you are to use is labeled with a 60-centimeter scale that
can be used to position the sliding fret. The sound produced by stringed
instruments are actually determined by three factors: (1) the size of the
string, (2) the tension on the string, and (3) the length of the string. These
activities primarily address the third factor, the length of the string.
On the monochord the size of the string is set when the string is placed
on the instrument. The string on the monochord has a diameter of 0.019
inches. The tension is controlled by turning the tuner on the right end of
the instrument. Loosening the string will lower the pitch. If the string is too
loose the sound is more like noise than music; however, if the string is too
tight, the string may break. Once the instrument is tuned, frequent
adjustments should not be necessary. However, if you must adjust the
tension on the string, turn the tuner in very small amounts to avoid
breaking the string. When a string breaks, it generally “flies” loose. Make
sure that it cannot hit you in the eye.
To play the monochord, place it on a table in front of you so that you can
read the numbers on the scale. (Since the sound board of the instrument
is rather limited, the table top will amplify the sound a little.) Gently pluck
the string with your left hand.
To change pitch, use the sliding fret to shorten the length of the string.
With your right hand, press the string down on the wooden part of the
movable fret, just to the right of the small metal bar so that the string makes
solid contact with the metal. Now pluck the string with your left hand. The
sound will be higher.
Experiment with playing different sounds on the monochord until you are
comfortable with the process.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
46
DRAFT
Student Notes
Tune 1
(6 notes – do, re, mi, fa, sol, do’)
do
do
do
re
mi
mi
re
mi
fa
sol
do’
do’
do’
sol
sol
sol
mi
mi
mi
do
do
do
sol
fa
mi
re
do
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
47
DRAFT
Student Notes
Tune 2
(6 notes – do, re, mi, fa, sol, la)
sol
la
sol
fa
mi
fa
sol
re
mi
fa
mi
fa
sol
sol
la
sol
fa
mi
fa
sol
re
sol
mi
do
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
48
DRAFT
Student Notes
Tune 3
(8 notes – do, re, mi, fa, sol, la, ti, do’)
mi
re
do
mi
re
do
sol
fa
fa
mi
sol
fa
fa
mi
sol
do’
do’
ti
la
ti
do’
sol
sol
sol
do’
do’
do’
ti
la
ti
do’
sol
sol
sol
sol
do’
do’
ti
la
ti
do’
sol
sol
sol
fa
mi
re
do
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
49
DRAFT
Student Notes
Tune 4
(6 notes – C, D, E, F, G, A)
C
C
G
G
A
A
G
F
F
E
E
D
D
C
G
G
F
F
E
E
D
G
G
F
F
E
E
D
C
C
G
G
A
A
G
F
F
E
E
D
D
C
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
50
DRAFT
Student Notes
Tune 5
(5 notes – D, E, F , G, A)
A
F#
A
A
F#
A
F#
G
A
G
F#
E
F#
G
A
D
D
D
D
D
E
F#
G
A
A
E
E
G
F#
E
D
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
51
DRAFT
Student Notes
Tune 6
(8 notes – D, E, F , G, G , A, B, C’)
D
G
D
G
D
G
A
B
G
C
C
G
A
B
D
G
D
G
D
G
A
B
G
G
A
A
A
B
A
A
A
G#
A
B
A
G
D
C’
C’
C’
G
G
A
A
B
E
F#
G
F#
G
E
D
G
B
C’
B
A
G
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
52
DRAFT
Student Notes
Tune 7
(6 notes – D, E, F , G, A, B)
G
G
B
B
B
A
A
F#
D
E
D
F#
A
A
B
A
F#
D
E
F#
F#
E
E
D
G
G
B
B
B
A
A
F#
D
E
D
F#
A
A
B
A
F#
D
E
F#
F#
E
E
D
Keyboards
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
53
DRAFT
Student Notes
C# D#
Db Eb
A#
Bb
A B C
D
E
F# G#
Gb Ab
F
G
C*
C
D
w
F G A B C
E
w
h
w
w
w
A
h
w
Figure 1
A B C1D1 E 1F 1 G 1A1 B1 C2 D2 E 2
octave
B
F * G*
A
D E
w
h
w
w
w
h
Figure 2
C4
F7 G7 A7 B 7 C8
"middle" C
Figure 3
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
54
DRAFT
Student Activity
Monochord Activity 1: Pythagorean Tuning
The Greek mathematician Pythagoras in the sixth century BC was the first
person known to apply mathematics to a musical scale. He noticed that
two vibrating strings of equal diameter and equal tension and with lengths
1
in the ratio :1 produced sounds that seemed very similar; the shorter
2
string appeared to be a higher version of the long one. He also found that
the sounds were very pleasing when the ratio of the lengths of the strings
2
3
was either :1or :1. These simple ratios actually defined the musical
3
4
intervals of an octave, a fifth, and a fourth. Using the syllables do, re, mi, fa,
sol, la, ti, do of a modern major scale, the long string plays do, the one that
3
2
is of the long string is fa, the one that is of the string is sol, and the
4
3
1
one that is
is the octave or the next do.
2
In the mathematics of the sixth century BC irrational numbers did not exist,
so all of the ratios for the musical scale were set as ratios between two
integers. The scale produced using the Pythagorean definitions is seldom
used for today’s music, so we will develop only enough of the
Pythagorean Scale to understand the concepts of tuning by measurement.
We will investigate the Pythagorean do, fa, and sol by locating these notes
on the monochord.
1.
Measure the length of the monochord string in centimeters.
______________
Play the open string and listen to the sound. We will define this
sound to be the “do” of our musical scale.
2.
Using the scale on your monochord place the sliding fret at exactly
the halfway point on the string.
How long is the new string (measured in centimeters)?
________________
3.
In both the older Pythagorean tuning and the modern well-tempered
tuning this note is one octave higher than the sound produced by
the open (unfretted) string.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
55
DRAFT
Student Activity
Every time that the string length is cut in half, a note one octave
higher is produced.
Place the fret at 15 cm and pluck the string. The new sound is two
octaves above the sound produced by the open string.
4.
What length string will produce a sound three octaves higher?
_____________
To produce a sound one octave below the open string the
monochord would have to be ___________ cm long.
5.
Locate the sliding fret so that the new length is
2
the length of the
3
open string.
How long is the new string? ________________________
Again pluck the string on the left. This should be the fifth of the major
scale or sol.
6.
Leave the moveable fret in the same location, but press down on the
string to the left of the fret and pluck the string on the right.
How long is the right side of the string? _______________
How does the length of the string on the right side compare to the
length on the left? _________________________
How does the pitch of the left side of the string compare to that of the
right side? _______________________________
7.
3
the length of the
4
open string. This is fa or the fourth note of the major scale.
Locate the sliding fret so that the new length is
How long is the new string? ____________________
8
Leave the sliding fret in the same location, but press down on the
string on the left side of the fret.
How long is the string on the right? ____________________
The new length is what fraction of the open string?
_______________________
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
56
DRAFT
9.
Student Activity
How does the pitch of this note compare to the note produced by
the open string?
________________________________________
Play do, fa, sol, and the high do for your teacher.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
57
Student Activity
Monochord Activity 2:
The Well-Tempered Scale
(Part 1)
The Pythagorean Scale for the simple music of the 6th century BC
provides an excellent means for tuning an instrument; however, for
today’s complex music, it is not very satisfactory. With an instrument tuned
to a Pythagorean scale playing songs in different keys requires retuning
the instrument because the ratio between the string lengths between two
consecutive half-steps varies. To avoid this problem our modern western
music is tuned to the well-tempered scale. The well-tempered tuning,
which is based on irrational numbers rather than rational ones, divides the
interval of an octave into twelve half-steps (13 notes counting the bottom
note of the octave) in which the ratio between the string lengths of any two
consecutive half-steps is always the same. On a piano keyboard the
interval between two adjacent notes is a half-step; therefore, the ratios
between the lengths of any two adjacent strings in the piano will always be
the same. The mathematical sequence produced by the lengths of the
strings in a well-tempered scale is geometric. Thus the string lengths of the
well-tempered scale form a geometric sequence.
If an open string is 60 cm long and the octave is 30 cm long then we must
find a multiplier that can be used 12 times to produce 30 from 60.
1
60
2
3
4
5
6
7
8
9
10
60x 60xx
11
12
13
60x 11
30
Use a calculator to find the required multiplier.
value of x ______________________
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
58
Student Activity
In order to play music on the monochords, we must calculate the string
lengths needed for a chromatic scale and then choose the ones we need
for a major scale.
(The multiplier that you calculated for the geometric sequence was actually
1
.)
12
2
The simplest way to do these calculations is to use the recursive feature
on the calculator. (TI directions)
(1) Measure the length of the open string on the monochord.
_________________
(2) On your calculator type in
{1, measurement of open string}
and press enter. This is your seed value. The 1 will provide a note
counter.
(3) Type in the procedure line:
{Ans(1)+1, Ans(2)* ((1/ 2)^ (1/ 12)) }
and press enter.
(4) Continue the iteration and fill in the chart on the next page.
(5) Check your calculations by playing the major scale and the chromatic
scale.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
59
Student Activity
In order to play music on the monochords, we must calculate the string
lengths needed for a chromatic scale and then choose the ones we need
for a major scale.
(The multiplier that you calculated for the geometric sequence was actually
1
.)
12
2
The simplest way to do these calculations is to use the recursive feature
on the calculator. (Casio directions)
(1) Measure the length of the open string on the monochord.
_________________
(2) On your calculator type in
{1, measurement of open string} [EXE].
This is your seed value. The 1 will provide a note counter.
(3) Type in the procedure lines:
List Ans[1] à A [SHIFT][EXE]
List Ans[2] à B [SHIFT][EXE]
{A+1,B* ((1/ 2)^(1/ 12)) }
and press [EXE]. Repeatedly pressing [EXE] will continue the iteration.
(4) Continue the iteration and fill in the chart on the next page.
(5) Check your calculations by playing the major scale and the chromatic
scale.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
60
Student Activity
String Lengths for the Monochord
Half-Steps
Major Scale
1
do
String Length
2
3
re
4
5
mi
6
fa
7
8
sol
9
10
la
11
12
ti
13
do
Play the notes for a major scale on the monochord. Remember that the
major scale uses only 8 of the 13 notes - whole step, whole step, half step,
whole step, whole step, whole step, and half step.
Play the thirteen notes of the chromatic scale.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
61
Student Activity
Monochord Activity 2:
Naming Musical Notes
(Part 2)
Music notes are named in a rather unusual fashion in that any two notes
1
that are an octave apart (the strings are in a ratio of 1: ) have the same
2
name. The only letters used are A, B, C, D, E, F, and G. (Sharps
and
flats
are used to raise (sharp) or lower (flat) a note by a half-step.
To help distinguish notes of the same name for people who are not
reading music, subscripts are used to identify the particular note. People
who tune piano and organs frequently use this system. A piano has 88
keys; the lowest note is an A and the highest note is a C. The lowest A
and B have no subscripts; however, the next A that occurs is labeled A1 ,
the next is A 2 , and so forth up the keyboard. The lowest C is C1 . The
same procedure is used on each of the different notes.
A B C1D1 E 1F 1 G 1A1 B1 C2 D2 E 2
octave
C4
F7 G7 A7 B 7 C8
"middle" C
Since the space between two adjacent notes is a half-space, the space
between a white note and a black note is therefore a half step. The space
between two white notes that a have a black note between them is a
whole step, and the space between two white notes without a black note
between them is a half step.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
62
Student Activity
The musical scale that uses all 13 notes is referred to as a chromatic
scale; however, most of the popular music that we listen to is based on
what is known as a major scale. This scale is, of course, a subset of the
chromatic scale. It always consists of the sequence whole step, whole
step, half step, whole step, whole step, whole step, and half step. This
arrangement is the same no matter what key is used as the beginning of
the scale. The C scale illustrated on the left in the diagram on the next
page uses only the white notes on a piano; however, the A scale, shown
on the right, needs an F , a G , and a C in order to have the correct
sequence of 2 whole steps, 1 half step, 3 whole steps, and a half step.
C#
C
D E
w
w
F G A B C
h
w
w
w
h
A
B
w
F# G#
A
D E
w
h
w
w
w
h
Using the String Lengths for the Monochord chart that you just completed
with the string lengths using the solfège names (do, re, mi, fa, sol, la, ti, and
do’), copy the string lengths on the following chart. The open string on the
monochord can be tuned to middle C so consider its string length as 60
cm. You will have to continue the mathematical sequence to get the last
few notes of this chart. Now we can play music together using both ways
of identifying the notes.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
63
Student Activity
Half-Steps
C
String Length
60 cm
C-sharp or D-flat
D
D-sharp or E-flat
E
F
F-sharp or G-flat
G
G-sharp or A-flat
A
A-sharp or B-flat
B
C’
C’-sharp or D’-flat
D’
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
64
Student Activity
Monochord Activity 3: Playing a Tune Together
Introduction
In order to create music with instruments playing together all instruments
must agree on the sound for a particular note. For stringed instruments,
tuning is done by adjusting the tension on the open string.
A tuning fork, a pitch pipe, or a piano can be used to establish the basic
tone; however, the musicians must adjust their instruments to produce the
correct sounds. Experienced musicians do this by adjusting the tones that
they are producing while listening to the correct sound. When two strings
play sounds that are almost, but not quiet the same, we hear a “beat”
between the two sounds. One of the strings is tightened or loosened until
this “beat” goes away. The beat is actually a point in the sound curve
where the sounds cancel each other out to produce no sound at all.
Mathematically, tuning instruments is pulling the two sound curves into
alignment.
Since you are not introducing a “real” instrument into the “orchestra” so
there will be no need to pay attention to the absolute frequencies. Tune
the monochords relative to each other. (If there is only one monochord, it
won’t matter what the absolute frequency is and it won’t have to be tuned.)
This of course means that references to “C” may not really be a “C”. As
long as the monochords are not playing tunes with real instruments, this
will not matter.
When turning the tuner be extremely careful to turn only a small
amount at a time. A small amount can make a large a difference
in the sound and strings break easily. Keep your face away from
the string so that you do not get hit in the eye by a breaking
string.
If using multiple monochords, make sure that the open strings on each of
the monochords produces the same sound as all the others.
For the first performance, name the notes in a major scale only as do, re,
mi, fa, sol, la, ti, and do(high). (You may have learned to use this system in
elementary school music classes.)
Each group is responsible for playing a note or two. Refer to the
measurement chart that you created in the last activity, and find the
measurement for the note(s) to which you are assigned. Make sure that
you know where to put the movable fret for each note and that you know
the name(s) of your note(s).
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
65
Student Activity
Everyone in each group must be able to play the notes because the song
will be repeated until everyone in the class has participated in the song.
The symbol do’ indicates the high do which is the octave above the open
string.
Note: Since the rhythm is not indicated, the conductor will have to
improvise.
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
66
Student Activity Answers
Monochord Activity 1: Pythagorean Tuning
Answers
1.
Measure the length of the monochord string in centimeters. 60 cm
2.
How long is the new string (measured in centimeters)? 30 cm
4.
What length string will produce a sound three octaves higher? 7.5 cm
To produce a sound one octave below the open string the monochord would
have to be 120 cm long.
5.
How long is the new string? 40 cm
6.
How long is the right side of the string? 20 cm
How does the length of the string on the right side compare to the length on
the left? twice as long
How does the pitch of the left side of the string compare to that of the right
side? 1 octave higher
7.
How long is the new string? 45 cm
8
How long is the string on the right? 15 cm
The new length is what fraction of the open string?
1
4
How does the pitch of this note compare to the note produced by the open
string? 2 octaves higher
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
67
Student Activity Answers
Monochord Activity 2: The Well-Tempered Scale
Answers
(Part 1)
1
2
3
4
5
6
7
8
9
10
11
12
13
60
60x
60x 2
60x 3
60x 4
60x 5
60x 6
60x 7
60x 8
60x 9
60x 10
60x 11
30
Use a calculator to find the required multiplier.
 1
 
1
 1   12 
value of x =  
= 12 = 0.9438743127...
2
2
(1)
Measure the length of the open string on the monochord. 60 cm
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords
68
Student Activity Answers
String Lengths for the Monochord
Half-Steps
Major Scale
String Length
1
do
60 cm
2
3
56.6 cm
re
4
53.5 cm
50.5 cm
5
mi
47.6 cm
6
fa
44.9 cm
7
8
42.4 cm
sol
9
10
40.0 cm
37.8 cm
la
11
35.7 cm
33.7 cm
12
ti
31.9 cm
13
do
30.0 cm
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords – Teacher Edition
69
Student Activity Answers
Naming Musical Notes (Part 2)
Half-Steps
String Length
C
60 cm
C-sharp or D-flat
56.6 cm
D
53.5 cm
D-sharp or E-flat
50.5 cm
E
47.6 cm
F
44.9 cm
F-sharp or G-flat
42.4 cm
G
40.0 cm
G-sharp or A-flat
37.8 cm
A
35.7 cm
A-sharp or B-flat
33.7 cm
B
31.9 cm
C’
30.0 cm
C’-sharp or D’-flat
28.3 cm
D’
26.7 cm
Mathematical Models with Applications, Fine Arts Module
Music Unit, Monochords – Teacher Edition
70