Math and Music
Ronda and Will Webber
NWMI Spring Meeting 2002
The
connections between math and music are numerous. We will start by looking at basic counting principles and end the
day by looking at waves created by sounds.
NOTE VALUES AND COUNTING
To start, lets
break music down into the basic units or “notes.” Without giving these notes actual sounds, we can discuss their
time values as ratios to one another.
In other words, each note tells us how many counts it gets compared to
the other note types. The following
chart shows different notes and rests, their names, and their associated beat
values or counts. Notice that the
Quarter note is given the base count of 1.
This will be discussed more in the time signature section.
NAME NOTE REST DURATION
Whole 4
beats
Half 2
beats
Quarter 1
beat
Eighth ½
beat
Sixteenth ¼
beat
Thirty-second 1/8
beat
Note:
eighth notes (as well as 16th, 32nd, etc notes)
can be tied together instead of using the flags. Example: or for 16th notes:
In order to
make music easier to read, musicians place the notes into groupings of
time. A vertical line called the “bar
line” separates these groupings. Each
group is called a “measure.” Depending
on the “feel” of the music, these measures can contain the most common number
of 4 beats, or any other number (2, 3, 6, being the usual ones) of beats.
*****Practice
number 1:
1) Using only whole,
half and quarter notes, how many different ways can a measure be filled given
that it needs four beats and the quarter note gets the beat? Write the notes and bar lines to show the
different combinations.
2) Thinking
about question one what happens if we change a measure to contain only two
beats with the quarter note still getting the beat?
3) Still
thinking about question one, what happens if the measure only contains two
beats, but the eighth note gets the beat?
4) Using only
quarter and eighth notes, how many different ways can a measure be filled given
that it needs four beats and the quarter note gets the beat? Hint:
a quarter note can start on the half beat this is referred to as a
“syncopated” rhythm. (Write out the
combinations OR use mathematics to find the correct number)
5) How many
beats does each of the following measures contain?
a) Quarter note gets the beat
b) Half note gets the beat
c) Eighth note gets the beat
a_________ a__________ a__________
b_________ b__________ b__________
c_________ c__________ c__________
TIME
SIGNATURES
The most
common time signature in Western music is denoted by a
C or
|
4 |
|
|
The two fours signify the number of
beats and the type of note that gets the beat.
Notice that there is no line between the two fours. This is not a fraction.
The “Common”
time signature can also be classified as a quadruple time (meter) signature
since it has four beats. Other
quadruple times often seen in music are:
|
4 |
|
|
Signifying
that there are four beats per measure and that the eighth note gets the beat.
|
4 |
|
|
Signifying
that there are four beats per measure and that the half note gets the beat.
Duple meter
implies that there are two beats per measure.
This is often used in marches where the left/right is emphasized. Since four is a multiple of two, the
quadruple time signatures can be thought of as duple meters for the sake of
note durations, but the emphasis on the overall melody is different). Common
duple time signatures are:
|
2 |
|
|
Signifying two
beats per measure, half note gets the beat.
|
2 |
|
|
Signifying two
beats per measure, quarter note gets the beat.
|
2 |
|
|
Signifying two
beats per measure, eighth note gets the beat.
Triple meter
is often used in music for dances (think waltz). This meter contains three beats per measure and is commonly
denoted as:
|
3 |
|
|
Signifying
three beats per measure, half note gets the beat.
|
3 |
|
|
Signifying
three beats per measure, quarter note gets the beat.
|
3 |
|
|
Signifying
three beats per measure, eighth note gets the beat.
MORE NOTE
VALUES
Given that
time signatures can emphasize even and odd numbers, the original note values do
not lend themselves very well to filling in all the measures. Since the original note values were all
even, signatures like
|
3 |
|
|
do not have a
“single” note that can fill the measure.
To solve this
problem, musicians have two options.
The first is to simply “tie” notes together. Placing an arc over the notes that are to be tied does this. The arc signifies that the notes are to act
as one.
Example:
This
notation implies three beats where the quarter note gets the beat. This notation is most commonly used when a
note value is to be held over a bar line.
The second
option for musicians is to use a dot.
The dot tells the musician to hold the note for one and a half times its
usual length.
Example: Since a half note
usually gets 2 beats the dot makes it add
one beat more for a total of three
beats. This notation is used when the
note will be used within one measure.
*****Practice
number 2:
1) Given the
following time signatures, state the time duration value for each note
underneath each note, and then write the total number of beats given. Place the bar lines where needed.
|
2 |
|
|
____
____ ____ ____
____ ____ Total: _________
|
3 |
|
|
____ ____
____ ____ ____
____ ____ Total: _________
|
4 |
|
|
____
____ ___ ___
___ ___ __ __ __ __ __ ___
___ ___
Total: _________
|
8 |
|
|
___
___ ___ ____
___ ___ ____
___ ___ ___
Total: _________
MORE TIME
SIGNATURES
Another time
signature that is commonly heard is the compound signature of
|
6 |
|
|
This has six
beats per measure with the eighth note getting the beat. Because six is a multiple of three, this signature
looks like a triple meter; however, it has accents on the first and fourth
eighth notes, which gives it a feeling of duple time. Many people recognize this meter by saying “trip-el-et” along
with the music.
Other common
compound time signatures are:
Duples:
|
6 |
|
|
|
6 |
|
|
|
6 |
|
|
Triples:
|
9 |
|
|
|
|
9 |
|
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|
|
9 |
|
|
Quadruples:
|
12 |
|
|
|
12 |
|
|
|
|
12 |
|
|
Note since compounds
look like the simple ones, the accents are the deciding feature.
Example:
|
3 |
|
|
|
6 |
|
|
SPECIAL NOTES
Sometimes a musician
wants a note length to be different then the ones given. Most often, this is seen when a quarter note
is divided into three equal notes, or in
|
3 |
|
|
time, a
quarter is divided in half.
This is referred
to as “borrowed division” and is denoted with the appropriate number above the
connected notes.
3
Example
= 1, then also equals 1.
*****Practice
number 3:
1) State the
number of beats to hold each note under each note. Check to see that your note values add to the correct number of
beats for the measures by finding the total.
|
3 |
|
|
_________________
Total:
|
3 |
|
|
_________________
Total:
|
3 |
|
|
_________________
Total:
|
2 |
|
|
_____________________
Total:
|
3 |
|
|
_____________________
Total:
|
6 |
|
|
_________________
Total:
2) Keeping the
same rhythm, rewrite each set of notes with bar lines to match the given time
signature. You may need to substitute
notes so that only ties are held over the bar lines.
|
2 |
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|
3 |
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THE SOUNDS
OF MUSIC
Pythagoras noticed
long ago that the pitch of a string changed as the length of string
changed. This led him to discover
several different relationships, namely the ratios between different pitches. In the western world, the note “A 440”
denotes the pitch that vibrates at 440hz.
Using this number the octave vibrates at 880hz or a ratio of 1:2.
*****Knowing
that low integer ratios produce “good” tones, what other ratios might be used?
_________________
The fifth note
of Pythagoras’ scale was 1.5 times the frequency of his base note. The fourth note of his scale was 4/3 times
the frequency of his base note. The
distance between the fourth and fifth was defined as a whole note with the
ratio of 8:9.
*****Show the
math to get the 8:9 ratio____________________
As new
instruments were developed, ranges of notes were extended, and musicians wanted
to transpose between keys. This pressed
the need for a new tuning system.
*****Using
mathematics, show why a piano that was tuned starting on A440 using a whole
note ratio of 8:9 would not produce “good” note tones over a wide range of
notes.
Several different tunings were tried,
but the most common one today is know as the “tempered” scale. Basically, the original octave ratio of 1:2 is
maintained, and then the other ratios are all modified by an equal amount to
give 12 different tones. The new fifth
is slightly lower in frequency then the true fifth and the new fourth is
slightly higher.
*****Given that the octave is a 1:2n
ratio, and that there are 12 notes per octave, find the multiplier that would
produce the geometric sequence for an octave.
The piano keyboard is a great example
of the chromatic scale. This scale
consists of 12 notes before repeating the octave (black and white keys). Each note is ½ step apart and is shown in
the chart below.
NOTE: In the chromatic scale, the “fourth” is the
fifth ½ step above the base note, and the “fifth” is the seventh ½ step above
the base note.
As pointed out earlier, Standard Middle A is 440 Hertz. In
the equal-tempered chromatic scale shown in the table below, there is a
definite mathematical relationship between two adjacent notes. The ratio of the
frequency of the higher note and the adjacent lower note is a constant (1.059463).
For example, the frequency of each note in the middle scale is mathematically
related to each other as follows:
|
|
Vocal note |
|
Lower note
(Frequency) |
x |
Constant |
= |
Freq. (vibe/sec) |
|
A3 |
La |
|
|
|
|
|
220.0 Hz |
|
A#3 |
|
= |
220 |
x |
1.059463 |
= |
233.1 Hz |
|
B3 |
Ti |
= |
233.1 |
x |
1.059463 |
= |
246.9 Hz |
|
C4 |
Do |
= |
246.9 |
x |
1.059463 |
= |
261.6 Hz |
|
C#4 |
|
= |
261.6 |
x |
1.059463 |
= |
277.2 Hz |
|
D4 |
Re |
= |
277.2 |
x |
1.059463 |
= |
293.7 Hz |
|
D#4 |
|
= |
293.7 |
x |
1.059463 |
= |
311.1 Hz |
|
E4 |
Mi |
= |
311.1 |
x |
1.059463 |
= |
329.6 Hz |
|
F4 |
Fa |
= |
329.6 |
x |
1.059463 |
= |
349.2 Hz |
|
F#4 |
|
= |
349.2 |
x |
1.059463 |
= |
370.0 Hz |
|
G4 |
So |
= |
370.0 |
x |
1.059463 |
= |
392.0 Hz |
|
G#4 |
|
= |
392.0 |
x |
1.059463 |
= |
415.3 Hz |
|
A4 |
La |
= |
415.3 |
x |
1.059463 |
= |
440.0
Hz |
Chart taken from http://tqjunior.thinkquest.org/4116/Music/music.htm
With this “new” system of tuning, the
relations known as the “circle of fifths” and “circle of fourths” became an important
part of music theory.
*****Problem:
Start on a given base note and go up to
the fifth. Rename this fifth as the
base note and go up to the fifth.
Continue doing this to see what happens.
____________________________________________________________________
Now try starting on the same base note
and go up by the fourth.
____________________________________________________________________
What happened?
____________________________________________________________________
____________________________________________________________________
Just looking at the letters, the fifth
steps and the fourth steps should show a cyclic pattern. For Pythagoras, this pattern was not used
because the tunings would be so far off as to be unpleasant to the ears. With the new system, the circles allow
transpositions of music and in fact can be placed together on a circle to show
the sameness between the two:
Notice that the notes inside the circle are the exact same notes as
the ones on the outside just with a different name. This is only true because of the even-tempered tuning.
Bass line beats quarters, melody in half and
whole notes
4:4 time with nice syncopation (Love song)
2:3:3 time signatures (space)
1&2&3&4& listening to the
under line
4:4 with emphasis on 2 and 4
Counting eighths and emphasis on 2 and 4
6:8 underline with 4:4 melody
4:4:4:3 with slippage