Download Up and Down the Scale

Set 7
What’s that scale??
1
Note
 Grades should be available on some computer somewhere.
The numbers are based on the total number of correct
answers, so 100% = 30. When I review the numbers, this
may change.
 We will get your individual results to you shortly .. Be
forgiving, I don’t have the foggiest idea how to do this stuff
yet.
 Now .. Back to music.
2
Remember Helmholtz’s Results
Note from Middle C
C
D
E
Frequency
264
297
330
F
G
A
B
352
396
440
496
3
Today
 Look at how this scale developed. It is mostly
arithmetic.
 This material is in Measured Tones.
 Readings: Chapter 1 pages 1-11
o Read pages 12-16 for the “flavor”
o Chapter 2 – All: 17-36 Don’t worry about the
musical notation.
 Today is a religious holiday for many, so no
clickers.
4
Last time we messed with this stuff.
Tone
Compare the results
From these two sources.
5
Violin
6
The Violin
L
We will make some
measurements based
On these lengths.
7
Play an octave on one string
• Volunteer to watch where the finger winds
up on the finger board.
• Measure the length of the string.
• How close is it to ½ the length?
8
Let’s Listen to the Violin
1) Let’s listen to the instrument, this time a real one.







9
The parts
One tone alone .. E on A string
E on the E string
Both together (the same?)
A Fifth A+E open strings
Consecutive pairs of fifths – open strings.
A second? Third? Fourth? Seventh?
The ratios of these lengths
Should be ratios of integers
If the two strings, when struck
At the same time, should sound
“good” together.
10
Remember this argument?
For the same “x” the
restoring force is double
because the angle is
double.
The “mass” is about half
because we only have
half of the string
vibrating.
11
Pythagoras
 Noticed that the sound of half of a string played against the
sound of a second full string, both with the same original
tone, sounded well together.
 This was called the octave (we discussed this last time).
 He then noticed that a very melodious tone also came when
the string was divided into 1/3 – 2/3.
 When the larger portion of the string was played against the
original length, it was called the fifth.
 In particular, the tone was “a fifth above the original tone”.
12
For the same “x” the
restoring force is double
because the angle is
double.
So…
F  kx
1
f 
2
k
m
The “mass” is about half
because we only have
half of the string
vibrating.
k doubles
m -> m/2
f doubles!
13
1
f 
2
1

2
2f
k
1

m
2
4k
1

m
2
2k
m/2
k
 4
m
Octave
1
0.5
f
0.001
0.002
0.003
0.004
0.005
0.001
0.002
0.003
0.004
0.005
0.001
0.002
0.003
0.004
0.005
-0.5
-1
1
0.5
2f
-0.5
-1
1.5
1
0.5
SUM
-0.5
-1
-1.5
14
The sum has the same basic periodicity as
The original tone. Sounds the “same”
Time 
The keyboard – a reference
The Octave
Next Octave
Sounds the “same”
15
Middle C
The Octave
 12 tones per octave. Why 12? … soon. Played sequentially,
one hears the “chromatic” scale.
 Each tone is separated by a “semitione”
 Also “half tone” or “half step”.
 Whole Tone = 2 semitones
16
Properties of the octave
 Two tones, one octave apart, sound well when played together.
 In fact, they almost sound like the same note!
 A tone one octave higher than another tone, has double its
frequency.
 Other combinations of tones that sound well have frequency ratios
that are ratios of whole numbers (integers).
 It was believed olden times, that this last property makes music
“perfect” and was therefore a gift from the gods, not to be screwed
with.
 This allowed Pythagoras to create and understand the musical
scale.
17
The Octave
 As we determine the appropriate notes in a scale, we will
make use of the fact that two tones an octave apart are
equivalent.
 We can therefore determine all of the equivalent tones by
doubling or halving the frequency.
 This process is used to build up the scale.
18
f  200 Hz
1
T
sec
200
f  300 Hz
1
T
sec
300
19
Part II
Scaling the Scale
20
Calendar
 The next examination will be on Friday, October 17th.
This is a one session delay from what is announced in the
syllabus.
 Today we continue building the scale.
 Then we return to the textbook to talk about energy,
momentum and some properties of gas (our atmosphere) so
we can deal with exactly what sound waves are.
 Let’s do a quick clicker review of the last class.
21
READING ASSIGNMENT
 Textbook
pp 313-320
324-325 (beats)
Measured Tones
Chapter 4 – pp 86-97
22
Fifth
C
f
G
C
1.5f
2f
A fifth is a span of 5 whole tones on the piano.
It also spans 7 semitones.
23
Let’s look at the “fifth”
 Formed with 2/3 of the original length.
 Considered to be a “perfect” sound because of the small
number ratio in lengths.
 We can form many of the notes of a scale using this ratio.
 The scale so formed sounds great but has problems.
2/3 L
m=2/3 M (smaller)
k=3/2 K
24
(larger)
The Perfect Fifth … Sounds Good!
f2/3
1

2
(3 / 2)k
1

(2 / 3)m 2
(3  3)k 3
 f2/3
(2  2)m 2
f 2 / 3  1.5 f
frequency
f
25
1.5f
fifth
2f
Octave
Other Fifths – also pretty good!
Beethoven’s Fifth
26
C
G
C
5
1 2 3
The Intervals:
1 2
3 4
fifth
f
4
fourth
1.5f
2f
 The fifth is 7 semitones above the fundamental tone, f.
 Since f and 2f are an octave apart, the interval from G to C
should also be melodic.
 This interval consists of five (5) semitones.
 This “special interval” is referred to as a FOURTH.
 Let’s see how much of a scale we can create using these two
musical intervals.
27
F  kx
1/4
3/4
1
f 
2
k
m
reference
1
f fourth 
2
(4 / 3)k
1

(3 / 4)m 2
16k 4
 f0
9m 3
This is a nice ratio of small integers that
will also harmonize with the cosmos.
28
OK … Let’s build a scale!
29
Pythagorean Fifths
Scaling the Scale
C
f
G
1.5f
 We start with Middle C at frequency f (264 Hz )
 We will actually add the numbers later.
 First tone is a fifth: 1.5f G
 Last tone is the octave: 2fC above Middle C.
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C
2f
P’s 5
Ratio
1/1
4/3
3/2
2/1
Decimal
1.000
1.3333
1.5000
2.000
 Question: Are there any other intervals between 1f and 2f
that correspond to singable intervals?
 Pythagoras Rule: Take an existing ratio.
 Multiply by 1.5 to get a fifth above the ratio.
 If the number is greater than 2, reduce it by an octave (divide by
2)
 If the number is less than 1, increase it by an octave by doubling
the number.
31
Another tone:
3 3 9
   2.25
2 2 4
Too Big! Divide by 2
9
 1.125
8
32
More of the same …
3 3 3 27
  
2 2 2 8
Too Big
27 1 27
 
 1.688
8 2 16
33
So Far
34
From C
Ratio
Frequency
264
1.000
264
1.125
297
1.333
353.3
1.500
396
1.688
445.6
2.000
528
C
D
E
F
G
A
B
We could start with the A below middle C and get the 440 right.
264
297
330
352
396
440
496
Tones together
 We discussed that a scale should be made up of tones that
sound well together.
 Even for a scale that is put together as we have just done,
some tones will sound a bit bad together; but not terrible.
 Let’s see why some of the better combinations sound well.
35
The original sound A:440 Hz.
1
0.5
0.005
-0.5
-1
36
0.01
0.015
0.02
time
The Octave: 440 + 880
1.5
1
0.5
0.005
0.01
0.015
0.02
-0.5
-1
-1.5
A PERIODIC sound and our brains
accept this as a “nice” tone.
37
The fifth
1.5
1
0.5
0.005
-0.5
-1
-1.5
38
0.01
0.015
0.02
The Third
1.125 f0
2
1
0.005
-1
-2
39
0.01
0.015
0.02
Longer period of time
2
1
0.02
-1
-2
40
0.04
0.06
0.08
A New Phenomenon
T~0.0195 seconds
estimate
2
1
0.02
-1
-2
41
T  0.0195 sec
1
f   51 Hz
T
f 0  440 Hz
f1  440  1.125  495 Hz
f1  f 0  55 Hz
1 1
T 
 0.018 sec
f 55
42
This phenomenon is called BEATS
2
1
0.02
0.04
0.06
-1
-2
43
The beat frequency between two similar frequencies
is found to be the difference between the frequencies
0.08
Min
44
Max
45
Beats
 Low beat frequencies (1-20 Hz) can be heard and
recognized.
 Faster beat frequencies can be annoying.
 Two frequencies an octave apart but off by a few Hz. will
also display beats (difference between the frequencies as
well) but they are harder to hear and somewhat
unpleasant to the ear.
46
Problems
 The system of fifths to generate a scale works fairly well BUT
 if you start on a different note (F instead of C), the frequencies
of the same notes will differ by a slight amount.
 this means that an instrument usually must be tuned for a
particular starting mote (key).
 Modulation doesn’t work well.
 One interesting problem is the octave over a large range.
47
The Octave Problem
 Seven octaves represents a frequency range of 27=128
 The same distance is covered by 12 fifths: (3/2)12=129.75
 Some people can hear this difference … a problem,
 Many other tones wind up being slightly different.
48
Problems..
 You can create scales using different sets of “primitive”





combinations … thirds, sixths.
Each yields a specific scale.
They are not the same (read chapter 1 in MT).
One can’t change “keys” easily using these schemes.
Something had to be done.
Solution: Equal Tempered Scales.
 The frequency difference between two consecutive semitones is
set to be:
49
12
2


Keeps the octave exactly
correct
Screws up all of the other
intervals
◦ But we can’t easily hear the
difference

One tuning will work for all
keys
50
(fourths, fifths
and sixths)
Interval
Ratio to Fundamental
Just Scale
Ratio to Fundamental
Equal Temperament
Unison
1.0000
1.0000
Minor Second
25/24 = 1.0417
1.05946
Major Second
9/8 = 1.1250
1.12246
Minor Third
6/5 = 1.2000
1.18921
Major Third
5/4 = 1.2500
1.25992
Fourth
4/3 = 1.3333
1.33483
Diminished Fifth
45/32 = 1.4063
1.41421
Fifth
3/2 = 1.5000
1.49831
Minor Sixth
8/5 = 1.6000
1.58740
Major Sixth
5/3 = 1.6667
1.68179
Minor Seventh
9/5 = 1.8000
1.78180
Major Seventh
15/8 = 1.8750
1.88775
Octave
2.0000
2.0000
51
Back for some physics
52