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Set 7What’s that
scale??
1
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NoteGrades 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.
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Remember Helmholtz’s Results
Note from Middle C Frequency
C 264
D 297
E 330
F 352
G 396
A 440
B 496
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Today
4
Look at how this scale developed. It is mostly arithmetic.
This material is in Measured Tones.Readings: Chapter 1 pages 1-11
oRead 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.
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5
Tone
Compare the resultsFrom these two sources.
Last time we messed with this stuff.
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Violin
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The Violin
7
LWe will make somemeasurements basedOn these lengths.
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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?
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Let’s Listen to the ViolinLet’s Listen to the Violin
9
1) Let’s listen to the instrument, this time a real one.
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?
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10
The ratios of these lengthsShould be ratios of integers If the two strings, when struck At the same time, should sound“good” together.
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Remember this argument?
11
For the same “x” therestoring force is doublebecause the angle is double.
The “mass” is about halfbecause we only havehalf of the stringvibrating.
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PythagorasNoticed 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”.
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So…
13
m
kf
kxF
2
1
For the same “x” therestoring force is doublebecause the angle is double.
The “mass” is about halfbecause we only havehalf of the stringvibrating.
k doubles
m -> m/2
f doubles!f
m
k
m
k
m
k
m
kf
2
42
14
2
1
2/
2
2
1
2
1
Octave
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Octave
14
0.001 0.002 0.003 0.004 0.005
-1
-0.5
0.5
1
0.001 0.002 0.003 0.004 0.005
-1
-0.5
0.5
1
0.001 0.002 0.003 0.004 0.005
-1.5
-1
-0.5
0.5
1
1.5
f
2f
SUM
Time The sum has the same basic periodicity asThe original tone. Sounds the “same”
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The keyboard – a reference
15
The Octave Next Octave
Sounds the “same”
Middle C
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The Octave
12 tones per octaveoctave. 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
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Properties of the octaveProperties of the octave
17
Two tones, one octave apart, sound well when played together.
In fact, they almost sound like the same notethe 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 PythagorasPythagoras to create and understand the musical scale.
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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.
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19
sec 200
1
200
T
Hzf
sec 300
1
300
T
Hzf
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Scaling the Scale
Part II
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CalendarThe next examination will be on Friday, October Friday, October
1717thth. 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.
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Textbookpp 313-320324-325 (beats)
Measured TonesChapter 4 – pp 86-97
READING ASSIGNMENT
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Fifth
23
C G C
f 1.5f 2f
A fifth is a span of 5 whole tones on the piano.It also spans 7 semitones.
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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.
24
2/3 Lm=2/3 M (smaller)
k=3/2 K (larger)
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The Perfect Fifth … Sounds Good!
25
ff
fm
k
m
kf
5.1
2
3
)22(
)33(
2
1
)3/2(
)2/3(
2
1
3/2
3/23/2
frequency
f 1.5f 2f fifth Octave
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Other Fifths – also pretty good!
26
Beethoven’s Fifth
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The Intervals:
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
C G C
f 1.5f 2f
fifthfourth
1 2 3 4 5 1 2 3 4
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28
m
kf
kxF
2
1
1/4 3/4
reference
03
4
9
16
2
1
)4/3(
)3/4(
2
1f
m
k
m
kf fourth
This is a nice ratio of small integers that will also harmonize with the cosmos.
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OK … Let’s build a scale!
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Pythagorean FifthsScaling the Scale
We start with Middle C at frequency f (264 Hz )We will actually add the numbers later.
First tone is a fifth: 1.5f GLast tone is the octave: 2fC above Middle
C.
30
C G C
f 1.5f 2f
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P’s 5
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
Ratio 1/1 4/3 3/2 2/1
Decimal 1.000 1.3333 1.5000 2.000
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Another tone:
32
125.18
9
2by Divide Big! Too
25.24
9
2
3
2
3
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More of the same …
33
688.116
27
2
1
8
27
Big Too8
27
2
3
2
3
2
3
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So Far
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
34
C 264
D 297
E 330
F 352
G 396
A 440
B 496
We could start with the A below middle C and get the 440 right.
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Tones togetherWe 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.
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The original sound A:440 Hz.
36
0.005 0.01 0.015 0.02
-1
-0.5
0.5
1
time
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The Octave: 440 + 880
37
0.005 0.01 0.015 0.02
-1.5
-1
-0.5
0.5
1
1.5
A PERIODIC sound and our brainsaccept this as a “nice” tone.
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The fifth
38
0.005 0.01 0.015 0.02
-1.5
-1
-0.5
0.5
1
1.5
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The Third1.125 f0
39
0.005 0.01 0.015 0.02
-2
-1
1
2
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Longer period of time
40
0.02 0.04 0.06 0.08
-2
-1
1
2
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41
A New Phenomenon
0.02 0.04 0.06 0.08
-2
-1
1
2
T~0.0195 secondsestimate
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42
sec 018.055
1
f
1T
Hz 55
Hz 495125.1440f
Hz 440f
Hz 511
sec 0195.0
01
1
0
ff
Tf
T
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This phenomenon is called BEATS
43
0.02 0.04 0.06 0.08
-2
-1
1
2
The beat frequency between two similar frequenciesis found to be the difference between the frequenciesthe difference between the frequencies
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44
MaxMin
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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.
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ProblemsThe system of fifths to generate a scale
works fairly well BUTif 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.
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The Octave ProblemSeven octaves represents a frequency range of
27=128The same distance is covered by 12 fifths:
(3/2)12=129.75Some people can hear this difference … a problem,Many other tones wind up being slightly different.
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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:
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12 2
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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
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51
IntervalRatio to Fundamental
Just ScaleRatio to FundamentalEqual 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
(fourths, fifths and sixths)
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Back for some physics
52