What’s that scale?? 1 Note Grades should be available on some computer somewhere. The numbers are...

What’s that scale??

1

NoteGrades should be available on some

computer somewhere. The numbers are based on the total number of correct answers, so 100% = 40. 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 either.

Let’s click…

2

The test was? (any answer = 2 points)

A. EasyB. OkC. HardD. Very HardE. Extremely difficult!!!

3

How much did you collect for a hitman?

A. $500B. $1,000C. $10,000D. $50,000E. It doesn’t matter … you are still here.

4

However you did, the test was

A. FairB. UnfairC. Very unfair

5

Important Definitions From last week ..

7

Tf

or

periodfrequency

1

1

The PERIOD, T is the time it takes to go from one condition to the next time that exact condition is repeated.The frequency, the number of oscillations per second, is given by:

Example:

If T=2 seconds

F=1/2 (sec-1)=0.5 per second

Question

8

What is a tone and how do you prove it??

9

Remember Helmholtz ? Today

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

Time in milli-seconds

"air

spee

d"

- re

lati

ve

The SINE curve

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

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Let’s start looking 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.

The Guitar Strings - Review

12

Consider Two Situations

13

For the same “x” therestoring force is doublebecause the angle is double.

The “mass” is about halfbecause we only havehalf of the stringvibrating.

So…

m

kf

kxF

2

1

14



k doubles

m -> m/2

f doubles!

Guitar

Pressing the fret that is in the middle of the string doubles the frequency~Walla … the octave

In general … the frequency is proportional to the length of the string.

The violin works in a similar way.

15

Now …. lets look briefly at the MONOCHORD

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What did Pythagoras do?

17

He compared sounds from different pressure points and listened to see which sounded the best.

Octave

18

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

SUMSUM

Time The sum has the same basic periodicity asThe original tone. Sounds the “same”

Violin works in the same way!

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The Violin

20

LWe will make somemeasurements basedOn these lengths.

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?

21

Let’s Listen to the ViolinLet’s Listen to the Violin

22

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?

23

The ratios of these lengthsShould be ratios of integers If the two strings, when struck At the same time, should sound“good” together.

Remember this argument?

24



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…

26

m

kf

kxF

2

1



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

The keyboard – a reference

27

The Octave Next Octave

Sounds the “same”

Middle C

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

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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.

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.

30

31

sec 200

1

200

T

Hzf

sec 300

1

300

T

Hzf

Fifth

32

C G C

f 1.5f 2f

A fifth is a span of 5 whole tones on the piano.It also spans 7 semitones.

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.

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2/3 Lm=2/3 M (smaller)

k=3/2 K (larger)

The Perfect Fifth … Sounds Good!

34

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

Other Fifths – also pretty good!

35

Beethoven’s Fifth

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.36

C G C

f 1.5f 2f

fifthfourth

1 2 3 4 5 1 2 3 4

37

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.

OK … Let’s build a scale!

38

Pythagorean FifthsScaling the Scale

Fifths and fourths sound good together so we try to make a scale with as many of these harmonies as possible.

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.

39

C G C

f 1.5f 2f

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.

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Ratio 1/1 4/3 3/2 2/1

Decimal 1.000 1.3333 1.5000 2.000

Another tone:

41

125.18

9

2by Divide Big! Too

25.24

9

2

3

2

3

More of the same …

42

688.116

27

2

1

8

27

Big Too8

27

2

3

2

3

2

3

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

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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.

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.

45

0.005 0.01 0.015 0.02

-1

-0.5

0.5

1

time

The Octave: 440 + 880

46

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.

The fifth

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0.005 0.01 0.015 0.02

-1.5

-1

-0.5

0.5

1

1.5

The Third1.125 f0

48

0.005 0.01 0.015 0.02

-2

-1

1

2

Longer period of time

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0.02 0.04 0.06 0.08

-2

-1

1

2

50

A New Phenomenon

0.02 0.04 0.06 0.08

-2

-1

1

2

T~0.0195 secondsestimate

51

sec 018.055

1

f

1T

Hz 55

Hz 495125.1440f

Hz 440f

Hz 511

sec 0195.0

01

1

0

ff

Tf

T

This phenomenon is called BEATS

52

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

53

MaxMin

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.

55

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.

57

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:

58

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

59

60

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)

Back for some physics

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