Set 6Let there be

music

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Wow! We covered 50 slides last time! And you didn't

shoot me!!

Stuff

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As of Saturday morning, the grades have yet to be posted on myUCF site.

Today we continue with our musical interlude.

We still have to cover two basic Physics concepts:EnergyMomentum

We will return to these topics later.

Last time

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We looked at strings, how they vibrate and mentioned the factors that determine the vibrational frequency of a string.

We also remembered the Helmholtz result (next slide) that each note on the musical scale had a specific frequency.

But were those specific frequencies selected?Why not different ones??Why these PARTICULAR Why these PARTICULAR frequencies??frequencies??

Helmholtz’s ResultsNote from Middle C Frequency

C 264

D 297

E 330

F 352

G 396

A 440

B 496

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Our immediate plan

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Apply modern electronic methods, with the assistance of a guest violinist, to answer these questions.

Apply these methods to the understanding of1) The scale progression2) The development of chords

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Tone

Compare the resultsFrom these two sources.

We can study tones with electronics

Oscilloscope

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http://commons.wikimedia.org/wiki/Main_Page

One More Tool

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Tone

Signal GeneratorElectrical

In using these modern tools

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1. We postpone understanding how some of these tools work until later in the semester.

2. We must develop some kind of strategy to convince us that this approach is appropriate.

One more thing

• These days, the tone generation and the oscilloscope can be “created” on a computer.

• This will often be our approach.

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The Violin

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LWe will make somemeasurements basedOn these lengths.

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

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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?

Guitar Tuning

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String Scientific pitch

Helmholtz pitch

Interval from middle C

Frequency

first E4 e'major third above

329.63 Hz

second B3 bminor second below

246.94 Hz

third G3 gperfect fourth below

196.00 Hz

fourth D3 dminor seventh below

146.83 Hz

fifth A2 Aminor tenth below

110 Hz

sixth E2 Eminor thirteenth below

82.41 Hz

Consider Two Situations

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

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

So…

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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

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2

1

2/

2

2

1

2

1

Octave

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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 keyboard – a reference

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The Octave Next Octave

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. This will be our next topic.

PythagorasPythagoras

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

PythagorasPythagoras

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Born in Samos, IoniaRemembered as a mathematician.Well educated; learned to play the lyre, read

poetry, and could recite Homer.Believer that ALL relations could be reduced

to number.All things are numbers; the whole cosmos is a

scale and a number.He developed the Pythagorean Theorm.

PythagorasPythagoras

Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly.

Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments.

He was a fine musician, and he used music as a means to help those who were ill.

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PythagorasPythagorasThe beliefs that Pythagoras held were [2]:

(1) that at its deepest level, reality is mathematical in nature,(2) that philosophy can be used for spiritual purification,(3) that the soul can rise to union with the divine,(4) that certain symbols have a mystical significance, and(5) that all brothers of the order should observe strict loyalty and secrecy.

So it is no surprise that he looked at the lengths of strings that sounded well together as a religious issue as well as a scientific issue. Luckily, in this case, it worked .. sort of.

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See you later ….

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