Lecture Set 07Lecture Set 07

October 4, 2004October 4, 2004

The physics of sounds from strings.The physics of sounds from strings.

Things to NoteThings to Note

• All lectures are currently posted on the website.– The solution to the exam is also posted on the

website.

• Some of the files are rather long.– I cut some of the lectures into two or three

parts to keel them small but some are still pretty big because of the photographs.

– Today, we continue with sounds from strings.

Last timeLast time• We listened to the sound generated by a

single stretched string.

• We noted:– The longer the string, the “lower” the tone.– The shorter the string, the “higher” the tone.– The more tension on the string, the “higher”

the tone.

An Interesting observationAn Interesting observation• When the string is reduced to half, it

produces a tone related to the original tone.

• The new tone is said to be an OCTAVE higher than the original.– The textbook calls this interval a “diapason”

which in Greek means “through all”.– I am stubborn and will call it an octave

anyway.

What are we doing?

Open string

Half length

D

D’

PLAY THEM TOGETHER AND THEY SOUND REALLY GOOD.PLAY THEM TOGETHER AND THEY SOUND REALLY GOOD.

What do we have so far??

D D D D D DL/4 L/2 L L/2 L/4 L/8

‘’D ‘D D D’ D’’ D’’’

So, they looked at another “simple” numerical division

DDL/3

NEW INTERVAL – THE FIFTH Length = 2/3 L

Also a fifthLength = 1/3

OCTAVE

A A

Observation

The FIFTH sounds good when played with the original tone.

Bass and tenors singing together will tend to sing a fifth apart.

Recall that men and women, singing the “same tone” together, tend to sing an OCTAVE apart.

Our scale now has “two” notes. A little shabby!

Differences/Intervals

The octave and the fifth represent an interval, or a “difference” between two reasonable consonant tones. Interval is D E F G A B C D E F G A ….= FIFTH

New IntervalThe FOURTH

The fourth

A D A

fourth

octive

3

4

(1/2)L

(2/3)L RatiosLength

2

1

L/2

L RatiosLength

These “Special” Ratios

All of these intervals sounded good to the ear.

The lengths corresponding to intervals that sounded good were found to have ratios expressible as small integers.

How perfect! God-like! MUST be true. The process can be continued to define

additional intervals.

The Pythagorean Musical Scale .. Developed by rules

The Pythagoreans believed that the laws of physics could be revealed NOT by observation but by pure thought.

So they thought about the results that we just mentioned and came up with the “rule” to form the various intervals of the scale.

The “RULE”

Take an existing ratio (lengths) and multiply or divide it by 3/2.

If the number that you get is greater than 2, then halve it.

If the number is less than 1, double it.

EXAMPLE : 1 x 2/3 produces the fifth.

NEXT: 1 x 3/2 = 3/2 .. Divide by 2 … = 3/4 = fourth

The rule basically raises or lowers a tone by a fifth

and then divides or multiplies by two to keep it in the

proper range.

So …

Then … The Pentatonic Scale

The septatonic scaleseptatonic scale (Our Own but starting with Re instead of Do)

By the way … we are cheating!

16th century

A 2000 year problem

SAME RATIOS!!!

The Music of the Spheres

The Music of the Spheres

KeplerLike Plato and Pythagoras, Kepler believed that the world was ruled by number. He tried hard to prove that the distances of the planets from the sun were given by an arrangement of Eucid's five regular solids; by doing so, he believed, he could demonstrate something of the order of the mind of God. But his faith in number went further: he believed that musical harmony, mathematically expressed, and the harmony of the spheres were one thing: 'I affirm and demonstrate that the movements (of the planets) are modulated according to harmonic proportions’.

Let’s talk some physics here.

What is observed

The string moves back and forth. It sets the air into some kind of motion. The air motion (we presume pressure) strikes

the ear and produces what we hear as sound. WE DO NOT YET KNOW THE DIFFERENCE

BETWEEN THE VARIOUS TONES OF THE SCALE IN TERMS OF WHAT IT IS THAT TRAVELS THROUGH THE AIR..

Enter Helmholtz ~1885

Siren

Air Stream

FREQUENCY f

second

events"" ofnumber f

Example

Suppose something has a frequency of 60 cycles per second.

One cycle per second is defined as the Hertz. So we have 60 hertz. In one second there are 60 events. Therefore ONE event takes 1/60th of a

second. That is defined as the PERIOD = T

Definition:

1

1

1

fT

fT

frequencyPeriod

The Helmholtz Resonator

A set of resonators

The Epiphany

“RESONANCE” TAKES PLACE ONLYWITH THE CORRECT HELMHOLTZRESONATOR.

SOUND HAS A FREQUENCY!

BUT HOW DOES IT MOVE TO US??

Waves

But What’s Moving??