Last time we left off with a mystery

what is the mathematical connection between the links and tensions of strings that sound good together?

Let's start with length

The simplest pattern I see is that strings with the same length sound good together

This is not too surprising since these strings are at the same tension and have the same length

They should make pretty much the same sound

Now what about our other pairs of strings that sound good together?

if we look at the third row of our table we see that our second string is fixed to a length of 60 centimeters

And the lengths of our first string that sound good with our second are 60 45 40 30 and 20 centimeters

Now what do these numbers have in common?

Well if we divide the length of our first string by the length of our second string

we see that the strings that sound good together do have something interesting in common

The ratio of their links all reduced to simple fractions

while the ratios of the lengths of strings that do not sound good together do not reduce to simple fractions

This was the first discovery of Pythagoras

String sound good together when the ratio of their lengths is simple

Now let's have a look at tension, maybe the same simple ratio rule applies

If we divide the tensions of our first string by our second

We see that the tensions that sound good together do result in relatively simple ratios

However some string tensions that don't sound good together such as three point six and one point eight kilograms also reduced to very simple ratios

So it seems that the hidden mathematical

relationship between string tension and sound must be a bit deeper

If we have a closer look at the tensions of strings. That sound good together. We may see a hint of a deeper pattern

Notice that quite a few of our good sounding tension ratios are exactly equal to the ratios of perfect squares

Following this hunch let's take the square root of our tension ratios

remarkably this simple transformation snaps our problem into focus

Our new square roots of ratios are simple when our string combination sound good together

and complex when our strings don't sound good together

This was Pythagoras is second remarkable discovery

string sound good together when the square root of the ratio of their tensions is simple

This information was helpful for early instrument makers,

but what's really interesting here is how these discoveries change the way we humans think about the universe we live in

These discoveries along with another interesting Pythagorean discovery involving right triangles really got Pythagoras and his followers thinking

Why is it that mathematics is able to predict what we observe in the world around us?

Division square roots fractions, and even numbers kind of seem like human inventions

Why are they showing up so clearly in triangles and vibrating strings?

These mysteries were compelling enough to convince the pythagoreans of something that might sound a little far-fetched today

That our world is literally built from mathematics

This may sound ridiculous, but we don't actually have to look very far to find very bright modern day physicists who believe this

Now whether or not our universe is built from math is still open for debate

But what I think is really interesting here is what we do with this mystery

For the pythagoreans this was basically it

Why does mathematics show up in vibrating strings and triangles

because the universe is built from math mystery solved?

It would take a couple thousand more years for us humans to really start probing what I think is the more interesting question

How deep does the connection between mathematics and our universe go

If we can predict when two strings sound good together? What else can we predict?

In the case of our vibrating strings, what actually makes our strings sound good or bad together?

Can we use math to really understand what happens when a string vibrates?

Answering questions like this required one more piece of the scientific puzzle that the pythagoreans never really committed to

experimentation

It may seem obvious to us now,

but it took quite some time for us humans to accept what is perhaps the most important idea in all of science

if it disagrees with experiment?

It's wrong!

In that simple statement

is the key to science

It doesn't make a difference how beautiful your guest is it doesn't make any difference how smart you are

who made a guess or what his name is

If it disagrees with experiment

Wrong. That's all there is to it.

When this idea finally started to catch on in 16th century Europe

Scientists were finally able to dig into all kinds of mysteries including the vibrating string

The first real progress came in the form of an educated guess from the Italian scientist Geum Battista benedetti

Benedetti suggested that musical sounds travel through the air as a series of rapid pulses

and how high or low a note sounds to us. It's pitch is a direct result of how frequently these pulses arrive

So if Benedetti suggestion is true then the sound we hear from a vibrating string is a direct result

of how frequently the string moves back and forth

Also known as it's frequency of vibration

This idea raised all kinds of new questions

How is the strings frequency of vibration connected to its length and tension

does anything other than length and tension affect a strings frequency?

How can we measure the frequency of real strings when they vibrate back and forth way too quickly for us to see?

The late 1500s around 50 years after Benedetti guessed that pitch was a direct result of frequency

the great Italian scientist Galileo Galilei turned his attention to these questions

guided by the work of his father Vincenzo

Galileo made some well-informed guesses at our first two questions. That would ultimately turn out to be correct

However, Galileo also claimed that since strings vibrate too fast for us to see

actually measuring their frequency was impossible

Meaning that he had no experimental means to test his guesses

but fortunately for us Galileo was wrong

within 30 years of Galileo's work the French priest and scientist Myron Marcin did measure the frequency of vibrating strings

and was able to experimentally confirm Galileo's guesses

Marcin was even able to use his methods to compute the frequency of pipe organ notes with around 90% accuracy

So how did mersin do what Galileo said was impossible?

What do you think?

Given the technology available in the early 1600s. How would you try to figure out how the rate of vibration of a string its frequency?

depends on its length and tension

How would you prove Galileo wrong?

For a small hint and some other cool stuff check out the PDF linked in the description below

Good luck, and thanks for watching