Hello, everyone.
Thanks to everyone for watching this video.
My name is Qi Dang.
I am an amateur musician.
Today's video is titled temperaments
and it is going to provide you the Pythagorean tuning and equal temperament.
I am going to give you an overview of what I am covering today's video. 
First at all, it is an introduction of sound frequency.
And then focus on the Pythagorean tuning and twelve-tone equal temperament before wrapping up with conclusions.
The sound frequency that human beings can hear is from 20Hz to 20,000Hz.
In such range, how should we name the different pitches?
We can try to pick out a few frequencies, and give them the name "Do Re Mi Fa So La Ti".
If we just divide the frequency range from 20Hz to 20,000Hz among the seven frequencies,
and name them "Do Re Mi Fa So La Ti".
However, the "Do Re Mi Fa So La Ti" we hear maybe like this.
This sound is terrible.
Fortunately, in the ancient Greek, Pythagoras and his follows proposed the "octave".
They thought about the relationship between music and mathematics.
Why do some notes sound better together?
Why do some notes sound very similar? 
The ancient Greeks did like to think in terms of proportions.
If you have a string and you stop it halfway along, you are produced an octave.
So that is the simplest ratio, want giving you the octave which is one of the perfect consonants.
If you stop the string two thirds of the way along,
you produce a fifth which is the next simplest ratio two to three, and is the other perfect consonants.
The formula for the fundamental frequency is equal to 1 over 2L times the root of F over mu,
where L is the length, F is the force, and mu is the mass per unit.
Actually, the Greeks didn't know all that we know about science and frequencies, and the way sound waves work.
But they did have stringed instruments, and they could experiment with strings.
They made this discovery that if you divide the string up in these simple ratios, you get the most consonant sounds.
Pythagoras and his followers were obsessed by numbers and their relationship with the real world.
So, when they made this discovery that the fifth, that's the most consonant interval together with the octave
correspond to a vibrating string stopped in this simple ratio of two to three.
They decided to base their whole system of tuning on the fifth, and I want to show you now, how this works.
Let's see if we can construct the scale of C major Pythagorean tuning, we are going to start at C, and we will just choose a frequency. 
The frequency is x. That is going to be our starting point.
Now, on the Pythagorean scale, the first thing to do is to raise that by a fifth which means multiplying by 3 over 2, so that get to the note.
How do we produce our next note on the Pythagorean scale?
We start from this note, and we raise it by another fifth that means multiplying by 3 over 2 again,
but now we’ve multiplied by 3 over 2 times 3 over 2, that’s 9 over 4. And 9 over 4 is bigger than 2, so it pushes it into the next octave.
We don't want that, and we want to be in this octave.
How do we get the corresponding note in this octave from the one we've ended up in?
In the octave, we have to divide by 2, because all the frequencies in this octave are half the frequencies in this one.
Okay, we've construct to the Pythagorean scale of C major based on repeated use of fifths.
The frequencies of seven notes are: x, 9/8x, 81/64x, 729/512x, 3/2x, 27/16x, 243/128x. 
If you play the seven notes using piano, like this.
If you change the order, like this.
In the Renaissance, musicians might think that only seven notes are too monotonous, so they want to add new notes.
According to Pythagorean tuning method, the frequency multiplies 3 over 2.
If the new frequency is higher than 2, then the new frequency should be dividing by 2.
Therefore, the frequencies of twelve notes are shown in this screen.
The thirteenth note is 531441/262144x,
and it's 2.027x which is close to 2x, but it is not 2x.
This note cannot return to the octave, what should we do?
The answer is that we can't solve the problem.
A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable.
If a whole number of perfect fifths is to close with the octave,
then one of the intervals that is equivalent to a fifth must have a different width than the other fifths.
The interval between the adjacent notes, it is not 3 over 2, and it's 262144 over 177147.
We name this combination as "wolf interval" or "wolf fifth". It sounds like the trembling piano.
This problem is solved until the 18th century.
Before that, when composers wrote their music, they had to be careful to avoid the combination that caused the wolf interval.
As a result, they could only write certain tones or chords, and not arbitrarily transpose.
In the 18th century, musicians decided to change their ideas.
Since there was no way to make all the interval combinations sound harmonious.
Otherwise, we will change it to make all the combinations only have a little dissonance, and the listeners may not notice it.
Then, we have equal temperament.
An equal temperament is a tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps.
This means the ratio of the frequencies of any adjacent pair of notes is the same,
which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.
In the twelve-tone equal temperament, the distance between two adjacent steps of the scale is the same interval.
Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications.
Specially, the smallest interval in an equal-tempered scale is the ratio that is equal to the 12th root of 2.
Let's listen to a piece of music played with different temperament systems. The first is the Pythagorean tuning.
Then it is equal temperament.
From conclusions what we learned,
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3/2.
Twelve-tone equal temperament is the musical system that divides the octave into 12 parts,
all of which are equally-tempered on a logarithmic scale, with a ratio equal to the 12th root of 2.
Okay, thanks to everyone for watching my video today
and I do hope you found the video informative
and that you learned somethings from it.
If you do have any questions, then you can write comments and messages.
I am happy to answer any questions.
Thanks for your watching!
