Monday, June 30, 2008

"Mind" -> The Scale Of Music Part III

We are going to return to our friend Pythagoras, and therefore I thought that I would start off this post with a nice photo of his bust.

You probably know him for a² + b² = c² , but we learned in the last post, he is also credited with the unique way of subdividing a string to give us an musical scale, or what is called Pythagorean Tuning.

However, if we are to trust the archaeologists, we will find out that although he was credited with the scale, there were many before him that had actually developed it. While this may be strictly true, we will use the common vernacular to allow Pythagorean Tuning to be a particular type of tuning.

In the last post, I spent a lot of time talking about how your could grab a guitar, and recreate some experiments to figure out how to find notes that are pretty close to our notes. However, once you get into the math of music, you will find out that saying things like "divide the string down by 1/3" is fine for the first step. In the second step, we want to understand what it means from a mathematical standpoint what the dividing of the string does.

So imagine that you have a string and it is going up and down 100 times per second. This note would be below middle C on the piano, since you should be able to remember middle C is 261.626 beats per second (or hertz). Now, if you cut this string in half at the same tension, you will get a string that vibrates at 200 hertz, or twice as fast. As we stated, twice as fast means an octave higher. From a mathematical standpoint, we look at these two notes as a series of ratios. The lower note is at 100 and the top note is at 200. Thus the ratio of the vibrations are at 2:1, or you get 2 vibrations on the top note for every vibration on the bottom note. Since the bottom note is the base (or fundamental or tonic note), we write this as 2:1 or as 2/1. If you Google scales, you will see this in many descriptions.

What happens if we cut down the string by 1/3? I never took you through the math of why a string vibrates twice as fast when it was cut in half, as this probably was obvious. What happens when we cut a string by a different amount? Well it obviously won't vibrate twice as fast, so you will intuitively grasp that the range of the vibration is between 1 and 2. What we will find out, with all the right instrumentation, is that the vibration of the string is proportional with the length of the string. If you want to make the string vibrate twice as fast, cut it in half. If you want to make the string vibrate three times as fast, cut it by three, and let only a third vibrate.

As long as we are here in our post, we are going to take a bit of a side division. We've been talking a lot about "dividing down" a string. However, let's talk about this a bit more. There are three ways to influence how fast a string vibrates.

1. It's length
2. It's tension
3. It's width

Given two strings of equal length and tension, the heavier one will sound lower. If you look at a piano or a guitar or other stringed instruments, you'll see that they really use both length and width to change the note. Why do they do this?

The problem with thick strings are that they don't sound very good at short lengths. The problem with thin strings is that they would have to be very, very long to sound good at low notes. The best sounding instruments are those that can vary all three factors to get the best sound. Some instruments like guitars can only vary two factors. Therefore, guitars will never have the same range as a piano.

In the next video, I'll try and demonstrate this on the guitar. In the video, I am playing a G note two ways. I am plucking the G string (3rd string from the bottom) and I am also plucking the E string only made very, very short by holding down the string. They both are the same note. However, the short G string is a bit dull. The nice long G string is quite a bit fuller. (And yes, I did think that the term G-string was funny in this post.) However, when you play the two strings together, you really can't hear the difference.



The last thing I'll leave you with is that the reason that the the short and thick string sounds less interesting than the long string is because of something called "overtones." I don't want you to know about overtones yet. All I want you to remember is that overtones are good stuff and make music sound more interesting.

Now, let's get this post back on track. Instead of talking about cutting down a string for a higher note, we were talking about how we derived our scale from Pythagoras.

What we have done is cut the string by 1/3, which leaves 2/3 of the string left. The math is pretty simple, to find out how much faster the string vibrates, you invert the fraction of the length of the string to make it 3/2, and you get 1.5. So the cut down string vibrates exactly 150% faster than the tonic or base note. However, we normally leave the number in the ratio of 3/2. Trim off 1/3 of the string, and you get a string that vibrates 3/2 faster.

If you remember that the the fourth note came from trimming off 1/4 of the string, you'll realize that you have 3/4 of the string left. As in the example above, the speed of the string is 4/3 or 1.333.... times faster. Now this series of threes never stops, so it is really necessary to keep the fraction of 4/3.

The above is a pretty important point, because from now on in this series of posts, we'll be explaining notes in terms of how much faster it vibrates than the tonic (or root or fundamental) note. However, you can remember that just because we are describing it this way, it still goes back to how you cut the string. So, we are talking about math, but it is all about the length of the string.

So, now we want to go back to our friendly Pythagorean tuning. If you remember from the last post, I said that they trimmed the string down by 1/3 and got another note. This note was exactly a musical fifth above the starting note. In this case, lets pretend he started at Do (or we can call it C). He went up a 5th, which is just another way of saying that he shortened the string by a third. Just a couple of paragraphs ago, we found out that this will vibrate 3/2 faster than the base note. This note is now So (or we can give it the name G).

The next step would be to go up another fifth, and we'd find Re (or D). The problem is that this is out of our current octave. We vibrated 3/2 faster, then another 3/2 faster. This means that we are now vibrating 9/4 faster than our base note, or 2.25 times faster. Remember that an octave is just 2 times faster. So, how do we get it back into the octave?

This is very simple from a mathematical perspective. You take the note, and drop it by an octave. So what we need to do is cut the vibration in half. You can easily do this by dividing the number by 2. 9/4 divided by 2 is 9/8 (to divide, you multiple 9/4 times 1/2). Thus D, in the octave, is 9/8 times faster than the base note. By doing this, we have gotten our note back into the octave.

If you continue to do this procedure, you will come up with all the notes of the scale: Do, Ra, Me, So, La, Ti. This almost gives you your scale.

The one problem is Fa. In our division, we get all the notes except for Fa. The last note we will find is Ti. However, dividing the string again gives you a funny sounding note, which we'll address in a second. So rather than dealing with a bunch of funny sounding notes, we simply make the string 1/3 longer. This makes the string vibrate 1/3 slower, or the vibration speed is 2/3 of the tonic note. By doing this, we find a note below our starting note. Now, to bring it up an octave, we multiply by 2 and we get a note 4/3 faster than our tonic (or C in our case) note.

This then became the basis of western music. All is solved.

Or is it?

Remember to get this scale, we have been going up a fifth, then bringing the note back down to the original music octave. As I stated above, an odd thing happens when you get to Ti. A fifth above Ti leads you to another note. A note that doesn't fit into the scale. We'll talk about the impact of this note later, but for now, we'll just go ahead and plot where this lands.

The note, a fifth above Ti, is a little higher than Fa. It is like a Fa+ note. We would call it today as F# (in our special case). We've been calling things the solfege names. However, solfege couldn't handle this note at first. Here is a new note cropping out of nowhere. Later in life, we did give it a name. If the close to note was Fa, then the slightly higher note was Fi. (A note a little higher than Do is Di, and similar So has Si. Pretty creative, huh?)

If you keep this up, going up fifth, then bringing it back down, you will eventually get to Fa (or F in our case). The next fifth up from Fa is Do (or C). So, we can also find Do this way.

By the way, by doing the 12 division method you will eventually land very close to Do. How many times do you need to do this routine of going up a fifth, then bring it back down before you get back to Do (or C in our case)?

The answer shouldn't surprise you. The answer is the number of keys that we have on on our scale. You need to do it 13 times. In what almost sounds magical, we have simply divided the string by 1/3 until we have gotten back our original note.

Well, almost gotten back our original note.

Remember I said that after doing this 13 times you are very close? Well you are left 23 cents away from perfect. This left over bit is called the Pythagorean comma. In other words, left overs are called commas. There are alternative tunings, and they also made have commas, so you best remember the word.

Now, what is a cent? We'll need to get to this later on, but simply think that the distance between any two notes is like a dollar. How many cents (think century) in a dollar? Obviously 100 of them. Therefore, there are 100 cents in between each note on a piano. 25 cents is enough to be noticed. Normally, a few cents are not. So what is the easiest way to close the circle? Well you could have 1 note that is 25 cents apart on your keyboard. However, the better way is to simply have the 25 cents scattered in all the notes that you just found.

This "mixing in" of a little remainder is called "tempered scale" and was to revolutionize music. However, we have not explored enough of the Pythagorean scale yet.

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