Saturday, June 28, 2008

"Mind" -> The Scale Of Music Part II

As discussed earlier, the roots for Western music derives from the Greeks and specifically the Lyre, which was a four string instrument in its simplest form. However, I doubt if you have a lyre in your house, but if you do have a guitar, I suggest you grab it now.

Guitars are about as close as you can get to a Lyre. If you look at a guitar it will have 6 strings, but for today we are only going to use one of them. The biggest string is called the E string. If you look at the guitar held by the young girl at the right, it is the string to the far left on the fret board. The surface of the guitar neck (or fretboard) has a series of frets or bars on it. These frets make it very easy to nicely subdivided the string. By holding your finger behind a fret, the string is clamped at that point and the vibrating string can be cut to various lengths. If you look at the fret board, you can she it has two dots on it. These dots are special.

If I pick up my guitar, I can measure how long the string is, for the section that vibrates. On my guitar it is 65.4mm long. If you remember in the last section, we said that an octave is when you played a note then played another note that vibrated twice as fast as the first note. So how do you make a note vibrate twice as fast? Well you can do two thing, you can take a string and stretch it harder, but this is difficult to do quickly and accurately. The easier thing is to cut the length of the string in half.

This is what we'll do with our guitar. So we measure out half of the string, and we find it is 32.7mm. Guess where this position is on our fret board? It is at the two dots.

The guitar makers clearly mark the octave point so you can quickly find it when you are learning to play. You want to raise the note by one octave? Just hold down the string at the two dots.

Let go back to the ancient Greeks and Pythagoras, who is credited with popularizing the first scale. Pythagoras found the string just the same same that we did and thought about dividing it to get different notes. However, instead of dividing it by two, which sounded very boring, since this was just an octave, he asked himself what were important numbers.

Pythagoras had actually created a cult of numbers. It might seem strange today, but his followers thought integer numbers as magic. (Another post in the future could be given over to this.) In many senses, we should not be surprised at this because Christianity does the exact thing. Number 7 has a very important meaning. Number 3 has a very important meaning. These number reappear on the Bible on a very consistent basis, and they mean something in and of themselves. Pythagoras and his followers simply made the numbers idols.

Pythagoras thought certain numbers had power. The powerful numbers? These were the numbers 1, 2, 3, 4, and 5. So, let's go back to our guitar string. We can lop off 1/2 of the guitar string to get our octave. That is very straight forward. If you listen to the string cut in half, the sound is "boring." As we stated before, half a string is just an octave, and an octave brings nothing new to the table.

So, let's do something more interesting. Since 1 over 3 is a magic fraction, let's cut the length of the vibrating string by 1/3, thus leaving two thirds to vibrate, and we'll have a new note.

If you take that guitar that we had and times the length by 2/3 you will get a length of 43.6mm. If you measure out 43.6 on most guitars, you will find that you will land on another dot, which may be marked on the side of the guitar. If you play this note at this fret you will find out that it is what we call in music "a fifth." In other words, when you cut down the string by one third, you get a note that makes up a fifth with the original note.

Let's go back to our other post. Do you remember that I said that the original scale was Do, Re, Mi, Fa, So, La, Ti and Do?

In our case, the open string note is "Do." We have just found out that Pythagoras thought that the "magic fraction" gave us So. To our ear this is a very pleasing sound. It just sounds like the two notes should "go" together. There is a reason for why these notes go together, which are called overtones, which we'll look at later.

These notes so much so blend together, we say in music theory that if the open string (or base of our Ionian or better termed "diatonic" scale) is the tonic note (think of the tonic note as the base note) and the dominant note is So. The dominant name can be a little confusing. However, it is used so much, it would be good to really try and remember this. The base note is call the tonic. The fifth note (or the magic cutting of the string down by one third) is the dominant note. Now, remember that dividing the string length by 1/3 gives us a fifth! This is a little less confusing than it sounds because once you know how to divide the length of the string down by a third, you never think of it that way.

Whew, got it? Then let's go to the next step.

We've cut our string down by the magic third. What would be another good fraction? How about one over four. So, we divide our string by three quarters, and this will come to be 49 cm on our guitar. You will find another dot here. This is a fourth on our musical scale. In our scale, we have found Fa. In our music in the west, we find that this interval is very attractive from an auditory sense. Many songs are drawn toward this note. Not as much as the fifth, however. So what will name this special note in our diatonic scale? It isn't quite dominant, so we'll call it Subdominant. Now, things may get a little confusing. This dividing stuff: it seems to give you a specific note; it seems to give you a ratio. I'm even bringing up names like dominant and subdominant. Once you get into music, you'll find that there is always multiple ways of calling the same thing a different name. My only advice is to get used to it.

Now, we have a bunch of other notes to do. The question is do these fall in line with all the other notes that we created. We've show that it is very easy to find the fourth and fifth, but what about the other notes.

This is where things get quite a bit more sticky. If you grab your guitar and do what I did, it would seem very obvious. You've divided the string down by 1/3, 1/4, so the next division should be 1/5. Once we get into the heart of this, we'll find out that the right interval is 17 over 81. This is very, very close to 1/5 (just .09 away).

Why is it 17/81? We'll go into the finer details later, but the conventional wisdom is Pythagoras created the scale by taking a string and dividing it down by one third. (Remember, we found the dominate note "So" this way.) Then they think that he took this string and divided it down by one third again. Thus the string length was 2/3 times 2/3 or 4/9ths. He kept dividing this string down by 1/3 until he got back to an octave, and if you counted all the times you would need to divide the string to get an octave, you would find it took you 8 times. So you have another note, but this note is very, very high. So high, that it doesn't fit in the scale. So, he decided to lower the note by octaves. This is very simple. He simple divided the vibrations by 4 to slow it down and get it closer to the original note.

Now, this is the theory. However, if you go back to the art from Greece, you will find many pictures of lyres. It is obvious to me that they didn't have any real ability to divide a long string the way that it is supposed to have been accomplished. The Greeks may have eventually gotten to a place were they carefully derived the scale from continuously dividing a string by one third, but my guess is that they found Do, Me, Fa, and So by simply dividing by a fifth, a fourth and a third.

This is obvious to more than just me, because I'm "just" explaining it. This type of fraction based tuning is called "Just Intonation." Just intonation is built off of the idea that all notes are based on common fractions. So we have most of the divisions, but we are missing Ra, La, and Ti. The only fraction that is a bit "odd" is the Ti fraction.

It was only later that somebody came back and built up the continuous dividing of the string by one third. The idea of "Just Intonation" was very popular, and many people played in this scale for many hundreds of years. The problem is that it sounds "wrong" if you modulate out of this scale into another key.

So what was it? Are the classic roots of our music just intonation or Pythagorean tuning? Clearly, from the literature, there were people that understood and played in both.

The root of the problem is that we really don't have the right evidence. There are no MP3 from ancient times, and most instruments don't have enough left on them to determine the pitch. If you are looking at the fossilized lyre from Greece, you will have no idea of the string tension. At the same time, many popular instruments were reed based. Virtually impossible to understand how these were tuned.

We are very fortunately, however, to have a pretty good idea of how these were played. You can see this in the picture that I clipped from the web. In this case, a lyre is being held. The man is hold a plectrum (like a guitar pick), and he uses his other hand to quiet certain notes.

As a side note to this, probably the best way of surmising the tuning would be to get a nice wind instrument that we dig up out of the ground, if it were perfectly preserved. Since there is no stringing, you would guess that it would be much easier to figure out the tuning.

We are fortunate enough to find out that such instruments exist in the Jiahu bone flutes. If you look at the pictures below, you will see the actual flutes that I'm talking about.

In an architecture dig in the China region of Jiahu, a bunch of flutes were found from 9000 years ago. They were in remarkably good shape, and extensive analyzing of the flutes was done. My take away from the research is that you can make an argument that some of these these flute could do a very workable Pentatonic scale (we'll cover this later, but this is the classic Chinese sounding scale that can be tied into the Western Scale), and others of the flutes simply were not able to pass for the normal pentatonic scale. The issue is that we'll never know because we weren't there

Now, Matt, in his comments to my last post, said that he heard that all music world wide has in common octaves, fifths, and fourths. He asked me if I believed this was true. On the face of it, the answer is obviously no. I mentioned the slendro scale, which is used in Indonesian music. If you youtube slendro, you can eventually find some singing in this style. It simply sounds off key. However, there is an argument as to why the western diatonic (Ionian) scale will win out from a sheer aesthetic reason. However, this is quite a few posts away. I think we've crammed enough information into our brains for this post.

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