Sunday, July 13, 2008

"Mind" -> The Scale Of Music Part VI

So far we have been examining the scale. I think most shocking thing about the scale is understanding that the structure doesn't spread the notes equally across the keyboard. The most important thing about the scale is that it is lumpy regardless if it sounds smooth.

As a review, any major scale is made up of the following: whole step, whole step, half step, whole step, whole step, whole step, half step. You will often see this abreviated as ws, ws, hs, ws, ws, ws, hs. To review this, a tetrachord is a ws, ws, hs. So a scale is simply two tetrachords separated by a whole step.

The two most popular instruments today is guitar derived instruments and piano derived instruments. There is little doubt that the guitar player is much more close to his instrument. They are lighter, cheaper, the are tuned by the player, and they are more microtonial. On a piano you are always going to get the same short term notes. On the guitar you can bend notes to get a greater degree of note variation.

It is also impossible to hide these note gaps on the guitar. If you go back to my last post, I showed you a keyboard, and I removed a note to show you the parallel structure of the octave. The piano is very deceptive, since it base based on the idea that there was a "perfect" scale. Since that scale had half notes in it, we wanted to hide the half notes. Therefore, all the keys are white, and on the piano the half step note looks just like a full step, in our C scale.

However, trying to hide it really wasn't in the cards for guitar. The guitar is a much more simple instrument. A fret board and a way of tightening strings. On the guitar it is very obvious because you must jump over a fret to get a full step. Now, there is a beauty about the guitar. Since all the steps are equally spaced as half steps, you can't really "hide" a whole or half step. You know exactly where the half steps and the wholes steps are. This gives a the very beginning guitar player a problem if he is being compared with the piano player out the gate playing in C. While the piano player only has seven even spaced keys to press, the guitar player has to remember that he only jumps a half step between the E note and the F note. However, as soon as the piano player gets off the C scale, suddenly he needs to start using all those black notes notes.

C scales has no black note.
G has one black note.
D has two black notes
A has three black notes

You continue this special assortment of black notes all the way through the 12 major scales. So, just to get the major scales down you have to memorize 12 different patterns. If you throw the natural minor scale, harmonic minor, and melodic minor on top of the major, you will have 48 different keys that you will need to memorize.

Let me say that again.

12 starting notes
4 different scales (and the melodic minor actually is two different scales, different on up and down)

= 48 different scales

Each scale is going to have different fingering.

Now, let's look at our friendly guitar. Once you have the fingering down for a C scale on a guitar, you have the fingering figured out for all the other scales also. In the picture on the left from the ScaleFinder Website, they will show you the fingering for any scale that you want. In this case, I took two pictures of a guitar with the scales on it. You can see that the D scale is just two frets above the C scale. So, the entire scale moves right on the fret board. Nothing changes in your fingering, other than the fret board is a bit smaller because you moved closer to the bottom of the guitar.

Unlike the piano, there is no major fingering change. To get all the fingering in a guitar, you need to learn one major scale and three minor scales. Then all you need to do to start another scale is to move to a different starting point.

Therefore, it would seem that the guitar would be the superior instrument. Right? It has a much easier to learn keyboard that you can move to any scale.

However, the guitar has some serious problems. The biggest problem is that it takes two hands to play it. One hand must hold down the right note, and the other one adds the power to the string to make the noise. The piano transfers the energy from pressing a key into a mechanism that hits the string to make a noise. This is a fundamental issue with the guitar. You can only pick two or three notes at any one time, so the music is simply not as dense as a nice piano piece.

Well, almost.

Guitarists have longed for the density capability of the piano. With the advent of amplification, it was found out that you could make a nice sound by simply tapping a string. So rather than having to hold and pluck, the guitarist simply holds down a string, and that striking motion will generate a sound. Now, the sound is very quiet, but with amplification, it can be made loud.

If you youtube "guitar tapping" you will see people doing just this. The interesting thing about guitar tapping is that it takes a guitar and turns it into a keyboard. These guys are simply pushing down on the string like a keyboardist would push down on a key.

The one problem with a guitar is that keyboards are very abstracted from the signal. This is why keyboards are the popular choice for synthesizers. As soon as you get to a synth, pressing a key doesn't trigger a hammer to hit a string. It triggers a piece of software. Often that software may sound like a cello, a square wave, or trumpet. It may even sound like a guitar.

Sunday, July 06, 2008

"Mind" -> The Scale Of Music Part VI

We've covered a lot of ground, and on many of these subjects, we've hardly have scratched the surface. On just the subject of scales, some people spend the rest of their lives thinking about this. Once you discover that our basis for a scale is made up, you can change your assumptions and create a whole separate music system. So we have the field of micro-tonal music, where some strange people are pulling out new bases for scales, and creating new music. It is my observation of this music is that this music is horrible, since we are already programmed to a musical scale early in life. You can no more hear a new musical scale than you can hear phonemes once your brain has gone past plasticity.

So, what have we found out up to now? What we have found out is that there are 12 candidate notes, and the major cultures selected notes for their scale. We've found out that their are pentatonic scale and heptatonic scales.

In Western music, we picked the heptatonic scale that we call "The Major Scale." Now, most of Western Music is based around fourths and fifths.

If you remembered the previous posts, we found out that the fifth was very popular. It vibrates 150 percent faster than the base note. The fourth is a fifth below the root that has been shifted up one octave. While the fourth and fifth isn't in every culture, it is certainly in ours. Due to something called overtones, these sound pleasent to our ears.

So lets draw out our scale again:


The gap between C and F is a fourth. The gap between C and G is a fifth. The gap between F and G is called a whole note. When we look at the whole note, it is found almost always between any two notes on a Major Scale. This would be great, but there is one issue with the whole step. There is not an even number of whole steps between the tonic note and the dominate note (between C and G). You need to go three and one half steps to go from the tonic note to the dominant note.

Now, this is a very big deal. Remember that the reason that we have three and a half notes is because there is a whole step between the forth and fifth note on the scale. These three notes are the notes that sounds good.

However our nice scale system falls apart below the subdominant note (F in our case), because we need 2.5 notes to get back to the tonic note. The answer is very simple. We need to take two whole steps and one half step. We've been writing about the major scale and the major scale picked that the step between 3 and 4 would be the half step. If you go back to our post on cutting the string down to find different notes

So, we find out that the major scale goes like this to the fourth

C to D (whole step), then D to E (whole step), then E to F (half step). This is a very important series of notes. This is so important that we've given it a name, the "tetrachord" and western music is built around it. A tetrachord is a chord made up of 2.5 steps to get to a fourth.

We say that the major scale, up to the fifth is as follows: Whole, Whole, Half, Whole, or 3.5 steps.

Once we have gotten to the dominant note (G in our case), we need to climb all the way up to C.

How much higher is the C than the G note? It is a fourth higher. Now, we just found out that the fourth is made up of 2.5 steps and is called a tetrachord. We are going to reuse our tetrachort, and use the exact same interval steps in our last fourth.

G to A (whole step), A to B (whole step), and finally B to C (half step).

The octave, in this system, is built around two tetrachords separated by a whole note.

CDEF is a tetrachord made up of whole, whole, half steps.
GABC is a tetrachort made up of whole, whol, half steps.

There is a space of 1 whole note between these two tetrachords.

Let's look at a piano keyboard. You can see that I've labeled the keys on the keyboard.

You may have remembered, if you look piano lessons at any time, that if you start on C, the scale is just the "white notes." I remember when I was a kid, the other kids saying things like, "Wow, the black notes, I don't even know what those things do."

Once you start to understand the scale, you'll start to understand that each key on the keyboard is spaced exactly the same amount apart (on a decibel scale, that it, which we won't get into now.) Thus any time that you cross a line, you will go up exactly a half step. The interesting this about the scale, as we've just learned, is that the scale for Western Civilization has a couple of tricky things about it. Again, look at the photo. To go from the E note to the F note is just a half note because it only crosses one line. Even if you are extremely musical, most people do not hear just a half note between E and F. What they hear is the next step on the scale. This is an auditory illusion. While it may sound as if each step is equal, in reality, the jump from E to F (or in our other terminology the move from the submediant--the third note--to the subdominant--the fourth note) is just a half of step.

Now we are going to take that exact same picture that we showed above to graphically point out what I've stated before. A scale is just two tetrachords. There is a tetrachord between C and F. Then there is a whole note--which I've removed from the picture. The gap from G back to C is a fourth, and without the note in the way, you can see that it is a mirror image of the bottom of the scale. So, in our music, the octave goes as follows:

Whole Step, Whole Step, Half Step

Whole Step Breather

Repeat the Whole Step, Whole Step, Half Step sequence.

Once you start thinking about this, it becomes very strange. If you were going to logically put together a keyboard, and all the notes were exactly half steps apart, why wouldn't you simply place each key exactly a half step apart? If this keyboard, instead of being a nice white key for F, you would simply make the F a black key and continue to alternative White and Black keys.

This exact keyboard and others have been made. These are called the bilinear keyboard, and a bit more strange, but heralded when introduced, the Janko keyboard, looked at this.

In both cases, the keyboard failed because of resistance by the public. We'll look at this in a future post.

Wednesday, July 02, 2008

"Mind" -> The Scale Of Music Part V

In the last post, we continued to take a look at ways that we found the scale. The most important thing that we learned is that the two biggest cultures in the world, the Chinese and the West, figured out that there was a "candidate list" of tones. From these many tones, the various cultures picked their scales. We found out that the Chinese were ahead of the west in discovering that if you continued to divide the string, by a third, 53 times, you would finally come up with a division that brought you back very, very close to the original note. This was called "Mercator Comma" in the west. (A comma is just the remainder from not quite matching up.)

We could address the fact of the comma now, but we are going to leave this on the back burner for just a little bit. What we are going to look at now is this whole idea of "picking" notes out of a candidate list. As we found out, for reasons that we'll explore later, both the Chinese and the West have the 12 different tones to pick from, but the Chinese went pentatonic (5 notes) and the west went heptatonic (7 notes). However, when I say "picked," it really doesn't describe what happened.

From everything that we can see, every modern culture has music. As the music theory was being formed it fell into the culture in which is was being made. This is a great mystery to me. If you think about it, what gives us boundaries are the instruments that we make. If you want to know what a C sounds like, you will go over to your piano and play a C. Your instrument then becomes the basis for all of our music. At least this is what happens for most people.

As I talked about a bit in my last post, there are a few very remarkable people that have what is called perfect pitch. Perfect pitch is the ability to simply remember what a C sounds like. In science this ability is know better as absolute pitch.

Let's examine this a bit more, only we are going to use the analogy of colors.

If I showed you red, you will remember that it is red. If I come back the next day and show you a red card, you will go "that's red." Music is not like that. The key for music is that most people need somebody to play a note and then the note becomes the reference for the rest of our music.

The equivalent in colors would be like the following:

Maybe you took a bit of color lessons when you were young. A woman would come to your house and she would take you to the family color player. A color player is an instrument that throws light on a wall and has keys like on a piano. There are 12 different colors that are used over and over. After you use the 12 colors, it has another set of keys that are higher on the keyboard. However, the higher colors are the same as the lower colors, only a bit brighter. So you have 12 colors repeating on the keyboard.

She taught you how to push the keys so that certain series of colors would flow across the wall. The flow of lights are very pleasing to the eye, and words are said with the flow of light to make it even more memorable. Good words with good light is very nice and pleasing to most people. As you age, you really don't have time for doing the colors yourself. Therefore, you log onto the iColor store and download series of pleasing color sequences from color groups. Each color sequence costs just .99 per download.

Now, your walking with a new friend of yours, and you see a tree. He says, "What color is this?"

"I don't know," you answer.

"It's definitely green," he answers.

You have no idea of how he knows that it is green. So you grab a leaf, and take it back to your house. You go to your color player, and press what you know is the green key. Once you have the green on the wall, you hold it up to the color on the wall, and sure enough, the colors match.

"How did you know it was green?" you ask.

"I been able to do this since I've been small," he answers.

Perfect pitch is just like this. You can be phenomenally talented in music and not have perfect pitch (or absolute pitch. However, I have read a few description of people with perfect pitch and they describe that they can hear the notes of raindrops falling. The note of a church bell ringing. Every sound that comes out, the gurgling of the fish tank, or the squeal of a blender, all of these sounds will have a fundamental note. If you have perfect pitch, then you are going to quickly understand which note that it is.

If you do not have perfect pitch, then it is only through training do you start to figure out that it is all based off of blending the colors together from a certain base color. The base color is our tonic note (or color). But here is the rub, I would suggest that having perfect pitch may be very helpful for an individual to compose in a culture, but it doesn't allow the culture to expand its musical horizons. Perfect picture can create a fence, and not having a fence around your pitch will allow more diversity.

Let's go back to the time of the Chinese and their 12 Lu scale, and remember that the Chinese population has a much higher percentage of people with perfect pitch. This means that once the scale has been decided (through whatever culture factors created it), there are more people inside of the culture that can hear somebody that diverges from it. They quickly say, "Hey, that's out of tune." Now, combine that ability to see "outside the lines" with a culture that is much more respectful of tradition and being "inside the lines," and you will get a culture that locks in to a tradition and has a tough time changing.

It would be a little like having a GPS in your car, and a great respect for efficiency and traffic patterns. You would always drive to your destination in the exact same way. If you started to get off course, your GPS (or absolute pitch) would quickly remind you that "you are off course, please turn around."

In the Western Culture, we have no GPS in our car. We have a general sense of where we want to go, but if things are off a bit, we really don't notice that quickly. As a matter of fact, we find out that we find things on our journey. We have discovered new stopping off points and restaurants. How many times have you discovered new stuff because you were lost?

In Western music, we have gotten lost a lot. And because of this, we have polyphony. We have transposition. We have all types of minor scales: natural minor, melodic minor, and harmonic minor.

I believe that in Western Culture, the inability of our ears to hear perfect notes allowed more creativity in picking different scales and different harmonies. In Chinese culture, with a much higher percentage of people having perfect pitch, you had a more rigorous establishment of the defined scales. They were less free, in some sense, to evolve their music.

Tuesday, July 01, 2008

"Mind" -> The Scale Of Music Part IV

We have really been focusing on theory and not on practical issues. However, before we leave the system of scales, we are going just a little bit farther and today we want to discuss the two great civilizations that drove what we call technology: The West and China.

Before we get there, let's do just a little bit of review of where we have been already.

We found out that our scale has seven notes. In the course of this series of posts, I have called out these notes as several different names. For the most part, I have called them

Do or C
Re or D
Me or E
Fa or F
So or G
La or A
Ti or B

This is the basis of all Western music. These series of notes make up the diatonic scale, or the Ionian scale. The key to the diatonic scale is the tonic note. This is call Do or C in our example. If C is called the tonic, the perfect fifth note, which is Sol or G in our example, is call the dominant note, since this is an extremely attractive note in most music. We often build to this note. If we almost get to this note, we call this the subdominant note.

So we can take our scale and find a couple of key intervals.

Do (or C) is the tonic note
Fa (or F) is the subdominant note
So (of G) is the dominant note

Now, there is a secret about the subdominant note. That secret is that it is very much like the dominant note only in reverse. If we layout the scale, we'll see that the dominant note is 5 steps above the C. However, we know that on most all instruments, we can find any note and go below this note. The subdominant note is exactly one fifth below C.

See here:


What I have done is layout a series of notes. You can see that if we started playing at F on our scale, we would need to go up five steps to get to C. Once we're at C, we need to go up 5 notes to get to G. So in any scale, the note that is five below the dominant note is called the subdominant note. However, since we can always double the frequency of the note, and make it go up an octave, the subdominant note is in two places. One place is a perfect fifth below the current note. The other place is a perfect fourth above the current note. You will also note, pun intended, the dominant note is both a perfect fourth below the fifth and a perfect fifth above.

So the fourth is a fifth, and the fifth is a fourth. Sounds confusing? Well is it is a bit. When we find out about overtones, we'll see that these types of relationships are extremely important. Music basically goes around and around in a circle.

However, we don't want this post just to focus on review, we wanted to talk a bit more about the derivation of scales.

We understand that seven notes is the derived scale that we use in Western Music. Let's review what we learned here.

The first tuning of 7 notes was probably done with Just Tuning.

Just tuning was when some musician somewhere took a string and started to cut it down. We found out that cutting it in half gave you an octave. An octave, for lack of a better word, is boring. If you cut the string by a third, you got a perfect fifth. If you cut the string by a fourth, you got a perfect fourth. If you cut the string by a fifth, you'll get a third (almost). We have just found the tonic, dominant and subdominant notes of the scale, and we've thrown in a third for good measure. For a variety of reason, cutting things into 3s, 4s and 5s is obvious. After you do this, you also find out that the sounds just "sound" like they are working with each other. (Up coming lesson on overtones to address this fact.)

So finding Do, Me, Fa, So on any stringed instrument seems pretty obvious. However, we have just found 4 notes. There are 3 missing in action.

I have been calling our 7 note scale as the Ionian or diatonic scale. The diantonic scale is just a special case of having 7 notes. Many cultures will pick different numbers of notes in there scale. However, we have a special name for all scales that have 7 notes. All of these scales are call Heptatonic scales.

There are a few more cases of these types of scales.

Heptatonic means 7 notes
Hexatonic means 6 notes (you might recongize this as the blues scale)
Pentatonic means 5 notes (this is the basis of a lot historical Chinese music)

Let's write a bit about the pentatonic scale. This is the scale of a billion Chinese. We were fortunate enough in the West to have a more complex musical system than many different cultures. While I am calling our scale "heptatonic" in reality is is chromatic. (We'll get to this later.) Our current musical instruments can replicate closely many different musical systems.

On the Pentatonic scales, they come in two types:

1. Anhemitonic
2. Hemitonic

Sound like big words? Yes, but they have simple meanings. An anhemitonic scale simply means that you can pick 5 notes off of our diatonic scale, and play this music just find. If you were playing Chinese music on a piano, you would only use C, D, E, G, and A.

So, this seems to also answers Matt's question about some cultures without fifths in their music. All we are missing the F out of this scale. Therefore, Chinese music doesn't have a fifth, right?

Well there are two answers for this. The first answer goes back to our discussion on dominant and subdominant. Do you remember that the subdominant note is just a fifth, only it is a fifth below C? Well in music, anytime that you have a fourth, you also really have a fifth. So, one could argue this either way. You can say that Chinese music has no fifth, or you could simply say that the fifth is the one below the tonic cord. This is why I referred to Indonesian music. Their scale really lands on the cracks.

One thing that you will see about all these scales, they are bits and pieces of the western scale. For the most part we can play the Western music on these scales. You may ask why.

The reason is that the Chinese also did the trick of subdividing the string muliple times until it wrapped around. Once they got all the possible notes, they then selected just five notes, from all the possible notes, to play their music. The ancient writings of China describes creating 12 bamboo pipes, or 12 lü.

So, we have both the West and the East run into the same problem. We want to find a few notes, and a few other notes come along for the ride. What both cultures eventually ended up doing was to say, "Well there are really 12 notes that are candidates for selection. However, we don't want all of those candidates. We'll pick some from this selection. These 12 notes are called the chromatic scale, and if you go to a piano, you will find there are 12 notes between any two octave notes.

But here is the rub to plague music students for all times. The problem with having all these notes is that you actually don't end up using all of them when you are playing a piece of music. We'll have to talk about this a bit more later, but this is an important thought.

What is important is that both the Chinese and the West said "there are 12 candidates to pick from in any musical scale." They pick their playing scale from the same type of chromatic scale that we did. I find this one of the most fascinating ideas in the world. If would appear that two separate cultures, once they had explored music and figured out overtones, would both come up with a candidate list for scales. The problem is that the two different cultures picked two different list of possible candidates.

At first though, you may think the Chinese didn't have a good understanding of what they were doing. However, they seemed to be far ahead of the West.

Do you remember that we talked about Pythagoras comma? This is the bit left over when you subdivid the string to find different notes. After 12 times, the next division gets you very, very close to your original note. For the West, we said "we'll good enough, who cares about a little left over." We then went on to build a bunch of music around these 12 notes. But not the Chinese. They were curious about the left over bit.

Ching Fang, around 50 years before the birth of Christ, said that he wondered if he kept subdividing the string if he couldn't get much closer to the original C. He went on to calculate that if you divided the string 53 times, you could get exceptionally close. Since he was Chinese, we never recognized his contribution. We recognize Nicholoas Mercator's reinvention of this technique 1600 years later. We now call this Mercator's comma.

What drives a culture to different musical scales? This is unclear. However, I do believe that it is tied back into the language. The Chinese language, mandarin, is tone based language. If you are ever listening to mandarin being spoken, you will quickly be familiar that there is almost a "sing song" quality to it. Unlike western language, if you say the exact same word with a rising tone or a falling tone, it changes the meaning of the word. The joke is that the word for donkey and Mom is the same. Western's forget to say the right pitch, and they'll say that they miss their donkey. (Instead of Mom.)

Because of this, speaker of Mandarin really have much better training of scales, or a concept of perfect (or absolute) pitch. If you had to sing everything, you have a bit of an idea of what this would do to your singing skills. It would be very great.

Therefore, the Chinese-Mandarin speaker have a phenomenally high percentage of people with perfect pitch when compared to Western Cultures, as reported by Diana Deutsche, who studies these types of things.

Confusingly, this ability to deal with pitch created a less diverse scale. We'll discuss this in our next post.

In both Western and Chinese music, we recognized that there were many notes. In both great civilizations, a subset of 12 notes were taken to come up with our scales. The difference is that the West picked 7 notes, and the Chinese picked 5.