We are happy to announce that we have now a Pizzicato office for US and Canada. It is owned by Blair Ashby and located in Denver, Colorado. Please read the full article below, in the next section.
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Let us continue with our analysis of music, with the purpose of finding new ways to approach music composition.
In our previous articles, we have reduced the structural parts of music to six basic elements: Rhythm, Melody, Harmony, Sound, Effects and Form. Then we have started analyzing Rhythm with our first musical viewpoint, which is orderliness.
Let us remember our practical definition of it:
The orderliness of a system is increased by the presence of similarities and decreased by the presence of differences, amongst its component parts.
By listening to some examples, we found out that orderliness is a relative quantity and also that each person has his own degree of tolerance, within which music is agreeable to him, and outside which music gets too difficult to understand and so, is rejected.
How can we use this to compose music?
Well, nature has some basic rules that you can find in many fields of experiences. One of them, very important in music, will help us in analyzing music.
The phenomenon of harmonics
Very early in music history, a link was established between music and mathematics.
The phenomenon of harmonics is basically simple to understand. It is based on the following natural sequence of integer numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, ...
The idea is that a quantity of 2 has some similarity with a quantity of one. It is exactly the double. Anyone can easily divide a cake in two equal parts. You can then see the full cake and see the two halves and say it has some order in it. The two parts are equal and nicely fit into one full cake. This is harmonic 2 of the cake situation :-)
But if you are married and have one child, you should think about harmonic 3 and cut the cake into 3 equal parts, which is in practice more difficult to divide so that nobody gets a smaller part than the other two.
Now, harmonic 4 is much easier to relate to harmonic 1 (which is the full cake), as you can first divide it in two and then divide each part also in two. In other words, you first create harmonic 2. Then you consider each part as a whole and create harmonic 2 for each one separately. There is a double relationship there, as harmonic 4 of the full cake is equal to harmonic 2 of half a cake.
Using harmonic 5 is much more difficult, as you may not rely on lower harmonics. 5 has nothing in common with 4, nor 3, nor 2. You must create the harmonic 5 from the full cake (harmonic 1).
Harmonic 6 can be constructed in two ways. You first divide the cake in 2 parts (harmonic 2) and then you cut each part into 3 (harmonic 3 of harmonic 2). Or, you divide the cake in 3 (harmonic 3) then you divide each part in 2 (harmonic 2 of harmonic 3). It is the same as saying that 6 = 2 x 3 = 3 x 2.
Harmonic 7 is even more difficult. It has no relationship to either 2, 3, 4, 5 or 6. So again, you must estimate it with no other reference than the full cake (harmonic 1). You may help yourself by thinking 'it is a bit less than 6 equal parts', or 'a bit more than 8 equal parts'.
For harmonic 8, it is again quite easy, as 8 = 2 x 2 x 2. So you can divide it into 2, then again into 2 then again into 2. This means that it is harmonic 2 of harmonic 2 of harmonic 2 of the full cake.
When you follow this process and want to divide the cake into other numbers of parts, you will see that each time you can break the harmonic into multiples, it is easier, and when you have a number that does not divide itself into smaller numbers, it is more difficult.
Perception of harmonics - Orderliness
What do we mean by 'easier' or 'more difficult', when cutting a cake into different parts?
It is the relative easiness by which we recognize and perceive the equal divisions of one unit.
Read that line again, as you might underestimate what you can do with it in music.
Divisions by two are the most natural and easy of all. The world is full of them. Look at how most building windows are designed with two equal parts, how tables, furniture, houses are designed and decorated with this duality in mind. You can find it in architecture, sculpture, painting, drawing,...
As an example, you can easily observe that in the following pattern, there is one green box for every two yellow boxes:
You do not need to think a long time or start to count the various boxes. It just appears so obviously that it seems natural to you. You see in it as an ordered schema. The degree of order is quite high and you perceive it instantly.
A division by three is also quite natural, although a little more "odd" than by two. You can however perceive it quite easily that the following pattern is organized by three:
When you increase the number of divisions, this instantaneousness of perception decreases fast. Look at the following pattern:
There is less chance that you perceive at once the presence of 7 yellow boxes per green box. You have to count the boxes and then you can know the relationship between the yellow and the green boxes. Notice that you can train yourself to recognize such patterns faster, without counting.
Now, as we have a larger number, it becomes very difficult to establish the relationship between the boxes without counting them one by one:
This pattern has 21 yellow boxes for each green box.
Now, is there any relationship between that and the concept of orderliness? Are we speaking about the same thing?
You can notice that each one of the above patterns give some impression of orderliness. As we defined it above, orderliness is increased by the presence of similarities. In all the above figures, the yellow boxes are all similar to each other. The green boxes are all similar to each other. This increases the orderliness.
Now, the fact that a series of yellow boxes are displayed next to a series of green boxes decreases orderliness, as it introduces differences. However, this difference is decreased by the fact that they are well aligned next to each other and that both are colored boxes. If the green boxes were drawn as circles, the differences would be more important.
Someone looking at these pictures - but also at any art form - will try to establish and perceive similarities as best as he can, within his own frame of reference and knowledge.
Looking at the above pictures, the next natural concept that appears is 'there is so much yellow boxes for every green box'. In the first two pictures, the ratio of 2 and 3 appears quite obvious. For ratios of 7 and 21, it is much less obvious, but the fact that they are all the same width still keeps some orderliness there.
A person with a very limited tolerance band of order may appreciate the two first patterns and reject the other two. As one's tolerance band of order increases, one can appreciate the last picture, even without having determined the exact ratio between the two lines of boxes. He just assumes there is a relationship and he is satisfied with that. Seeing the picture, he knows that there must be an exact ratio in there, but he does not care to measure it exactly.
By establishing a ratio, or by assuming there is one, he has found and established the relationship between the two different lines of boxes. He is then more satisfied with his observation. He can make the picture have more meaning for him; he can understand the picture better, so he can appreciate it better.
Well, seems like we are coming closer to something related to the experience of art, aren't we?
At this point, we could advance the following theory:
The appreciation of art is proportional to the degree of order you can perceive in it, to the meaning you can assign to it and to how much you can understand of it.
We all know that appreciation of art is quite subjective, so this observation fits well with the above proposition, as it depends on how the person will perceive the orderliness of a piece of art.
Cakes and graphic patterns are fine, but what about music?
Maybe you thought I was digressing here with cakes and graphics... How do we apply this to music?
Harmonics apply broadly to music. Any instrument produces a sound that is a combination of harmonics of its main pitch, that is, a multiple of the frequency of the note by 2, 3, 4,... From these ratios of numbers, we can deduce chords, cadences, scales and much more. We will come back to this when studying harmony, melody and sounds. This application of harmonics is used in many music courses.
However, there is a second application of harmonics in music, but it is not so often explained or studied in terms of harmonics. It is however a direct application of the above explanation to the construction of a rhythmic pattern.
Each of the above pictures could be translated into a simple rhythmic pattern. The widths of the boxes become the duration between beats and the colors become different instrument sounds. Here are examples. In each case, you first have a percussion rhythmic pattern with two instruments, then an example with melodic instruments.
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