As we live in an age where anything goes, you will see paintings that range from rigidly structured compositions to ones that have no apparent compositional structure at all. If you look at Leonardo's Last Supper painting you can see the basic perspective underlying this work.
Now take a closer look at this - it is even more complex than is first apparent. Below is an incredibly elaborate, perspective, underlay, which doesn't even take into account the figures themselves!
In contrast, lets take a look at this drip painting which at first glance appears to be far more complicated and structureless.
This is also a complex image, however, by comparison, it is incredibly random - or is it? It might be argued here that the black lines provide a sort of structure to this work. Whether that was planned or accidental, we might never know. In fact if you focus on any one colour, there is a kind of energy that dances you around the painting. It is primarily a painting about line and energy. It is the line that is doing 90% of the work here as there are no flat planes of colour like those we see in the Last Supper painting.
You can understand therefore that structure in a composition may take different forms.
There are three main compositional elements in painting -
Line - the chief element of beauty in a work
Notan - Light & Dark values
Colour - Good colour is dependent upon good Notan
LINE - The term line refers to the boundaries of shapes. The line in fact creates the shape. it refers to the way in which spaces and lines interact. The harmony of combined lines creates Line-Beauty.
NOTAN - The word NOTAN is a Japanese word which means dark/light. It relates to the massing of different value areas, some light some dark.
When we talk about Notan-Beauty this refers to the harmony which results in pleasing combinations of dark and light spaces that can occur in nature, photographs, paintings or buildings.
This is regardless of whether they contain colour or not.
Also to be noted is that there is a fine distinction between Notan - an element of Universal Beauty and the the appearance of Light and Shadow - these are not the same thing.
The Merriam-Webster Dictionary defines Notan as - the combination of lights and darks especially as used in Japanese art : the design or pattern of a work of art as seen in flat areas of dark and light values only.
COLOUR - The term COLOUR refers to a quality of light. Painting good colour will be dependent on good NOTAN, which in turn is dependent on good spacing.
If these three things, Line, Notan and Colour are the building blocks of composition then it makes sense to begin by becoming familiar with line and spacing before introducing either Notan or Colour.
That was the simple definition. Now within these parameters we can further break composition down into the following main areas.
OPPOSITION
TRANSITION
SUBORDINATION
REPETITION
SYMMETRY
I will continue this discussion at a later date, and examine each of these subjects separately.