Arch Maps Theory Things   William Wieland
An arch map is a tool for learning about the structure of a composition.
Music may be described as sound and silence organized in time, rhythm being the artistic organization of time. Rhythm is hierarchical. It occurs on a number of levels simultaneously. The lowest level is the smallest rhythmic unit in a piece or implied by the piece. The highest hierarchical level is the entire work itself. (Silence exists before and after a piece.) Any number of intermediate levels may be present in a composition.
Unfortunately, not all levels can be measured by the same yardstick (or meterstick, if you come from Europe – note that even metaphorically we cannot use the same measuring device!). It is fairly easy for us to hear the hierarchical structure of a measure of music in clear triple meter (3/4): the first beat is stronger than the next two. It is also not too difficult to hear the structure at the phrase level; we are usually able to recognize two-, three-, and four-bar phrases without much trouble. However, if the phrase groupings are more complex or if we are asked to remember longer spans of time, it becomes more difficult to make precise statements about how long the levels last and how they are subdivided into smaller groups. At this juncture it is useful to turn to some kind of graphing approach to aid us in recalling the hierarchical temporal structure.
In the process of constructing the graph, we work between the note-to-note level and the conception of the composition as a whole. We learn the work by understanding how the distinguishable parts form the whole. Ideally this process leaves a residue of the structure in our mind. The graph is really only a record of the process by which it was created. Creating it is a way to get into the piece and to begin to see the real issues that might foster a good discussion and a deeper understanding of the work.
Creating an arch map can be described as simply dividing the music into phrases and subphrases as well as the larger complexes of these. Although rules for the divisions can be formed, we usually need to resort to them only in the most complex and ambiguous situations. It is always best to begin by using your musical intuition. As an example of how to create a graph, we will use the melody of the “Aria” from the Notebook for Anna Magdalena Bach. Here are the first eight bars:

Aria Melody
As we familiarize ourselves with the melody, it becomes rather clear that it divides into two phrases of four measures each. After we have recognized this division, we might ask, “What are the musical factors which create it?” Perhaps the most obvious factor is the repetition of the motive from the first bar of the piece in measure 5. In singing the melody, when we get to measure 5, we can remember back to the first bar, and recall that the same motive served as the opening gesture of the piece. This parallelism encourages the listener to hear a division between the end of measure 4 and beginning of measure 5.
Another aid in hearing this division is the melodic cadence in measure 4 created by the use of the scale degrees 3-2-1. (Cadence comes from the Latin word cadere which means “to fall”.) also notice that, in general, the first four bars form a curve with the ends bounded by the note G. the coordination of these musical elements makes the phrase division quite clear.
An arch map uses a time line as its foundation or point of reference. This line may be divided into whatever units are most useful for the piece under investigation or for what one wishes to demonstrate. Usually, the basic unit is the bar.

Time Line
Arches are drawn to show the division of time.

2 phrases
I like to use graph paper when I create an arch map. WW
Graph Paper   Lighter Graph Paper   Even Lighter Graph Paper
Notice that the first arch ends at the end of the fourth bar and that the second arch begins at the beginning of the fifth bar, neatly representing the exact location of the division in the music that we have heard. The place where the two arches meet is called a seam.
Returning to the “Aria”, we may wish to show the smaller divisions in the temporal hierarchy. The first four-bar phrase seems to divide easily into two two-measure phrases. Perhaps the main reason for this is that the human brain seems to prefer symmetrical divisions when no conflicting evidence exists. The division is enhanced by the presence of a new rhythmic motive in measure 3. The repeat of that motive encourages us to hear a one-measure subdivision of the second two-measure phrase. The first two-bar phrase does not divide so easily. Thus an arch map of the first eight measures of the melody might look like this:

Arch Map
Notice that the final step was to create an arch over the whole eight-measure span making clear that it is the highest level of this particular hierarchy. Notice also that if an area is to be subdivided into arches (such as mm. 3-4) it is completely subdivided. In this way one avoids situations such as this:

Incorrect:   Incorrect Arch Map         Correct:   Correct Arch Map
As you can see, the map allows us to see the structure of the melody at a glance and can also be used as an aid to remember specific information about it. Given the fact that the excerpt is short, this map may not seem to represent a significant advantage over the original musical notation itself; it is in longer time spans that the technique becomes most useful, but the essential principles are still those described above.
From Dr. Mike Pisaro