I just started reading Audacious Euphony: Chromaticism and the Triads Second Nature pretty early on it mentioned Schenkerian graphs and put them in the same group of analysis as Roman Numeral analysis. I've never used them before so I was curious what are Schenkerian graphs and how are they used to analyze music?
Schenkerian Analysis is a form of musical-structural analysis that iteratively reduces the structure of a piece of music to an increasingly simple melodic line. The piece is viewed as consisting of multiple "layers" -- foreground, middle-ground, and background -- each consisting of successively less detail.
At each stage of the analysis process, a pseudo-score is produced which gives a hierarchical representation of the importance of each note within that level. The resulting scores can be called Schenkerian graphs. An important thing to realize within these scores is that the temporal/rhythmic element of music (durations of notes) are more or less removed. Instead, the way that the notes are drawn (whether empty or filled, whether stemmed or not, and whether beamed or phrased or not) indicates the relative prominence of the note within the analysis, and its grouping with other notes. When seen through a Schenkerian Analysis, much of the detail in a piece of music becomes seen as prolongations of more fundamental underlying progressions.
This page is one of several that gives more detail on how to do a Schenkerian Analysis.
The purpose of a Schenkerian graph is not so much to describe the hierarchical importance of given notes at a given level, than to show directions and distant relations. The graph should always be read with the score itself. The graph above, from wikipedia, to some extant explains the score, but the score at the same time explains the graph. The graph shows important lines, here the one descending Bb-Ab-Gb, the one doubling this at the upper third, the less important one descending from the high Gb down to Db, etc. But these lines must then be found in the score. In this case, it happens that all the notes of the graph are also in the score. But that is not always the case, because at times notes in the score (or in the graph) are in lieu of other notes in the graph (or in the score).
The ultimate point is to show how, starting from a simple graph that could serve for almost any piece, successive layers contribute to creating a piece that is unique. This is shown first by "going down" from the piece to the graph, but one should not forget the second stage, climbing up, from the graph, back to the piece itself.