Motivation

The principal motivation is to determine whether it is truly the case that melodic phrases tend to exhibit an arch-shaped pitch contour. NOTE: We already know that this is not a universal feature of phrases. For example, the beginning of "Joy to the World" shows a U-shaped or convex contour. The question is whether there is a disproportionate tendency to have

In addition, there is an ancilliary motivation: To determine whether it is truly the case that whole melodies tend to exhibit an arch-shaped pitch contour.

Population

The ideal population would be the totality of all melodic phrases (for the first question), and the totality of all melodies (for the second question). For practical purposes, the population for this study is the totality of all Western European folksongs.

Sample

For the first question, the sample should consist of a random selection of Western European folksong phrases. The sample consists of 6,251 folksongs from the Essen Folksong Collection. This is not a random sample.

The sample size is nice and large. But note that there are two possible problems. First, all of the works come from a single computer collection. If Helmut Schafrath (the original collector) tended to favor certain kinds of works, then the sample is not random. In fact, nearly 80% of the works in this database are German. A second problem is that the phrases are not independent. All phrases in a folksong were studied.

Main Theoretical Concepts

"Melodic Arch". Just what is a "melodic arch". How do we operationally define this?

Operational Definitions

This study uses three different operational definitions of a "melodic arch."

1. Aggregate Phrase Contour. A semitone-based averaging of all phrases of a given length. (This operational definition, allows us to graphically display the average pitch shape of many phrases of a given number of notes.)
2. Contour Classification. Each phrase is conceived of three pitch points: (1) the first pitch, (2) the last pitch, (3) the average of all the other pitches. With these three points, 9 contour types can be defined: up-up (ascending), down-down (descending), up-down (concave), down-up (convex), flat-up (horizontal-ascending), flat-down (horizontal-descending), up-flat (ascending-horizontal), down-flat (descending-horizontal), and flat-flat (horizontal). (Classifying phrases in this way results in many more convex phrases than concave phrases -- plus many more descending phrases than ascending phrases.)
3. Contour Classification (Method 2). A variant of the above is also offered. Each phrase is conceived of three pitch points: (1) the average of the first 1/3 of all pitches, (2) the average of the middle 1/3 of all pitches, (3) the average of the last 1/3 of all pitches. (The classification results are the same as above: there are many more convex phrases than concave phrases, plus many more descending phrases than ascending phrases.)

Tests

1. Combinations of Phrases. Nearly 50 percent of all phrases were classified as either ascending or descending.

Chi-Square Test

Conclusions