Melodic Chains

Hofmann-Engl [4,5] has proposed a much more complex model of melody that incorporates more information than a limited system such as the Parsons Code. In fact, this scheme is arguably more precise than traditional music notation.

In this system, each note in a melody is represented by three elements: one that represents pitch, one that represents duration, and one that represents loudness. Hofmann-Engl defines these in terms of cognitive measurements, or intersubjective values, and replaces the terms pitch, duration and loudness with his own terms meloton, chronoton, and dynamon, as defined by his cognitive hypothesis. However, for the purposes of the present discussion, we will continue use the more conventional terms.

A sequence of notes is then represented as a melodic chain, and each melodic chain can be thought of as incorporating a pitch chain, duration chain, and loudness chain. Each chain includes a single parameter, the atomic beat, that specifies the time unit represented by each atomic element in the chain.

Rather than attempt to represent the full melodic chain, it is normally represented as one of its consituent components. In this text I will use Cp for the pitch chain, Cd for the duration (or rhythmic) chain, and Cl for the loudness chain:

$\displaystyle C_p = [p_1,p_1,p_1;p_2;p_3,p_3,p_3;p_4, \dots ] (1/16)$

$\displaystyle C_d = [d_1,d_1,d_1;d_2;d_3,d_3,d_3;d_4, \dots ] (1/16)$

$\displaystyle C_l = [l_1,l_1,l_1;l_2;l_3,l_3,l_3;l_4, \dots ] (1/16)$

Semicolons separate articulated notes, while a comma separates all other entries. Rests are defined as having a pitch and loudness of zero.

In this representation, N atomic elements are stored for each note of duration N atomic beats, so that all three types of chains carry the full information about duration. In this sense each note represented by a rhythmic chain can be thought of as having both its a width and a height equal to the length of the note.

Note also that Hofmann-Engl defines the pitch to be the logarithm of the perceived fundamental frequency of a note. This will be important when we consider similarity measures using this model.

While equating loudness information with pitch and duration information is appealing, Western music notation does not allow for as detailed a specification of loudness as Hofmann-Engl's model implies. This forces him to estimate what he considers reasonable values for the loudness chains in his examples. As a result it seems likely that similarity measurements cannot apply the same importance to loudness information as they do to pitch and duration.