Derler 1: ein System zur Klassifikation von Rhythmen
The system for classifying musical rhythms, named after its originator, Rudolf Derler, was developed between 1978–80. It is based on a combination of two numeric sizes, pulse number and impulse number, of which the first approximates the function of a bar, while the second approximates the function of accents or beats in a bar. The combination of both numeric sets takes place by means of permutation. The permutation of the Derler System begins thus, that all impulses within a pulse number are brought next to one another at the beginning. A remaining set of pulses exists after the last impulse, until the end of the pulse number. This remaining set is used to carry out the Derler permutation: with each cycle of the pulse number-impulse number combination, the respective last impulse of a pulse value is deferred outwards into the remaining set. This procedure is repeated as often as necessary, until the end of the pulse number is reached. The same procedure is carried on in a likewise fashion with the next inner impulse. The procedure lasts as long as it takes all impulses to be placed directly next to one another, at the opposite end of the pulse number. On the way from the beginning to the end of this process, the permutation runs through all rhythms which are capable of being produced in the context of their pulse number- impulse number combination. The number of rhythms produced is called the rhythmic amount.
The rhythms of the Derler system are limited by two rules of redundancy: all rhythmical figures which are similar to one another in transposition and transformation are not listed. The actual rhythmic amount is thereby reduced to PI-permutations of distinct rhythmic figures only; these are designated as equivalent classes. The most important equivalent class is at the end of a PI-permutation: it is called the final rhythm. Determined by the numerical and proportional relation of the two basic values – pulse number and impulse number, the final rhythms of their permutations have a very definite character. Certain PI combinations create rhythmic figures as a final rhythm, which are identical to the meters of European music. Other PI combinations create rhythmic figures which correspond to the Timelines in African music.
The article develops the vocabulary of the Derler system and demonstrates its procedures and results with selected examples. The factual segment is left at the above designated place, and the article moves on to a discussion of African Timelines. It begins with their history of discovery, history of notation, history of nomenclature and history of interpretation. In connection with the presentation of their complementary nature, the ties to the Derler system are produced again. It is clear therefore, that the Derler system offers a possibility for mathematically determining those rhythmic functional formulas which, in the meantime, are termed as timelines in African musicological research.
This theme is expanded upon to include music with uneven pulse numbers (bars) in the next section of the article. In this area likewise, equivalent classes become recognizable, as to final rhythms with timeline character.
They appear as either classical or optional versions in all music with uneven bars. On the basis of this observation, it is then attempted to systematize altogether the rhythmic movement behaviour of contemporary music cultures. Three fundamental behavioural forms are identified and named: rhythmic symmetry, rhythmic asymmetry and Aksak rhythm.
In closing, it is attempted for the first time on hand of a surveyed sample, to sketch the distribution of these behaviour models worldwide, as well as furnishing historical proof for the discovered rhythm system on hand of classical, middle ages and modern theorists of European and non-European origin. In doing so, especially the existence of asymmetrical and Aksak rhythm is referred to in the realm of old European music.