The grouping of intervals is synonymous with music notation and the manner in which these groups are read and interpreted is an integral part of music making. Notation grouping calls for an effective procedure for solving this particular mathematical problem in a finite number of steps. It may be that this procedure, identified as an algorithm, can also be an effective practice procedure for interpretation, rhythm and technical accuracy.

The concept of grouping seems a necessity given the continuous flow of sounds that make up a piece of music. Multiples of sounds become intervals from which the full gambit of music is in place. Combinations of groups in the form of intervals leads to shape within the flow of music as compared to a single sound.

Shape is a fundamental part of music and perhaps requires more attention than it receives in the preparation of music for performance. Algorithmic procedure places a high value on the mathematical content and also complements the vital aspect of rhythm. There may be an opportunity to research the benefits of algorithmic practice as an alternative to the traditional practice methods in use at this time.

Rhythmic variation and interpretation is dependent on the use of shapes in groups, beginning with two sounds onward. The more extended the group, the more need to manage the flow of intervals. These groups are represented numerically, beginning from a group of 2 notes. Groups with an even number of notes, such as 2, 4, 6, 8, and 10 represent multiples of 2s and do not present difficulty in management. However, the odd numbers pose a slightly different approach to accommodate the extra note. In the case of the 3 group, it may be read as a double and one (2+1) or one plus two, bearing in mind that there is no detachment of the extra note from the rhythmic flow of the group. By and large, groups can be read in combinations of twos and ones. The four group can be 2 + 2; the 5-group, 4 + 1; 6-group, 2 + 2+ 2; 7-group, 2 + 2 + 2 + 1; 8-group, 2 + 2 + 2 + 2; and the 9-group, 2 + 2 + 2 + 2 + 1. And so on for even larger groups.

The groupings need to be complete in their mathematical representation to create and master the shapes. Finally we still have a technical task presented by the intervals comprising the group. Now you have the challenge of intervals and shape fully exposed for you to use your algorithmic practice. Each group needs to be mentally embedded in a flowing manner so that the group becomes an integral part of the piece.

Grouping of intervals will reliably inform you as to which interval or intervals remain to be mastered, step by step, before the passages are safe for performance. Algorithmic practice has the effect of elimination of rote practice by working step by step through the intervals bearing in mind some intervals are more difficult than others, which is well demonstrated in contemporary compositions. However, groups are usually a combination of both varieties of harder and easier intervals which inexorably leads to technical roadblocks. Based on present practice methodology, if all the intervals in the groups are given the same amount of practice time, then the easier intervals will get easier which in turn makes the hard intervals more difficult. Knowing where the difficult intervals are is extremely facilitative to a sound technique.

In this approach, the close relationship between shape, rhythm and interpretation is brought to attention. The feature of shape introduces the potential for interpretation and nuance in performance. Groupings come in many shapes and sizes which need to fit their allocated space in the composition. Every note can be a ‘pearl’ leading to flowing and accurate groups. Subjective interpretations are then appropriate for extension to the fluency and accuracy of performance at your level of skill.

The methodology outlined in this paper describes the solving of particular problems through the use of algorithms. Groups of any number are practiced in a necessary set of logical and finite steps allowing for a conclusion of accurate and fluent technical freedom in performance. Should it prove to be a vexed question in the minds of some aspiring music makers, then it may be a teaching aid.

The acceptance of the notion that intervals are the basis of music making may well lead to the logical conclusion that the practice of grouping intervals as such, is axiomatic.

2016