Abstract:

Mathematics and music share, as known, deep connections. Some of these are well known: for instance the construction of the Pythagorean scale in terms of simple ratios between whole numbers; or the tempered scale, in which the height of the succession of the twelve semitones in the range of an octave is a geometric progression whose reason is the twelfth root of two. Other points of agreement between the two arts/disciplines are less known, probably because the process of teaching/learning requires interdisciplinary skills (mathematical and musical) whose coexistence is not particularly widespread. Required skills Include the broad class of symmetries found both in the formal structure of Western music (scales, modes, tones, etc..) or in specific musical compositions (examples include the various works of Bach). These symmetries (such as uniform translations of a given interval - or transpositions - inversion of the temporal and/or harmonic order of a succession of semitones) have a deep correlation with a fundamental field of mathematics: the theory of groups.
In this context, the present work proposes a multimedia tutorial, realized mainly in Java language, designed to illustrate the main types of symmetry commonly noticeable in the Western musical paradigm. The multimedia nature of the tutorial is that it enables the learner to explore different examples of the proposed symmetries, by various sensory points of view, integrating the traditional representation (staff), the representation by means of sound’s spectrogram and the direct perception of sequences of sounds used .
Particular emphasis is given out in the tutorial to a scheme well-known to musicians: the circle of fifths. This argument makes it possible to highlight a direct relationship between musical symmetry and mathematical properties, as the group underlying the circle of fifths is the cyclic group Z12. This correlation is interactively illustrated in the tutorial either by using the metaphor of the analog clock with twelve hours on the face or by projecting the points of a constant pitch helix on a plane perpendicular to its axis.
Finally, it should be clear that multimedia can be used at different levels, since the hyper textual path presents links that lead to sections devoted to deepen the more technical mathematical aspects.

Keywords:

music, multimedia tutorial, circle of fifths.