Medieties - Mean Value Proportion Chains

In this chapter (Medieties - Mean Value Proportion Chains) the Babylonian (then the most important other historical medieties) are presented first and they are described primarily by means of the proportion chain representations. Afterwards, formalisms are developed between the mean magnitudes and the corresponding division parameters, which lead, especially in the hyperbola of Archytas, to remarkable connections between mean value proportions and geometry. In the first place one gains the symmetries of Babylonian-chains of proportions as a result of this from which the ‘Harmonia perfecta maxima’ has emerged. Now the focus of this chapter will be the generalization of the Harmonia perfecta from the historical canon 6 – 8 – 9 – 12 to the Harmonia perfecta abstracta (diatonica) as a universal musical symmetry principle of the diatonic and its abstract generalizations. The special role of the geometric mean will be highlighted in this context.