Proportion Sequences of Babylonian Medieties

In this chapter (Proportion Series of Babylonian Medieties) we consider the process of perpetual averaging – formations and in this way obtain whole families of babylonian medieties chains as nested iterations. This can be done both in the interior of the interval [a, b] and in its exterior. It leads us to the following three types of iterated mean-value-proportion chains: the families of iterated babylonian medieties as nested iterations of babylonian proportion chains in the interior of the interval [a, b], the families of ‘contra-medieties’ as a bipartite series of iterated proportion chains belonging to the exponential division parameter series (a/b)n the families of the third and higher proportions as interpolation of progressing mean proportions into the exterior of the interval [a, b]. Hence, chains of proportions of arbitrarily great length are created. For these, too, a harmonia perfecta maxima infinita can be developed, which describes the symmetries running to infinity and thus provides an inner order for the processes of boundless – and partly microtonal – interval generations.