The Chaos

In the beginning every scale had equal significance. However this led to random wandering between scales resulting in complete chaos.

The concept of a scale is inconsistently defined in music theory but a good starting point to consider might be "a series of notes differing in pitch according to a specific scheme (usually within an octave)" [1].

This informal definition leaves a lot to desire so a better model should be developed. If we model the spectrum of sounds using the strictly positive real line where each note is denoted by a distinct frequency value in Hertz then there are infinitely many differing pitches and an even greater size of infinity of sets of notes. Going on no further information this does not allow you to solve the problem of finding a scale to use for a given situation.

It is reasonable to modify the model such that only cyclic sets of notes are permitted. Where the cycle is defined by some continuous mapping from the real line onto some continuous cyclic set such as the unit circle in the complex plain. Because of the strong sound of doubling a note's frequency to get the octave a reasonable mapping might map any real number x to e^(2*pi*i*log₂(x/a)) where a is some fixed real number which will act as a reference frequency such as 440, which is a solution for f to the equation f(2x)=f(x).

The restriction of a scale to cyclic sets of notes does not eliminate the problem of having infinitely many scales to choose from. It is therefore reasonable to ensure that the number of notes in the cycle is finite. Along with the octave, another strong sound is the perfect fifth which corresponds to a scaling of the frequency by 3/2. What is remarkable is that an ascent of 12 perfect fifths is only slightly greater than an ascent of 7 octaves by a scale of 3^12/2^19, which is a canonical form of (3/2)^12/2^7. It is worth noting that a ratio of 2^(7/12), which is the solution for r of r^12=2^7, would be required to reach exactly seven octaves in 12 equal scalings which is irrational hence it was a reasonable assumption to use the real line rather than the rational numbers only in the initial model. Such adjustment of the seven perfect fifths to fit into an octave is the art of temperament which had a range of solutions an happens naturally in a performance with varying degrees of deliberation.

By assuming that a temperament can be selected for a particular performance we have constructed a model of a scale consisting of 12 possible notes in 2^12=4097 distinct combinations. Selecting a suitable scale without any further information is still an extremely daunting task.

Here are some thoughts worth considering:

• Is the set of useful scales a null set or is there enough space to measure the number of useful scales giving an evenly distributed random scale in a continuous model a chance of sounding good? Can this be answered by considering the intervals between the perfect and even temperaments along the 12 sequence ascent and then consideration the various concepts of a scale still to be defined?
  
• Is m=12 and n=19 the solution to inf|1-3^m/2^n| where m,n are strictly positive integers? Does this equation with 3 and 2 replaced by other prime numbers give rise to other good sounding sound cycles?