Just a couple of *notes* before you send me protest e-mails. Firstly, this is not a historical presentation. The aim is to shine light where there was darkness in a field which remains mysterious to many. So some discoveries which took centuries to refine are here presented as faits accomplis without apology to any historical figures who may feel left out. Secondly there are no octaves in this discussion. (An octave is what you get when the frequency of a note is doubled.) Which means that when we write fractional frequency ratios, all powers of 2 (4, 64, 4096, etc) are completely interchangeable without prejudice to the argument. Finally, except where germane to the issue, I will use a default set of 12 notenames: c db d eb e f gb g ab a bb b. The reason may become clear shortly.