rationalizing the natural scale

Here again is the table of the frequencies of the steps in the natural scale and the gaps between them, both expressed as binary logarithms:

Here's how that looks as a 1-dimensional graph:

1        2          3    4        5         6         7    1    steps in log(2)
0    .1    .2    .3    .4    .5    .6    .7    .8    .9    1.0  log(2)

• note that the gaps can be put into two categories:
  • 1. ~ 0.16
  • 2. 0.093

• note that the category 1 gaps are roughly half the size of those in category 2
  • this suggests that we could insert a new step in the category 1 gaps, halfway between the two frequencies in each gap, thereby creating 12 roughly equidistant steps

    0    .1    .2    .3    .4    .5    .6    .7    .8    .9    1.0  scale in log(2)
    1   .    2    .     3    4   .    5    .    6    .    7    1  steps in log(2)
    

    (. = new step)

  • 7 of these 12 steps have the same pitch as a natural scale step

• note that the gap unit is one 12-step scale step

• note the pattern of the gaps: 2, 2, 1, 2, 2, 2, 1