Pythagoras’ Comma

In Problem 1 (Six Notes all Nice Ratios ) we explored a six point scale : shortest length to double that, with four lengths in between.

In this problem the scale now has 12 points : shortest to double, with ten lengths in between.

We are going to look for a closed and complete scale. The question is this : If we work in fractions, where 1 is the lowest note and 1/2 is the highest, what would the other notes need to be so that intervals between any pair of notes were fractions from elsewhere in the set? For example if the set included the fractions 2/3, 3/4 and 8/9, then the lengths 8/9 and 3/4 themselves make an interval of 2/3 , and 2/3 is in the set.

Here's one way people tried to find a suitable set : Using an understanding that 1:2 and 2:3 were good ratios, they started with a length and then kept reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range, while still in theory connecting to an agreeable note.

After shifting down twelve times, which required seven doublings, they were almost back at the start note - but not quite.

So in fact they'd failed to get a closed set. So the question is:

How much were they off by?

This discrepancy is called "Pythagoras Comma".

So we've got a 12 point scale of sorts but it lacks the beauty of forming a closed and complete set within which notes relate to each other as a sequence of 1:2 and 2:3 shifts.