Pythagorasand#8217; Comma
Problem
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.
Getting Started
Hint : What fraction is 2/3 of 2/3 of 2/3 . . . twelve times ?
And you've had to double 7 times, so what's the fraction now?
Maybe work in decimals and give the error as a percentage.
Student Solutions
Here's how Julian from Wilson's School reasoned :
The first note in the scale is 1
The second note is 1 $\times$ 2/3 = 2/3
The third is 2/3 $\times$2/3 = 4/9, but that is less than 1/2, so we multiply it by 2, giving the answer 8/9
The fourth is 8/9 $\times$2/3 = 16/27
This process needs to be repeated 12 times in total.
That means that 1 will be multiplied by 2/3 12 times, so we can do the following calculation:
$(\frac{2}{3})^{12} = 0.0077073466$ (10dp)
We are also know that 7 doublings are required, so 0.0077073466... $\times$ 128 = 0.9865403685 (10dp)
Therefore, they were off by 1 - 0.9865403685... = 0.0134596315 (10dp)
Thanks Julian
Teachers' Resources
The topic of music can make a good connection between science and mathematics
The nature of sound and the working of the ear are rich areas of applied mathematics.
The ratio emphasis follows from harmonics or overtones and rests on ideas like lowest common multiple.
One teacher has offered the following comment : This Stage 4 work makes practice in multiplying and dividing fractions purposeful. And in school invites collaboration between the music and mathematics departments.