Button to move back towards 1/1

[davidryan59]

At the moment, the SPACE bar may be tuned to a large fraction such as 2^10 3^-7 5^2 7^-1 11^4 13^-3 and it is very difficult to know how to get back to 1/1.

There is musical value in having a button that takes you on a predictable step back towards the origin 1/1. It needs to be deterministic (no random selection of prime powers) so that you can get the same tune every time.

Probably the best way to do it is to calculate the largest prime power factor (in the example above, these are 2^10, 3^7, 5^2, 7^1, 11^4, 13^3) and make a suitable (2, 3, p) fraction mapping the p-exponent 1 step closer to the origin. After a finite number of key presses, you will arrive back at the origin 1/1. Since after each step the highest prime power may change, the interval sequence might be pleasant and interesting.

Examples of suitable steps might be:
3/4, 8/9, 9/8, 4/3
4/5, 5/6, 6/5, 5/4
6/7, 7/8, 8/7, 7/6
11/12, 12/11
12/13, 13/12
...

Suggest that:

• 4/3 is the largest step encountered
• The smallest step encountered should be somewhere between 13/12 and 24/23
• 2/1 can be made from 4/3, 4/3, 9/8
• 3/2 can be made from 4/3, 9/8
• It may be useful to compare each prime p against 1/1, 9/8, 4/3, 3/2, 16/9, 2/1
• There ought to be a function on Peo which returns the prime p that is the largest prime factor. (This might be a callback, perhaps a peo.reduce(...) function.)

[davidryan59]

peo.reduce is now available in Peo, and also through ji-rcn packages.