User:Akselai/RES comma

Recursive equal step (RES) commas are a sequence of commas {a₁, a₂, ...} generated by an initial interval a₀ and a set of intervals {b₁, b₂, ..., b_k}. They are constructed by repeatedly comparing stacks of an interval with each of the intervals, and finding the smallest difference between them.

Mathematical Definition

By convention, all intervals are in fraction form (instead of monzo form). Suppose S is a finite set of intervals {b_k} and a is an interval (generically a^m ∉ S for all m ∈ Z). Then for all i > 0,

a₀ = a

a_i+1 = min_{m ∈ Z, 1 ≤ j ≤ k} ‖a_i^m/b_k‖

where ‖x‖ = exp|log(x)| denotes the multiplicative absolute value, which turns an interval into an ascending interval.

We can write a_n as RES_n(a; S).

Worked out example

We start with a = 5/4, S = {3/2, 2/1}. The closest power of a to 3/2 is (5/4)², and the closest power of a to 2/1 is (5/4)³. Since (2/1)/(5/4)³ < (5/4)²/(3/2), we have a₁ = (2/1)/(5/4)³ = 128/125, the augmented comma.

With the value of a₁, we can find a₂. The closest power of a₁ to 3/2 is (128/125)¹⁷, and the closest power of a₁ to 2/1 is (128/125)²⁹. Since (128/125)¹⁷/(3/2) < (2/1)/(128/125)²⁹, we have a₂ = (3/2)/(128/125)¹⁷ = [-120 1 51⟩.

We can repeat this process to get a₃, a₄ and so forth.

The complexity of the comma after 2 steps is already very large. However, [-120 1 51⟩ is not small compared to other commas with a similar complexity; it is larger than 3.9 cents, which is roughly the size of [24 -21 4⟩, the vulture comma.

More examples

a = 5/4, S = {3/2, 2/1}

n	RESn(a; S)	Size (cents)
0	5/4	386.31
1	128/125	41.059
2	[-120 1 51⟩	3.9544
3	[36361 -303 -15453⟩	1.8144
4	[14071708 -117262 -5980311⟩	0.21180

a = 7/5, S = {6/5, 3/2, 7/4, 2/1}

n	RESn(a; S)	Size (cents)
0	7/5	582.51
1	50/49	34.976
2	[-8 1 -19 18⟩	0.86075
3	[11153 -1394 26486 -25092⟩	0.10909

In general, the more elements S contains, the smaller the comma after a fixed number of steps. The monzos coordinates appear to grow at the rate of factorial (or x^x). However, the rate of diminishing of RES commas is an unsolved problem. Nevertheless, we have an absolute upper bound of 2^-na₀, because the distance from an interval to two equal steps cannot exceed half the distance between the two equal steps.

Miscellaneous

It is possible for an RES comma to be 1/1. Consider a modified version of the above example. We have a = 5/4, but this time S = {[-120 1 51⟩, 3/2, 2/1}. We get a₁ = 128/125 and a₂ = [-120 1 51⟩ again. But in the case of a₃, since a₂ is already in the set of intervals, no other interval in the set is going to be closer than a power of a₂, so a₃ = a₂/a₂ = 1/1. Since all the powers of 1/1 are 1/1, the closest interval is just the smallest interval in the set, so a₄ = [-120 1 51⟩, a₅ = 1/1, and so on.