Module:Limits

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This module depends on the following other modules: Module: ET Module: Rational

Todo: add documentation

---

local rat = require('Module:Rational')
local p = {}

-- multiply the ratio by a power of `equave` so that it lies within [1; equave)
local function canonical(a, equave)
	equave = equave or 2
	if type(a) == 'number' then
		a = rat.new(a)
	end
	local b = rat.copy(a)
	while rat.lt(b, 1) do
		b = rat.mul(b, equave)
	end
	while rat.geq(b, equave) do
		b = rat.div(b, equave)
	end
	return b
end

function p.limit_modulo_equave(q, equave)
	equave = equave or 2
	local ratios = {}
	for n = 1, q, 2 do
		for m = 1, q, 2 do
			local a = rat.new(n, m)
			a = canonical(a, equave)
			local key = rat.as_ratio(a)
			ratios[key] = a
		end
	end
	return ratios
end

function p.additively_consistent(equave, size, ratios, distinct)
	distinct = distinct or false
	local function approximate(a)
		return math.floor(size * math.log(rat.as_float(a)) / math.log(rat.as_float(equave)) + 0.5)
	end
	if distinct then
		local approx_set = {}
		for a_key, a in pairs(ratios) do
			local a_approx = approximate(a)
			if approx_set[a_approx] then
				return false
			end
			approx_set[a_approx] = true
		end
	end
	for a_key, a in pairs(ratios) do
		local a_approx = approximate(a)
		for b_key, b in pairs(ratios) do
			local b_approx = approximate(b)
			
			local c = rat.mul(a, b)
			local c_approx = approximate(c)
			
			c = canonical(c, equave)
			local c_key = rat.as_ratio(c)
			if ratios[c_key] then
				if c_approx ~= a_approx + b_approx then
					return false
				end
			end
		end
	end
	return true
end

function p.consistency_limit(size, equave, distinct)
	equave = equave or 2
	distinct = distinct or false
	local n = 1
	local last_n = 1
	while true do
		if rat.is_int(rat.div(n, equave)) then
			n = n + 1
		else
			local ratios = p.limit_modulo_equave(n, equave)
			local consistent = p.additively_consistent(equave, size, ratios, distinct)
			if not consistent then
				return last_n
			end
			last_n = n
			n = n + 1
		end
	end
	return n
end

return p