Module:Harmonics in equal

---

Documentation transcluded from /doc
Note: Do not invoke this module directly; use the corresponding template instead: Template:Harmonics in equal.

Calculates approximations for harmonics in equal-step tunings and present them in form of a table.

---

local p = {}
local utils = require("Module:Utils")

-- direct mapping
local function map(p, steps, num, denom)
	local s = math.log(p) / math.log(num/denom)
	return math.floor(s * steps + 0.5)
end
-- check consistency for 9, 15 and 21
local function check_consistency(steps, num, denom)
	local p3 = map(3, steps, num, denom)
	local p5 = map(5, steps, num, denom)
	local p7 = map(7, steps, num, denom)
	local p9 = map(9, steps, num, denom)
	local p15 = map(15, steps, num, denom)
	local p21 = map(21, steps, num, denom)
	
	return (p9 == 2 * p3) and (p15 == p3 + p5) and (p21 == p3 + p7)
end

local function approx(steps, num, denom, intervals, title, prec, reduction, collapsed)
	local thead
	if collapsed then
		thead = "{| class=\"wikitable center-all mw-collapsible mw-collapsed\"\n"
	else
		thead = "{| class=\"wikitable center-all\"\n"
	end
	local tpri = {"! colspan=\"2\" | Harmonic "}
	local tabs = {"! rowspan=\"2\" | Error \n! Absolute ([[cent|¢]]) "}
	local trel = {"! [[Relative interval error|Relative]] ([[relative cent|%]]) "}
	local tdeg
	if reduction then
		tdeg = {"! colspan=\"2\" | Steps<br>([[octave reduction|reduced]])"}
	else
		tdeg = {"! colspan=\"2\" | Step"}
	end
	local fmt_abs = string.format(" %%+.%df", prec)
	local fmt_rel = " %+.1f"
	local equave = math.log(num/denom) / math.log(2)
	for _, p in pairs(intervals) do
		s = math.log(p) / math.log(num/denom)
		v = s * steps
		ev = math.floor(v + 0.5)
		table.insert(tpri, "" .. p)
		table.insert(tabs, "" .. string.gsub(string.format(fmt_abs, 1200 * equave * (ev - v ) / steps), "%-", "-"))
		table.insert(trel, "" .. string.gsub(string.format(fmt_rel, 100 * (ev - v)), "%-", "-"))
		if reduction then
			table.insert(tdeg, "" .. ev .. "<br />(".. ev % steps .. ")")
		else
			table.insert(tdeg, "" .. ev)
		end
	end
	local titleMarkup = ""
	if title then
		titleMarkup = "|-\n|+ style=\"font-size: 105%; white-space: nowrap;\" | " .. title .. "\n"
	end
	
	thead = thead ..
		titleMarkup ..
		"|-\n" ..
		table.concat(tpri, "\n!") .. "\n" ..
		"|-\n" ..
		table.concat(tabs, "\n|") .. "\n" ..
		"|-\n" ..
		table.concat(trel, "\n|") .. "\n" ..
		"|-\n" ..
		table.concat(tdeg, "\n|") .. "\n" ..
		"|}"
	
	return thead
end

local intervals = {}

intervals.prime = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229 }
intervals.odd = {3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127 }
intervals.integer = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128 }

-- evaluate input on error use default
local function eval_num_arg(input, def_value)
	local result = input
	if type(input) ~= "number" then
		result = def_value
		if type(input) == "string" then
			input = input:match("^%s*(.-)%s*$")
			if string.len(input) > 0 then
				result = tonumber(input)
			end
		end
	end
	return result
end

-- calculate default precision
local function prec_by_equal(steps, num, denom)
	return math.floor(10 * math.log(steps * 1.9 * math.log(2) / math.log(num / denom)) / math.log(10)) / 10
end

function p.harmonics_in_equal (frame)
	-- optional number of steps, default: 12
	local steps = eval_num_arg(frame.args["steps"], 12)
	-- numerator, default: 2
	local num = eval_num_arg(frame.args["num"], 2)
	-- denominator, default: 1
	local denom = eval_num_arg(frame.args["denom"], 1)
	-- optional number of columns, default: 11
	local columns = eval_num_arg(frame.args["columns"], 11)
	-- optional start column, default: start with prime 2
	local start = eval_num_arg(frame.args["start"], 1)
	-- option intervals
	local select_intervals = "integer"

	local name = steps .. "ed" .. num .. "/" .. denom
	if denom == 1 then
		if num == 2 then
			name = steps .. "edo"
			select_intervals = "odd"
			if check_consistency(steps, num, denom) then
				-- select_intervals = "prime_no2"
				select_intervals = "prime"
			end
		elseif num == 3 then
			name = steps .. "edt"
		else
			name = steps .. "ed" .. num
		end
	end
	
	if num == 3 and denom == 2 then
		name = steps .. "edf"
	end
	
	-- override default intervals
	if frame.args["intervals"] and string.len(frame.args["intervals"]) > 0 then
		select_intervals = frame.args["intervals"]
	end
	
	if select_intervals == "odd" then
		title_intervals = "odd harmonics"
	elseif select_intervals == "integer" then
		title_intervals = "harmonics"
	else
		title_intervals = "prime harmonics"
	end
	
	local title = frame.args["title"]
	if title == nil or string.len(title) == 0 then
		title = "Approximation of ".. title_intervals .. " in " .. name
	end
	-- optional precision for abs error, default about 3 digits
	local prec = eval_num_arg(frame.args["prec"], prec_by_equal(steps, num, denom))
	local reduction = true
	if steps == 1 then
		reduction = false	
	end
	local collapsed = utils.value_provided(frame.args["collapsed"])
	
	return approx(steps, num, denom, {unpack(intervals[select_intervals], start, start + columns - 1)}, title, prec, reduction, collapsed)
end

return p