モジュール:Q-odd-limit intervals

---

このドキュメントは /doc から参照読み込みされたものです。
注意: このモジュールを直接呼び出さないでください。代わりに次の対応するテンプレートを使ってください。テンプレート:Q-odd-limit intervals

与えられた奇数リミットの音程の与えられた平均律による近似を計算し、テーブル形式にします。

---

bit32 = require("bit32")
utils = require("Module:Utils")
ET = require("Module:ET")
limits = require("Module:Limits")

local p = {}

local PRIME_LIST = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61}

local function table_filter(t, thres)
	for i = 1, #t do
		if t[i] > thres then
			return {unpack(t, 1, i - 1)}
		end
	end
	return t
end

local function inner_product(val, monzo)
	local result = 0
	for i = 1, #val do
		result = result + val[i] * monzo[i]
	end
	return result
end

local function monzo2ratio(monzo, subgroup)
	local num = 1
	local den = 1
	for i = 1, #subgroup do
		if monzo[i] > 0 then
			num = num * subgroup[i]^monzo[i]
		elseif monzo[i] < 0 then
			den = den * subgroup[i]^(-monzo[i])
		end
	end
	return {num = num, den = den}
end

local function ratio2monzo(ratio, subgroup)
	local monzo = {}
	for i = 1, #subgroup do
		monzo[i] = 0
		while ratio.num % subgroup[i] == 0 do
			monzo[i] = monzo[i] + 1
			ratio.num = ratio.num / subgroup[i]
		end while ratio.den % subgroup[i] == 0 do
			monzo[i] = monzo[i] - 1
			ratio.den = ratio.den / subgroup[i]
		end
	end
	return monzo
end

local function monzo2cent(monzo, subgroup)
	local jip = {}
	for i = 1, #subgroup do
		jip[i] = 1200 * utils._log(subgroup[i], 2)
	end
	return inner_product(jip, monzo)
end

local function ratio_8ve_reduction(ratio)
	local oct = math.floor(utils._log(ratio.num / ratio.den, 2))
	if oct > 0 then
		ratio.den = ratio.den * 2^oct
	elseif oct < 0 then
		ratio.num = ratio.num * 2^(-oct)
	end
	return ratio
end

local function odd_limit_monzo_list_gen(limit)
	local monzo_list = {}
	local subgroup = table_filter(PRIME_LIST, limit)
	for num = 1, limit, 2 do
		for den = 1, num, 2 do
			if utils._gcd(num, den) == 1 then
				ratio = ratio_8ve_reduction({num = num, den = den})
				table.insert(monzo_list, ratio2monzo(ratio, subgroup))
			end
		end
	end
	return monzo_list
end

local function find_error(val, subgroup, monzo_list)
	local step_size = 1200 / val[1]
	local true_size
	local approx_size
	local error_list = {}
	for i = 1, #monzo_list do
		ratio = monzo2ratio(monzo_list[i], subgroup)
		comp = {2 * ratio.den, ratio.num}
		true_size = monzo2cent(monzo_list[i], subgroup)
		approx_size = step_size * inner_product(val, monzo_list[i])
		nearest_size = math.floor(true_size / step_size + 0.5)
		error_abs_direct = math.abs(nearest_size * step_size - true_size)
		error_abs_val = math.abs(approx_size - true_size)
		error_rel_direct = 100 * error_abs_direct / step_size
		error_rel_val = 100 * error_abs_val / step_size
		error_list[i] = {
			ratio = ratio, 
			comp = comp, 
			error_abs_direct = error_abs_direct,
			error_abs_val = error_abs_val, 
			error_rel_direct = error_rel_direct,
			error_rel_val = error_rel_val
		}
	end
	table.sort(error_list, function(a, b) return a.error_abs_val < b.error_abs_val end)
	return error_list
end

local function approx(steps, subgroup, monzo_list, t_title, tag, approx_t)
	local t_body_direct = {}
	local errlist_direct = {}
	local t_body_val = {}
	local errlist_val = {}
	local val = {}
	local bgclr

	for i = 1, #subgroup do
		val[i] = utils._round_dec(steps*utils._log(subgroup[i], 2))
	end
	error_list = find_error(val, subgroup, monzo_list)
	for i = 1, #error_list do
		ratiocomp = string.format("%d/%d, %d/%d", error_list[i].ratio.num, error_list[i].ratio.den, 2 * error_list[i].ratio.den, error_list[i].ratio.num)
		eav = error_list[i].error_abs_val
		erv = error_list[i].error_rel_val
		ead = error_list[i].error_abs_direct
		erd = error_list[i].error_rel_direct
		error_abs_val = string.format("%.3f", eav)
		error_rel_val = string.format("%.1f", erv)
		error_abs_direct = string.format("%.3f", ead)
		error_rel_direct = string.format("%.1f", erd)
		if bit32.band(error_list[i].ratio.den, error_list[i].ratio.den - 1) == 0 and utils.table_contains(subgroup, error_list[i].ratio.num) then -- check power of 2 for den and prime for num
			ratiocomp = "'''" .. ratiocomp .. "'''"
			error_abs_val = "'''" .. error_abs_val .. "'''"
			error_rel_val = "'''" .. error_rel_val .. "'''"
			error_abs_direct = "'''" .. error_abs_direct .. "'''"
			error_rel_direct = "'''" .. error_rel_direct .. "'''"
			bgclr = ""
		end
		if error_list[i].error_rel_val > 50 then
			ratiocomp = "''" .. ratiocomp .. "''"
			error_abs_val = "''" .. error_abs_val .. "''"
			error_rel_val = "''" .. error_rel_val .. "''"
			error_abs_direct = "''" .. error_abs_direct .. "''"
			error_rel_direct = "''" .. error_rel_direct .. "''"
			bgclr = " style=\"background-color: #c0c0c0;\""
		else
			bgclr = ""
		end
		
		errlist_direct[i] = {
			bgclr = bgclr,
			ratiocomp = ratiocomp,
			ead = ead,
			erd = erd,
			error_abs_direct = error_abs_direct,
			error_rel_direct = error_rel_direct,
		}
		
		errlist_val[i] = {
			bgclr = bgclr,
			ratiocomp = ratiocomp,
			eav = eav,
			erv = erv,
			error_abs_val = error_abs_val,
			error_rel_val = error_rel_val,
		}
	end
	
	table.sort(errlist_direct, function(a, b) return a.ead < b.ead end)
	
	for i = 1, #error_list do
		t_body_direct[i] = "|-" .. errlist_direct[i].bgclr .. string.format("\n| %s || %s || %s\n", errlist_direct[i].ratiocomp, errlist_direct[i].error_abs_direct, errlist_direct[i].error_rel_direct)
		t_body_val[i] = "|-" .. errlist_val[i].bgclr .. string.format("\n| %s || %s || %s\n", errlist_val[i].ratiocomp, errlist_val[i].error_abs_val, errlist_val[i].error_rel_val)
	end
	
	local t_head_0 = "{| class=\"wikitable center-all mw-collapsible mw-collapsed sortable\"\n"
		.. "|+ style=\"font-size: 105%; white-space: nowrap;\" | "
	local t_head_1 = "|-\n"
		.. "! class=\"unsortable\" | 音程と補音程 !! 誤差 (絶対, [[セント|¢]]) !! 誤差 (相対, [[相対セント|%]])\n"
	local t_foot = "\n|}"
	
	if tag ~= "none" then
		tag_direct = "（直接近似, 一貫性の無いものも含む）"
		tag_val = "（パテントヴァルによるマッピング）"
	else
		tag_direct = ""
		tag_val = ""
	end
	
	if approx_t == "direct" then
		local out_str_direct = t_head_0 .. t_title .. tag_direct .. "\n" .. t_head_1 .. table.concat(t_body_direct, "\n") .. t_foot
		return out_str_direct
	elseif approx_t == "val" then
		local out_str_val = t_head_0 .. t_title .. tag_val .. "\n" .. t_head_1 .. table.concat(t_body_val, "\n") .. t_foot
		return out_str_val
	end
end

-- local function prec_by_equal (steps)
-- 	return math.floor (math.log (steps*1.9)/math.log (10))
-- end

function p.q_odd_limit_intervals(frame)
	local steps = tonumber(frame.args["steps"])
	local limit = math.max(tonumber(frame.args["limit"]), 2)
	local constcy = limits.consistency_limit(ET.parse(steps .. "平均律"), false, 43)
-- 	local prec = tonumber (frame.args['prec']) or prec_by_equal (steps)
	local note = frame.args["note"]
	local title = frame.args["title"]
	local header = string.lower(frame.args["header"]:gsub("%s+", ""))
	local apx = string.lower(frame.args["apx"]:gsub("%s+", ""))
	local tag
	
	if not utils.value_provided(title) then
		title = string.format(steps .. "平均律内の" .. "%d奇数リミット音程", limit)
	end
	
	local subgroup = table_filter(PRIME_LIST, limit)
	local monzo_list = odd_limit_monzo_list_gen(limit)
	local out_str
	if header ~= "none" then
		if constcy >= limit then
			apx = "val"
		end
		out_str = "以下の表は、"
			.. steps .. "平均律で" .. limit .. "奇数リミット音程がどのように表されるかを示している。素数倍音は'''太字で'''"
			.. ((constcy >= limit) and {"示す。\n\n" .. steps .. "平均律は" .. limit .. "奇数リミットにおいて[[一貫性|一貫的]]であるため、直接近似とパテントヴァルを介したマッピングが一致する。"} or {"、非[[一貫性|一貫的]]な音程は''斜体''で示す。"})[1]
			.. "\n\n"
		-- out_str = "The following "
		--	.. string.gsub("table show", "()", {[((apx == "direct" or apx == "val") and {11} or {6})[1]] = "s"})
		--	.. " how [[" .. limit .. "-odd-limit]] intervals are represented in "
		--	.. steps .. "edo. Prime harmonics are in '''bold'''"
		--	.. ((constcy >= limit) and {".\n\nAs " .. steps .. "edo is consistent in the " .. limit .. "-odd-limit, the mappings by direct approximation and through the patent val are identical."} or {"; inconsistent intervals are in ''italics''."})[1]
		--	.. "\n\n"
		if constcy >= limit then
			tag = "none"
		end
		if note ~= "" then
			out_str = out_str .. note .. "\n\n"
		end
		if steps == 12 then
			-- out_str = out_str .. "Note that, since the [[cent]] was defined in terms of 12edo, the absolute and relative errors for 12edo are identical.\n\n"
			out_str = out_str .. "[[セント]]は12平均律をもとに定義されているので、12平均律の絶対誤差と相対誤差は同じであることに注意。\n\n"
		end
	else
		tag = string.lower(frame.args["tag"]:gsub("%s+", ""))
		out_str = ""
	end
	
	if utils.value_provided(apx) then
		out_str = out_str .. approx(steps, subgroup, monzo_list, title, tag, apx)
	else
		out_str = out_str .. approx(steps, subgroup, monzo_list, title, tag, "direct") .. "\n\n" .. approx(steps, subgroup, monzo_list, title, tag, "val")
	end
	return out_str
end

return p