Module:Harmonics in equal

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

Do not use this module directly, use the template instead: Template:Harmonics in equal.

---

Documentation transcluded from /doc

---

local p = {}

-- direct mapping
local function map(p, steps, num, denom)
	local s = math.log(p) / math.log(num/denom)
	return math.floor(s*steps + .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 = ' %+.0f'
	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 + .5)
		table.insert(tpri, '' .. p)
		table.insert(tabs, string.format(fmt_abs, 1200 * equave * (ev - v ) / steps))
		table.insert(trel, 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=white-space:nowrap| ' .. title .. '\n'
	end
	
	return 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' ..
		'|}'
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 }
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 }
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}

-- 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(math.log(steps*1.9*math.log(2)/math.log(num/denom))/math.log(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
	
	title_intervals = "prime harmonics"
	if select_intervals == "odd" then
		title_intervals = "odd harmonics"
	elseif select_intervals == "integer" then
		title_intervals = "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 = eval_num_arg(frame.args['collapsed'], nil)
	
	return approx( steps, num, denom, {unpack(intervals[select_intervals], start, start+columns-1)}, title, prec, reduction, collapsed)
end

return p;