Module:Temperament data

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Note: Do not invoke this module directly; use the corresponding template instead: Template:Temperament data.

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

Todo: add documentation

---

local rat = require('Module:Rational')
local p = {}
-- Utility / matrix functions
local function gcd(a,b)
  if type(a) == "number" and type(b) == "number" and 
        a == math.floor(a) and b == math.floor(b) then
    if b == 0 then
      return a
    else
      return gcd(b, a % b) -- tail recursion
    end
  else
    error("Invalid argument to gcd (" .. tostring(a) .. "," .. 
          tostring(b) .. ")", 2)
  end
end


local function matadd(a, b)
	local result = {}
	for i = 1, #a  do
		result[i] = {}
		for j = 1, #(b[1]) do
			result[i][j] = a[i][j] + b[i][j]
		end
	end
	return result
end

local function matsub(a, b)
	local result = {}
	for i = 1, #a  do
		result[i] = {}
		for j = 1, #(b[1]) do
			result[i][j] = a[i][j] - b[i][j]
		end
	end
	return result
end

local function matmul(a, b)
	local result = {}
	for i = 1, #a  do
		result[i] = {}
		for j = 1, #(b[1]) do
			result[i][j] = 0
			for k = 1, #(a[1]) do
				result[i][j] = result[i][j] + (a[i][k] * b[k][j])
			end
		end
	end
	return result
end


local function scalarmatmul(a, b)
	local result = {}
	for i = 1, #a  do
		result[i] = {}
		for j = 1, #(a[1]) do
			result[i][j] = a[i][j] * b
		end
	end
	return result
end

local function matinv(a)
	dbl_identity = {}
	for i = 1, #a do
		dbl_identity[i] = {}
		for j = 1, #a do
			if i == j then
				dbl_identity[i][j] = 2
			else
				dbl_identity[i][j] = 0
			end
		end
	end

	xn = scalarmatmul(a, 0.000001)
	
	for i = 1, 30 do
		xn = matmul(xn, matsub(dbl_identity, matmul(a, xn)))
	end
	return xn
end

local function transpose(a)
	local result = {}
	for i = 1, #a[1] do
		result[i] = {}
		for j = 1, #a do
			result[i][j] = a[j][i]
		end
	end
	return result
end

local function pseudoinv(a)
	return matmul(transpose(a), matinv(matmul(a, transpose(a))))
end


-- Actual temperament-related functions start here
-- Generator list (e.g. 2/1, 3/2 for meantone) is needed so you can input irregular mappings like {{5,8,12},{7,11,16}} for meantone
-- and still have the outputted generators be ~2/1 and ~3/2
-- (this will be important later so people using the template can just input an ET list instead of having to figure out the mapping)
-- Generators are passed as monzos in the specified subgroup here

function p.get_te_generator(subgroup, mapping, gens)
	local w = {}
	for i = 1, #subgroup do
		w[i] = {}
		for j = 1, #subgroup do
			if i == j then
				w[i][j] = math.log(2)/math.log(subgroup[i])
			else
				w[i][j] = 0
			end
		end
	end
	
	local jw = {{}}
	for i = 1, #subgroup do
		jw[1][i] = 1
	end
	local vw = matmul(mapping, w)
	local g = matmul(jw, pseudoinv(vw))
	
	local mapping_cols = #mapping[1]
	local tempered_subgroup = {}
	
	for i = 1, #subgroup do
		tempered_subgroup[i] = 0
		for j = 1, #mapping do
			tempered_subgroup[i] = tempered_subgroup[i] + g[1][j] * mapping[j][i]
		end
	end
	
	return tempered_subgroup
end





return p