Module:No-1s limits
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Documentation for this module may be created at Module:No-1s limits/doc
local rat = require('Module:Rational')
local ET = require('Module:ET')
local p = {}
-- compute all positive ratios n/m with n and m <= q modulo powers of equave
-- previous: already computed ratios for q-1
function p.limit_modulo_equave(q, equave, previous)
equave = equave or 2
local ratios = {}
if rat.eq(rat.modulo_mul(rat.new(q, 1), equave), 1) then return ratios end
if previous then
for n = 2, q do
if not rat.eq(rat.modulo_mul(rat.new(n, 1), equave), 1) then
local a = rat.new(n, q)
a = rat.modulo_mul(a, equave)
local a_key = rat.as_ratio(a)
local b = rat.new(q, n)
b = rat.modulo_mul(b, equave)
local b_key = rat.as_ratio(b)
if previous[a_key] == nil then
ratios[a_key] = a
end
if previous[b_key] == nil then
ratios[b_key] = b
end
end
end
else
for n = 2, q do
if not rat.eq(rat.modulo_mul(rat.new(n, 1), equave), 1) then
for m = 2, q do
if not rat.eq(rat.modulo_mul(rat.new(n, 1), equave), 1) then
local a = rat.new(n, m)
a = rat.modulo_mul(a, equave)
local key = rat.as_ratio(a)
ratios[key] = a
end
end
end
end
end
return ratios
end
-- compute q-integer limit
-- if a function `norm` and a number `max_norm` are provided, the output will be additionally restricted
function p.integer_limit(q, norm, max_norm)
local check_norm = type(norm) == 'function' and type(max_norm) == 'number'
local ratios = {}
for n = 1, q do
for m = 1, q do
local a = rat.new(n, m)
if not check_norm or norm(a) <= max_norm then
local key = rat.as_ratio(a)
ratios[key] = a
end
end
end
return ratios
end
-- check additive consistency for a set of ratios (modulo powers of equave):
-- approx(a*b) = approx(a) + approx(b) forall a, b: a, b, ab in ratios
-- `distinct`: whether distinct ratios are required to be mapped to distinct approximations
-- `previous`: already computed ratios for the previous iteraton
function p.additively_consistent(et, ratios, distinct, previous)
distinct = distinct or false
previous = previous or {}
if distinct then
local approx_set = {}
for a_key, a in pairs(previous) do
local a_approx = ET.approximate(et, rat.as_float(a)) % et.size
if approx_set[a_approx] then
if not rat.eq(rat.modulo_mul(rat.div(a, approx_set[a_approx]), et.equave), 1) then
mw.log(a_key .. ' -> ' .. a_approx .. ': conflict!')
return false
end
end
approx_set[a_approx] = a
mw.log(a_key .. ' -> ' .. a_approx)
end
for a_key, a in pairs(ratios) do
local a_approx = ET.approximate(et, rat.as_float(a)) % et.size
if approx_set[a_approx] then
if not rat.eq(rat.modulo_mul(rat.div(a, approx_set[a_approx]), et.equave), 1) then
mw.log(a_key .. ' -> ' .. a_approx .. ': conflict!')
return false
end
end
approx_set[a_approx] = a
mw.log(a_key .. ' -> ' .. a_approx)
end
end
if type(distinct) == 'number' then
return true
end
local previous_ordered = {}
for a_key, a in pairs(previous) do
table.insert(previous_ordered, a)
end
local ratios_ordered = {}
for a_key, a in pairs(ratios) do
table.insert(ratios_ordered, a)
end
for i, a in ipairs(ratios_ordered) do
local a_approx = ET.approximate(et, rat.as_float(a))
for j, b in ipairs(previous_ordered) do
local b_approx = ET.approximate(et, rat.as_float(b))
local c = rat.mul(a, b)
local c_approx = ET.approximate(et, rat.as_float(c))
c = rat.modulo_mul(c, et.equave)
local c_key = rat.as_ratio(c)
if previous[c_key] or ratios[c_key] then
if c_approx ~= a_approx + b_approx then
mw.log('a = ' .. rat.as_ratio(a) .. '; b = ' .. rat.as_ratio(b) .. '; ab = ' .. c_key)
mw.log(a_approx .. ' + ' .. b_approx .. ' != ' .. c_approx)
return false
end
end
end
for j, b in ipairs(ratios_ordered) do
if i <= j then
local b_approx = ET.approximate(et, rat.as_float(b))
local c = rat.mul(a, b)
local c_approx = ET.approximate(et, rat.as_float(c))
c = rat.modulo_mul(c, et.equave)
local c_key = rat.as_ratio(c)
if previous[c_key] or ratios[c_key] then
if c_approx ~= a_approx + b_approx then
mw.log('a = ' .. rat.as_ratio(a) .. '; b = ' .. rat.as_ratio(b) .. '; ab = ' .. c_key)
mw.log(a_approx .. ' + ' .. b_approx .. ' != ' .. c_approx)
return false
end
end
end
end
end
return true
end
-- find additive consistency limit
-- returns nil when at least `max_n`
-- `distinct`: whether distinct ratios are required to be mapped to distinct approximations
-- - if an integer, it is the regular consistency limit already known
function p.consistency_limit(et, distinct, max_n)
if et.size == 0 then
-- the answer is known already
return '∞'
end
max_n = max_n or 1/0
distinct = distinct or false
local n = 1
local last_n = 1
local previous = {}
while true do
if type(distinct) == 'number' and n > distinct then
return last_n
end
local ratios = p.limit_modulo_equave(n, et.equave, previous)
for key, ratio in pairs(ratios) do
mw.log('step ' .. n .. ': ' .. key)
end
if next(ratios) ~= nil then
local consistent = p.additively_consistent(et, ratios, distinct, previous)
if not consistent then
mw.log('Not consistent at step ' .. n .. ', returning ' .. last_n)
return last_n
end
for key, ratio in pairs(ratios) do
previous[key] = ratio
end
last_n = n
end
n = n + 1
if n > max_n then
return nil
end
end
end
-- find vals generate no-1s consistency intervals
-- returns string with wart notation
-- limit must meet no-1s odd-limit consistency
function p.vals(et, limit)
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43}
local wart = 'abcdefghijklmn'
local patent_val = {}
local wart_dir = {}
local minprime
local val = {}
local rets = {}
for i = 1, #primes do
patent_val[i] = ET.approximate(et, primes[i])
if ET.cents(et, patent_val[i]) < math.log(primes[i]) / math.log(2) * 1200 then
wart_dir[i] = -1
else
wart_dir[i] = 1
end
end
if et.equave == 2 then
minprime = 3
else
minprime = 2
end
local previ = 1
for i = 1, #primes do
local ip = primes[i]
if ip > limit then break end
if ip == et.equave then
val[i] = patent_val[i]
elseif ip == minprime then
val[i] = patent_val[i]
previ = i
else
val[i] = val[previ] + ET.approximate(et, ip / primes[previ])
previ = i
end
end
rets[1] = ''
if et.equave ~= 2 then
local i = table.find(primes, et.equave)
if i then
rets[1] = rets[1] .. string.sub(wart, i, i)
else
rets[1] = rets[1] .. 'q'
end
end
rets[1] = rets[1] .. et.size
if limit < minprime * minprime then
local nonpatent = false
local binded = false
local bias = 0
for i = 1, #val do
local ip = primes[i]
if ip == et.equave then
elseif ip == minprime then
bias = wart_dir[i]
else
if val[i] ~= patent_val[i] then
nonpatent = true
elseif wart_dir[i] ~= bias then
binded = true
end
end
end
if nonpatent or not binded then
rets[2] = '' .. rets[1]
for i = 1, #val do
local ip = primes[i]
local difference = (val[i] - patent_val[i]) * wart_dir[i]
if difference > 0 then
rets[1] = rets[1] .. string.rep(string.sub(wart, i, i), difference * 2)
elseif difference < 0 then
rets[1] = rets[1] .. string.rep(string.sub(wart, i, i), -difference * 2 - 1)
end
end
for i = 1, #val do
local ip = primes[i]
if ip ~= et.equave then val[i] = val[i] - bias end
local difference = (val[i] - patent_val[i]) * wart_dir[i]
if difference > 0 then
rets[2] = rets[2] .. string.rep(string.sub(wart, i, i), difference * 2)
elseif difference < 0 then
rets[2] = rets[2] .. string.rep(string.sub(wart, i, i), -difference * 2 - 1)
end
end
if bias > 0 then rets = {rets[2], rets[1]} end
end
else
for i = 1, #val do
local ip = primes[i]
if ip * minprime <= limit then
if val[i] ~= patent_val[i] then
return '(error)'
end
end
local difference = (val[i] - patent_val[i]) * wart_dir[i]
if difference > 0 then
rets[1] = rets[1] .. string.rep(string.sub(wart, i, i), difference * 2)
elseif difference < 0 then
rets[1] = rets[1] .. string.rep(string.sub(wart, i, i), -difference * 2 - 1)
end
end
end
return table.concat (rets, ", ")
end
function p.for_small_edos (frame)
local lines = tonumber (frame.args['lines'])
local t_body = {}
local val = {}
for i = 1, lines do
local et = ET.parse('' .. i .. 'edo')
local limit = p.consistency_limit(et, false, 43)
local vals = p.vals(et, limit)
t_body[i] = string.format ("|-\n| %s\n| %s\n| %s", i, limit, vals)
end
return "{| class=\"wikitable center-all mw-collapsible mw-collapsed\"\n" ..
"|-\n" ..
"! EDO\n" ..
"! No-1s consistency limit\n" ..
"! Associated vals\n" ..
table.concat (t_body, "\n") ..
"\n|}"
end
return p