complete lists of monzismas

Hi there,

I've just extended the nlimits program to
be able to show the results using arbitrary
precision arithmetic - well up to a
fixed maximum actually, but set it to
40,000 digits.

So here is a search for monzismas
(following the usual convention of naming
a class of intervals by the name of one
of the first one found of the same size)

First a short list, less than one cent,
and up to quotient of 10^20:

command line:
(commas_etc.txt) 1 1 1e20 6 60 -2 0 1.0 2 3 5

Ratios differing from 1/1:
by a comma or diesis etc in range 0 to 1 cents
and in the form:
2^a * 3^b * 5^c

Max quotient to look for: 1e+020

[53 -10, -16> = 9007199254740992 : 9010162353515625 = -0.56943
[1 -27, 18> = 7629394531250 : 7625597484987 = 0.861826
[54 -37, 2> = 450359962737049600 : 450283905890997363 = 0.292396

(10773 values tested, 3 values found)

Then a longer list, up to 10^100, but requiring it to
be within 0.3 cents so at least as close as a monzisma:

command line:
(commas_etc.txt) 1 1 1e100 6 60 -2 0 0.3 2 3 5

Ratios differing from 1/1:
by a comma or diesis etc in range 0 to 0.3 cents
and in the form:
2^a * 3^b * 5^c

Max quotient to look for: 1e+100

[234 -7, -96> = 27606985387162255149739023449108101809804435888681546220650096895197184 :
27603820795493645094967205956375255748724839577334932982921600341796875 = 0.198463
[90 15, -49> = 17763086495282268024161967871623168 : 17763568394002504646778106689453125 = -0.0469664
[144 -22, -47> = 22300745198530623141535718272648361505980416 : 22297583945629639856633730232715606689453125 = 0.245429
[180 30, -98> = 315527241838879287681003245623571079738007668199398203029858994356224 :
315544362088404722164691426113114491869282574043609201908111572265625 = -0.0939328
[54 -37, 2> = 450359962737049600 : 450283905890997363 = 0.292396
[107 -47, -14> = 162259276829213363391578010288128 : 162285243890121480027996826171875 = -0.277035
[17 -62, 35> = 381469726562500000000000000000 : 381520424476945831628649898809 = -0.230068
[251 -69, -61> = 3618502788666131106986593281521497120414687020801267626233049500247285301248 :
3618568848776963314868646215645086282319820014663491747342050075531005859375 = -0.0316055
[73 77, -84> = 51704256436926231056548749215693807357721577836111615492096 :
51698788284564229679463043254372678347863256931304931640625 = 0.183102
[161 -84, -12> = 2923003274661805836407369665432566039311865085952 : 2922977339492680612451840826835216578535400390625 = 0.0153609
[163 92, -133> = 918427179263375612497045544745479384735430961095852508095104436308294151034903409650594480128 :
918354961579912115600575419704879435795832466228193376178712270530013483949005603790283203125 = 0.136136
[71 -99, 37> = 171798691840000000000000000000000000000000000000 : 171792506910670443678820376588540424234035840667 = 0.0623273
[305 -106, -59> = 65185151242703554760590262029100101153646988597309960020356494379340201592426774597868716032 :
65175332598511232702200746749593760591384089462170699225129766318787005729973316192626953125 = 0.26079
[19 114, -86> = 1292388115393055295535123767426518253869322213731913416835072 :
1292469707114105741986576081359316958696581423282623291015625 = -0.109294
[109 129, -135> = 22956801879201681976690597845282233120100417067004083019003286109860166873505729190113990148096 :
22958874039497802890014385492621985894895811655704834404467806763250337098725140094757080078125 = -0.15626
[268 -131, -26> = 474284397516047136454946754595585670566993857190463750305618264096412179005177856 :
474356090424868085430657218781319266517924834775101706304576914012432098388671875 = -0.261674
[178 -146, 23> = 4567192616659071619386515102238384436424789196800000000000000000000000 :
4567759074507740406477787437675267212178680251724974985372646979033929 = -0.214707
[88 -161, 72> = 65536000000000000000000000000000000000000000000000000000000000000000000000000 :
65542350158517637872691969508970705427701150314738255642438471845988797065603 = -0.167741
[322 -168, -24> = 8543948143683640329580086824678208458410818089426611079788166431288878903122562200091848347746304 :
8543796527187709452365478117234643266819859489103112524152573223541825961022005140781402587890625 = 0.0307219
[2 176, -121> = 3761844347944019374779739640524225271625916118636799838222299542960846288544841383684 :
3761581922631320025499956919111186169019729781670680068828005460090935230255126953125 = 0.120775
[232 -183, 25> = 2056880696651507552693711478196688131228419832041974829185761280000000000000000000000000 :
2056788397238392593160822648456620167261469637864670311548042275113842708261892026529227 = 0.0776883
[142 -198, 74> = 29514790517935282585600000000000000000000000000000000000000000000000000000000000000000000000000 :
29512665430652752148753480226197736314359272517043832886063884637676943433478020332709411004889 = 0.124655

(277305 values tested, 22 values found)

And here is the updated program:

http://www.robertinventor.com/nlimits.exe

C-code if interested, e.g. to run on another platform:
http://www.robertinventor.com/nlimits3.c

Uses nothing windows specific so should compile
easily in e.g. unix / linux.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> So here is a search for monzismas
> (following the usual convention of naming
> a class of intervals by the name of one
> of the first one found of the same size)

How are you generating cantidates for this search? It seems to me you
are testing too many.

If you take, for instance, 1171, 1783 and 2513 equal, and invert the
matrix of the corresponding 5-vals, you get commas |-90 -15 49>,
|-17 62 -35>, |54 -37 2>. Any 5-limit interval is a linear combination
with integer coefficients of these, and if you restrict the exponents
to a certain range you are looking at commas which are within that
range in terms of how many steps of 1171, 1783 and 2513 it takes to
get to them. If you look at the list you already have see what range
you could have searched and gotten these same results; pulling out a
few at random, |1 -27 18> maps to [1 1 2], [180 30 -98] to [-2 0 0]
(a square, so I wouldn't have counted it myself) |88 -161 72> to
[0 -2 1] and so forth. Searching 11^3 values, with exponents from
-5 to 5, will obviously be more than sufficient; a 5-limit comma would
need to either be relatively (compared to what we are looking at)
large or ultra complex not to fall into that range, but already
32805/32768 is small enough that this would catch it.

Hi Gene,

Thanks for the tip.

I just searched the entire lattice,
only restricted by the need for
the linear combination in the
logs to add up to the log of
the desired result.

Interesting to know that there
is a way to speed it up.
I will ask you to explain how
it works if I need to
do that. May be needed, you never know.

With these searches anyway right
now it seems plenty fast enough,
but you never know for the future...
I remember when working on xenharmonic
bridges it slowed down a bit at times.
You need to have many factors and a high quotient
for it to slow down. Even a million candidate
tests isn't really a great deal to do
nowadays. But a hundred million is
a fair number, and certainly a billion is.

On a modern p.c. multiplication is as fast as
addition and each candidate test
requires quite a feew of those,
not even particularly optimised
for numbers of additions and multiplications
- but I did optimise the logs by
finding al those in advance as
taking exp or log is very slow
even on a modern p.c. (compared
with addition and multiplication
that is).

Robert