The powers of 3 in binary

I recently discovered some interesting properties of the powers of 3
rendered in binary, as well as other powers of odd numbers. Does
anyone here have an idea if there is a predictable pattern to the
digits, eventually? And where should i look for more information about
such things?

11
1001
11011
1010001
11110011
1011011001
100010001011
1100110100001
100110011100011
1110011010101001
101011001111111011
10000001101111110001
110000101001111010011
10010001111101101111001
110110101111001001101011
10100100001101011101000001
111101100101000010111000011
10111000101111001000101001001
1000101010001101011001111011011
11001111110101000001101110010001

The powers of 5:

101
11001
1111101
1001110001
110000110101
11110100001001
10011000100101101
1011111010111100001
111011100110101100101
100101010000001011111001
10111010010000111011011101
1110100011010100101001010001
1001000110000100111001110010101

The powers of 7:

111
110001
101010111
100101100001
100000110100111
11100101110010001
11001001000011110111
10101111111011011000001
10011001111011111101000111
10000110101100011101011110001
1110101110110111001110010010111

/Ö

"A New Kind of Science" by Stephen Wolfram covers these pretty well.
There is a pattern, but it's hard to find. It has to do with taking
the numbers 2 or more at a time and treating them as coordinates on a
(hyper) torus, the points will clump together in certain dimensions.

Check out http://planetmath.org/ and http://mathworld.wolfram.com/

On 1/5/07, Mats Öljare <oljare@...> wrote:
> I recently discovered some interesting properties of the powers of 3
> rendered in binary, as well as other powers of odd numbers. Does
> anyone here have an idea if there is a predictable pattern to the
> digits, eventually? And where should i look for more information about
> such things?
>
> 11
> 1001
> 11011
> 1010001
> 11110011
> 1011011001
> 100010001011
> 1100110100001
> 100110011100011
> 1110011010101001
> 101011001111111011
> 10000001101111110001
> 110000101001111010011
> 10010001111101101111001
> 110110101111001001101011
> 10100100001101011101000001
> 111101100101000010111000011
> 10111000101111001000101001001
> 1000101010001101011001111011011
> 11001111110101000001101110010001
>
> The powers of 5:
>
> 101
> 11001
> 1111101
> 1001110001
> 110000110101
> 11110100001001
> 10011000100101101
> 1011111010111100001
> 111011100110101100101
> 100101010000001011111001
> 10111010010000111011011101
> 1110100011010100101001010001
> 1001000110000100111001110010101
>
> The powers of 7:
>
> 111
> 110001
> 101010111
> 100101100001
> 100000110100111
> 11100101110010001
> 11001001000011110111
> 10101111111011011000001
> 10011001111011111101000111
> 10000110101100011101011110001
> 1110101110110111001110010010111
>
> /Ö
>
>
>
> Meta Tuning meta-info:
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--
--TRISTAN
Dreaming of Eden is a Comic with no Pictures
http://dreamingofeden.smackjeeves.com

> "A New Kind of Science" by Stephen Wolfram covers these pretty well.
> There is a pattern, but it's hard to find. It has to do with taking
> the numbers 2 or more at a time and treating them as coordinates on a
> (hyper) torus, the points will clump together in certain dimensions.

Heh, i'm not so surprised that he had done something on it, as he also
did a lot of work on one dimensional cellular automatas which were
discussed here earlier. I really wonder what would happen if you
rendered it to 10,000 digits or more and played it back as a audio
waveform...

--- In metatuning@yahoogroups.com, Mats Öljare <oljare@...> wrote:
>
> > "A New Kind of Science" by Stephen Wolfram covers these pretty well.
> > There is a pattern, but it's hard to find. It has to do with taking
> > the numbers 2 or more at a time and treating them as coordinates on a
> > (hyper) torus, the points will clump together in certain dimensions.
>
> Heh, i'm not so surprised that he had done something on it, as he also
> did a lot of work on one dimensional cellular automatas which were
> discussed here earlier. I really wonder what would happen if you
> rendered it to 10,000 digits or more and played it back as a audio
> waveform...

That depends on the bit length of a single sample, and whether you are
doing stereo or not....mono would be better for hearing pattern I
would think. i've tried stuff like that before (using the golden
string, related to phi), and it's basically going to give you noise.

OTOH, it might be more interesting to hear a really fast tempo
modulation, where 1 is an upper pitch and 0 is a lower, and think of
it as a square wave LFO that goes up and down at about 16hz or
so---could we hear some rhythmic structures at high speed emerge that
wouldn't be apparant at the local level?

-A.

Both Robert Walker and I independently and a bit together have worked with Quasi periodic patterns based on Penrose Tiles.
The unfortunate thing is that while they never repeat as soon as you stop they can be considered a sample of one of a larger period patterns. So you have to run them forever to really get that they might repeat in sections then vary but never really repeat as a whole. The periodic patterns resemble the rings of the horograms and one can extend them as long as possible/desirable Both David Canwright and I have articles in Xenharmonikon on such rhythmic patterns. One can easily change long and short to high and low.

Aaron Krister Johnson wrote:
> --- In metatuning@yahoogroups.com, Mats �ljare <oljare@...> wrote:
> >>> "A New Kind of Science" by Stephen Wolfram covers these pretty well.
>>> There is a pattern, but it's hard to find. It has to do with taking
>>> the numbers 2 or more at a time and treating them as coordinates on a
>>> (hyper) torus, the points will clump together in certain dimensions.
>>> >> Heh, i'm not so surprised that he had done something on it, as he also
>> did a lot of work on one dimensional cellular automatas which were
>> discussed here earlier. I really wonder what would happen if you
>> rendered it to 10,000 digits or more and played it back as a audio
>> waveform...
>> >
> That depends on the bit length of a single sample, and whether you are
> doing stereo or not....mono would be better for hearing pattern I
> would think. i've tried stuff like that before (using the golden
> string, related to phi), and it's basically going to give you noise.
>
> OTOH, it might be more interesting to hear a really fast tempo
> modulation, where 1 is an upper pitch and 0 is a lower, and think of
> it as a square wave LFO that goes up and down at about 16hz or
> so---could we hear some rhythmic structures at high speed emerge that
> wouldn't be apparant at the local level?
>
> -A.
>
>
>
>
> Meta Tuning meta-info:
>
> To unsubscribe, send an email to:
> metatuning-unsubscribe@yahoogroups.com
>
> Web page is http://groups.yahoo.com/groups/metatuning/
>
> To post to the list, send to
> metatuning@yahoogroups.com
>
> You don't have to be a member to post.
>
> > Yahoo! Groups Links
>
>
>
>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

As a disclaimer i should add that i find such way for thinking myself now pretty foreign.
While not in the case of those on these list, I do find too much music 'scientific sounding' .
i prefer to music with a fair amount of dirt or at least have a fair amount of dust around.
i will pass on mud although i would love to capture what it feels like drying on my hands.

Kraig Grady wrote:
> Both Robert Walker and I independently and a bit together have worked > with Quasi periodic patterns based on Penrose Tiles.
> The unfortunate thing is that while they never repeat as soon as you > stop they can be considered a sample of one of a larger period patterns. > So you have to run them forever to really get that they might repeat in > sections then vary but never really repeat as a whole. The periodic > patterns resemble the rings of the horograms and one can extend them as > long as possible/desirable Both David Canwright and I have articles in > Xenharmonikon on such rhythmic patterns. One can easily change long and > short to high and low.
>
> Aaron Krister Johnson wrote:
> >> --- In metatuning@yahoogroups.com, Mats �ljare <oljare@...> wrote:
>> >> >>>> "A New Kind of Science" by Stephen Wolfram covers these pretty well.
>>>> There is a pattern, but it's hard to find. It has to do with taking
>>>> the numbers 2 or more at a time and treating them as coordinates on a
>>>> (hyper) torus, the points will clump together in certain dimensions.
>>>> >>>> >>> Heh, i'm not so surprised that he had done something on it, as he also
>>> did a lot of work on one dimensional cellular automatas which were
>>> discussed here earlier. I really wonder what would happen if you
>>> rendered it to 10,000 digits or more and played it back as a audio
>>> waveform...
>>> >>> >> That depends on the bit length of a single sample, and whether you are
>> doing stereo or not....mono would be better for hearing pattern I
>> would think. i've tried stuff like that before (using the golden
>> string, related to phi), and it's basically going to give you noise.
>>
>> OTOH, it might be more interesting to hear a really fast tempo
>> modulation, where 1 is an upper pitch and 0 is a lower, and think of
>> it as a square wave LFO that goes up and down at about 16hz or
>> so---could we hear some rhythmic structures at high speed emerge that
>> wouldn't be apparant at the local level?
>>
>> -A.
>>
>>
>>
>>
>> Meta Tuning meta-info:
>>
>> To unsubscribe, send an email to:
>> metatuning-unsubscribe@yahoogroups.com
>>
>> Web page is http://groups.yahoo.com/groups/metatuning/
>>
>> To post to the list, send to
>> metatuning@yahoogroups.com
>>
>> You don't have to be a member to post.
>>
>> >> Yahoo! Groups Links
>>
>>
>>
>>
>>
>> >> >
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

But does anyone have an idea what kind of free software i could use to
calculate that many digits?

On 19/01/07, Mats Öljare <oljare@...> wrote:
>
> But does anyone have an idea what kind of free software i could use to
> calculate that many digits?

Python, bc, Java, what's the problem?

Graham

--- In metatuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> On 19/01/07, Mats Öljare <oljare@...> wrote:
> >
> > But does anyone have an idea what kind of free software i could use to
> > calculate that many digits?
>
> Python, bc, Java, what's the problem?
>
>
> Graham
>

Well, how would i go about with making that calculation?

On 19/01/07, Mats Öljare <oljare@...> wrote:
>
> --- In metatuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
> >
> > On 19/01/07, Mats Öljare <oljare@...> wrote:
> > >
> > > But does anyone have an idea what kind of free software i could use to
> > > calculate that many digits?
> >
> > Python, bc, Java, what's the problem?
> >
>
> Well, how would i go about with making that calculation?

With bc, and a suitable Unix environment (might even work in Windows):

echo 'obase=2;3^150'|bc
11010110011011101000101011100110001100101000010001110101101000011001\
00101000110011000110110100010110101110011100010001010011000000000111\
11111110111101110011110110000011001100000101101110011010010010111110\
1101100011110001101110001000011001

is the binary representation of three to the power 150. Note the end
of line formatting that you might have to strip out.

In Python it's more complicated because there's no function to convert
a number to binary (that I can find). But this does the trick:

>>> def bin(n):
... assert n >= 0, "only works for positive numbers"
... i, result = n, []
... while i:
... result.append("01"[i&1])
... i = i >> 1
... result.reverse()
... result = ''.join(result)
... assert int(result, 2)==n
... return result
...
>>> print bin(3**150)
1101011001101110100010101110011000110010100001000111010110100001100100101000110011000110110100010110101110011100010001010011000000000111111111101111011100111101100000110011000001011011100110100100101111101101100011110001101110001000011001
>>>

In my tests, the Python's orders of magnitude faster. I don't know
why but it may be significant if you want audio length samples.

Graham