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Since Modulo seems to have a significant role in the Collatz Conjecture for one reason or another, I wondered what would happen if the Conjecture was put to music (I was inspired by this interesting YouTube video). Shuffling about the internet, I came across this web tool called Piano Pencil Code Gym that would play any sequence of letters.

I wrote some not very impressive code (and for 3x+5 defaulted to using an Excel spreadsheet with the =Mod function) to translate the trajectories into mod 7 numbers and then lazily used Word's replace feature to convert those into letters. I then copied and pasted the whole thing into the web tool and listened to the result. After playing around with some 3x+1 and 3x+5 trajectories, the immediate thought came about:

Is it possible to find any music melody somewhere in any given 3x+b rule, where b is an odd integer?

The tunes can be repetitive (For $3x+1$, it seems to alternate between ...D B A D G G G B A whatever... and ...E F E F E F C C E F something...), but different rules such as 3x+5 may mix it up a bit. Since the scale is really basic, I suspected the possibility of every song melody appearing by coincidence was possible. I have no idea where to start with trying to answer this question other than playing some different Collatz Rules and hearing what happens.


Here are some sequences I already generated. Feel free to copy and paste them into the web tool.

Just as a heads up: I assigned A=1 and G=0. If you re-assign the mod numbers so C=1 for example, the music sounds different.

$3x+1$ trajectories of 1-39 (odd seeds only)

p=new Piano
p.lt 90

#1#
p.play "D3 B2 A2"

#3#
p.play "C C E B A D B A"

#5#
p.play "E B A D B A"

#7#
p.play "G A D F C C E F E F C E B A D B A"

#9#
p.play "B G G G A D F C C E F E F C E B A D B A"

#11#
p.play "D F C C E F E F C E B A D B A"

#13#
p.play "F E F C E B A D B A"

#15#
p.play "A D B G G A D F C E F C E B A D B A"

#17#
p.play "C C E F E F C E B A D B A"

#19#
p.play "E B A D B A D F C C E F E F C E B A D B A"

#21#
p.play "G A D B A D B A"

#23#
p.play "B G G A D F C E F C E B A D B A"

#25#
p.play "D F C E B A D B A D F C C E F E F C E B A D2 B2 A3"

#27#

p.play "F E F E F C C E B A D B G G A D B G G G A D F C E B A D B G G G A D F C C E B A D F C E F E F E F E F C C E B A D F C C E B A D B G G A D B A D B G G A D F C E F C C E F E F C C E F C E B A D B G G A D F C E F C E B A D B A"

#29#

p.play "A D B A D F C C E F E F C E B A D B A"

#31#

p.play "C C E B A D B G G A D B G G G A D F C E B A D B G G G A D F C C E B A D F C E F E F E F E F C C E B A D F C C E B A D B G G A D B A D B G G A D F C E F C C E F E F C C E F C E B A D B G G A D F C E F C E B A D B A"

#33#

p.play "E B A D F C E B A D B A D F C C E F E F C E B A D B A"

#35#

p.play "G A D F C E F C E B A D B A"

#37#

p.play "B G G G G G A D F C C E F E F C E B A D B A"

#39#

p.play "D F C C E B A D F C C E F C E B A D B A D F C C E F E F C E B A D2 B2 A3"

$230631$, A number with a really long Collatz trajectory. I found in the 3x+1 Completeness and Gamma Records Table from Eric Roosendaal's page. (http://www.ericr.nl/wondrous/comprecs.html)

It is a proud 442 steps long:

p.play "B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 C1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 G1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 F1/2 E1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 G1/2 A1/2 D1/2 B1/2 A1/2 D1/2 F1/2 C1/2 E1/2 F1/2 E1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 F1/2 C1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 E1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 F1/2 C1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 F1/2 C1/2 C1/2 E1/2 F1/2 E1/2 F1/2 C1/2 C1/2 E1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 G1/2 G1/2 A1/2 D1/2 F1/2 C1/2 E1/2 F1/2 C1/2 E1/2 B1/2 A1/2 D1/2 B1/2 A1/2"

Warning: This trajectory is 4 times longer than 27's. If you delete all of the 1/2's, you are in for 5 whole minutes of Collatz Music.


Here are some $3x+5$ trajectories (odd seeds only):

#1#

p.play "A A D B A"

#3#

p.play "C G G E F B A D C E F C G G E F C E"

#5#

p.play "E F C E"

#7#

p.play "G E F B A D C E F C G G E F C E"

#9#

p.play "B D B A D B A"


#some random numbers...#

#75#

p.play "E F C G G E F B A D B A D C E F C E F C E"

#12347#

p.play "F B A A D C E F C G G E F C E F C E F C E F C G G E F B A D B A D B D B D B D B D B D B D B A D C E F C G G G E F C E F C G G G G G G E F C G G G E F C G G E F B A A D C E F C E F B A D C E F C E F B A D C E F C G G E F C E"
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1 Answer 1

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It contains all binary strings as prefixes, (taking $1$ as $(3n + 1)/2$ and $0$ as $n/2$) I suppose, since if you take $n = m + 2 ^ k$ then $n$ has the exactly same prefix as $m$ and it repeats at least every $2 ^ k$ and the $n$ sequence has to be different from $m$ at some point.

[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1], [0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1], [1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1], [0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1], [1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0], [1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1], [0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1], [0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1]

That's for 1-50. Considering all the sequences as never ending (Keeps repeating 1-2-1-2)

If you see that every $k$ bit prefix repeat only and exactly after $2 ^ k$

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