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Tune Smithy Melodies for other sequences in the OEIS

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Van Eck's sequence (A181391 in the OEIS) as a melody with Tune Smithy

The sequence is here: A181391

Played with Tune Smithy, http://tunesmithy.com

Neil Sloane describes this sequence at the end of this video:


and the start of this video:

HOW THE SEQUENCE IS DEFINED

You look for the last number in the sequence so far. If it is the first occurrence of that number, set the next number to 0. Otherwise, set the next number in the sequence to the number of terms since it's last occurrence.

Example:

0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, when you get to this point, the last number is 5, and it repeats another 5 which was 4 terms earlier.. So the next number in the sequence is 4. 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4

But now this is the first 4 in the sequence so far, so the next number is 0. 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, The last 0 was 5 terms earlier, so the next number is 5 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, The last 5 was 3 terms earlier so the next number is 3 0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3 and so it goes on.

ABOUT THE TUNING

The tuning is a pentatonic scale.

Because the numbers go so high in this sequence eventually, I set the scale to "wrap around". I did it so that the pitches go:

1/1 5/4 3/2 5/3 2/1 1/1 ...

That's how you get the octave above the 1/1 as well as the 1/1 both played, and why you get the same note played by numbers that are 6 apart so 0, 6, 12, ... play the same note, the 1/1, and 1, 7, 13, ... play the 5/4, and so on. 5, 11, 17, ... play the 2/1.

Gijswijt's sequence (A090822 in the OEIS) as a melody with Tune Smithy

The sequence is here: A090822. The first 5 occurs at 10^(10^23) - that's the number consisting of 1 followed by 10 to the power 23 zeroes - so you are never going to see a 5 in practice, if you play it all the way from the start, no matter how fast you play it and how long you wait.

Played with Tune Smithy

HOW THE SEQUENCE IS DEFINED

The nth number in the sequence gives the number of repeating blocks in the sequence so far, where you are allowed to ignore the beginning of the sequence, as much of it as you like, but it must then repeat exactly from then on.

So for instance, in:

1, 1, 2, 1, 1, 2, 2, 2, 3,

the first few 2s and the 3 are found like this:
(1 ; 1 ) 2
1 1 2 (1 ; 1) 2
(1, 1, 2 : 1, 1, 2), 2
1, 1, 2, 1, 1, (2 ; 2) 2
1, 1, 2, 1, 1,( 2 ; 2 ; 2), 3,
where the repeating blocks are shown in brackets separated by colons.

This is the first "interesting" 3:
1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, (2, 2, 2, 3 : 2, 2, 2, 3 : 2, 2, 2, 3), 3,

ABOUT THE TUNING

It's just playing the notes of a major chord, so far but in a harmonic series based "just intonation" tuning.

3/2 2/1 5/2 6/2

with the 1/1 set to C (never plays the 1/1 because there is no 0 in the sequence).

If it ever got to a 5 then it would play a 7/2 because it is actually set to play a harmonic series (starting with the second harmonic as the 1/1 because the result is more interesting that way), but we know it is never going to get there in a practical time span :).

You may feel as though you hear a "4" quietly in the gaps between the other notes sometimes before you hear it for real, It's not played at all, no echo. Instead, you are hearing a harmonic of one of the other notes - it's an octave above the 3/2.


More about the number 10^(10^2)

It's a number consisting of 1 followed by 100,000,000,000,000,000,000,000 zeroes.

You could write this number out even on the surface of the Earth as a decimal expansion though the individual digits would have to be much smaller than millimeter sized. I make it that each zero would be about 40 microns across if you inscribed this number on the available land surface of the Earth (ignoring oceans) :). In other words, the digits would be about a 25th of a millimeter in height, a little taller as they are narrower than they are high but smaller than a 10th of a millimeter surely in an easily readable font.

So, it's not like a googolplex, too large to even write out within the observable universe in decimal even using one atom for each digit - but it is pretty vast :).

For mathematicians, though it is vast, yet this isn't regarded as a remarkably enormous number nowadays. Far far larger numbers have been used in maths, some too large to write out within the observable universe even using exponentiation, such as Grahams number.

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