Katabatak Math-An Exploration In Pure Number Theory

[Turtle]

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Edited September 21, 2013 by Turtle

[Tormod]

I'm in for the ride...except it's midnight here in Norway so I'll catch up later. Maybe this will be the first official Hypography composition?

[maddog]

I'm in for the ride...except it's midnight here in Norway so I'll catch up later. Maybe this will be the first official Hypography composition?

 

I curious to know what Fractals would sound like... ;)

 

Maybe try a symphony as a combination of the two... ;)

 

Maddog

[Tormod]

Before I ask you to fill in the rest of the squares table for K(n), (I see some of you already have)

 

Gulp, how did you see that? :)

[Tormod]

Okay. I also got 4 (or rather, 121, which I then assume is 1+2+1 = 4).

 

Now please take a step back and point out to me why this is important. Remember I am a tourist who just stopped by for the ride with little mathematical insight.

[Tormod]

Aaargh. My brain is boiling. I need to reread this a couple of times...

[pgrmdave]

This is fun!

[maddog]

This is starting to look like numeralogy... :)

 

Maddog

[Tormod]

I'll go through it again.

 

Maddog - this is a fun exercise and I don't care what this is called - if it can be turned into a piece of music in the end then that's a worthwhile timekiller for me (plus I might actually learn something new). :hyper:

[Tormod]

I am printing out your graphs and trying to see how this can be turned into something I have in my mind at the moment. Don't expect miracles though...but I think this lends itself well to electronic music.

[Tormod]

I'll grab my flashlight and see what strange creatures I might find. :(

[Tormod]

You crack me up Turtle. How the heck am I going to find some order in this.

 

Oh never mind. I'll find my own way. Any specific wishes for instruments you'd like in my piece? :hihi:

[Tormod]

I have been dabbling a bit with the drawings and thoughts here with the aim of finding some basis for a musical interpretation.

 

The first problem I'm having is that the regular western scale has 13 semitones, so using a base 10 means I lose some notes. I therefore interpret this using only the notes of a major scale.

 

Starting at the root key of C and using the n^2 (1 4 9 7) gives me C F D B. These notes are quite allright but do not provide a lot of stability (no fifth, for example).

 

Now, this is easily remedied if I am allowed to use the basic triad chords based on those notes, because I'd get the following:

 

C major (1 3 5)

F major (4 6 1)

d minor (2 4 6)

B diminished minor (7 2 4)

 

which means I add the 5 and 6 to the series. But that may be cheating?

 

Chordwise this gives me a progression of I - IV - ii - vii(dim), or C - F - d - bdim.

 

The base notes (1 4 9 7) also allows me to twist this a bit. For example, I can lean on the F chord as a base, giving me a sequence of I - vi - iv(dim) - V (F - d - bdim - C), or a lydian progression. This might be more promising since the V chord naturally falls back to F. This still sounds a bit weird to me but I'll play around with it and see how I can work it out.

 

Among other and more severe problems is - what role does the Sierpinsky Sponge play in the Katabatak function?

 

And my limited mathematical insight means that I simply choke on stuff like this:

"Observe the first, K(n^(6m-4))" (from post 29). Can you simplify?

[Tormod]

Okay, Siepinski sponge disregarded. It is very cool, though. :eek:

 

I may have confused some when I wrote that I added the 5th and 6th notes yet I also list a 2nd - however, with musical notation a 9 is simply the second note above the octave (so in the key of C the note D is both 2 and 9). The 9 is usually only used in seventh chords, ie chords where more than the fundamental triad is used.

 

Using this same logic (ie, counting in thirds above the root, as in 1 3 5 7 9 11 13), the 5th does not have a higher number (there is no "12th" note in the triad system), but the 6th can also be seen as the 13th note. This note is not a member of the k(n^2) function, though, and k(13)=4 which we already have as the F note. So the 3rd and 6th notes in the Ionian (ie, standard major scale) are not allowed.

 

However, in the F Lydian scale (which is simply a C Major scale played from F to F) these notes (ie, E and A) would constitute a 7 and a 3, respectively, and as such are not really important notes.

 

This whole post may appear to be a lot of hogwash but I am simply trying to set some limitations on what I can and cannot do, so that I can build a framework for a composition. In music liberties must always be taken to create some interesting twists but math and music are closely related.

[Tormod]

Ah...you just gave me an excellent idea with that chart. I'll be back...

[Qfwfq]

Here's an old trick, explainable by Number Theory, in fact by what Turtle has been taking all through in this thread. Especially in the days before pocket calculators, it used to quite often spread amongst students as a way of checking arithmetic. Add two numbers:

 

2719853

3524876

_______

6244729

 

Then, consider the Ks of the two addends: K(2719853) = 8 and K(3524176) = 8 and the K of their sum: K(8 + 8) = 7.

 

Now, roll of the drum, compare it with the K of the sum: K(6244729) = _... :naughty: Coincidence? Try the same with multiplication.

 

Of course, wrong arithmetic might chance on passing this test; given three random numbers the odds are 1 in 9, not a huge probability but a relevant one.

[Qfwfq]

To avoid the cave tourists a couple of possible but not so obvious pitfalls:

 

If you want to check a division, regard it as a multiplcation instead:

 

a / b = c <==> c * b = a

 

If you want to check a raised to the power b, don't take the K of the exponent! Raise K(a) to the power b, not to the power K(:naughty:.

 

I'll let Turtle continue according to his plans...