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Analysis of 1-Liner Music
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Bit Music | Analysis of 1-liner C music

Copyright(c) 2011 by Christopher Abad
20 GOTO 10 |

This project was inspired by two youtube videos:

	"Experimental one-line algorithmic music - the 2nd iteration"

	"Experimental music from very short C programs"

The musical generation of the 1-liner C program is of the form:

	unsigned char transform(unsigned long t);

	for(t = 0; t < (1 << N); t++) {

This also assumes tha output is used to feed a PCM audio device at 8000hz 8-bit mono.

This process can alternatively be described in many ways. It would first be useful to look at the loop iterator (t) in a specific
novel way.  What at first appears to be a simple monotonically increasing sequence may prove to be more useful when written as
a set of binary vectors {b_n} each representing a single-bit slice of (t) over the entire sequence of iterations.

	  t  =  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15 ... 2^N-1
	b_0  |  0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1 ...   1
	b_1  |  0   0   1   1   0   0   1   1   0   0   1   1   0   0   1   1 ...   1
	b_2  |  0   0   0   0   1   1   1   1   0   0   0   0   1   1   1   1 ...   1
	b_n  |  0 x (2^n) ...   1 x (2^n) ...                                       1

Taken independantly, each bit as an indepedant sequence is a square wave. Furthermore, each subsequent period is halving, and
is incidentally orthagonal when taken as an entire set. These features readily form a very useful linear algebraic basis.

The signal frequency series is (don't forget nyquist):

	f_n = frequency(b_n) (hz)
	f_n = 8000 / 2^(n+1) (hz)
	f_n = { 4000, 2000, 1000, 500, 250, 125, 62.5, 31.25, 15.625, ... } . (hz)

After about f_9, it is useful to consider the period of the vector.

	p_n = 1 / f_n (s)
	p_n = { ..., 0.128, 0.256, 0.512, 1.024, 2.048, 4.096, 8.192, 16.384, ... } . (s)

To select a tone and duration, it is simply a matter of selecting the corresponding bit vectors and composing the two in some manner.
Selction is via an AND mask.

To select the 1000hz tone at bit vector b_2, calculate function b_2(t) = (t>>2)&1
To select a note duration of approximately 1/4th second we calculate:

       2^(n+1) = 8000 * 1/4
   n+1 log_2 2 = log_2 2000
           n+1 = log_2 2000
             n = log_2 2000 - 1
             n ~ 11 - 1 ~ 10

Then we select b_10 and calculate function b_10(t) = (t>>10)&1

An entire b_n vector could be composed by b_n = { b_n(0), ..., b_n(N-1) }

Now, the signals can be composed by simple binary amplitude modulation (AM) using either multiplication operation (*) or the bitwise AND (&)
operation (provided that the bits are correctly aligned of course). We did happen to align them correctly above. This correct bitwise alignment
could be done via normal matrix row-swap operations on the binary vector (t) instead of the bitwise shift operation (>>). Additionally, we
should select a volume level. Assuming the output is unsigned 8-bit, we can simply shift the modulated signal by up to 7 positions to the left.

So we get the final signal generation function:

	b_2_t = (t>>2) & 1   // select tone signal
	b_10_t = (t>>10) & 1 // select duration signal
	o = b_10_t & b_2_t   // modulate duration signal by tone signal
	o <<= 6              // select volume

Reduction of the above produces this compact form:

	o = ( (t >> 2) & (t >> 10) & 1 ) << 6

Further reduction:

	o = (t << 4) & (t >> 4) & 64

The same process can be described as a dot-product operation with (s), the selection vector, operating on (t) as s.t,
and then another dot-product operation (c), the composition vector, also operating on (t) as c.t, and then finally the
volume scalar (v), so that the complete modulation is v(s.t)(c.t).

At this point, this becomes an interesting point of study for Kolmogorov complexity and the computation of music.

Just given about 9 tones and 8 time intervals, it would be easy to sequence somewhat novel audio scores, but we're going to continue
the breakdown of this basis to improve upon these numeric tools.

Given that the initial basis {b_n} is orthagonal, but of row rank significantly lower than its column rank, it would be of moderate interest
to see if a complete linear independant orthagonal basis of rank 2^N could be generated from the original basis in some simple way. Given this,
then in an albiet rather trival manner, any sequence of length 2^N could be managed in a rather long "1-liner". But then provided sufficient
knowledge in the theory of data compression, an optimization between a minimal vector subset and a maximal signal approximation could be reached
using only the most basic of algebraic operations.

Construction of a complete orthagonal basis of rank 2^N inituitively can be imagined to be by time-scaling of any of the original basis. One
additional fact to remember is that each of the square-waves are self-similar and only differ in their time-scale. Also, provided that the total
time interval of a complete sequence aligns on the period boundary of every basis vector, then the complete basis will retain orthagonality.
So only a single representative vector from the basis is needed to generate an entire basis of rank 2^N provided a simple time-scaling function
can be determined. Firstly, selection of the basis vector with minimal period is simply:

	b_0(t) = (t & 1)

To scale a function in range by a constant factor of (k) would be k*f(n), while to scale a function in domain by the same factor would be f(n/k).
So then it should follow that the period of the basis vector b_0 can be expanded simply by b_0(t/(k+1)) = (floor(t/(k+1)) & 1). The original b_0 square
wave is an odd function, analogous to the sine function, so therefore given k = { 0, ..., 2^N-1 } then the complete basis can be generated by:

	b'_k = { b_0(2 * t / (k + 1)) } = { floor(2 * t / (k + 1)) & 1 }

With the frequency of each basis vector:

	f'_k = frequency(b'_k) (hz)
	f'_k = (8000/k) mod 8000 (hz)
	f'_k = { 0, 4000, 2666, 2000, 1600, 1333, 1142, 888, 800, 727, 666, ... } . (hz)

Unfortunately, unlike the DST, the discrete implementations of this basis are not orthagonal because of aliasing problems at period boundaries
of many of the square waves.  When the wave frequency does not divide the number of discrete samples in the signal, then an imbalance in the
distribution of the high and low states of the signal. Even tho this does not mean the basis is not linear independant, the basis loses some of
the simplicities in decomposition calculations. Orthagonalization of this basis can be performed at the cost of significant complexity. A famous
compromise between the hilbert square-wave basis and the advantages of orthagonality is the "Haar Wavelet".

Trying to revise the original modulation scheme with a target frequency of 2600hz, we get the following:

	k = 8000/2600 ~ 3.08
	b'_3.08 = b_0(2*t/(3.08+1)) = floor(2*t/(3.08+1))&1 = floor(t*2/4.08)&1 // select tone
	b_10_t = (t>>10) & 1 // select duration signal
	o = b_10_t & b'3.08  // modulate duration signal by tone signal
	o <<= 6              // select volume

One may notice that when a frequency that is significantly misaligned with the total signal period is selected, many artifacts arise
in the output signal.

Reduction of the above produces this compact form:

	o = ( ((t>>10)&1) & (((int)floor(t*2.0/4.08))&1) ) << 6

Given a number of output signals o = {o_k}, these signals can then be composed together with a few possible
methods, namely binary XOR (^), binary OR (|) and arithmetic addition (+), each with various auditory effects.
Namely, addition runs the risk of clipping as the mean signal power approaches the range limit of the output
channel (in this case 8-bit or 255), OR may saturate all of the output bits if the signal becomes too complex,
and XOR may cancel out common-mode signals of similar power.

	o' =  o_0 + o_1 + ... + O_n
	o' =  o_0 ^ o_1 ^ ... ^ O_n
	o' =  o_0 | o_1 | ... | O_n

More shit to come, obviously.
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