Trig Waves and Divisibility

Finally uncovered some old matlab/octave generated graphs from my prime number research (2008).  I don't want to take the time to fully detail each formula/function, but the images are interesting to study, and you can probably figure them out if you really want to.

This first graph is the one that guitar harmonics reminded me of in an earlier post...  It's the absolute value of sine/cosine waves increasing in period and amplitude by integer amounts.

As you can see, taking the absolute value makes it much easier to manage, because you can cut the period in half, and you don't have to mess with those annoying negative numbers.

This graph is increasing sine waves, with the asymptotes showing how the trend would continue if more and more waves were plotted. In this graph, N=3.

You can see that the waves all equal zero whenever they reach a number that is divisible by each waves period.  More waves coincide at integers that are highly divisible, and an obvious pattern emerges, and repeats.

Note how the asymptotes alternate +/- (positive/negative).

This next graph is essentially the same function, but this time, N=50.  Although this looks much more chaotic, the same symmtry is still underlying. 

(the first three asymptotes are identical to the graph above... note how much more they've "filled-in" with the resulting waves...)

I haven't decided which I like better, sine or cosine - they each have their advantages when dealing with the prime numbers and divisibility.  Taking the absolute value of the waves chosen makes all of the asymptotes positive, as you can see in this graph:

This graph shows the exact same information as before, but now we are restricted to positive numbers.

I only used the increasing amplitude in my graphs to make the the symmetry easier to see.  Modifying the formula to give each wave equal amplitude results in this representation of the symmetry:

It's tougher to discern where the waves of each period start and stop.  I've filled in black dots for all of the prime numbers up to 60, and included the circles to show the symmetry of the spacing of the waves.  Remember, when the waves all coincide at the axis, the number is composite (divisible by other numbers - not prime).  This is a visual representation of the Seive of Eratosthenes made much quicker with symmetric waves.  Wherever the waves do not touch (up to a limit) will be the prime numbers.

So an easy way to see where the waves all line up is to take the SUM of the values of each wave at each integer.  Using the equal amplitude waves and taking the SUM results in this crazy graph:

This is the sum of equal amplitude cosine waves up to 50 partial sums (kind of reminds me of the stock market, but that's another post for later). The points in the graph where it seems to dip or have peaks and valleys is just another representation of the prime number symmtry.  Each major dip happens at a highly divisible number (e.g. 24, 30, 36, 48, 60).  A cool thing happens when you take sums of symmetric functions:  the sums also retain a similar symmetry.

I guess that's enough on this topic for now.  All of these graphs and functions are from research I did in 2008, so I've got plenty more to cover.  I'll bring on another installment soon enough!

quaternions - hamilton

       

Stumbled upon this image... pretty cool.  Description below is roughly from Wikipedia:

Quaternions are a number system used to extend the “complex numbers”. Hamilton defined a quaternion as the quotient of two “directed lines” in a three-dimensional spaceor equivalently as the quotient of two vectors.Quaternions can also be represented as the sum of a scalar and a vector.
Directly applies to mechanics in 3-dimensional space - (physical existence as we know it)

Guitar Harmonics and Math/Physics

http://en.wikipedia.org/wiki/Guitar_harmonics

Found this nice layout of the harmonics on a guitar fretboard, and noticed it is identical to the wave-forms I've been using to try to work on the Goldbach Conjecture and other problems with prime numbers/divisibility.

I find it amazing that the physical phenomenon that we hear as music is essentially just waves of tones/frequencies that directly relate to the divisibility of integers.  I've always liked the quote:  "Music is the pleasure the human mind experiences from counting without being aware that it is counting." (Gottfried Leibniz).  With this layout of the harmonics, you can easily see the symmetry that arises when multiple waves are overlapped. 
Why does that matter?  Well, I guess it depends on your perspective of our universe... is it infinitely divisible (yielding smaller and smaller particles as you get smaller and smaller, never reaching a "smallest" particle) or is the universe finite (if so there would have to be some "smallest" particle, from which all other matter is created)? 

If finite, then each larger and large particle could possibly be defined in terms of these "smallest" particles.  However, there could be some order to the crazy quantum mess that we currently understand to be random.  Look under the red domes in the picture above... when all of the waves are overlapped, and their wave-lengths are not coinciding, it looks pretty chaotic, but once you zoom out, you can see that the apparent chaos is actually just a small portion of a fully symmetric structure.

If the universe is truly infinite, and infinitely divisible, then questions of continuity and differentiability come into play.  Most of our physics is contingent upon equations involving limits and infinite sums, which would require this infinite divisibility.  But we can't ever really know at the smallest or largest scales how "accurate" our math/science is. 

I believe the same symmetries that exist in music are also present in mathematics (discrete and continuous), and that it is those symmetries which we have detected as reproducible patterns in our physical world (yielding functions and formulas based on our observations).  We may not have the exact formulas or functions to yield perfectly accurate results, but at the same rate, our approximations of objects like spheres and circles would never come close to the "real" shape, which can't exist in our universe as we understand it - but even though the shape can't physically exist, we can figure out an exact formula for that shape to base our approximations on. 

It is the same with the laws of physics.  Our formulas and functions are just trying to explain the perfect shape that we see should be there, but that we may never acheive due to our finite limitations.

I wonder if Music Theory has ever been directly applied to quantum mechanics and particle physics?  Sounds like a blog for another day...  will do some research and see!

seeing the chords on top of scale patterns

The image below is showing the major scale pattern in the key of G.
If you want to try to play the 7 possible triads within G major, this chart makes it easy to see which chords fit in where!


The I, IV, and V chords are all major, and the II, III, and VI chords are minor.  The VII chord is a minor, but with a flatted fifth - since F# minor would have a C# in it.  G major doesn't have C# at all, so the chord gets a diminished kind of sound.

So far, looking at the chords this way on top of the major scale has really helped to pick out the arpeggios when I'm in a scale.  The chart helps to see where to aim in the scale when you're in a major key that matches a chord progression (for example, if the song has a part with the chord pattern: G D Am C, you could focus your lead guitar's soloing to the I, V, II, and IV notes shown in the image above).

These patterns aren't just restricted to the key of G - you can slide this left or right by any amount of frets, and the patterns remain the same - it is just the key of the major scale that changes. 

Many songs don't stay in the same key signature for long, but knowing where to play when the key signature is well-defined is critical if you want to clean up your soloing and improvisation while jamming.

Enjoy!

TIPS: 

- Try to fill in the names of the notes on the above image - each chord is only 3 notes, repeated over and over again in the patterns.  By doing this, you'll be learning the next level of detail (which note is which, and where octaves of each note can be found) - and once you are comfortable with that, playing scales along with any chord progression can be mastered!

- Practice these same patterns in key signatures other than G major.  Just slide the whole major scale pattern to match the fret you wish to be your root note - everything else stays the same!

My Recording/Jamming Setup

Here's a quick post to show off my "studio".

The picture is a little rough, but shows the basic idea.

So each of my 4 tracks (piano/synth, guitar, bass, and vocals) all go directly to the mixer.
From there I can adjust their "send" levels for how much of each track's sound is to be processed with effects.
I can also adjust the main levels for each track from the mixer too.
Then from the mixer, all tracks are sent to the amplifier depending on their FX Send Level.
The amplifier lets me add reverb, chorus, or overdrive/distortion at the press of a built-in foot pedal.
With these effects applied, I can then sample any of my four tracks either at the same time or independently, and have the effects included on the loop!

Finally, the loop pedal returns the FX mix signal back to the mixer for final levels, and from there it can be sent out to a recording device, or to a PA if not recording.

The only problem I've run into with this setup is if I want to record and listen at the same time.  There is a slight delay (just a millisecond), but it adds up when trying to sample multiple tracks.

If I choose not to record, and just loop directly to speakers, this results in no delay between live and loop.

So I still need to find a way to listen to what I'm playing/sampling/looping while I'm recording that avoids this annoying millisecond delay.  Until then, it's a great setup for practice, but still not quite ready to deploy on-stage... working on it...  :)

Quadrilateral Integration

The integral can be defined as the area between a curve (continuous and differentiable function) and the axis on some given interval.

If you want to find the area between two curves that lie on the same interval, it turns out you can subtract one from the other.  So the area in the picture here is _a∫^bF(x) - G(x) dx.
But what if the curves are on different intervals?  How can one determine a function to calculate the area?
A paper I wrote 2/24/2008 explores the area between curves that lie on different intervals.
It does require that for every point in the intervals on which the functions lie, the functions are able to have a line drawn between each other without intersecting.  So each point on curve F essentially must be "in view" of another point on the curve G.  The area between F (on [α,β]) and G (on [a,b]) is:

?AREA? = _α∫^βf(x)dx – _a∫^bg(x)dx + (1/2) * [(α – a)*(f(α) + g(a)) + (b – β)*(f(β) + g(b))]

I have the intervals switched in the picture, but the formula is correct for what I've typed in the line just above... (just put F on (alpha,beta), and G on (a,b) instead of what the picture shows...)

I was going to post the whole paper, proofs and all, but the formatting is weird transferring from MSWord to blog.  So I've taken screen shots of the main points of the 11 page doc.  I've cut out most of the proofs, but leave a comment if you'd like to see them. 

There is a lot of weird notation, but I tried to make it easy to follow.


*** Anything preceded by *** is extra notes outside the paper screen shots...

 *** The proof of this first theorem in mirrors the proof provided at the website: http://www.geometryalgorithms.com/ but they have deleted the original link since this paper was written.

*** Here are the terms I defined to try to explain the quadrilateral integration process.

 *** So the rest of the paper goes on to prove that the limit of this quad sum can be determined... but it does rely on this following theorem.

*** This next theorem is finding the area between a curve on one interval and another interval on the axis, and there is an example to illustrate.



 

*** Next, I find formulas to describe the lines that would form the boundary of the area between F and G.

 *** The picture for example two is a little tough to make out, but I wanted to show the lines C and D intersecting with the X axis. 

*** Theorem 5 is the main point of the paper.  This is the formula shown at the ?AREA? = part...




*** Look at this part of that formula...  (1/2) * [(α – a)*(f(α) + g(a)) + (b – β)*(f(β) + g(b))]
*** split it up...                                      (1/2)[(α – a)*(f(α) + g(a))] + (1/2)[(b – β)*(f(β) + g(b))]
*** a litle more...             (1/2)(α – a)*f(α) + 
***                                  (1/2)(α – a)*g(a) +
***                                 (1/2)(b – β)*f(β)  +
***                                 (1/2)(b – β)*g(b)
*** each of these segments is half the area of the green highlighted rectangles in this picture...

***  Using these rectangles, this would be the quick way to determine the area geometrically if the integrals of F and G were known already.  I found it pretty amazing that adding up infinitely many quadrilaterals and taking the limit gives a formula that is essentially identical to the geometrical method.

*** The following corollaries can be derived from the above theorems. 
*** There are a couple examples to illustrate what a few of the corollaries are saying.

 ***  I thought it was interesting that the quadrilateral integration can be done with regards to either axis (X or Y), so long as each F and G have inverses...

 

*** this last corollary is just to point out the connection to an existing discovery...

 *** Well that's the end of the paper....  Pretty much the whole thing now that I've gone back and looked at it, just missing the proofs (comment if you want to see them).

If you have MATLAB, below is the code I wrote to draw examples/compute values
Before you run the code, you have to input the missing values for alpha, beta, a, b; the functions F and G; and then an additional 4 values depending on which functions F and G you choose (in the second FOR loop).


The code will plot the graphs, show the partitions, and then calculate the area based on the partition size chosen.  It is the limit of this value as N reaches infinity that is the actual area between the curves.Quadrilateral Integration (DSB 2008):

alpha = %  the value desired for alpha;
beta = %  the value desired for beta;
a = %  the value desired for a;
b = %  the value desired for b;

     delta = beta - alpha;
     DELTA = b - a;
     x = 0:.1:max(b,beta)+1;

y = %  This is f(x);
y(2,:) = %  This is g(x);

plot(x,y);

n = input('partition size n = ');
     inc = delta/n;

%          this first for loop draws the partition lines%          on the plotted functions.

for i=0:n
     xfi = alpha + i*inc;
     xgi = (DELTA/delta)*(xfi - alpha) + a;
     line([xfi,xgi],[,]);
     i=i+1;
     end

area = 0; %  initialize to zero at the start

%    this second for loop calculates the area.

for i=1:n
xfi = alpha + i*inc;
xfj = alpha + (i-1)*inc;
xgi = (DELTA/delta)*(xfi - alpha) + a;
xgj = (DELTA/delta)*(xfj - alpha) + a;

x1 = xgi;
x2 = xfi;
x3 = xfj;
x4 = xgj;

y1 = % set equal to g(x1) with the actual formula...example, if G(x1) = x1, y1 = x1;
y2 = %  set equal to f(x2)   actual formula... example, if F(x2) = x2+1, y2 = x2+1;  y3 = % set equal to f(x3);
y4 = % set equal to g(x4);

A = (x1*y2-x2*y1)+(x2*y3-x3*y2)+      (x3*y4-x4*y3)+(x4*y1-x1*y4);

area = area + A/2
i=i+1;
end

One, Four, Five!

Here's a way to find out what key a song is in while playing along (and a little music theory)...

Just remember that for any Major key, the first (I), fourth (IV), and fifth (V) notes are always going to be major.  So consider the key of C.

C Scale:     C    D    E     F    G    A    B
Number:     I     ii     iii     IV   V   vi    vii

So we have the One, Four, Five for C major:  C F G

Lets say you're listening to a song, and you can tell it has an A major chord in it.
Using this method, you can quickly find out which key the song is in, and then you'll know which scale to play to fit in.

There are three places this A major chord could fit into the scale, either the first, fourth or fifth position...

1 4 5                   I   ii     iii      IV  V   vi      vii                1 4 5
A _ _   A major (A  Bm  C#m  D  E   F#m  G#m) ==> A D E   is the 1,4,5 for A major
_ A _   E major (E  F#m G#m  A  B  C#m  D#m) ==> E A B  " " " " " " " " " " " E major
_ _ A   D major (D  Em   F#m  G  A  Bm    C#m) ==> D G A " " " " " " " " " " " D major

So from here, you'd want to try to find more chords in the song's progression, and you can narrow down this short list.

Similarly, if all you can find is a minor chord in the song, there will be 4 places in the scale the chord could fit.

Let's say Dm, now we're dealing with 2nd, 3rd, 6th, and 7th positions in the scale (takes a little more time and memorization than the major chord/scale).

2    3 6 7                            I    ii       iii      IV  V   vi      vii                 1  4  5
Dm _ _ _  ==> C major   (C   Dm   Em   F    G   Am   Bm)   ==>    C F G
_ Dm _ _  ==> Bb major  (Bb Cm   Dm   Eb  F   Gm   Am)   ==>   Bb Eb F
_ _ Dm _  ==> F major    (F   Gm   Am   Bb  C   Dm   Em)   ==>  F Bb C
_ _ _ Dm  ==> E major    (E   F#m G#m  A   B   C#m Dm)   ==>    E A B


Practice Tips: 
Try to do the exercise above for another chord other than A major or Dm. 
Try to figure out the chords to a song by ear.

Also,
Play the scale of the key you want.
Then, try to play the chords at each note of the scale. 
Try to mix up which chords are major or minor as you go up the scale, and see how it sounds if you change from the 1,4,5 pattern shown above.  (MAJOR minor minor MAJOR MAJOR minor minor)

Example:
play first the notes: C D E F G A B
then play the chords: C Dm Em F G Am Bm
now see how it sounds to play these chords: C D E F G A B (all major)
or try any other variation from 1,4,5 being major.