just wanted to post a quick follow-up to the entry I had about cosine waves... taking the cosine value to a higher power just helps in math terms to guarantee the values are in an acceptable range...
when actually trying to program these functions, it is best to make use of the "floor" function to scrap any values that are between 0 and 1... this helps retain only the 1 values from the waves in the SUM, and is less taxing on the computation.
so rather than SUM(Cosine(pi*n/k)^(2^n), for k = {1,n}), utilizing SUM(FLOOR(Cosine(pi*n/k))) will yield the same results as the sigma/divisor functions.
that's all, just wanted to throw that out there in case anyone has read this and actually wants to implement it!
more details on the FLOOR function and its applications can be found here: http://en.wikipedia.org/wiki/Floor_and_ceiling_functions
Friday, August 10, 2012
Wednesday, August 1, 2012
scribbles from 2009 - sigma and divisor functions
More prime stuff
Took photos of my notes rather than typing out the formulas. Playing around with the cosine waves led me to this summation in 2009, Using cosine squared keeps the function even and continuous (the absolute value of cosine has non-differentiable points, or cusps).
Description from Wikipedia (http://en.wikipedia.org/wiki/Divisor_function):
"...
The sum of positive divisors function σx(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n, or
..."
Here are the first two of these sigma functions as determined by my cosine representation, the divisor function
δ(n), and the sum of divisors function σ(n):
These equations should probably have a '~' over the equals signs... By way of construction, these waves only equal 1 when N is divisible by the K value. I took the cosine squared function to the Nth power because it would be positive, even (and symmetric), retain all 1 values, and minimize any values that were less than 1 (making use of the fact: if 0 < x < 1, then lim [x^n] = 0 as n à ∞).
Here is another clip from my notes where I've represented sigma function with a slightly different ordering:
The sigma and divisor functions have so many applications throughout mathematics and they are related to many unsolved problems. I'm still looking for a closed-form equation, but haven't found the right path to go down... Lots of "rabbit trails" that seem promising at first, but end up winding in circles. I keep thinking there might just be some trig identity that helps collapse sums of waves, but haven't found it yet.
Kind of related, from 2008:
I also found a conjecture of mine concerning the amount of prime numbers between powers of 2... No proof yet, but computer tests look promising as the "i" value increases to infinity, the error value decreases to 0...
Simply put, let π(n) be the number of prime numbers less than or equal to n.
If...
C = π(2^n) = the number of primes up to 2^n
A = π(2^(n-1)) and...
B = π(2^(n-2)) ...
Then C < 3A-2B
The number of primes up to 2^n is less than three times the primes up to 2^n-1 minus twice the primes up to 2^n-2.
From my notes, I had written, "the number of primes in between 2^i and 2^(i+1) is less than 2 times the number of primes between 2^(i-1) and 2^i."
Equation form:
π(2^(i+1)) - π(2^i) < 2 * [ π(2^(i)) - π(2^(i-1)) ]
==>
π(2^(i+1)) - π(2^i) < [ 2*π(2^(i)) - 2*π(2^(i-1)) ]
==>
π(2^(i+1)) < 3*π(2^(i)) - 2*π(2^(i-1))
==>
π(2^i) < 3*π(2^(i-1)) - 2*π(2^(i-2))
So after a little manipulation, and shifting the index, you get the same inequality as shown in the conjecture... I've tried studying recursion with these kinds of inequalities, but found that the accuracy decreases as you expand further..
I've found that the conjecture fails when i=5, but couldn't compute high enough to find another exception to the rule. Here is that example:
π(2^5) = π(32) = 11 prime numbers up to 32: (2,3,5,7,11,13,17,19,23,29,31)
but
π(2^4) = π(16) = 6 prime numbers up to 16: (2,3,5,7,11,13)
π(2^3) = π(8) = 4 prime numbers up to 8: (2,3,5,7)
3 * π(2^4) - 2 * π(2&3) = 3*6 - 2*4 = 10 < 11...
So π(32) is not less than 3*π(16) - 2*π(8) as the conjecture suggests it should be, it's off by 1...
That's the only hiccup that I've found in testing this conjecture. I have yet to find another value for 'i' which causes the inequality to fail, but that's only as far as my computer can count for now - better programming will yield better results. It's very possible that way down the number line, some huge power of 2 exists that has more primes than the estimate... but that's why it's still a conjecture!
Thanks for reading - that's more than enough prime numbers for today :)
Monday, July 30, 2012
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.
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!
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 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:
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!
Sunday, July 29, 2012
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)
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)
Thursday, July 19, 2012
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!
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!
Sunday, October 16, 2011
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!
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!
Tuesday, September 6, 2011
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... :)
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... :)
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