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...  :)

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.

Easy Octaves on Guitar

In standard tuning, there are easy locations to jump up an octave for whatever you wish to play.

This picture shows where notes can jump up an octave:

The red/blue dots represent octaves of each other as you move left to right going up in pitch.  So for example, 2nd fret on low E string is an octave below 4th fret on D string, which is an octave below 7th fret on B string...

The significance of this layout is that any pattern you wish to play on two strings that are next to each other (string pairs low E & A, D & G, then B and high E), can also be played one or two octaves higher using the exact same pattern, but just by shifting over a few frets and down 2 strings.  I've tried to show this in the following picture:

So the green box is around each of these 2-string pairs, and the pattern repeats at each level. 

Try it out!  Doesn't matter which fret you start on so long as the spacing is the same as shown on the chart.

I've been playing around with different patterns and switching from octave to octave rapidly, makes for some good effects/sounds.

Have fun!

Scales on Complete Fretboard

Here is a work in progress.  I am trying to find a comprehensive way to map out scales for the entire fretboard. 



fretboard scale diagrams

 
I created this by smushing all of the individual patterns for each scale onto one fretboard layout.  The root notes are marked by boxes around the dots.  Some of the notes for Harmonic and Melodic minor actually get used by two patterns, so these are marked as unfilled dots. 

So then for whatever key you want to play a scale in, just line up the frets on the guitar to the boxes and dots on the diagrams.  If you run out of room, no need to worry, the patterns for these scales repeat every 12 frets, so just pick up back at the beginning when you can't go any further on the fretboard.

This is a great tool for jamming and practicing.  Just pick a key, pick a scale, and go nuts!

First Post

Hello,

Thanks for coming across my blog!  I've never blogged before, but I hope to create a great archive of my favorite discoveries...  I will be posting my findings related to Music, Math, and Martial Arts.

I've been playing music all my life.  Started learning piano at the age of 4 from my mother, who is a piano teacher in Tega Cay, South Carolina.  I had listened to my father play various styles of guitar all my life as well, and when he got me a Fender Squire electric in high school, I started to learn every scale, chord, and "lick" I could wrap my head around.  My father also has a few guitar students in Tega Cay.  Now I am always striving to learn new technique and styles.  I used to perform in a band called Juniper Spring playing piano and singing.  We would cover songs from Phish, Talking Heads, The Beatles, the Grateful Dead, and many more...

I've had a lifelong interest in mathematics, and am always looking for new areas of research to discover and learn new, amazing truths.  My main interests are numerical analysis, number theory, and mathematical applications to physics.  I received my degree, Bachelor of Science in Mathematics, from the University of Florida in 2008.

By far, martial arts is my favorite sport and activity.  Very few practices help you gain such incredible levels of physical fitness, self discipline, and medidation.  I have obtained second degree black belt in Tae Kwon Do (or Tang Soo Do), and have also spent years studying other styles including Shotokan and Wado Ryu.  I am always eager to learn new techniques and styles from any source possible.

So this blog will be my outlet for anything I come across as I continue my education and discoveries in these fields. 

Thanks for reading!  I hope to start posting very soon...