Brainfart Gem Discovered While Editing

An e-book version of "Chaos In Boxes" is currently in the editing phase. While I was editing, a footnoted brainfart got a bit of exploration, and an exciting new observation is the result.

Ready?
Here it is:

QUESTION: What do you get when you subtract the Major Scale from the Chromatic Scale?

PROCEDURE: The C Major Scale contains the notes C, D, E, F, G, A, B. The Chromatic scale contains all of these, plus C#, D#, F#, G#, A#. Those notes make up the F# Major Pentatonic Scale.

ANSWER: the Chromatic Scale minus any chosen Major Scale results in the Major Pentatonic Scale whose root is a tritone away from the root of the chosen Major Scale.

At the risk of attracting ridicule from the establishment, I'm sticking my chest out real far while putting a big old "Quod Erat Demonstrandum" on this document.

Q.E.D.
Booyah, baby!

Sean Luciw,
authour, "Chaos In Boxes: twisted adventures in music theory"

Root 2 connections

The brainfart on page 47 of my book "Chaos In Boxes" describes a hairy scenario involving the Square Root of 2; it was the first irrational number to be discovered by Hippasus* and resulted in his assassination by his fellow Pythagoreans, who believed all numbers to be expressible as the ratio of two integers!

(This came about during the Chapter "Quincunx And The Legend Of The Chromatic Transformation Matrix Starflower", wherein I was inspired to compare a semitone to a fifth.)

Since then, I've noticed a few connections concerning Root 2.

A musical interval between two notes can be described by dividing their frequencies; it so happens that Root 2 (which equals 1.414213...) is exactly the ratio for the infamous Tritone in Equal Temperament. This seems somehow suitable or ironic, considering that:

a) the Tritone was once condemned as "evil" by some religions - echoing in my mind the Pythagorean's rejection of Hippasus

b) the Tritone ultimately holds the key to undeniably powerful harmony through the resolution of a dominant chord towards its parent tonic chord - echoing in my mind the importance of Hippasus' discovery of irrational numbers

I was quite thrilled to notice this.

But wait, there's more...

As I was sitting at a friend's living room jamming along with cable radio, I switched to the jazz station. It was channel 414. I had to wonder if someone at the radio station chose channel 414 as their station number because of the significance of the number 1.414. After all, a very common technique in jazz is Tritone substitution, a.k.a. the "Sub V". This is based on the fact that a dominant 7th chord is extremely similar to the same type of chord a Tritone away, sharing most of the same tones.

Just today I learned that ISO paper sizes (A0, A1, A2, A3, A4, etc.) all have an aspect of Root 2! (The big advantage of this proportion is that you can cut or fold such a rectangle in half and it will still have the same aspect ratio, enabling scaling without cutoffs or margins.)

I feel inspired to find more Root 2 connections; I thought I would find something in astronomy but haven't found anything yet. I guess I'll just have to wait for the next brainfart to dawn on me; that's just the way it goes sometimes.

* note: The Hippasus scenario happened in the 5th century B.C., but there exists a Babylonian clay tablet depicting Root 2 which is 1100 years older. Others may have probably discovered irrational numbers at various points in history.