Sunday, August 2, 2015

Introducing the Polytonic Ratios

The following is a summary of the starting point for my crop circle research. I realized early into my research in 2010 that someone else had already noticed the patterns I was seeing in the crop circle designs. His name was Professor Gerald Hawkins, and he was a pioneer in the study of the geometric properties that appeared in these designs. As an astronomer, however, he should have left the theoretical mathematics to mathematicians, and his "crop circle theorems" are nothing more than the application of a well known formula I will explain below. That being said, I am very grateful for the work of Professor Hawkins, because not many credible scientists even take the subject seriously.

Circles Inside, Circles Outside

A regular polygon is a 2-dimensional closed shape, consisting only of straight lines, that is both equilateral and equiangular. This means all the sides have the same length, and all angles have the same measure.  A regular polygon can either be convex, with all sides “bulging outward,” or shaped like a star (with a mixture of concave and convex sides).  The most common examples of convex polygons are the triangle and square, and the most common examples of star polygons are the pentagram and hexagram.

Every regular polygon can be both inscribed and circumscribed by a circle. The circle that is tangent to all sides of the polygon is said to inscribe the polygon. This unique circle is called the incircle, and its radius is referred to as the inradius (r) or apothem of the polygon. 

circumscribed triangle with inradius r
Incircle with inradius “r”

Likewise, a circle can be drawn around the polygon, passing through all of its vertices, and this circle is said to circumscribe the polygon. This is referred to as the circumcircle, with a radius called the circumradius (R).

Circumcircle with circumradius “R”
Circumcircle with circumradius “R”
The formula for the circumradius (R) of any polygon (“n-gon”) can be expressed in terms of its inradius (r) and number of sides (n) as follows 

Suppose we want to compare the size of the circle that nestles the interior of the shape with the circle that caresses the outside.
Inradius r and Circumradius R

Dividing both sides of the equation by r gives us a generalization for the ratio of circumradius/inradius:
This formula tells us the ratio of circumradius/inradius is constant for each regular polygon, and based on a periodic function (cosine) that is used to describe light and sound waves. Equivalent ratios are produced by looking at circumference or diameter, because they are both proportional to the radius, which means they grow at the same rate.

The ratio of areas of the circumcircle and incircle form a second set of ratios that is the square of the first. The formula to calculate these values is the square of the circumradius/inradius formula above.
I have named the ratios defined by these two formulas as the polytonic ratios, and polygons surrounded by circles as encapsulated polygons. Each shape has two interval ratios that are unique to that polygon and independent of scale. The values for both the linear and the square polytonic ratios are given in the table below. The red bold ratios are the ones referenced in Gerald Hawkins’ Theorems.

Table 1 – Linear and Square Polytonic Ratios
Looking at the table of polytonic ratios, it is easy to see why Professor Hawkins picked out the values of 2, 4, and 4/3. I know the trigonometry is scary for some, so let's look at some pictures. This formula gives us a way to figure out which polygon (if any) fits between any two given circles.

Table 2 - The First 12 Encapsulated Polygons

A Look Ahead...

As you might guess, many of the above rings compose key elements in the design of the crop circles. Rings can be combined with each other, producing a multiplicative effect of the polytonic ratios which resembles the way musical diatonic ratios are combined.There is much to be explored here, but for now I will leave you with a polytonic representation of the perfect fifth formed with two hexagons and a heptagon. It's actually not a perfect fifth ratio (1.50), but it's close (1.48)

A Polytonic Perfect Fifth

Monday, January 5, 2015

Circles as Musical (Diatonic) Ratios

A Short Lesson in Music Theory

Pythagoras was the first one credited with realizing that a string could be cut at certain ratios to produce harmonically pleasing sounds. This idea undoubtedly predates Pythagoras, as it is the basis for all musical theory, and he likely encountered it in Egypt or one of the many places he traveled in his youth. Pythagoras constructed his scale using only perfect fifths (3/2) and octaves (2/1). Mathematically speaking, it means he used only powers of 2 and 3 to generate the interval ratios.

The Pythagorean tuning system produces perfectly consonant fifths, but presents several issues for musicians. For example, enharmonic notes such as A and Gare considered to be different notes with distinct frequencies. In the equal temper tuning we use today, enharmonic pairs are treated as the same note because they have the same interval ratios and frequencies. 

The real issue with this system is that the circle of fifths does not close. That means that as you progress through the notes of the scale, you won't end up on the same note an octave above where you started. Some adjustment needed to be made in order to close the circle, and that adjustment was to lop off the last interval and make it smaller then the rest. Thus, the Pythagorean tuning system consists of 11 intervals of perfectly just fifths, and 1 slightly shorter, badly-tuned interval known as the wolf interval. This wolf interval is the direct result of forcing the circle of fifths, which is actually spiral in nature, to be a circle.

The next evolution in musical theory was to introduce other prime numbers into the construction of musical intervals. In addition to using powers of 2 and 3, as in the Pythagorean tuning, the number 5 was introduced in a system we now refer to as a 5-limit just tuning. Pythagorean tuning is sometimes referred to as 3-limit, and other tuning systems such as 7-limit and 11-limit also exist, but are less common. All of them seek to represent harmonic ratios using the smallest integers possible.

There are some people today who argue that the 3-limit tuning, despite its dissonant wolf interval, is superior to the 5-limit tuning system, so let’s compare them a little:

  • 3-limit has consonant fourths and fifths, but dissonant major and minor thirds
  • 5-limit has consonant major and minor thirds, but some dissonant fifths
  • 3-limit tuning lives on today mostly in the form of 2-dimensional isomorphic keyboards. More will be said about these later, but for now it will be noted that the way to eliminate the Wolf interval is to add a spatial dimension to our musical instruments.  
  • 5-limit tuning became popular in the late Middle Ages because chords and triads are based on three notes, which means major and minor thirds need to be consonant in order to achieve a certain complexity in the music. Chords are the basis for most of the music we are familiar with today.
The 5-limit tuning was used until the late 1500s, when equal temper tuning was discovered and a method for calculating 12 equal musical intervals was developed. Today, in the West, we use a twelve-tone equal temperament tuning system, or 12-TET. Each note in the chromatic scale is the same distance apart, and that distance is 2^(1/12). It is my opinion that by moving away from pure ratios and into a logarithmic-based tuning system, we have done our culture a disservice. This will be the subject of a future blog post.

For now, I want to focus on 5-limit tuning, and explore it a bit further… The following table shows what the notes in the chromatic scale look like. Diatonic ratios are generally considered to be only those intervals found in a major scale, although some places on the web you will see a much broader use, or overuse of the term. Diatonic ratios are in red in the table below. The key of C has been chosen for the Note column because it is the one I am most familiar with.

5-Limit Tuning System

5-limit tuning also has its own issues, and its own wolf intervals. As shown in the table above, the notes F# and Gare extremely close to each other, but not equal in ratio/pitch. By seeking to use just intonation and produce pure harmonic ratios, equal interval sizes are sacrificed. This means that each of the interval types, except for the octaves, has three or even four different sizes. 5-limit tuning uses a combination of justly tuned fifths and non-just fifths of several sizes to close the circle of fifths. It's a complex and imperfect system.


Musical Ratios in 2-Dimensions - Circles

Up to this point we've been looking at musical ratios in terms of strings or lines, which are theoretically 1-Dimensional. Now imagine you were to fix each string at one end and spin them around to trace out a circle. The relative radii (or diameters) of the circles would be in the same ratio as the strings.This provides a 2-Dimensional model of musical intervals that in many ways bridge the gap between music and geometry. 

5-Limit Chromatic Scale Using Circle Radii

Note the slim interval between A4 (1.406, F#) and d5 (1.422, G). In an equal temperament system, these two notes are equivalent, and the spacing of the circles would be logarithmically equal.

It turns out there are a lot of ways to represent music using 2-Dimensional figures, but circles are the most natural element for this visual representation of sound. Because circles are defined by one number (radius), they can be thought of as a 2-D embodiment of number itself. For more information, see my previous blog post Circles as Numerical Sequences.

The chromatic scale represented by the circle radii above shows all the notes stacked on top of each other. Each of the 7 major diatonic notes are shown individually in the table below. It demonstrates the relative sizes of the intervals.

 5-Limit Diatonic Ratios Represented as Circle Radii

By using circles to represent musical ratios, we can:
  1. Express ratios in terms of linear measurements like radius, diameter or circumference
  2. Express ratios in terms of square measurements like area
  3. Change the positions of the circles to create an interplay of ratios using linear and square measurements
Expressing ratios in terms of radius and area are fairly straightforward, but once you add another variable representing the positions of each circle, a whole new level of complexity develops. The figure below demonstrates the relationship between the octave, fourth and fifth more elegantly than any 1-D representation ever could...

Octave, Fourth and Fifth - The Circular Dance

The red circles are in a perfect fourth ratio (4/3) and the blue circles are in a perfect fifth (3/2) ratio. There are also three pairs of circles in octave ratios (2/1, 4/2, 6/3)  and three other perfect fifth pairs (3/1, 6/1, 6/4), some spanning more than one octave.

I've seen designs like this appear in crop fields around the world, most notably in the UK. It appears in many forms, but has the same underlying structure. For example, the following design appeared near Liddington Castle in the UK in June of 2010.

Liddington Castle crop circle near Swindon, UK - June 2nd, 2010.