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 G♯ are 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 G♭ are 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.

Circles as Numerical Sequences

Mathematics is a universal language, and our best way to communicate with those who are very different from us. We see increasingly complex mathematical designs appear in crop fields worldwide every year - in grass, wheat, barley, corn, snow and ice (I call them all “crop circles” for simplicity’s sake). Even the seemly simple ones hide a treasure of mathematics if you dig below the surface. These crop circle designs have an underlying order and beauty that even the mathematically challenged can appreciate. 

It is my belief that these beautiful designs encode data, because they contain blocks of objects that repeat, and there are rules about how the blocks can be assembled. This basically describes how language works, so it’s probable that these crop circle designs are using a geometric-based language. I think of it as a geometric object-oriented programming language that may be self-executing.

Let me clarify here that I am only talking about non-people-made (NPM) crop circle designs. The designs made by people don’t follow the same geometric rules, and have problems with proportion, scaling, and placement of the designs with respect to tram lines. Some of you may be asking – if they’re not made by people, then who? My answer to that is - an intelligent being who knows more than I. 

The Concept of Number

With the idea that math is universal, let’s start with something basic like the concept of number. In the book Contact by Carl Sagan, a radio transmitter on earth picked up a non-terrestrial signal with a series of beeps and spaces that communicated intelligent knowledge of prime numbers. Suppose we wanted to do something similar, except instead of using sound we wanted to use a visual medium to transmit the first 7 prime numbers.   

The best option for communicating numbers using geometric figures would depend on how many spatial dimensions you have available to communicate, and from which dimension the figures would be viewed from.

1-D

In one dimension (1-D), the best option to visually represent number is a sequence of lines and spaces. A line of length 2 would be followed by a space, then a line of length 3, followed by a space, etc… Note this would need to be viewed from a 2-D perspective in order to decode the numbers. 

Linear 1-D Representation of Prime Numbers

2-D

In two dimensions (2-D), the best option to visually represent numbers is a sequence of circles, placed concentrically or within a 2-D coordinate system. Radial lines on a polar coordinate system, triangles or squares also provide other options, but circles are superior because of their simplicity. Lines alone pose difficulty because of their 1-D nature. The line would need to be thick enough to be seen, but then a second dimension is introduced as well as a second number indicating the width. Using equilateral triangles or squares to represent pure numbers is also possible, but practically speaking, circles are still required to construct a perfect 60 or 90 degree angle.

Circles are the best 2-D choice to represent number because:

• A circle is defined by one number – its radius
•  A circle consists of one continuous line
•  All points on the circle are the same distance from the center, which makes a circle resilient to distortion
• Circles are the easiest geometric figure to construct

The simplest, most elegant way to represent prime numbers in 2-D is with a sequence of concentric circles with radii (or diameters) measurements equal to the prime numbers. Does this look familiar to anyone?

3-D

In three dimensions (3-D), the best option to visually represent numbers is a sequence of spheres, placed concentrically or within a 3-D coordinate system. It would be impossible, however, for a being living in 3-D to measure the relative sizes of nested spheres, so this type of encoding is not ideal for us here in 3-space.

Conclusion

The circle is the best 2-D geometric representation of pure number, and the best visual representation of numbers for those of us living in 3 dimensions.

Now we have a starting point for our crop circle language. The next logical step in decoding is to look at the sequences actually being generated by the designs in our crops. Then comes the tricky task of nailing down the positions of the design elements using a 2-D (polar?) coordinate system.

There is much work to be done… Are you on board? Are you ready?

Tibetan Monks and the Geometry of Levitating Stones

In addition to the ancient accounts of large stones being levitated or “walked into place,” there are also a few modern accounts that provide evidence for such a phenomenon. One of the most compelling cases is that of Henry Kjellson, a Swedish engineer who in his 1961 book The Lost Techniques detailed the visit of his friend Dr. Jarl to a Tibetan monastery. While at the monastery, Dr. Jarl witnessed Tibetan monks levitating large stones from the ground up to a cliff approximately 250m high.

This account of stone levitation is remarkable because of the scientific observations and measurements that were documented by Jarl (or Kjellson using a pseudonym). The detailed measurements allow for an analysis of geometric proportions used by the monks. The following excerpt comes from Kjeller:

In the middle of the meadow, about 250 meters from the cliff, was a polished slab of rock with a bowl like cavity in the centre. The bowl had a diameter of one meter and a depth of 15 centimeters. A block of stone was maneuvered into this cavity by Yak oxen. The block was one meter wide and one and one-half meters long. Then 19 musical instruments were set in an arc of 90 degrees at a distance of 63 meters from the stone slab. The radius of 63 meters was measured out accurately. The musical instruments consisted of 13 drums and six trumpets. (Ragdons).

Eight drums had a cross-section of one meter, and a length of one and one-half meters. Four drums were medium size with a cross-section of 0.7 meter and a length of one meter. The only small drum had a cross-section of 0.2 meters and a length of 0.3 meters. All the trumpets were the same size. They had a length of 3.12 meters and an opening of 0.3 meters. The big drums and all the trumpets were fixed on mounts which could be adjusted with staffs in the direction of the slab of stone. 

The big drums were made of 3mm thick sheet iron, and had a weight of 150 kg. They were built in five sections. All the drums were open at one end, while the other end had a bottom of metal, on which the monks beat with big leather clubs. Behind each instrument was a row of monks. The situation is demonstrated in the following diagram: 

When the stone was in position the monk behind the small drum gave a signal to start the concert. The small drum had a very sharp sound, and could be heard even with the other instruments making a terrible din. All the monks were singing and chanting a prayer, slowly increasing the tempo of this unbelievable noise. During the first four minutes nothing happened, then as the speed of the drumming, and the noise, increased, the big stone block started to rock and sway, and suddenly it took off into the air with an increasing speed in the direction of the platform in front of the cave hole 250 meters high. After three minutes of ascent it landed on the platform.

Continuously they brought new blocks to the meadow, and the monks using this method, transported 5 to 6 blocks per hour on a parabolic flight track approximately 500 meters long and 250 meters high. From time to time a stone split, and the monks moved the split stones away.

Personally, I find the account very compelling because there are a lot of details, as well as a group of measurements that can be analyzed regardless of whether the whole thing is fiction or not. The geometry can speak for itself. From the side view, the ground and cliff form two sides of nested right triangles as shown below. The green arc represents the flight path of the stone blocks.

       

The distance from the instruments to the stone (63m) is about 1/4 the distance from the stone to the cliff base (250m). Consider the base of the cliff as the center of two circles with the two triangle legs as radii. The circle diameters as well as the triangle legs are in the ratio of 5:4 as shown visually below.

       

These two circles in a linear ratio of 5:4 are seen repeatedly in crop circles and megalithic monuments around the world. One significance of this set of circles is that they provide an easy approximation for squaring the circle. In the above figure, the red square has an area equal (within 0.2%) to that of the green circle. I have often wondered why squaring the circle would play such a prominent role in crop circles, ancient architecture and monuments. Recently, I have come to realize that the angles encoded within this geometric structure may play a key role in non-linear resonance because they are an embodiment of the golden ratio, phi (ɸ). This is Nature’s fractal.

The angle formed by the musicians, point of levitation, and base of the cliff (51.38°) is very close to the angle seen in a vertical cross-section of the Great Pyramid of Giza (51.83°). Cultural myths allege that the stones used to build this pyramid were levitated into place, so the geometric similarity is interesting to note.

The vertical cross-section of the Great Pyramid of Giza appears to be composed of two triangles known as Kepler triangles. A Kepler triangle is the only right triangle with sides in a geometric progression according to the golden ratio (1, sqrt(ɸ), ɸ). Geometrically speaking, Kepler’s triangle is the embodiment of the golden ratio in more ways than the “golden triangle” some readers may be familiar with.

The question remains about the geometric intent of the Tibetan Monks, assuming the story is true. Were the monks using a 5:4 ratio to approximate the geometric progression of the golden ratio? A comparison of the triangles in question show how close they all are.

The connection between Kepler triangles and resonance will the subject of a future blog post.

Drum and Horn Proportions

The description tells us that there were 13 drums and 6 Ragdons, which are the long, skinny trumpets still in use at Buddhist monasteries today. The drums came in 3 sizes, and it is assumed that Kjellson’s measurement for “cross-section” represents the diameter of the head of the drum. Then we have the following drum sizes:

Drum	Diameter	Length	Number
Large (L)	1.0 m	1.5 m	8
Medium (M)	.7 m	1.0 m	4
Small (S)	.2 m	.3 m	1

Upon further inspection, the proportions of the drums appear to be based on powers of 2 and 3, within a small margin of error. The number of drums is also based on powers of 2.

Drum	Diameter	Length	Number
Large (L)	1	3/2	8
Medium (M)	2/3	1	4
Small (S)	24/34	23/33	1

The medium drum is 2/3 the size of the large drum (in diameter and length), and the small drum is 8/27 times smaller than the medium drum. Based on these ratios, the small drum would be significantly higher in pitch than the other two drums. It seems logical that the small drum was the one leading the beat, since it could be heard above all the other instruments according to Jarl’s account. 

The ragdon trumpet produces a very low frequency sound, one I would describe as somewhat “farty.” It’s similar in frequency to a tuba, but more piercing, and with less “umpah.” With a length of 3.12m and an opening of 0.3m, the length/width ratio is about 10.4.  If I was sticking with the powers of 2 and 3, that would be close to 3⁴/2³ = 81/8 ≈ 10.125. 

From a music theory point of view, it makes perfect sense to use the ratio 3/2 for proportioning instruments, because it represents one of the most important musical intervals, the perfect fifth (P5). The musical system supposedly developed by Pythagoras (but likely borrowed from the Babylonians) relied entirely on powers of 2 and 3. This 3-limit tuning system was the dominant music theory used up until the Renaissance where it was supplanted by an equal-temper tuning system that allowed for easier key changes. 

Conclusions

I tend to believe that this account of Tibetan monks levitating stones is genuine. It is a lot of detail to fake, and the geometry is just too familiar. I see this Kepler triangle, this weird 51-ish° angle everywhere, in conjunction with geometric progressions of the golden ratio. The central angle of a septagon, the 7-sided regular polygon that plays a prominent role in sacred geometry, is 51.42°. Perhaps if it was named “sonic geometry” instead of “sacred geometry” more people would take it seriously and start analyzing the fractal nature of sound.