Introduction

Unlike previous chapters this chapter will be published in installments (much like a blog), but since there is experimentation involved there maybe revisions of previous installments along with more duplications of material in new installments.
This chapter will divert momentarily from the Windings discussed in the previous chapters and instead focus on two new but connected ideas:

1. The elusive "seven-ness" that exists in the icosahedron and dodecahedron as seven geometric structures that will be termed "Syzygies". A Syzygy consists of a pair of new geometric structures inherent in both the icosahedron and dodecahedron that will be individually termed as "Waveforms".  Later installments of this chapter will then connect these Syzygies with the seven spheres of ancient cosmology: Saturn, Jupiter, Mars, the sun, Venus, Mercury and the moon as they have been associated with seven Major Arcana cards. 

2. Demonstration of a musical side to the Platonic solids that utilizes the twelve notes of the modern music even tempered scale: A, A sharp, B, C, C sharp, D, D sharp, E, F, F sharp, G,  and G sharp. These twelve notes when applied to the vertexes/faces of the respective icosahedron/dodecahedron show that these two solids via the Syzygies have a natural tuning that is a reflection of the diatonic semitone relationship as best exemplified by the seven white keys to five black keys of the piano keyboard:

This musical side is really the origin of this entire work that was started several decades ago and will play a "programming" part in this Tarot computer/philosophical machine. "The Music of the Spheres" part of this chapter's title refers to the fact that the icosahedron and dodecahedron (as well as the other three Platonic solids) will fit inside a sphere so that all their vertexes will touch the sphere's surface. This plays off the ancient concept that the movements of the planets and stars produce music, even if the "music" is of the variety that is a visual/geometric analog to sound. This synthesis is amplified once the Platonic idea mentioned in both chapter 1 and chapter 3 of the dodecahedron symbolizing the cosmos is also taken into consideration. Once it is understood by the reader how the icosahedron/dodecahedron both "plays" and "is played" through and by the scales and intervals produced via the Syzygies; the result really is a "Music of the Spheres."  Music theory will be kept to a minimum as much as possible by utilizing links to Wikipedia and other online resources included for the reader to follow if need be. Also, this text will try to keep utilizing what is generally know as "the modern twelve-tone equal tempered scale," where the vibration-per-second difference between all twelve half-steps (A-A sharp, A sharp-B, B-C, C-C sharp ect.) is the same.

The Waveform

Previous chapters studied these Platonic solids by sequentially connecting the faces/vertexes of the respective icosahedron/dodecahedron in a structure termed "Winding."  However, in order to both flush out both the seven-ness and demonstrate the music connection, the inverse context of sequentially connecting the twelve vertexes of the icosahedron instead of its faces and likewise the faces of the dodecahedron instead of vertexes will be used. The resulting structure of connected icosahedron vertexes/dodecahedron faces will be termed "Waveform," because of its resemblance to periodic waveforms:

In this case though, it as if the periodic waveform was wrapped around a cylinder. As in previous chapters the icosahedron will be focused on initially because its structures are simpler. All Waveform images will use exaggerated edges for clarity:

Icosahedron with Waveform using Exaggerated Edges	Basic Waveform of an Icosahedron

Side View	Front View	Top View

Because edges are used to connect the icosahedron's vertexes, the Waveform has a resemblance to the ISH_Winding_0 described and utilized back in Chapter 2. However, the icosahedron's Waveform has two major differences from the ISH_Winding.

1. The Waveform is made up of alternating 60 and 108 degree angles. The ISH_Winding on the other hand was constructed of four 108 degree angles clumped in the center as two pairs and eight 60 degree angles clumped in pairs of four each on the sides.

2. Unlike the ISH_Winding_0 the Waveform does not have a mirror opposite. This is reflected in the fact that the Waveform's angles of the same degree alternate in handedness ("handedness" will be demonstrated in an up coming appendix), while the ISH_Winding's angles are of uncertain handedness, or rather, their handedness is defined by the handedness of the ISH_Winding_0 itself.

-ISH_Winding_0 Angles	Waveform Angles  L: Left  R: Right	+ISH_Winding_0 Angles

Four parallel equilateral triangles can be drawn connecting connecting angles of the same handedness and angle:

Four Equilateral Triangles

The Tone Waveform and First Musical Results

Now starting with note 'A' at the 108 degree angle vertex nearest the viewer, the twelve notes of the modern even-tempered scale will be applied to the twelve vertexes of the icosahedron Waveform (below left). A solid triangle has been added at the top connecting the vertexes labeled A sharp, D and F sharp as a device to help keep the viewer oriented, as well as a way to keep track of the Waveforms as they are manipulated in reference to the icosahedron. This triangle will be known as the "Tone Triangle". While other note-to-vertex sequences will be studied later, this is the sequence that will be used for now. The resulting labeled Waveform will be called the "Tone Waveform":

Tone Waveform	Rotate Tone Waveform

One of the first musical results of this labeling (or any sequential labeling for that matter) is that the notes are automatically arranged in groups of major third intervals (two whole steps) based on the handedness and degree of the angle. The image below color codes these major third interval groups based on coloring of the previous "Four Equilateral Triangles" image. It includes the appropriate enharmonic name for the note in white parentheses along with an arrow to the major third interval for which the note was renamed:

Major Thirds Intervals with Enharmonics (top to bottom): D - (B flat), D - F sharp, and F sharp - A sharp Enharmonic: A sharp = B flat B - D sharp, (E flat) - G, and  G - B Enharmonic: D sharp = E flat  (D flat)  - F, F - A, and A - C sharp Enharmonic: C sharp = D flat  E - G sharp, G sharp - (B sharp), and C - E Enharmonic: C = B sharp

As the above images show there is a recognition of enharmonics in this system,  and it will probably play a useful part in the "programming" (especially as it presents an opportunity to jump out of the system as a singer's voice). However,  for simplification purposes for an already complex system, enharmonic notation may not always be used in all the illustrations; so music theorists, please be flexible!

Second Musical Result

The second musical result is a three dimensional expression of what is known in music theory as the "Circle of either Fifths or Fourth Intervals". A Fifth interval consists of three and half steps, while a fourth interval is two and half steps. The two intervals are reciprocal in that what is a fifth interval ascending is a fourth interval descending and visa-versa . The music scale image below demonstrates this as 'A' is three and half steps above 'D' ascending while 'A' an octave lower is two and half step. Likewise, 'G' is two and half steps above 'D' and three and half as an octave below.

"Circle" is a key word here, because as the visual circumference of circle returns to its beginning point, so does the aural "circle of fifth/fourth intervals" return to its beginning note or more correctly the same note label such as "A to A" or "C to C". The difference between the visual and aural circles is that the aural circles' beginning and end notes may be separated by more than one octave depending on the number of fourth to fifth intervals used in the scale. In the example below a proportion of seven fourth intervals to five fifth intervals keeps the scale in one octave and returns to the exact same "A" with the same vibrations-per-second as the initial "A". Since this chapter is linking the visual/geometric with the aural, then for the duration a note's actual vibrations-per-second is not taken into account. Only the note label is used.

Play Circle of Fifths and Fourths Scale

This continuity of fourth/fifths can be duplicated inside the Tone Waveform as the images below show. The image on the right shows the degrees of what will be termed "Fifths Tone Waveform".

	Rotate Fifths Waveform

The top view on the far right will prove useful for creating sigils when subdivided into smaller sections.

Side View of Fifths Tone Waveform	Front View of Fifths Tone Waveform	Top View of Fifths Tone Waveform

One remarkable relationship between the Fifths Tone Waveform and the Tone Waveform is that the two construct a series of rectangles together. Using a slightly different perspective, the Waveforms' parallel legs and edges are shown in six different colors below. In the images the two Waveforms are shown with one pair of opposite parallel edges/lengths at a time, while in the center are the 2D circle analog rectangles color matched to the 3D images with the same notes labeling the vertexes:

The Seven Basic Syzygies

The icosahedron contains a maximum of twenty Waveforms, because each of their GaugeTriangles is matched to one of the icosahedron's  twenty faces. Seven Syzygies are revealed when those nineteen new Waveforms are studied in relation to the single Tone Waveform. "Gauge Waveform " will be the term for those new nineteen Waveforms. These nineteen Gauge Waveforms to Tone Waveform relationships can be best seen using seven colors that highlight seven basic positions the Gauge Triangles have in relation to the white Tone Triangle. In the images below the icosahedron's faces have been colored top-down as blue, yellow, purple, green, red, orange and black. The music note labeling can act as an aid for relative positions. While it can be seen that there are seven basic Gauge Triangle Positions, it can also be seen that six of the colors blue, yellow, purple, green, red and orange exist in threes with black existing just once. This seeming disproportion will be explained later on.

Right Side Perspective View	Left Side Bottom Perspective View
Front View	Back View

Now since each of these colored faces represent a Gauge Waveform, it becomes necessary to establish a labeling system to differentiate Gauge Waveforms/Syzygies of the same color. So by incorporating top and bottom views below, the Gauge Waveforms/Syzygies are labeled according to their Gauge Triangle center positions (shown in aqua color) using a simple number line and then incrementing and decrementing right and left respectively with 0 being the center. Although there is only one of both the Tone Waveform and Gauge Waveform VII, they also have been included for consistency.  The labled Tone Waveform has been removed for clarity. Notice how more than one Triangle Face center share the same position on the number line.

Top View with Gauge/Tone Triangle Number Line Placement	Bottom View with Gauge Triangle Number Line Placement

The seven Syzygies can be summed up in the glyph image below as seven different relationships between the White Tone Waveform and the other seven Gauge Waveform as symbolized by their Gauge Triangles.

Seven Basic Syzygy Relationships

All Syzygies produce their own scale or melody because its Gauge Waveform connects (or plays!) the Tone Waveform's labeled vertexes in a new sequence that can be traced by the reader. However, in this Basic section only one melody/scale example is demonstrated for each Syzygy type!  Each example melody/scale is shown on a music staff as well as being playable on a mp3 file. For the moment, these scales will be cataloged starting with four criteria:
1. The Syzygy type of the same color will be designated by Roman numerals I to VII and used thoughout this work.

2. The number line system of -3 to +3 described above will be used to diffentiate between Syzygies of the same type.

3. The initial note or home tone used to start the scale. The following examples will start on "A".

4. Degree of the angle designated by the initial note, which will be for the moment either 60 or 108.

 Example Syzygy images that show both the Tone and Gauge Waveforms have them slightly offset for easier comparison.

Also, the labled Tone Waveform is shown twice in the following images. First it is shown in normal size bent around the icosahedron and then echoed one-third size above its Tone Triangle.  All the Syzygies are also individually demonstrated with one-third size colored Gauge Waveforms hovering over their corresponding Gauge Triangles on the icosahedron.  

Syzygy VII will be demonstrated first:

Perspective View   Front View Tone Waveform and Gauge Waveform VII

Top View     Bottom View Gauge Waveform VII

Syzygy VII   Gauge Waveform VII

Play Syzygy Scale

Syzygies I:

Front View Gauge Waveforms I	Top View Gauge Waveforms I	Syzygy I +1
Gauge Waveform I -2	Gauge Waveform I 0	Gauge Waveform I +2

Play Syzygy Scale

Syzygies II:

Syzygy II -3	Front View Gauge Waveforms II	Top View Gauge Waveforms II
Gauge Waveform II -3	Gauge Waveform II +1	Gauge Waveform II +2

Play Syzygy Scale

Syzygies III:

Top View Gauge Waveforms III	Front View Gauge Waveforms III	Syzygy III +3
Gauge Waveform III -2	Gauge Waveform III -1	Gauge Waveform III +3

Play Syzygy Scale

Syzygies IV:

Syzygy IV -3	Top View Gauge Waveforms IV	Front View Gauge Waveforms IV
Gauge Waveform IV -3	Gauge Waveform IV +1	Gauge Waveform IV +2

Play Syzygy Scale

Syzygies V:

Front View Gauge Waveforms V	Top View Gauge Waveforms V	Syzygy V +3
Gauge Waveform V -2	Gauge Waveform V -1	Gauge Waveform V +3

Play Syzygy Scale

Syzygies VI:
The scale demonstrated here is only tentative for comparison with the other Syzygy scales. As this chapter progresses the basic scale to be identified with this Syzygy will be altered.

Syzygy VI -2	Front View Gauge Waveforms VI	Bottom View Gauge Waveforms VI
Gauge Waveform VI -2	Gauge Waveform VI 0	Gauge Waveform VI +2

Play Syzygy Scale

Animations of the Syzygies rotating on a vertical green axis (shown in the Tone Waveform Reference image) can be compared below. It should be noted that this rotation will take the Syzygies I to VI examples through their respective number line positions. Also, the music notation is a fixed constant, so it is removed during the animation.

Rotate Syzygy I	Tone Waveform Reference	Rotate Syzygy II	Rotate Syzygy III

Rotate Syzygy IV	Rotate Syzygy V	Rotate Syzygy VI	Rotate Syzygy VII

To be continued!