Randomness in Music With the 12 Tone Technique

Randomness of the 12 Tone Technique Applies to our Math

Randomness of the 12 Tone Technique Also Applies to our  math.  The initial proponent of this technique was Arnold Schoenberg (1874–1951), Austrian-American composer. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note[3] through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key.

Randomness of Music and Numbers

Our modern use of numbers parallels the 12 tone technique. The 12 tone technique is best described as willful randomness.  Antiquity thought of numbers one to nine as belonging to a system. It was called the 3 x 3 number square.  In our music that is set in the circle of fifths, this is called a key signature.  The numerical key signature of the ancients  was the vehicle of the number square. They favored 7 primary number squares. This could equate with 7 key signatures. The simplest and first was 3 x 3. Their favored squares ranged from 3 x 3  to 9 x 9. They did use higher numeric squares. However, the basic 7 were most common. Sacred prayers in Judaism coded higher number squares. Two favored ones were 13 x 13 and 17 x 17. I have blogs on this subject on DSOworks.com. Some 10,000 years ago, and maybe further back in time, all numbers belonged to unified systems. They were also connected  to words. For example, “order” could be 264. Each symbol of the ancients represented a letter and a number. There were no separate letters and numbers. Their unity called by a Greek name, gematria.  Look it up online. At one time there was no  randomness. You can sample ancient unity on the 3 x 3 number square picture below.

With the ancients, randomness is lacking.
Every number relates to the others in a meaningful way in the Neolithic times.
  • Any two opposite numbers around the perimeter total 10. Examples are 4 + 6= 10 ; 9 + 1 = 10; etc.
  • The average of any two opposite numbers around the perimeter is 5. Five is the core number.
  • Each number contributes to a perimeter whose total around #5 equals 40.
  • The total of all the numbers on the square is 45. . (That equals the sum of the  numbers  from 1 to 9).
  • Each number is set so that any row of three totals 15. This is true vertically, horizontally or diagonally.

The high level of organization of numbers in antiquity is staggering. Today, with our modern sciences, we totally lack such an organizing system for our numbers. I believe that result is  social conflict. The genius of Arnold Schoenberg made a powerful musical statement as to where our culture was heading. Let us return to the way of the ancients. Reviving number squares is what many of my blogs are about. Enjoy the illuminating sample of the 12 tone technique below!

 

 

Mexican Sun Pyramid has a blueprint

Mexican Sun Pyramid Draws on Platonic Codes

Mexican Sun Pyramid Draws on Platonic Codes. Yes, there is an ancient temple plan. It dates back to Neolithic times. Plato used it. It was employed all around the world in antiquity. I discovered it in a vision while hiking around the loop by Oquaga Lake.  The same Ancient Temple Plan also graces the Mexican Sun Pyramid.  I discovered it by means a vision while hiking around Oquaga Lake. The special character of the lake nature is described in my book of poetry on DSOworks.com: The Oquaga Spirit Speaks. It can be downloaded.

I was walking down the main road by Scott’s golf course when I had the insight that forms the basis of this blog.

Teotihuacán
 Pyramid of the  Sun (click to see picture source)

Coordinates: 19°41′33″N 98°50′37.68″W

Add the length of  the perimeters of the four  triangles ABC, ADC, ABD and CDB.  Each one totals 2520′ ( 738 + 738 + 1044 = 2520) .  Four triangles = 10,080. Next, use pi as 22/7. A circle around diameter of length 10,080 = 31,680.

Mexican Sun Pyramid Parallels Plato’s Writings

Plato’s Ideal City uses the same numbers in his desciption of the Ideal City.  He states in Laws V: “(1) Mark off 5040 allotments. They must be cut in two so that “each contains a near piece and a distant piece- joining the piece next to the city with the piece furthest off.”. That totals 10,080 markings of land (5040 on each side of the radius). That equals the total of the sum of perimeters around the 4 triangles of the Sun Pyramid as described above.  A circle drawn around diameter 10,080 has a circumference of 31,680 . As 10,080 x 22/7 = 31,680. The co-incidence goes even further. John Michell explains defines the area of each of its 2520 pairs of rings. The combined area of each of the pairs is 31,680. Both the Mexican Sun Pyramid and the Ideal City use the same number tradition of prehistory.  I refer my reader to The City of Revelation by John Michell pgs. 84-86

Conclusion: At one time there existed a peaceful formula for civilization. Due to some kind of cataclysm, everything was destroyed.  This possibility is discussed by Immanuel Velikovsky (/ˌvɛliˈkɒfski/; Russian: Иммануи́л Велико́вский; IPA: [ɪmənʊˈil vʲɪlʲɪˈkofskʲɪj]; 10 June [O.S. 29 May] 1895 – 17 November 1979) He was a Russian-Jewish independent scholar best known as the author of a number of controversial books reinterpreting the events of ancient history, in particular the US bestseller Worlds in Collision published in 1950.[1] Earlier, he played a role in the founding of the Hebrew University of Jerusalem in Israel, and was a psychiatrist and psychoanalyst.

Conclusion: I think that ancient ruins hold lost knowledge. Uncovering and understanding what’s in these ruins provides possible pathway way to peace. I think we are approaching another Golden Age. How good is that?

Image result for picture of Book cover of John Michell's City of Revelation
John Michell quotes Plato’s Laws V which one of my sources for this blog.

This is a candidate for one of the ancient burial sites by its yin yang characteristics.

Ancient Burial Sites Used the Perfect Fifth Ratio 3/2

Ancient Burial Sites Used the Perfect Fifth Ratio 3/2. Many Neolithic cultures placed the numbers of harmonious ratios of musical intervals into their buildings and environment.   How can musical intervals possibly apply to burial sites? What was the purpose of seeking harmonious intervals for interment? Where and when did this happen?

  1. The tradition belongs to  yin-yang concept of the ancient Chinese
  2. The ideal was the 3/2 ratio. Three parts yang to 2 parts yin. 3/2 defines the musical interval of a perfect fifth. The higher note vibrates 3 times; for 2 of the lower.
  3. The tradition characterizes ancient burial sites in China. I found what I thought was such a location in Wiki commons. It is pictured as the ALMATY, KAZAKHSTAN. See featured pictured above.
    railroad tracks interrupted the yin yang flow of ancient Chinese burial sites
    The natural flow of yin yang was thought to be interrupted by railroad tracks.

    The fifth has always been considered a perfect interval. In Western music, intervals are most commonly differences between notes of a diatonic scale. The smallest of these intervals is a semitone. In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2. There are only 3 perfect intervals in our scale system. They are the octave, fourth and fifth. They are called perfect for the following reason: They vibrate in whole number ratios from 1 to 4. They sound the most harmonious. Major and minor intervals vibrate with higher number integers. Note the following list:

    • The interval between C and D is a major 2nd (major second).
    • The interval between C and E is a major 3rd (major third).
    • The interval between C and F is a perfect 4th (perfect fourth).
    • The interval between C and G is a perfect 5th (perfect fifth).
    • The interval between C and A is a major 6th (major sixth).
    • The interval between C and B is a major 7th (major seventh).
    • The interval between C and C is a perfect 8th (perfect octave).

    Ancient Burial Sites share the 3 to 2 Perfect 5th ratio with other disciplines

    (1) Microbiotic cooking  uses the 3/2 ratio for healing. It advocates 3 foods that grow above the ground in addition to 2 that grow under.
    (2) Chinese geomancers detect yang and yin currents. Yang is the blue dragon, Yin is the white tiger. Yang current takes the path over steep mountains. Yin mainly flows over chains of low    hills. Most favored is where 2 streams meet surrounded by three parts yang and 2 parts yin.  That was the spot where Chinese ancient burial sites were built.

    Chinese believed that proper burial of ancestors controlled the course  of the surviving family’s fortune. Great dynasties are said to have arisen from proper placement of tombs. Also, the 1st action of a government facing rebellion was to destroy the family burial grounds of the revolutionary leaders.

    If Ancient Burial Sites are Beyond You, Here’s a Simple Musical Exercise to Help Your Health and Fortune

    Twinkle, Twinkle Little Star incessantly uses the interval of the perfect fifth. So does Baa, Baa Black Sheep. Sing the first 4 notes of each. With both nursery rhymes, the interval between the 2nd and 3rd notes is a perfect  fifth. You have your choice: (1) Sing the first four notes over and over, Or (2) simply and just sing the 2nd and 3rd notes over and over.  Another choice is take piano lessons. Play Mozart.

     

 

Musical Inversions parallel the 5 Platonic solids

Musical Inversions of Triads Run Parallel to Platonic Solids

Musical Inversions Run Parallel to Platonic Solids. Two concepts must be understood. (1) Inversions of triads. (2) The regular polyhedron property called duality. I will demonstrate musical inversions with the “C” major triad. For our purposes, every other note starting with “Middle “C” on the piano. That makes for C-E-G. These notes can be turned around, A.K.A. inverted. Then we have E-G-C and G-C-E.  Musical inversions once more returns us to C-E-G. They look and even sound different.  But they are still the same basic 3 tones.

Image result for Wikicommons illustration of the C major triad with inversions on the piano keyboard
The same C major triad in different inversions
Musical Inversions Parallel a Property called Duality Possessed by the Platonic Solids
The Circle of Fifths Fits the properties of the 5 Platonic Solids Like a hand fits a glove.

Now for the parallel property with the regular polyhedrons. First, we must look at a chart that defines their topological features. Note the octahedron-cube pair. The octahedron has 8 faces. The cube has 8 vertices. The octahedron has 6 vertices. The cube has 6 faces. Like musical inversion, the order changes from one to the next. We could also say the 2 geometrical figures are related like a musical inversion of the “C” triad.

Look at the next pair: The icosahedron has 20 faces. The dodecahedron has 12 faces. Next: The icosahedron has 12 vertices. The dodecahedron has 20 vertices.  Again we have a parallel to musical inversion. They may seem or look different. However, they simply re-arrange their topology but the same numbers.

The octahedron can be drawn inside the cube with vertices centered on each face of the cube (picture below). The same applies to the pair of the pair of the icosahedron and dodecahedron (picture below). Again, they are as closely related as inversions of the basic musical triad.

The grandest parallel between our music and the Platonic solids is found between the dodecahedron and our circle of fifths. Our circle of fifths has 12 basic key signatures (not counting enharmonic keys). Each one is located the distance of a musical fifth from the last one. The dodecahedron has 12 faces of pentagons (5 faces). You can superimpose the basic outline of the circle of fifths on a dodecahedron. Conclusion: over 2,500 years ago ancient civilizations thought of architecture as frozen music. Indeed, music and these 5 geometrical solids have strong parallels. To acquire the ability to gain such insights, I suggest musical instruction for our children. Plato said music should be mandatory study until the age of 30.

Cartesian coordinates
FigureTetrahedronOctahedronCubeIcosahedronDodecahedron
Faces4862012
Vertices46 (2 × 3)812 (4 × 3)20 (8 + 4 × 3)
Orientation
set
121212
Coordinates(1, 1, 1)
(1, −1, −1)
(−1, 1, −1)
(−1, −1, 1)
(−1, −1, −1)
(−1, 1, 1)
(1, −1, 1)
(1, 1, −1)
(±1, 0, 0)
(0, ±1, 0)
(0, 0, ±1)
(±1, ±1, ±1)(0, ±1, ±φ)
(±1, ±φ, 0)
φ, 0, ±1)
(0, ±φ, ±1)
φ, ±1, 0)
(±1, 0, ±φ)
(±1, ±1, ±1)
(0, ±1/φ, ±φ)
1/φ, ±φ, 0)
φ, 0, ±1/φ)
(±1, ±1, ±1)
(0, ±φ, ±1/φ)
φ, ±1/φ, 0)
1/φ, 0, ±φ)
ImageCubeAndStel.svgDual Cube-Octahedron.svgIcosahedron-golden-rectangles.svgCube in dodecahedron.png

 

Neolithic number eight is on the piano keyboard.

Neolithic Number Eight Permeates the Great Pyramid of Egypt

Neolithic Number Eight Permeates the Great Pyramid of Egypt. Also the modern piano keyboard. Here’s how.

  • First use of eight (8). The featured picture illustrates an octahedron.  It is a symmetrical, eight-faced, triangulated figure. All angles at their corners are 60°. Bisect the featured picture across the square at the center. The bisected octahedron then becomes two square based pyramids.  The above I call the positive. The below I call the negative. All square base pyramids imply an attached equal and opposite pyramid.  The mere existence of any square base pyramid, implies a counterpart. Granted, the Great Pyramid of Egypt has differing angles. It uses isosceles triangles.  But, the extra four reverse-faced pyramid is still implied. When they are joined, the square bases become internal. They literally disappear. There no longer is a separated square base. We have our first usage eight. As,  4 faces (postive)  + 4 (negative) faces = 8.

2nd Usage of Neolithic Number Eight

  • Image result for picture of the book cover by John Michell the View Over Atlantis
  •  Each side of the square base measures 440 shorter Egyptian cubits. Shorter cubits are 1.718…feet. A more encompassing measure is the Great Cubit. It measures 55 shorter Egyptian cubits. Thus each side of the Great Pyramid of Egypt is 8 Great Cubits. 440⁄ 8 = 55.  Reference John Michell, The View Over Atlantis. Therefore the Great Pyramid is 8 x 8 Great Cubits.

Neolithic Number Eight and Musical Octaves on the Piano Keyboard

  • Last, but not least. We will tie the Great Pyramid into concert note A-440 and its octaves. Its essential measures come from octaves of the concert note A 440. A higher octave doubles the vibrations per second. The lower octave cuts them in half. The lowest note on the 88-keyed piano is “A”. It vibrates 27.5 times per second. On the Steinway below, it is the furthest note to the left.

The musical keyboard of a Steinway concert grand piano

Here’s the connection. The height of the Great Pyramid is 275 cubits. Neolithic builders freely multiplied and divided by 10’s. This is because 10 ten was considered a synthetic number in antiquity. Reason: It totaled any two opposite numbers on the 3 x 3 number square. Diagram is below.  4 + 6 = 10. Or, 9 + 1 = 10. Etc. We now have the following:
  1.  The note A,  underneath Steinway’s name, vibrates 440/per second.
  2. The lowest note on the piano, also an “A” vibrates 27.5 /second.
  3. The length of any side of the square base on the pyramid is 440 cubits.
  4. The height of the truncated Great Pyramid of Egypt is 275 cubits

Image result for picture of the 3 x 3 number square on dsoworks.com

 

 

Suite Sonata or are Sonatas No Longer Sweets?

Suite Sonata or are Sonatas No Longer Sweet?  In my blog this means is the sonata form no longer sweet or in vogue? Let’s define our two featured terms. Firstly, I must state that by sonata, I mean the sonata form. Here are the two terms with definition:

  • Suite: In music, a suite (pronounce “sweet”) is a collection of short musical pieces which can be played one after another. The pieces are usually dance movements. The French word “suite” means “a sequence” of things, i.e. one thing following another. In the 17th century many composers such as Bach and Handel wrote suites. In the Baroque period, a sonata was for one or more instruments almost always with continuo. A continuo is mostly not used in the sonata form of the classical area. A continuo  means a continuous base line.
  • Suite Sonata - Which One? Answer is on the cover.
    Suites were the way for composers to go in the baroque era. They reappeared in the Romantic era.

 

  • Sonata form, also known as sonata-allegro form, is an organizational structure based on contrasting musical ideas. It consists of three main sections – exposition, development, and recapitulation – and sometimes includes an optional coda at the end. In the exposition, the main melodic ideas, or themes, are introduced.  After the Baroque period most works designated as sonatas specifically are performed by a solo instrument, most often a keyboard instrument, or by a solo instrument accompanied by a keyboard instrument. Quite frequently, the older baroque “sonata” was performed by a group of instruments. The term evolved through the history of music, designating a variety of forms until the Classical era, when it took on its own specific importance. 

The Sonata form was, in a way, a rebellion against the musical vehicle of the suite. Styles in fashion, furniture, music, manners etc, change in cycles. The earlier Beethoven sonatas used the sonata form. His later extended sonatas are more of the freer Romantic era. Most agree that Beethoven was the transition composer that launhced that Romantic era of music.

Suite Sonata or Are Sonatas no Longer Sweet?

I predict that styles, taste and music,  the Suite will rise above other forms. Suites are perfect form carrying beautiful melodies.  Each number in a suite can carry its own melody. This was the practice of the romantic era. The Holberg Suite by Grieg is such an example. As a composer, I love the form of sites. Here are 2 examples of my compositions:

  • The Dance of the Zodiac- with numbers for each of the 12 zodiac signs.
  • The Ringling Suite- inspired by paintings at the John Ringling Museum in Sarasota, Fl.
  • The Elemental Suite depicting the ancient belief in Earth, air, fire and water as elements.

Conclusion on Suite Sonata -The future will give sweets to the Suite. 

 
Musical Backbone of the Cosmos is the 3 x 3 number square

Musical Backbone of the Cosmos is a Number Square

Musical Backbone of the Cosmos is a Number Square.Musical unity of the cosmos issues forth from the 3 x 3 number square. Notice that 8 boxes surround the central box in the featured picture. Be it ancient or modern, the number 5 is the crux of tones or key signatures:

  • In antiquity 8 the tones of the scale were derived from a series of ascending  fifths. For example, “C” to “G” or “G” to “D”.
  • In modern times (today) key signatures are derived from a circle of 5ths. In music theory, the circle of fifths (or circle of fourths when approached backwards) is the relationship among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys. More specifically, it is a geometrical representation of relationships among the 12 pitch classes of the chromatic scale in pitch class space. *On ascending 5 tones to a new key signature, a new sharp is added that is 5 tones higher than the previous sharp. For example, G major has one sharp. It is an F#. D major has 2 sharps. It maintains the F# from the key of “G”.  D major adds new C#. Note: F#,G,A,B,C#.  Counting “F”, then  “C” is 5 letter names higher.
  • 16 Musicians are the musical backbone of peace.
    A true musical backbone of civilization is the symphony orchestra

Musical Backbone Set in the 3 x 3 Grid

Music comes in 8 bar segments. In the same way, on the featured picture boxes and their numbers, they are paired by opposites. For example, any boxed number and its opposite boxed number always totals ten. This includes 2 + 8. Or, 3 + 7, etc. Two paired boxes total 8 lines. This is at 4 lines per box.

  • In music the first 4 bars are called antecedent
  • The second 4 bars are called consequent

A complete musical song form typically uses 32 bars segments.  These are described by AABA. Each letter (A or B)represents 8 bars of music. Likewise, 8 boxes are encompassed by the circle on the featured picture. That makes for 32 lines: 4 (lines per square) x 8 squares = 32 lines.

Here is one for those who believe this blog is simply a “slight of hand.” Go around the perimeter. That is, the numbers surrounding the central “5”. Use two numbers at the time in either direction as follows. You get the same total. Here is the clockwise direction. 49 + 92 + 27 + 76 + 61 + 18 + 83 + 34 = 440. The ancients tuned to A-440. We tune to A-440.

Conclusion

A number of blogs on DSOworks  show the unity of the arts and sciences.  They demonstrate how this is effected  through the above featured number square. We know the way to peace. Why not follow the road signs? It’s that easy.

 

Neolithic number eight is on the piano keyboard.

Minute Waltz Glimpse of Chopin’s Genius

Minute Waltz Glimpse of Chopin’ Genius. When a genius creates, everything he or she does is great. Such is the piano music of Frederic Chopin. The Minute waltz has a touching story attached to it. It was inspired by a dog. The dog belonged to his muse and girlfriend, George Sand.

Minute Waltz as a Glimpse into Genius
Chopin’s Minute Waltz offers a rare look at Polish rhythmic complexity

The “Minute Waltz” is the nickname for the Waltz in D flat major, Op. 64, No. 1 by Frederic Chopin. It was written in 1847. It is a piece of music for the piano. It is sometimes called “The Waltz of the Little Dog” (French: Valse du petit chien). This is because Chopin was watching a little dog chase its tail when he wrote it.[1] The little dog was “Marquis”. He belonged to Chopin’s friend George Sand. Marquis had befriended Chopin. The composer mentioned Marquis in several of his letters. In one letter dated 25 November 1846, Chopin wrote: “Please thank Marquis for missing me and for sniffing at my door.”[2]

Related image

The waltz was published by Breitkopf & Härtel. It was the first of three waltzes in a collection of waltzes called Trois Valses, Op. 64. The publisher gave the waltz its popular nickname “Minute”. The tempo marking is Molto vivace (English: Very fast, very lively), but Chopin did not intend the waltz to be played in one minute as some believe. A typical performance will last between one and a half to two and a half minutes.[3][4]

The Complex Rhythms of the Minute Waltz Revealed

Just take a look at my 5 measure excerpt above for this:

  • The treble staff has the 2 beat motif of four eighth notes in measures 1 and 2.  The motif  is repeated many times during the waltz.
  • The scale that follows in has 8 eighth notes. They cover 4 beats.
  • Measures 4 and 5 have a dotted quarter note beginning each measure. The entails 1½ beats each.
  • Also in 4 and 5, following the dotted quarter are 3 eighth notes. Each 3 note phrase lasts for 1½ beats.
  • Finally, against all this melodic complexity, we find  a steady 1-2-3 beat in the left hand. It takes the form of “Bass-chord-chord.”

So Where Can I Hear David (this blogger) Play Chopin’s Minute waltz?

I am still booked six days a week through April 14 at the Gasparilla Inn. It is on the Florida isle of Boca Grande. There I get my choice of 2 vintage steinway Grand pianos. I played in the “living room” from 6:20 to 7:00 pm. Then I go in the dining room and play from 7 – 9 pm. See you there.

Chopin's Minute Waltz can be heard nightly at this setting on an exotic island
Gasparilla Inn where the music of Frederic Chopin is heard on vintage Steinway grands nightly as played by David Ohrenstein

 

A triad trinity is at the basis of any keysignature

Triad Trinity and Temples Play Tick-Tack-Toe?

 

Triad Trinity and Temples Play Tick-Tack-Toe. Musical Temples Become a  Reality in Tick Tack Toe Design. Here’s how: Right below is a blurry picture of the tick tack toe board blueprint that includes the basis of both the Holy Temples in Jerusalem. The First Temple based on the middle row of 3 vertical boxes. That corresponds to the pictured middle, vertical row of the C – E – G triad. On the 3 x 3 number square,  that relates to 9-5-1. The Second Temple was built to include the entire nine-boxed tick tack toe board. The 2nd temple extends the dimensions of the 1st to the left and right of 9-5-1. The First Temple was 60 x  20 cubits. According to the Tanach (Hebrew for Bible), he 2nd temple becomes 60 x 60 cubits.

Judaism and much of the beliefs of sacred  antiquity springs from this 3 x 3 board. It becomes apparent when the boxes are filled with numbers one to nine arranged so that any row of three totals 15. These 9 boxes, in a parallel way can also hold the  9 tones of the primary triads that define a key.The triad trinity  is named (Vertically left to right) tonic, dominant and sub dominant. In the key of “C”, the tonic is C-E-G. It  rests on the  central vertical row of the number square pictured  below.  Thus:

  • The triple-boxed shape of the 1st Temple is likened to the  middle 9-5-1 row of the 3 x 3 square.
  • The dominant of G-B-D occupies the same position as the 2-7-6 to the right.
  • The sub-dominant of F-A-C. occupies the same position as 4-3-8 to the left.

When used in combination, a triad trinity can be seen as outlining  the boundaries of the 2nd Temple. They also define the primary triads of C major.  The grid can hold the three primary triads of any key signature. These three key defining triads comprise 9 tones altogether.

Three main triads and Temples in Israel use the same form
Music as a part of service in the temple is quite fitting.

Related image

Triad Trinity is, Top to Bottom and Vertically Left to Right-  Sub Dominant, Tonic, Dominant

Three Main Triads and the Holy Temples in Jerusalem Use the Same Design
Our pattern of three primary triads is sampled here in C major. It uses the same design as the Jerusalem Temples

So Where else in the Grid Do We Find the Concept of  Musical Temples?

  1. The same grid that was the foundation of the  two Jerusalem Temples sets the basic diatonic musical interval of the fifth by vibrations per second. They are A-440 and E- 660.  Take the numbers two at the time. Go around the perimeter either way. Here is how to find the A-440: 49 + 92 + 27 + 76 + 61 + 18 + 83 + 34 = 440. Here is how to find the E- 660. You get the same sum either vertically or horizontally.  (49 + 35 +  81) + reversed as 94 + 53 + 18) + from the other side: (29 + 75 + 61)  + and again reversed (92 + 57 +16) = 660.
  2. The basic musical interval from which ancient and modern musical systems is the musical 5th. Our more modern music uses key signatures of the Circle of Fifths. Ancient music used individual tones derived by the actual fifth. The key or core number of the 3 x 3 number square is 5. Any other number placed in the center destroys its symmetry. This is the basis of its sacred order.

So How Did I Discover This Approach to the Trinity of Triads?

There is an American Indian spiritual presence on Oquaga Lake in the Catskill Mountains. For years I had been the piano player at Scott’s Oquaga Lake House. This spirit would accompany me on walks in wilderness. I call on it by the name of the Oquaga Spirit. On my product page of DSOworks I have some 80 of her poems. It is called, The Oquaga Spirit Speaks. I also have a free thumbnail of me reading the spirit’s poetry. It is on the front page. Here is a sample couplet: If it’s life you wish to live and enjoy to the marrow, then get thee a walking stick and hear the morning sparrow. 

Oquaga Spirit Speaks

3 Measuring Rods of the Great Pyramid
octahedron unifies space time

Octahedron Unifies Space Time in Ancient Cultures

Octahedron Unifies Space Time in Ancient Cultures. It does so from an Earthly viewpoint. First of all, what is an octahedron? It is one of the 5 regular polyhedrons. The other 4 are the tetrahedon, icosahedron, cube and dodecahedron. However you view any one of them, it is totally symmetrical. . Together they are also called the Five Platonic Solids. How is the octahedron identified? By its number corners, edges and faces. It has the following:

  • 8 faces
  • 6 corners
  • 12 edges
  • These total 26 topological features. See the featured picture above

Hexahedron.png
Cube
(dual polyhedron)

The octahedron has a non- identical twin brother (or sister). It is called a cube. They don’t look alike. But consider this. The cube has:

  • 8 corners
  • 12 edges
  •  6 faces

The twelve edges are the same in both. Whereas, the number of faces and corners trade places. They are as closely connected as twins. The octahedron pictured below contains a cube. The 6 corners of the octahedron have their points touching the center on the 6 faces of the cube.  For that reason, they are called dual polyhedrons.

File:Dual Cube-Octahedron.svg

So How is it That the Octahedron Unifies Space Time?

Unfortunately, the Egyptian Library at Alexandria was burned down. Its wisdom describing prehistory was destroyed.  Both the cube and octahedron were considered to be harmonious figures. This thought actually goes back to at least 11,000 B.C. Why harmonious? Because of the numerical relationship of its topology.

  • 12 is one-third greater than 8
  • 6 is one-third less than 8.
  • Eight is the number that defines the musical octave. That is the most harmonious and fundamental overtone of the entire overtone series. Guy Murchie thoroughly explains this in his two volumes of The Music of the Spheres.

How Does This Knowledge Date Back to Prehistoric Times?

The holiest sites of antiquity were designed as cubes or square base pyramids. The square base upright pyramid is found in the top half of the octahedron. Although the bottom half is not there, it is implied. As a cube, the Biblical Holy of Holies was set in back third of Solomon’s Temple. The total  rectangular perimeter of  the temple was 60 x 20 cubits.  The 20 x 20 cubit back  third becomes cubic. Also, the Ka-aba in Arabic literally means, cube. 

Much of the  world order of antiquity was destroyed. The cause was invaders from Afghanistan. The invaders were called Kurgans.  Riane Eisler discusses this her The Chalice and the Blade.

chalice-blade-cover
An award winning book by a great author

What was the purpose of these Holy Sites? – To spread harmony and peace throughout the world. This was effected by their geometric harmony. Since many were destroyed, war has ensued. In unity we find peace. In division we find war. The octahedron unifies space time. It defines space as a geometric form.  How does it define time? Each vertex of the regular triangles holds 60°. The 4 upper triangles of the octahedron have a total of 12 vertices. 12 x 60° = 720°. The lower 4 triangles total 12 vertices. They  also total 720°. The upper 4 triangles represent the 720 minutes in 12 hours of daytime at the equinox. The lower 4 triangles represent 720 minutes contained in 12 hours of nighttime also marked by the equinox.

Conclusion: Look for harmonious  models. Base civilization on  these models. Peace follows. The ancients did in through geometry. The same can also help us today.