Common Time is Anything but Common. The time signature (also known as meter signature,^{[1]}^{[2]} or measure signature)^{[3]} is a notational convention used in Westernmusical notation. Time signatures specify:

How many beats (pulses) are contained in each measure (bar).

The common signature (C) is an abbreviation of 4/4 meter.

Yet, when written for piano, which has a bass staff as well as treble, something unusual happens.

You see 4/4 for the treble staff; then you see 4 /4 for the bass staff (or common time for each staff). Subliminally that becomes four sets of four. What does that infer? Music draws on the 4 x 4 number square of Jupiter pictured below. The perimeter of numbers around the central four has four numbers in each direction of the compass.

Common Time Becomes Four Sets of Four

Where in antiquity do we find four sets of four? In the “magic square of Jupiter”. Jupiter, in astrology, is the bearer of good things as success in business and abundance.

Here is an ancient picture of the 4 x 4 number square. Note that four numbers are located on each side of the number square. Common time, as written for piano can be expressed by four fours: 4/4 and 4/4. At the center are four numbers: 7, 10, 6 and 11. The core of four creates the totality of of this magic square in a most special way:

Cross multiply the central numbers: (6 x 11) + (7 x 10) = 136

Next, add all the numbers in the number square from one to sixteen. They also total 136.

Now for a peek at a future post: add 136 to 136. The sum is 272. That probably is the most important number of all ancient civilizations! Plato refers to it as the grand number of harmony in antiquity. Revival of this knowledge is also a goal of Revivingantiquity.com

Five Bar Opening for Beethoven’s Fifth Symphony. Phrases have always come in four bar segments. Beethoven opens his 5th symphony with five bars. Look at the opening motif . We have two bars and a fermata (hold). Then we have three bars and a hold. The hold sign is called fermata. It looks like a bird’s eye. By arithmetic, 2 + 3 = 5. Was this an attempt by the Great Master to be cute? Did he think it was clever to start the Fifth with five bars of music? For the most likely answer, we must look into his Masonic roots.

Five Bar Opening: BEETHOVEN’S DELIBERATE USE OF THE FIBONACCI NUMBERS

Look at the red numbers: To the right of Leonardo Bonacci’s back, the highest red number is 55. However, he covers numbers 21 and 34. Each new number is the sum of the preceding two. We have 13 (visible) + 34 = 55. Then 21 + 34 = 55. So, let’s continue the series: 34 + 55 = 89. Next, 55 + 89 = 144. Next 89 + 144 = 233. The length of Beethoven’s opening section is exactly 233 bars. . Next 144 + 233 = 377. Beethoven’s development section is 377 bars. I think this was learned as a result of his Masonic association.

Why This Opening?

Beethoven, being the brilliant genius that he was, knew exactly what he was doing. When we listen to the symphony it sounds so natural; but can you imagine how he must have struggled to make the bar length come out right and still sound like that’s how it should be? Leonard Bernstein says of Beethoven and the 1st movement in The Joy of Music: “he will give away his life just to make sure that one note follows another inevitably.” In conclusion, I think that in addition to an even greater appreciation of Beethoven, we have graphic proof the relationship between music and numbers. This is why music lessons, theory and composition increase aptitude for mathematics. In no uncertain terms, music is a stimulus for success in every sense of the word!

External Link: I am also a composer, my wife a lyricist and book writer. Enjoy part of our brand new opera, now called Patra. We just showcased it in New York, before corona came.

“Octavian & Cleopatra” Imagine an operatic work that pours out incredible melodies, mesmerizes …link: Please share this post with as many as possible. It has relevance for today.

Musical Duration Dots Abound in Chopin Prelude #3. What do dots do to the duration of notes? In Western musical notation, a dotted note is a note with a small dot written after it. In modern practice, the first dot increases the duration of the basic note by half (the original note with an extra beam) of its original value. This means that a dotted note is equivalent to writing the basic note tied to a note of half the value. For instance, a dotted half note is equivalent to a half note tied to a quarter note. Subsequent dots add progressively halved value.Chopin uses this quite effectively in his Prelude #3. I call his use by the name of pyramided dots.That is because you can make the numbers 1,2, and 3 into a pyramid. See the illustrations below:

On the top treble clef, three dots are used twice:

The 1st beat of measure one.

Also, the first beat of measure three.

One the second treble/bass staff, two dots are then used twice:

On the third beat of the treble.

Single dots are used by Chopin in the 5th and 6th measures.

A triple-dotted note is a note with three dots written after it; its duration is 1^{7}⁄_{8} times its basic note value. Use of a triple-dotted note value is not common in the Baroque and Classical periods. It is quite common in the music of Richard Wagner and Anton Bruckner, especially in their brass parts.

What Effect do the Musical Duration Dots have on Chopin’s Prelude?

I feel Chopin’s compositions for the piano are highly innovative. My own teacher was Mischa Kottler. He studied with Alfred Cortôt. This was in the 1920’s. In turn Cortôt studied with a pupil of Chopin. I was taught to play a dotted note on the piano with more tone. Then certainly a double dot should be played with more tone. A triple dotted note should then have tone to the max. Tone, when played on the piano, should always sound pleasing. Ask me for a live demonstration of this Chopin Prelude on the newly refurbished 1924 Steinway Concert Grand at the Gasparilla Inn. I play for there for dining 6 nights weekly. See internal link:

Central Pillar and the 3 x 3 Number Square. In the featured picture the corner numbers on the Lo shu are 4,9,3 and 5. Remaining 5 remaining numbers are called the gnomon. These five are 2,7,6,1, and 8. There are three more possible corner/gnomon arrangements. That can be a subject for future blogs. The Tree of Life uses this one arrangement on the central pillar: The upper left corner v. its gnomon. Here are some instructions on how to read and compare the two systems. A vanished civilization knew what you are about to read. They enjoyed a Golden Age until they succumbed to a worldwide cataclysm. Certainly, Plato’s account of Atlantis fits this description. Allusions to this lost culture are found in the survival of ancient measurements. Below are a couple of my internal links.

Central Pillar and its Link to the 3 x 3 Number Square

First, what is a number square? It is found in recreational mathematics and combinatorial design. A magic square^{[1]} filled with distinct positive integers in the range . Each cell contains a different integer. The integers in each row, column and diagonal are equal.^{[2]} Now to compare the two systems:

On the number square, multiply the numbers of the upper left corner: 4 x 5 x 9 x 3 = 540.

Multiply the gnomon numbers: 2 x 7 x 6 x 1 x 8 = 672.

Next, look at the central pillar on the Tree of Life:

Multiply the Central numbers of the four emanations (circles):1 x 6 x 9 x 10 = 540. This duplicates 540 product of the upper left corner of the number square.

Take the central pillar numbers again. Add them: 1 + 6 + 9 + 10 = 26. Square 26 as 26² = 676. This is not the 672 gnomon product above. However, by rules of gematria, one can be added to each word or factor, in this case- circle, without essentially altering its meaning. We have 4 circles on the central pillar. Thus 672 + 4 = 676. We have congruence again. Gematria is explained in great depth by my favorite author, John Michell. His books are extremely rare and difficult to come by.

Three Square Code Structures Stonehenge. Life often offers us the polarity of opposites. Something stands in opposition to something else.

This tiny number square gives rise to the pillar arrangement at Stonehenge. It was used in countless ways by ancient historic and even prehistoric civilizations. Currently we are entering a new age. It will be marked by peace and plenty. This is also known as a Golden Age. The same arrangement that formed the Palestinian cubit and the Egyptian cubit also structured the pillar arrangement at Stonehenge. Some information I quote from my own internal link below:

Abu Mūsā Jābir ibn Hayyān explains how this number square was divided into various corners and gnomons. I show how the dotted points below were the options that were used for the appropriate Egyptian and Palestinian cubits. It was also used for builiding Stonehenge.

Multiply the 4 corner numbers: 5 x 7 x 1 x 6 = 210. The Palestinian cubit is 2.107 feet.

Multiply the remaining five numbers which are called the gnomon: 8 x 3 x 4 x 9 x 2 = 1728. The larger Egyptian cubit is 1.728 feet.

15th-century European portrait of “Geber”, Codici Ashburnhamiani 1166, Biblioteca Medicea Laurenziana, Florence

Three Square Code Uses Lower Right Corner and its Gnomon for Stonehenge

Numbers in a row are called a sequence. Sequence is also used in music and dance. Stonehenge uses 5,6,7, and 8:

5 x 6 = 30. This numbers inner stone circles.

7 x 8 = 56. This numbers the outer holes.

Plan of the central stone structure. The stones were dressed and fashioned with mortise and tenon joints before 30 were erected as a 108-foot (33 m) diameter circle of standing stones, with a ring of 30 lintel stones resting on top.

For fun, add 5 + 6 + 7 + 8 = 26. Now, let’s look at the two products and one sum we created from consecutive numbers 5,6,7 and 8: They are 30, 56 and 26. Any beginning student of chemistry knows these numbers define the most stable isotope of iron: (1) 30 neutrons. (2) atomic number 56 (3) Finally 56 – 30 = 26. That is the atomic number of iron. Why is this important? Iron is the ash of nuclear fusion on stars. When enough iron is at the star’s core, it explodes. This creates all the heavier elements than iron.

Here’s the big question: Does Stonehenge represent the cosmic stellar process of creation? Was this known in the distant past? Or, is this just another numerical co-incidence?

Plus one factor shared by Geometry and Math. This post applies to all of the prominent ancient number squares. For various reasons, they are also called magic squares. What characterizes these squares of numbers? In recreational mathematics and combinatorial design, a magic square^{[1]} is a {\displaystyle n\times n} square grid (where n is the number of cells on each side) filled with distinct positive integers in the range {\displaystyle 1,2,…,n^{2}}.

Each cell contains a different integer.

The sum of the integers in each row, column and diagonal is equal.^{[2]}

So Where is the Plus One Factor?

The plus one in number squares is one thing above and beyond obvious definition on the number square. Let’s look at the application in geometry first. The 3 by 3 number square is the smallest that can be constructed. It has 2 parallel lines that intersect two more parallel lines. Thus, 3 x 3 columns of vertical and horizontal numbers, are set on a 2 x 2 set of horizontal and vertical lines. Note: By custom, these parallel lines are not encased by a square. Surrounding these nine numbers by a square would then create 4 x 4 parallel lines. In a way, this hides the true nature of the number square in consideration. Number square perimeters should be left open ended as in the featured picture. I must confess, some of my earlier number square pictures are encased.

Here is the mathematical plus one factor: This number square contains the numbers 1 to 9. Any straight row of three totals 15. Opposite numbers total ten. Hence, the following sets of numbers total 10: 4 + 6; 3 + 7; 8 + 2; and 9 + 1. The plus one factor becomes the total of the opposite numbers- 10. Ten is above and beyond the nine. It is not even notated on the prime number square; being invisible, like the Deity. So how does this apply in religious antiquity?

The 1st of the 10 Commandments is the Plus One Factor

The 10 Commandments are thus modeled on the 3 x 3 number square. The 1st commandment is actually not a commandment. It is a statement of the presence of the Deity who is above and beyond everything. He is invisible, like the number 10. Compare it to the “colel” in numbers. That’s why in the 10 Commandments it becomes a statement as opposed the apparent nature of the other nine numbers.

Graphic Polarity Activation on the 3 x 3 square. The prime number square of every higher number square is the 3 x 3. The complete, traditional, square is pictured at the bottom of the featured picture. There are a number of possible arrangements, possibly seven. However, the understanding of the distant past begins with this traditional. Many blogs on DSOworks are about this number square. Check them out by key word. They are free to view. I currently have some 570 posts about two primary subjects: (1) Music (2) Number squares. Most of the posts on number squares about the 3 x 3. It is the simplest, yet the most complex. Both of these subjects were the backbone of a former and lost Golden Age.

The full set of rings, imaged as Saturn eclipsed the Sun from the vantage of the Cassini orbiter. It was 1.2 million km distant, on 19 July 2013. The brightness is exaggerated. Earth appears as a dot at 4 o’clock. Our planet can be seen between the G and E rings. Most important for this blog: The rings of Saturn are the most extensive ring system of any planet in the Solar System. Why is the important? There are three obvious perimeters around this number square. However, the number of possible rings goes to infinity.

Graphic Polarity Activation Instructions

Reference the traditional arrangement at the bottom as I point our a few of the possibilities. Also a have a number of internal links on infinity and this number square. Also, “perfect numbers” favor this number square. Ancients knew this. Today, our civilization does not. Of course, there once was a Golden Age. Today peace and plenty seem be elusive. Here are a couple of such links on DSOworks.com:

Graphic Polarity Activation on Isaiah 45:2-3New International Version (NIV)

^{2 }I will go before you and will level the mountains^{[a]}; I will break down gates of bronze and cut through bars of iron. ^{3 }I will give you hidden treasures, riches stored in secret places, so that you may know that I am the Lord, the God of Israel, who summons you by name.

How does Isaiah allude to graphic polarity activation with the above? Look at the chapter number. It is 45. Total the numbers 1 to 9 in the grid. The order does not matter. Sum = 45.

Next, look at the two verse numbers of the Scripture. They they are verses 2 and 3. The sum of the verse numbers define the central number of the number square. 2 + 3 = 5. “Five” activates the entire system of all number squares. This is the smallest and prototype of all number squares.

Breaking the gates of brass and cutting through the bars of iron allude to reading the numbers without the grid. See the internal links above.

Specific applications will be covered in future blogs.

Please credit an Indian spirit from the Lennie Lenape tribe for my knowledge of this number square. Her haunt is Oquaga Lake at Scott’s Oquaga Lake House. I was Scott’s piano player for some 15 summer seasons. There, I encountered the spirit on hikes in the Catskill Mountains.

Mexican Pyramid Squares the Circle. First. What is “square the circle?” Squaring the circle is a problem proposed by ancientgeometers. It is the challenge of constructing a square with the same area as a given circle. You may only use a finite number of steps with compass and straightedge. In 1882, the task was proven to be impossible. This was a consequence of the Lindemann–Weierstrass theorem. But wait a minute. Lindemann and Weierstrass did not consider R. Buckminister Fuller’s theories in conjunction with the Mexican Pyramid of the Sun.

Mexican Pyramid Really Spheres the Circle

So who was R. Buckminster Fuller?

R. BUCKMINSTER FULLER, 1895 – 1983

The Estate of R. Buckminster Fuller:

Hailed as “one of the greatest minds of our times,” R. Buckminster Fuller was renowned for his comprehensive perspective on the world’s problems. For more than five decades, he developed pioneering solutions that reflected his commitment to the potential of innovative design to create technology that does “more with less”. Born in Milton, Massachusetts, on July 12, 1895, Richard Buckminster Fuller belonged to a family noted for producing strong individualists. They were inclined toward activism and public service. Fuller developed an early understanding of nature during family excursions to Bear Island, Maine. He also became familiar with the principles of boat maintenance and construction. Below are a couple of internal links for further information.

Comparing the Mexican Pyramid of the Sun and Fuller’s Formula for Packing of Spheres

Packing of spheres in successive layers falls under a formula. It was discovered by R. Buckminster Fuller. It’s the number of the particular layer being considered, squared x 10 + 2. You can pinpoint how many spheres successivelyen circle a central sphere. Here are the 1st five examples.

For the 1st layer, 1² x 10 + 2 = 12 spheres.

The 2nd layer we have 2² x 10 + 2 = 42.

For the 3rd layer, 3² x 10 + 2 = 92.

The 4th layer is 4² x 10 + 2 = 162.

The 5th layer is 5² x 10 + 2 = 252.

Ancients looked at what things had in common by common numbers. This was regardless of unit of measure used. We can thus equate the packing of the 5th layer of spheres with the measure of the Mexican Sun Pyramid. Also, Plato’s Ideal City, in his Republic, had 2520 rings. Is this line of thought just fun? Perhaps. But also, perhaps there is no such thing as mere co-incidence?

Mayan Egyptian Connection Spans Atlantic Ocean. Who were the Mayans? The Maya developed their first civilization in the Preclassic period.^{[9]} Scholars continue to discuss when this era of Maya civilization began. Discoveries of Maya occupation at Cuello, Belize have been carbon dated to around 2600 BC.^{[10]} The civilization still had great endurance. Look back some 3,000 years. Mayan Tikal was around from 250 to 700 AD. At its height during the Late Classic, the Tikal city polity had expanded to have a population of well over 100,000.^{[37]}

Egyptian Mayan Connection

We have no written evidence of connection between the two cultures and their countries. But, we do have similarity by measurement. Many are too fixated on proof by written record. They forget that numbers and letters once shared the same symbol. This practice of using the same symbol for both is called gematria- a Greek word. Only our modern languages (except Hebrew) have separate symbols for both. In effect, when you made a measurement, you also could imply a (1) A word. (2) A phrase. (3) An entire sentence. (4) Even a larger unit of writing translated into numbers.

Great Court at Tikal

A surviving Mayan great court is at Tikal. It outlines a rectangle. East to West is 400 feet in length. North to South is 250 feet in width. The courtyard is flanked by platforms, large temples and pyramids. The ratio of the length to the width of the Tikal courtyard is 8:5. Note 400/250 = 8/5 = 1.6. Compare this to the Great Pyramid of Egypt. One side of the square base is smaller Egyptian 440 cubits of 1.71818…feet. Its truncated height is 275 of the same sized cubits. 440÷275 = 8/5 = 1.6.

Comparing the Great Pyramid’s Dates to the Mayan first civilization in the Preclassic period.^{[9]}

Based on a mark in an interior chamber naming the work gang. and a reference to the fourth dynasty Egyptian Pharaoh Khufu, Egyptologists believe that the pyramid was built around 2560 BC. As stated above, Mayan carbon dating goes back to 2600 B.C. The dimensions of the Mayan courtyard and of the Great Pyramid had the same ratio of 8 to 5. These numbers are part of the Fibonacci series:

Leonardo Bonacci who was born 600 years before Beethoven. The Fibonacci series, named after him. is colored in red. Beethoven used the Fibonacci numbers in composing his fifth symphony. In the internal link underneath, I relate Fibonacci numbers to ancient number squares. I don’t know that Fibonacci ever made the connection?

BEETHOVEN’S DELIBERATE USE OF THE FIBONACCI NUMBERS

Behind Leonardo Bonacci’s back, the highest red number is 55. Each new number is the sum of the preceding two. Number 34 precedes 55. So, let’s continue the series: 34 + 55 = 89. Next, 55 + 89 = 144. Next 89 + 144 = 233 (the length of Beethoven’s opening section). Next 144 + 233 = 377 (which the length of Beethoven’s development section). Beethoven, being the brilliant genius that he was, knew exactly what he was doing. Note in red numbers. Number five is the 1st separated number. It is not consecutive. Almost every composer uses phrases in 4 bars. The genius of Beethoven made the opening phrase 5 bars long on purpose. He knew what he was doing!

The Best Architecture and Music are at Least Based on Tradition

The Egyptian Mayan connection of the 8 to five ratio is found with Beethoven. Of course, in this symphony Beethoven uses many phrases also use 8 bar phrases. When we listen to his 5th symphony it sounds natural. Can you imagine how he must have struggled to make the bar length come out right? Leonard Bernstein says of Beethoven and the 1st movement in The Joy of Music: “he will give away his life just to make sure that one note follows another inevitably.” In conclusion, I think that in addition to an even greater appreciation of Beethoven, we have graphic proof the relationship between music, numbers and architecture in this post: The Mayan Egyptian Connection and even more. This is why music lessons, theory and composition increase aptitude for mathematics.

Merry Christmas Number Square Sums it all Up. December 25, Christmas Day, is the 359th day of the year (360th in leap years) in the Gregorian calendar. This blog is about the 7 x 7 magic number square. It could well be called the Merry Christmas Number square. First of all what is a number square?….. Magic number squares only require the sum of each row, column and diagonal of numbers equal the same sum. Second, what defines the 7 x 7 number square? Ancient civilizations were into number squares. Each of the 7 known planets had its own square of numbers. Also, seven was known as a virgin and prime number. It cannot be evenly divided by any other number other than itself.

Merry Christmas Number Square Sums it Up

The 7 x 7 number square was assigned to Venus. She was goddess of beauty. So was also connected to Greek goddess Athena. Because 7 was called a “virgin number,” it is associated with Mary, mother of Jesus.

Note the central number of the 7 x 7 square. It is number 25. That numbers Christmas Day.

Total all the numbers in the 7 x 7 number square. Those are the the sum of all the numbers from one to fifty. Their total is 1225. Read this as 12/25 and you have both Christmas month and day.

Below is an internal link to a fun story for the Holiday Season. Rubinoff, the famed violinist, literally brought holiday cheer all year round to children. I recently gave a concert under the auspices of the Ted Lewis Big Band Museum. The concert was under the baton of museum curator, Joseph Rubin in Circleville, Ohio. Since I was Rubinoff’s arranger and accompanist, Maestro Rubin asked me to play in the concert. Rubinoff spent a good part of his life giving concerts for children in public schools. In the “gag” picture, found in the internal link, comedian Jimmy Durante is playing Rubinoff’s violin (not very well, I might add) Rubinoff is clowning around at the piano. What fun!

Our best wishes for a happy holiday season- David and Sharon.