What exactly is Fibonacci Inversion?

Fibonacci Inversion is Like Musical Inversion of Intervals

Fibonacci Inversion is Like Musical Inversion of Intervals. . Inversion means to reverse the order, be it  of numbers or the two tones of a musical interval.  We also have melodic inversion. An example will be given by J.S. Bach. A unison inverts to an octave as 1 + 8 = 9. The second inverts to the seventh as 2 + 7 = 9. The third inverts to a sixth as  3 + 6 = 9. The fourth inverts to a fifth as 4 + 5 = 9. Inverting music is further discussed in my internal link.immediately below.

Music and Math Share the Rule of 9’s

Also, inversion also means turning the melodic intervals up-side-down.

Fibonacci inversion has a parallel in music
An example of melodic inversion from the fugue in D minor from J.S. Bach’s The Well-Tempered Clavier, Book 1.[1] Though they start on different pitches (A and E), the second highlighted melody is the upside-down version of the first highlighted melody. That is, when the first goes up, the second goes down the same number of diatonic steps (with some chromatic alteration); and when the first goes down, the second goes up the same number of steps.

Fibonacci Inversion is Also Like Inverted Triads

Image result for Wiki Commons illustration of C major triad and inversions
The same three basic notes are always there, but turned around. In order C-E-G; E-G-C, and G-C-E.

What are the Fibonacci numbers?

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones:[1][2]

{\displaystyle 1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }

For the Fibonacci  inversions, I take the 1st four numbers: 1,1,2, 3.  Each of these Fibonacci numbers with its inversion totals four. In similar fashion each musical interval with its total equals the same number. The number is different but the principle is the same.

The Fibonacci  inversion of “1′ becomes “3”. This happens for each “1”. The inversion of “2′ becomes 2. This number inverts to itself. The musical parallel is as an octave inverts to a unison.  Next, the inversion of “3” becomes “1”.

In order, the inverted numbers of  1,1,2,3,  are 3,3,2,1. Now we have to points to make (1) Ancient philosophers often either separated successive numbers and/or placed them together.(2) Ancient numbers squares give rise to the Fibonacci series. The internal link explains, in depth, how Fibonacci numbers dominate 4 x 4 number square.

Remarkable Foursome is a Mathematical Wonder

I keep within the ancient tradition of number squares for this next explanation.  Take the first four inverted  Fibonacci numbers, 3,3,2,1 – as a straight read. You have 3321. The is the numerical total of the 9 x 9 number square of the Moon. This square (with other ancient squares) is pictured below. It houses all the numbers from 1 to 81. Any two opposite numbers total 82. My page was copied from an earlier blog about the “Neolithic Periodic Chart”. Note the obvious vertical sequence of numbers on the periodic chart.  It is found on the diagonal on odd numbered squares. They are clearly reinforced in reinforced black ink. The numbers are 2,8,8,18,18,32,32, …

Hidden Periodic Chart Sequence Found in a set order

So what is my conclusion? Again,  there once was a former advanced civilization. It was based on number squares. Times were then peaceful and harmonious. Somehow it was destroyed. Could it have been the continent of Atlantis that Plato mentions in his writings?

Reversing polarities means going to the left or right from "C"

Reversing Polarities in Mathematics and Music

Reversing Polarities in Math and Music. First, let us define polarity:

  1. the relative orientation of poles; the direction of a magnetic or electric field.
    plural noun: polarities
    “the magnetic field peaks in strength immediately after switching polarity”
the state of having two opposite or contradictory tendencies, opinions, or aspects.
“the polarity between male and female”.
synonyms:differencedichotomyseparationoppositioncontradictionantithesisantagonism.
Reversing polarities is also found on this number square.
This number square acts as a bar magnet when it is cut in half (see illustration of bar magnet).
It is impossible to make magnetic monopoles from a bar magnet. If a bar magnet is cut in half, it is not the case that one half has the north pole and the other half has the south pole. Instead, each piece has its own north and south poles.

 

Reversing polarities as the subdominant and dominant are extremes.
The key of “C” is like the center of a bar magnet in this example.

Reversing Polarities in the Number Square

Many blogs on DSOworks are about this basic 3 x 3 number square. They are easy to access. I’ll use the bar figure of the numbers 9-5-1 for purposes of explanation. Taken as a straight read, any three numbers that cross the central 5 in a straight line is its own number backwards. It is just like a bar magnet. The number that always occurs is 1,110. Here, 951 + 159 = 1,110.

Next, let’s cut these numbers down the middle. We now have 95 and 15. This still has its North and South poles. This is like the split bar magnet on the left.  Note: 95 + 15 =110. Reversed- 59 and 51 =110. These numbers are smaller than the initial 159 and 951. However, they still have their poles. A theme on DSOworks.com is how this number square sets the cosmos in motion.

Reversing Polarities in Triads or Key Signature Relationships

The subdominant and dominant relationship mark the extremes in the poles of the keys. The tonic draws these two opposite keys together. Here’s how:

  1. The highest note of subdominant “F” chord is “C”. That is also the lowest note of the tonic triad.
  2. The highest note in the tonic triad example here is “G”. That now becomes the lowest note of the dominant “G” triad. This central “C” triad bonds the extremes together.
  3. Finally let’s cut the three letter names so “G” now becomes the central note. “C” is now set to the left. “D” is now to the right. This becomes like cutting the bar pole magnet in two new parts.

Conclusion: Polarity refers back to the 3 x 3 number square.  I would like to conclude a picture of  an emblem of the Lennie Lenape. They have a wonderful motto: “we are all family”. The Lenape (English: /ləˈnɑːpi/ or /ˈlɛnəpi/),[7] also called the Leni Lenape,[8] Lenni Lenape and Delaware people.[9] They are an indigenous people of the Northeastern Woodlands.  They live in Canada and the United States.[4]

Related image

Internet is the Keyword in Megalitic Times

common musical geometrical ratios

Common Musical Geometrical Ratios

Common Musical Geometrical Ratios. First, what is a ratio?

common musical geometrical ratios
Ratio example of intervals that make a perfect musical fourth.

Musically, in the diagram above: Every time a higher tone vibrates four times, the lower vibrates three. This creates the sound of a perfect fourth. All the perfect intervals and most harmonious tones of nature can be found at a bowling alley. Also, in the link below I explore the ratios of 6 to 5 found at Atlantis.  The size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as:

1:1 (unison),

2:1 (octave),

3:2 (perfect fifth),

4:3 (perfect fourth),

5:4 (major third),

6:5 (minor third).

Below are the only the Perfect Intervals found by bowling pins in an alley

  • The unison becomes the single, front standing pin.
  • The perfect octave is the 1st pin divided by the 2 pins in the 2nd row: 2:1 is the higher octave.
  • A perfect fifth is the ratio of the 3 pins in the third row divided by the two in the second: 3/2.
  • As mentioned, the 4 divided by the 3 makes the ratio of the perfect fourth.
 Ratios are often used to describe other items as: The ratio of width to height of standard-definition television.

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.[1]

Common Musical Geometrical Ratios of 5 and 6 were used by Atlanteans!

Clues about Atlantis are also found in the Temples on Malta
The ratio of the minor 3rd is 6 to 5.  It was the basis of a multitude of ancient measures. Read the internal link about Atlantis.  One of my books, The Ancient Engineers’ Philosophy: The Pinnacle of Thought in the Unified Culture of Ancient Builders, is placed in a triangle at a temple in Malta built circa 3500 B.C.

 

Clues in the Search for Atlantis Come With # 5 and #6.

When it comes to music, Atlantis lives!

Plato wrote of Atlantis in Timaeus that numbers 5 and 6  were prominently featured: People were gathered every 5th and 6th years alternately: Thus giving equal honor to odd and even numbers. The gathering of the population was for judgement and atonement.

Image result for free picture of Plato

 

Silbury Hill

Silbury Hill In England Tells Quite a Story by Measurement

Silbury Hill In England Tells Quite a Story by Measurement. How can you tell a story by measurement? At one time letters doubled as numbers. One symbol could represent both. This was called gematria– a Greek word. Numbers then can be understood as words, or even concepts.  Several monuments were built around  Avebury  in Wiltshire.  Silbury Hill was built as a  landmark Neolithic monument.

Image result for map of Silbury Hill
This famous hill is certainly a wonder of the ancient worlld

The hill has a unique latitude location: Divide the northern hemisphere into seven equal segments:

  • Karnak is found on the 2nd division.
  • Delphi on the third.
  • Silbury hill is on the 4th.
  • Its exterior angle, in turn, has the same latitude as the Gizeh plain. That is the location of the Great Pyramid

The  hill was developed in stages, over hundreds of years.  My primary source was Stonehenge and its Mysteries by Michael Balfour, Charles Scribner, NY, 1980. Much is also available on line:

Image result for picture of book Stonehenge and its Mysteries by Michael Balfour
This book also discusses Silbury Hill.
  • . Currently it forms a perfect circle. The diameter is 550 feet.
  • It was also originally a circle.  This was the 1st phase. The diameter was 120 feet. Circumference was 377 feet.

How Can these Silbury Hill Numbers Be Read?

Image result for DSOworks.com pictures of the 3 x 3 number square
The Master Code for Ancient Civilizations is Here

Our little “grain of mustard seed” has countless hidden codes. It has the potential to revive a Golden Age of Peace and Plenty. The hidden codes frame the Fibonacci series by sequences of fives. This smallest of number squares (3 x3) gives birth to the series. Next, here’s how 377 is a Fibonacci number. The series begins: 1,1,2,3,5 (the first number out of consecutive number sequence), 8,13, 21, 55, 89, 144, 233, 377, 610… Please note the Beethoven internal link. Beethoven uses the 377 as a sectional group of measures. The original circumference of Silbury was 377  feet.  With Beethoven, it is found in his Fifth Symphony. He deliberately made the opening 5 measures long. Usually musical thought comes in multiples of 4 bars.

Beethoven: His Fibonacci Fifth – DSO Works

{\clef treble \key c \minor \time 2/4 {r8 g'8[ g'8 g'8] | ees'2\fermata | r8 f'8[ f'8 f'8] | d'2~ | d'2\fermata | } }
Note the usual 4 bars, almost always used by composers of music,  becomes 5 bars in the hands of the Great Master, Ludwig van Beethoven!
.

Tens: Here’s the Formula on How Tens Grow into Infinity – DSO Works

Also be sure to read the above 2nd internal link. You’ll discover how repeated fives take hold of this number square by opposite pairs of numbers.

Back to Silbury and its original diameter: You can find 15 in  8 distinct straight line totals:  3 are vertical. 3 are horizontal, 2 are diagonal. 3 + 3 + 2 = 8. Next, the product of these eight:  8 x 15 = 120. The diameter of the 1st phase of the Silbury Hill was 120 feet.

The Next Phase of Silbury Hill also Draws on the 3 x 3 Number Square

Land ahoy! The expert a claims southern Britain was a series of islands linked by waterways, channels and swollen rivers, and that Stonehenge was effectively located on the coast. The mound (pictured) would have acted as a lighthouse and harbour for those travelling by boat

Finally,  the 2nd phase has a diameter of 550 feet. Like, I stated, repeated fives are part of a hidden code. The code is amply described in many of the 510 posts currently on DSOworks.com. That makes a circumference of 1728 feet. Note in the picture below how 1728 was traditionally used on the number square. This square has been historically divided into a 4 number corner: The left over 5 numbers was called its gnomon. Below is strong connection between the Great Pyramid and Silbury Hill. The circle around the truncated Great Pyramid and its mirrored underground image is 550 cubits. The circle around this diameter is 1728 cubits. The numbers employed by the Great Pyramid and Silbury Hill are the same. An illustration of the gnomon and corners used by both structures is given below. Incidentally, the corner numbers multiplied approximates the Palestianian cubit of 2.107 feet. 5 x 7 x 6 x 1 = 210. The larger Egyptian cubit is 1.728 feet. Again, this measure comes from the 5 numbered gnomon. Silbury Hill and the Great Pyramid were both built developed primarily to illustrate the various ways the 3 x 3 number square can be used.

Mirrored pyramid is implied by the Great Pyramid of Egypt.

 

 

 

Musical ornaments

Musical Ornaments – Those for and Those Against

Musical Ornaments  – Those For and Those Against. Everyone has opinions about the necessity of ornaments in music. I suppose the same could apply to the use of ornaments in fashion. At this point I venture a prediction: The use of set ornaments in music and in dress will return quite strongly. Richard Wagner commented on ornaments. He would tell musicians: “Pay attention to the small notes…The large ones will take care of themselves.”

Image result for Wikicmmopns a picture of Richard Wagner
Richard Wagner stressed the importance of grace notes and ornaments.

Nature of Musical Ornaments

Why, at one time, were ornaments belittled?  Some thought they were only needed because of  weaker harpsichord sounds. The modern piano, they thought, did not need reinforcement. Among those who held this opinion were Marmoutel, Le Couppey and Méreaux. Yet, both the voice and violin had rich ornamentation. They had the same volume in the past as they have today.

Image result for Wikicommons a picture of C.P.E. Bach
C.P.E. Bach seated at the keyboard.

C.P.E. Bach wrote a definitive manual playing keyboard instruments. While in Berlin, C.P.E. wrote, Versuch über die wahre Art das Clavier zu spielen (An Essay on the True Art of Playing Keyboard Instruments). “Both Haydn and Beethoven swore by it.”[9] By 1780, the book was in its third edition. It laid the foundation for the keyboard methods of Clementi and Cramer.[1]Bach presented his thoughts on the virtue of ornaments in his treatise. He believed that without ornamentation the best melody becomes empty and dull.

  1. He comments on how most composers use them profusely.
  2. On how they can connect notes.
  3. Ornaments can enliven music.
  4. They attach particular stress and importance to the notes they adorn.
  5. They make musical meaning clear: They can emphasize either sad or happy qualities.
  6. Ornaments can actually improve a mediocre composition.

 Musical Ornaments of J.S. Bach Kept Intact with my Own Arrangement of

The Boogie Man of the Opera

Robotic repeats lack tempo rubato

Robotic Repeats Lacking in Tempo Rubato

Robotic Repeats Lacking in Tempo Rubato. First of all, what is tempo rubato? Tempo rubato ([ˈtɛmpo ruˈbaːto]; “free in the presentation”, Italian for “stolen time”). It is a musical term referring to expressive and rhythmic freedom. It is done by slightly speeding up and then slowing down of the tempo of the music. This is totally at the discretion of the soloist or the conductor.

In this context I’ve featured a picture from the movie, Pink Panther. A Princess “Dala” receives a gift from her father.  It is the largest the largest diamond in the world. This huge pink gem has  a tiny discolored inclusion. It resembles a leaping panther. She escaped from her country with the diamond after a hostile takeover. Her country is called Lugash. 

During a costume party at Dala’s villa in Rome, Sir Charles and his nephew separately attempt to steal the diamond. Shockingly, they find it already missing from the safe. In the true Italian sense of the word, we have a series of “rubatos.” Ironically, Henry Mancini’s four-note theme from the Pink Panther, is played in strict tempo. No rubato.

 

Pink panther63.jpg

Robotic Repeats Avoided in Tempo Rubato

Speed and power are the gods of today. This is mostly accomplished under steady tempos. These “gods” were shunned in the past. In defining “rubato”, within the context of the beat, there is much give and take. Mozart and Chopin’s use of rubato added to their fame. Nothing was ever repeated the same way twice in this technique. On a repeat, you were expected to played it differently. Rubato is quite effective in slow, emotional music. It was used in romances, adagios and nocturnes. However, even in the 1600’s Johann Froberger recommended that a lament be played “without a steady beat.” There are other types of music lacking steady beat. My free sample below  of my own Dervish Dance illustrates another genre. Is is excerpted from DSOworks.com That is my website.

King David’s Dance is a Dervish Style Piano Composition

 

Ancient number groupings

Numerical Meaning by Delving into Words

Numerical Meaning by Delving into Words. Today, numbers are adjectives. They define the quantity of a noun. Here are some simple examples: She has one cat. He has three birds. Our neighbor has two trees in the front yard. In the remote past, numbers represented both qualities and quantities. Furthermore, in China a numerical tradition of the Lo Shu survives. It is described below. Are numbers quantities or are they a lot more?  That depends on you civilization.

Numerical meaning for modern man
The modern mind prefers seeing numbers as adjectives rather than nouns.

21st century man prefers numbers as adjectives.  One side works with music and math. The other deals with words. Currently, most become quite uncomfortable in trying  to see numbers as nouns. There are exceptions. Musicians give numbers more meaning. Some examples are working with  the 8 bar phrase. The 32 bar period had its own meaning. . Another is animation by 3/4 or by 2/4 time. The prime number square  is the 3 x 3. It is the featured picture. Genesis is all about secret codes of this number square. So is the Chinese Lo Shu. Likewise, Greek mythology used this square of numbers. I must ask, who is more primitive? Someone who understands and works with all its hidden codes, or those who are ignorant of them?

1st sentence of Genesis Archives – DSO Works

Lo Shu Square (simplified Chinese洛书traditional Chinese洛書pinyinluò shū; also written 雒書; literally: Luo (River) Book/Scroll), or the Nine Halls Diagram (simplified Chinese九宫图traditional Chinese九宮圖pinyinjiǔ gōng tú), is the unique normal magic square of order three. Lo Shu is part of the legacy of the most ancient Chinese mathematical and divinatory (Yi Jing 易經) traditions, and is an important emblem in Feng Shui (風水).   Feng Shui is the art of geomancy concerned with the placement of objects in relation to the flow of qi (氣) “natural energy”.

Delving into Numerical Meaning

When placed in balanced mathematical relationships, numbers are more than adjectives. Such is the case with the featured picture.  Please read the internal link (1st sentence of Genesis) before continuing. Now let’s do the following. The sum of the numbers 1 to 9 totals 45. Next 45² = 2025. In many ancient languages, letters doubled as numbers. There was no brain split between the two sides and two disciplines.  This was called gematria. Information is available online. Also, on this website I quote the Reverend John Michell. He and Robin Heath amply cover gematria in their works.

For our last thought for numerical meaning in this blog, take 2025. Again, this is 45².

  • In Greek add the gematria of 3 gods:  Zeus = 612. Apollo = 1060. Hermes = 353. 612 + 1060 + 353 = 2025.
  • Gematria of words were often pyramided. Other DSOworks.com  blogs deal with this topic. The four pyramided letters for “Torah” in Hebrew is 2023. Tav = 400,Tav + vav = 406. Tav + vav + reisch = 606. The full Hebrew word as tav, vav, reisch,  and hei = 611. Thus. 400 + 406 + 606 + 611 + 2 = 2025.  There  are 2 additional commandments added to the 611 by tradition.  With the + 2 we have: 2023 + 2 = 2025.  Again, this is 45².

There once was a golden age. Its guiding vision was the featured number square. This time of milk and honey can now be regained by the knowledge of how this basic number square functions..

Image result for pictures of book covers by John Michell
John Michell  and Robin Heath  quite descriptive  about the subject of gematria.
Earth Moon system by a number square

Earth Moon System and 27 x 27 Number Square

Earth Moon System and 27 x 27 Number Square. The barycenter  is the center of mass of two or more bodies that are orbiting each other. Earth and Moon have a shared barycenter. As such they are a dual planetary system.  Barycenters are an important concept astronomy and astrophysics. This is the case for the Earth–Moon system. The barycenter is located on average 2,902 mi from the Earth’s center. This is well within the planet’s radius of 3,963 miles.

Earth Moon system are are dual planetary system
Sharing a center of gravity within the Earth’s radius

phases of the Moon Archives – DSO Works

What do things have in common? That was the definition of wisdom in antiquity. Look at the internal link just above.  Above, this Moon link applies to our four types of musical triads.

Four types of musical triads parallel the phases of the Moon.

Earth Moon system and the 27 x 27 number square

Four Hundred Fortieth Post Defines Genesis   You might want to review the previous post.  My current post (441) is simply a continuation.  The common Earth based system for both is implied on the 27 x 27 number square. Here’s how:

  • Look at the core number of this number square. It is numbered three hundred and sixty-five. That defines the days of our solar year. Its core is the essence of any number square.
  • Look at the white diagonal from #352 to #378.  Each each lunation is approximately ​29 12 days (29 days, 12 hours, 44 minutes, 3 seconds, or 29.530588 days).  It is common for the months of a lunar calendar to alternate between 29 and 30 days. There are  twelve such lunations. The Lunar year is  is therefore 354 days, 8 hours, 48 minutes, 34 seconds (354.367056 days). Both 354 and 355 are included in this same diagonal.

Both the timing of the the Earth and Moon are found the the aforementioned diagonal line. The Earth is the main and central planet. The Moon has its center of gravity on the same diagonal. Likewise, Moon is part of the 365 day Earth based system.

At one time civilization was structured by number squares. It was a time of peace and prosperity. Does that describe the legend of Atlantis?  Plato describes in Critias what his grandfather was told by King Solon. Solon heard the story from the Egyptians. Atlantis was a mighty power based on an island in the Atlantic Ocean. Perhaps, over time, they also lost their interaction with number squares? Many blogs on DSOworks.com cover number square mathematics.

 

 

Operatic Broadway

Operatic Broadway – Blurring the Lines Has Precedent

Operatic Broadway – Blurring the Lines Has Precedent.  A number of modern musicals cross over into operatic territory. Opera is an art form in which singers and musicians perform a dramatic work. It  combines text (libretto) and musical score.  Opera usiually has usually in a theatrical setting.[1] Singers do two types of singing: recitative, a speech-inflected style[2] . The second are arias, a more melodic style. Opera incorporates many of the elements of spoken theatre, such as actingscenery, and costumes and sometimes includes dance. Traditionally, it is sung all the way through. Musical theater, on the other hand has featured songs. However, most of its book is spoken. Recently there has been more and more cross over between opera and musical theater. They include Rent, Les Mis and The Phantom of the Opera. 

The Atlanta Opera Lucia di Lammermoor finale

Blurring Musical Vocal Boundaries Has a Romantic Precedent

The oratorio dates back to the 1500’s. It reached a climax under hand of Handel. The Romantic movement of the 19th century revived his ideals. Like Handel, with the Romantic composers, half were written in a Biblical or religious vein. The other half was secular or historical. There was only one difference: Handle’s historical oratorios were limited to either classical Greek or ancient. Handel examples include Hercules, Semele, “Alexander’s Feast”, or Alceste. Romantic oratorios had a broader scope. Instrumental works took on more significance. Here are a couple of examples:

  • Berlioz’s Romeo and Juliet is somewhere between a symphony and a cantata.
  •   The Damnation of FaustOp. 24 is a work for four solo voices, full seven-part chorus, large children’s chorus and orchestra[1] by the French composer Hector Berlioz. He called it a “légende dramatique” (dramatic legend). It was first performed at the Opéra-Comique in Paris on 6 December 1846. It has been seen as a symphony, oratorio or opera.

Operatic Broadway is Simply Following in this Precedent of Mixed Tradition

Octavian and Cleopatra: a 2 Act Opera in English – DSO Works

I, blogger David, have been the composer of three such works, My book-writer lyricist has been my wife Sharon Lesley Ohrenstein.  Check out the internal link above for some quite exciting live examples. Sharon plays Cleopatra.  Contact us on DSOworks@gmail.com if you are interested in our up and coming works. We need a new sound for the new times we are entering. This translates into meaningful income.

Lesley and Ohrenstein’s “Octavian & Cleopatra” – YouTube

The youtube example below sets up our Operatic Broadway show.

https://www.youtube.com/watch?v=bhGHHXuBr8Q
Dec 12, 2007 – Uploaded by Rudder3218

“Octavian & Cleopatra” Imagine an operatic work that pours out incredible melodies, mesmerizes …

New Musical Theory Model

New Musical Theory Model Based on the Dodecahedron

New Musical Theory Model Based on the Dodecahedron. We know about the circle of fifths. A quick review: In music theory, the circle of fifths (or circle of fourths) is the relationship among the 12 tones of the chromatic scale.  It shows their corresponding key signatures.  Also found arethe associated major and minor keys. Musical tones use 7 letters of the alphabet. They are A-B-C-D-E-F-G. These letters repeat over and over to encompass the full key signature span.

Standard Musical Circle of Fifths
Circle of fifths show the relationship of major keys and their minor relationships.
 At the top of the circle, the key of C Major has no sharps or flats. Starting from the apex and proceed clockwise by ascending fifths.  The key of G has one sharp. The key of D has 2 sharps….. Proceed counterclockwise from the apex by descending fifths: The key of F has one flat. The key of B has 2 flats…… At the bottom of the circle, the sharp and flat keys overlap, showing pairs of enharmonically equivalent key signatures.

New Musical Theory Model in Featured Picture

Related image
The Dodecahedron is colored yellow. It it a 3 dimensional representation of the Circle of Fifths.

Mesolithic Times Had a Unique Standard

Click on the above internal link. It offers background on the 5 Platonic solids. The Platonic Solid we are concerned with here is the dodecahedron. In antiquity it symbolized the overall shape of the Universe. By this rational,we live in a musical Universe. As for the featured picture, note: The major keys are written in the center of each pentagon. Start in consecutive order (from C major) from its lower left point. The tone on the lower right completes the 5 tones. In the case of “C” major, it is the tone “G”. This lower right begins the next pentagon. It is completed at “D” …

  • Ascend 5 consecutive tones with each pentagon.
  • The fifth tone on the lower right of each pentagon defines the key signature name of the next pentagon. This goes for the complete cycle:  “G major, D major, A major…”
  • The 12 pentagons make the complete cycle at F major. “F” major’s lower right hand tone is “C”.  The outline of the dodecahedron is complete at this point.