Music and Math Share the Rule of 9’s


Music and math share the rule of 9’s.  I find this very appropriate because music and numbers also share in usage of the same side of the brain. Words, on the other hand, use other side of the brain.  I will first demonstrate the rule of 9’s by music. Then I will demonstrate it with numbers. Inversion means to reverse the order, be it  of numbers or the two tones of a musical interval sounding  at the same time.  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. Change in the quality ( major to minor intervals or diminished to augmented)) will be the subject of a future blog


Now let’s look at inverted numbers.  If someone is adding an entire column of numbers in a ledger, once in a while the digits in any one number might be mistakenly reversed.  For example, instead of writing 189, you write 198; or instead of writing 235, you write 532. This act is comparable to inverting or reversing musical tones. If what you expected the total to be, as opposed to what it is, differs by a multiple of nine, then you inverted or reversed the digits of the numbers in the manner that I have just demonstrated. Here is the proof: 198-189 = 9. With the second example, 532 – 235 = 297 When you divide 297 by 9, the quotient is 33. Let’s take a larger number: When you record 23,572 as 32,572 the difference is 9,000. That is also a multiple of nine.

In conclusion, since math and music are virtually twins, to study one without the other is like separating these twins, How sad! The study of mathematics must be complemented by the study of music.

Every child in school should be given the opportunity to learn music through piano lessons or musical programs at school.

The point I made above is amply demonstrated by Albert Einstein; a great mathematician who  played the violin. The fictional detective genius sleuth, Sherlock Holmes, also played the violin.  Arthur Conan Doyle realized the importance of music and how it is a part of superior intellect. One more word on the rule by 9’s. In the highest court of our land, the Supreme Court, we also have a rule by nine. The founding fathers of America were also brilliant.

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Albert Einstein is the prototype of the musical mathematician. He played the violin. In his writings he discusses how one the the best moments of his life was one he received a good review by a music critic for playing .

The 4 x 4 Magic Square and Music?

The 4 x 4 number square and music? The meaning of of number squares has degenerated to nothing more than a puzzle or a simple  curiosity. In antiquity the magic number square had a multitude of associations. They were of extreme importance. Ancient builders recognized 7 primary number squares. Each one was thought to invoke the power of one of the seven recognized planets. For example:  3 x 3 invoked Saturn.  5 x 5 called on Mars. The 4×4 magic square, pictured below, was said to invoke Jupiter. John Michell, in The View Over Atlantis, points out how the statue of Jupiter at Olympus was built by the numbers of this square.


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The wood carving above was done by Albrecht Durer in 1514. It is called Melancholia. The picture on the right is a blowup of a section of the same woodcarving of the angels wing against numbers one and fourteen. The angel is apparently depressed because she is overwhelmed by all her labors. The 4 x 4 square holds the cure for the angel’s melancholy mood. Gustav Holst, in his The Planets, refers to Jupiter as the bringer of Joviality. Don’t worry. Help is on the way for this angel!

Now, how can you invoke this number square through music and get rid of sadness and melancholy? King David was able to cure Saul’s melancholy with music:  If each square represents one measure, four measures (the top row) represents the smallest complete unit of musical form. Two phrases are often placed together in a “question-answer format”.  Hum the opening of Twinkle,Twinkle, Twinkle Little Star, how I wonder what you are? The first part is the question the second is the answer. Phrases are sometimes placed together to make a 16 bar musical double period. Dancing is often done to units 16 bar measures of music. Joining phrases in such a manner gives the listener or dancer a sense of symmetry and balance. That in turn brings happiness. Viva la music!


Our Music, Bodies, etc. Use the Same Numbers

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Our music, bodies, etc. use the same numbers. As Fibonacci  numbers develop by successive addition of numbers that are adjacent; man, music and even love, yes-love,  use the exact, same numbers.   When the larger number of the two in this series is divided by the adjacent smaller number; the ratio keeps getting closer to what is called the Golden Section or phi. This ratio never comes out evenly: It is 1.6180339… The Fibonacci numbers are named after an Italian mathematician.   Numbers that develop into this ratio, in order of size are: 1,1,2,3,5,8,13,21… The higher the numbers by successive addition,  the closer it comes to phi.

The Use of Phi in Music

The formula for the phi ratio is: square root of 5 + 1; whose total is then divided by two. The piano keyboard is set up by the Fibonacci series. We have two spacing of black keys. They are by 2’s and by 3’s for a total of 5. Taking C major as the prototype, from one “C” to the next is 8 tones on the white keys. Thus, from C to C’  in addition to the 1 octave we have, 2 black keys, then 3 black keys and 8 white keys. The total black and white keys are 13.


All tones have what’s called overtones that vibrate sympathetically with  the fundamental tone. The 1st overtone  is the octave of 8 notes. The second overtone is the 5th, Both overtones are Fibonacci numbers. Not co-incidentally, the Great Pyramid of Egypt uses the same numbers in its 5 to 8 ratio of its height to one length of the square base.


The factors of the Golden Section ( another name for phi) are 1,2 and 5 (as already described above). People have one torso, one head, two arms, two legs, five fingers on each hand and five toes on each foot. We are the mathematical manifestation of the factors of phi.   Most significantly, love ties the whole system together: The Hebraic verb for love (pronounced, Ahav) uses three Hebrew letters.  Spelled out in English they are: aleph, hei and beis. Aleph is also the Hebrew symbol for 1; hei is also the symbol for 5; and beis is also the symbol for 2. What is the lesson? Life and music are all about love.


Harmony’s Number


 Harmony’s Number is 272. My unpublished manuscript, entitled, The Sacred Engineers Philosophy, the Pinnacle of Thought in the Unified Culture of Ancient Builders, was placed on the etched stone triangle at the Mnajdra Temple on the Island of Malta. The treatise discusses the number codes that the ancient engineers used for building temples. These same codes defined the vibrations per second of the musical tones of the ancient diatonic scale. I merely rediscovered the codes that they used.

Harmony’s number is 272. What can this possibly mean?  In antiquity, measure and music shared a common bond. The scale in use by the ancient Greeks was called diatonic. They knew the vibrations per second of each tone (Refer to Issac Asimov’s On Physics for the list).  Middle “C” vibrated 264 times per second. “C”, and octave higher, vibrated twice as fast: 528 times per second.  “A” still vibrates in symphony orchestras today at 440/sec. “E” above middle “C” was 264 times per second, etc. Numbers are readily available.

In antiquity, harmony referred to a balance of forces that were able to co-exist in peace. Harmony, according to Webster,  is an agreement or an accord. In music Webster states that it is a simultaneous sounding of tones that are pleasing to the ear. When a pair is in accord, be it people or music, they are able to co-exist harmoniously. John Michell in The View Over Atlantis elaborates on how 272 was connected by gematria  (defined as the former unity  that letters and numbers shared). The example that Michell uses goes back to ancient Greece with Harmonia who was the wife of Cadmus. The gematria of her name in Greek equaled 272. Numbers, in antiquity, could represent words, things and forces. Many ancient languages did not have separate symbols for numbers. Letter doubled as numbers. You could therefore  total the numerical values of the letters in a single  word, phrase or even a lengthy sentence. Each had a sum. The writings of John Michell covers the subject amply.


Harmony’s number  is 272.  Question: In what possible way can 272 bind opposing forces as one? Answer: Numbers 5 and 6 were considered polar opposite numbers. Five was yin. Six was yang. The five pointed star was a symbol of man. The hexagon, with 6 sides, was connected to the cosmos and the inanimate. In the Medieval gardens and decorations the five strong petaled rose was thought of yin; while the six- petaled lily was yang (refer to John Michell in City of Revelation). Many crystalline forms, the snowflake, even the cells of a honeycomb are six sided…So how does 272 unite man and the cosmos through numbers 5 and 6? Answer:  272 results from basic arithmetic of numbers 5 (yin) and 6 (yang). As, 6 x 5/ 6 + 5 = 2.72727272… Or stated in another way, 30/11= 2.727272….


As harmony’s number in ancient Greece was 2.72; even older megalithic cultures all over the world used 2.72…. as it defined the 2.72 feet of the megalithic yard.  Its hidden existence was  discovered by professor Alexander Thom of Oxford  in the 1960’s  He found that it measured most of the ancient sites in England.  Among the sites that were built by the megalithic yard are (1 Karnak Temple in Egypt whose length of either side of the rectangular base was 440MY. (2) the placement of the Sphinx in relationship to the Great Pyramid which was 440 MY distant,  (3) the  perimeter around the foundation of Persepolis in Iran was a square that measured 4 x 440 MY. (4) Rhe rectangular lengths of Troy in Turkey (2 x 220MY) combine to equal 440 MY. (5) The Akapana of Tiahuanaco in Mexico had one length and one width of its rectangular foundation together span 440 MY.  “A” , at 440, exists in two systems: (1) the old diatonic note “A” above middle “C” and (2) the modern well tempered “A” of our current tuning system.   As far back as 3500 BC the megalithic yard is found on Malta at the Mnajdra Temple at the higher section of the North entrance.  A triangle is etched in this section. Gerald J. Formosa found its dimensions and most of the temples in Malta that were defined  by the MY. He wrote about it in, The  Megalithic Monuments of Malta. Yes, music married architecture. The minister was Mr. Megalithic Yard, and their child was called harmony!


A very significant use of 272 bonds Islam and Judaism. The three letter root of “Arab” in Hebrew is ayin (70), reisch (200), and  veis (2). Its gematria is 272. The word, Hebrew, uses the same letters, re-arranged as: ayin (70), beis (2),and reisch (200)= 272. In this manner, both cultures have an harmonic accord. Also, the two primary Hebrew names for God in the Torah equal 272 but in a hidden way.. Scholars have pyramided the letters of a word to come up with the hidden pyramided value.  Thus the Tetragrammaton becomes: yud (10)  +  yud, hei(15)  +  yud, hei,vav (21)  + the full name- yud, hei, vav, hei (26).  The total is 72.  Using the same technique for the other primary name, Elohim, the total is 200. Therefore, the sum of these two pyramided names also equals  272.  I can only conclude that if we allow it, the natural state of the Middle East and the world  is all about peace and harmony. One last 272 point. When you multiply the Hebraic factors of the pyramided names of the Lord (described above) as 72 X 200, you get 14, 400…….. 2.72  defines the number of feet in a megalithic yard, 14,400 defines the number of feet in the “megalithic mile” which is even found in China as a measure called the Pu.

Uniting Music and Measure

Uniting music and measure raises a basic question. How? I have briefly blogged about this in Stonehenge Was Built by Musical Tones.  I will be developing  a theme sporadically throughout my blogs which is:  How the numbers by which architecture was measured duplicated the numbers of musical tones of the ancient diatonic scale.  As one philosopher put it: Architecture is frozen music. The units of measure by which ancient buildings could be  measured  varied by the culture.  Numbers, however, were the same.   Units of measure could include the shorter (1.718′) or longer Egyptian cubits (1.728′). They could have used the Palestinian cubit (2.107′).  The megalithic yard (2.72′) was extremely popular. The Roman pace (2.433′) or even the 12 inch English foot were utilized.   In the realm of the old diatonic scale,  a number of authors document their numbers in terms of vibrations per second. Thus I was able to see how the numbers of measure and music happily correspond. Authors that document the old diatonic scale in terms of vibrations per second include Issac Asimov in On Physics and Guy Murchie in The Music of the Spheres.

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The Great Pyramid of Gizeh is the large pyramid on the left side.


A 175 to 176 ratio of measures were used by cultures across the face of antiquity. This is well documented in John Michell’s  scholarly treatise entitled Ancient Metrology. The ratio takes in consideration the diameter across the equatorial bulge (176)  and the polar diameter (175). Michell discusses how the northern latitude measure of the Egyptian foot was 1.152 feet (the 176 ratio) at fifty degrees latitude. While the southern measure was was 1.145 feet (175 ratio) at ten degrees latitude.


As I have already discussed in my blog about Stonehenge, diatonic “F” above middle “C”  on the piano vibrates at 352 times per second. The standard of measure in antiquity, as I’ve already stated,  is based on a 176 ratio to 175.  One hundred and seventy-six is one-half of 352. Musically, in terms of vibrations per second, it is exactly an octave lower than the 352-F. Now, if we take the shorter 175 ratio of measure of 1.152 feet; then the perimeter around the great Pyramid is 2.640 feet. Diatonic “C” vibrates 264 times per second.  This perimeter, in terms of this shorter Egyptian foot, is exactly ten times the number by which the old diatonic “C” vibrates.


Another diatonic musical tone is duplicated in measure at the Great Pyramid. When shorter Egyptian cubit of 1.718 feet is used to measure the perimeter around the Great Pyramid, then each side is 440 cubits. The note “A” vibrates to 440 times per second. This is the standard not only of the old diatonic scale, but also the well-tempered scale still in use. A-440 is the only tone that is being used from the ancient diatonic scale by musicians- at least in England and America. In this regard, an essential dimension of the Great Pyramid is alive and well; and is still being tuned to by at least some of our orchestras. Music and empire: No wonder King David was considered a musician first and was a king only later.

Melody and 3/4 Time are Returning

Melody and 3/4 time are returning. Music that is heavy on square, off beat rhythm and simple chant but lacks beauty of melody and rhythmic variety has overstayed its welcome. They have been like guests that just won’t leave.  These guests, by the way,  have names. I call them the three “R’s”:  Rock and Roll and Rap. Most of their rhythms are square . They dominate through 4/4 or 2/4 time meters.  Two of the guests, Rock and Roll, insist and the constant hypnotic repetition of the “upbeat,” beats two and four. These three guests tend to avoid triple meter like the plague. To them, the words and melody of the 1936 song “By Strauss” music by George and lyrics by Ira Gershwin represent a worst case scenario: Ira  Gershwin writes about how a melody in 3/4 time literally goes lilting through the house. By the way, you can enjoy By Strauss in the classic movie, the 1951 production of : An American in Paris. It is a showstopper It is danced with comedy and verve as only Gene Kelly can.  The movie was inspired by the 1928 orchestral work with the same title by George Gershwin. It also features some of Gershwin’s most loved songs. The climax is a 16  minute  ballet on the 1928 orchestral work.  By the way, the movie won 8 academy awards.

Detail from frontispiece to Thomas Wilson’s Correct Method of German and French Waltzing (1816), showing nine positions of the Waltz, clockwise from the left- where the musicians are seated (from Wikopedia)


Duple (2/4 and 4/4) meters have sharp angles. Just watch a conductor outlining the angular motions with his baton. Triple meter can be conducted with a baton going in circles. When  understanding time meters and their effects,  the alchemical “squaring of the circle” takes on new meaning: Duple meters are earthbound. The square is associated with the alchemical element of  of earth. The circle represents heaven. Squaring the circle means bringing heaven to earth. The circle represents spiritual qualities- Qualities that are beautiful but you cannot touch.  We need to paraphrase “squaring the circle.” Since the angular rhythms have been so prominent for the last 60 years, we need to “circle the square” with 3/4 time and melody.


 I recommend this needed return to melody written in three-four time meter. It will help with our spiritual survival which ultimately helps our physical survival.  This task is up to the composers.  One such popular waltz number is by Cat Stevens, Morning Has Broken. Norwegian Wood by the Beatles is another. You Light Up My life by Joe Brooks is a third example.  Schubert wrote a collection of waltzes. Brahms has a book of waltzes. Chopin, of course loved waltzes and gave the Viennese waltz a Polish twist.  Remember Strauss, the waltz king, which the Gershwins championed as I just wrote about above. The old minuet is in 3/4 time.  Let’s look to these rich traditions of the past. Then we can turn around to face the future, knowing that our lives will be happier as a result. Incidentally, the ancient Greeks had the same thoughts about music – that music affects character.  They called this musical property  by the name of “ethos”.

Stonehenge Was Built By Musical Tones


There’s music in the air as I, David Ohrenstein, believe that Stonehenge was built by musical tones.





Credit for photography: Digital Imaging Project of Mary Ann Sullivan, Bluffton College;

Stonehenge was built by musical tones in terms of the numbers found in the vibrations per second of an old musical scale.  This the personal theory of David Ohrenstein (myself). The numbers are derived from what is now called the Greek diatonic scale.  Stonehenge  translates the numbers of this scale, in terms of vibrations per second, into  numbers of  measure that were also used by former civilizations from around the globe.


There once was a blueprint that unified music with simple geometry. I discovered the  plan while playing the piano in the summer of 1993 for the Scott family at Scotts Oquaga Lake House in Deposit, New York. Years after, I discovered that the blueprint is preserved by a system of numbers called gematria: Letters used to double as  numbers. Therefore each letter, word, phrase   or sentence  had a numerical value. That is the essence of gematria. 

352 the Diatonic musical tone “F”

Here is the modus operandi of how Stonehenge was built by musical tones: The diatonic tone “F” (as per Issac Asimov in On Physics and Guy Murchie in The Music of the Spheresvibrated 352 times per second. This strongly ties into Judaism.  The most important Judaic prayer is the Shema Yisroel ,which declares the Oneness of the Creator. The two words (Deuteronomy 6:3) right before this six word declaration (Deuteronomy 6:4)  translate to “milk and honey” with gematria of 352. The meaning is simple: from milk and honey, which represent being loving, nurturing and caring, we achieve oneness or unity with the Creator. Now, take 352 as the diameter of a circle. Its circumference in whole numbers is 1106. When the 6 word declaration of the Shema Yisroel is spelled with the double “yud” for the Holy Name- as it is in the prayer books- the gematria of this phrase is also 1106. The message is: Milk and honey places us at one with our Creator.


 The diatonic tone “F” (352) is also found at Stonehenge as the average distance between the pillars (above picture) is 3.52 feet. The average of width of these same stones, facing the center,  is double that- 7.04 feet. Musically, when a number is doubled, it sounds one octave higher.  The average length of the lintels on top of the pillars is triple 3.52 or- 10.56 feet.  Stonehenge is but one example.Other diatonic tones can be found not only at Stonehenge; but at sites of all cultures and creeds.

  Here’s another example of how Stonehenge was built by musical tones:The diatonic tone “C” (264 vibrations per second) is also at Stonehenge and is expressed as  the measure of a distance. John Michell illustrates how the distance from the center of the station rectangle to the heel stone is 264 feet. The  station rectangle is made by the paths of the orbits of the Sun and Moon. It marks where their paths intersect at right angles.  The same number, 264, is  prominent in Judaism. The Hebrew word for the Passover ritual is Seder; which means order. The number behind the same musical tone (C) that marked the order of the paths of the Sun and Moon at Stonehenge, also denotes the order of the ritual at Passover.

                                                                                                                                LIFE, LOVE AND MUSIC

Measures expressed here are not an anachronism. John Michell, in The View Over Atlantis, writes about how the British 12 inch foot existed in the past.  Samples of the cubic inch of gold for  weight comparison are found at the British Museum. They are from ancient Egypt, Babylon and Greece.  I think it would be fitting for there be a new re-dedication to the kindness, love and nurturing that are expressed by the words of milk and honey in Deuteronomy 6:3.  To begin the work, which is substantial, we should dedicate ourselves to renovating sites that were once built by measures derived from tones of the diatonic scale. The rebuilding would offer all mankind a unified and hopeful vision of peace and co-operation; and lead us to the path of a new Golden Age of peace and plenty.