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Thread: Sacred Geometry and Composition

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    Senior Member Captainnumber36's Avatar
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    Default Sacred Geometry and Composition

    I feel as though Bach accomplished applying the golden ratio to music best. I wonder if there is anything special about this, finding musical patterns which already exist in nature.

    I also think it has to have great emotional depth. I used to write Bach off as lacking in that department, but he was just too lucid for me then.


    Thoughts?

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    Senior Member pianozach's Avatar
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    Yeah, music and math (and physics) have some relationships to each other. And, yes, Bach's music, especially his fugues took the relationships to a logical conclusion. His music includes patterns, structures, recursions and other precisely crafted features.

    And there are even hints of Fibonacci numbers and the Golden Ratio in Bach’s music:

    "Bach’s The Art of Fugue (BWV 1080), which was composed in the last decade of Bach’s life and was clearly designed as an ultimate expression of Bach’s “mathematical” style [and its] partner, The Musical Offering (BWV 1079), which has a similar objective and style, was named by musicologist Charles Rosen as the most significant piano work of the millennium.

    "Sylvestre and Costa tabulated the number of bars in each of the 19 movements of The Art of Fugue, then carefully analyzed different groupings of the movements. They found a number of intriguing patterns, including the following:

    • The total number of bars for counterpoint movements 1 through 7 is 602. Of these, 372 are in counterpoint movements 1 through 4 and 230 in counterpoint movements 5 through 7. Note that 602/372 = ø [the 'Golden Ratio, or ø = (1 + sqrt(5))/2 = 1.6180339887 . . .] very closely, and also 372/230 is very close to ø.

    • Counterpoint movements 8 through 14 (988 bars in total) can be divided into double and mirror fugues (377 bars) and triple fugues (611 in total). Note that 611/377 and 988/611 are each very close to ø.

    Note that in each case, the ratios mentioned above are as close as possible to ø as an integer ratio, given the respective denominator. Several other examples of this sort are given in the Sylvestre-Costa paper.

    "Sylvestre and Costa conclude, “we report a mathematical architecture of The Art of Fugue, based on bar counts, which shows that the whole work was conceived on the basis of the Fibonacci series and the golden ratio.”

    "However, as a word of caution, it should be kept in mind that the evidence cited by Sylvestre and Costa is a bit on the weak side. For example, their observation of the partial Fibonacci sequence 2,3,5,8,13 in the bar counts could be just a coincidence. Even the instances of the golden ratio could simply be due to Bach’s innate sense of natural balance, instead of deliberate numerical design (which itself is rather remarkable). Hopefully additional research in this arena will clarify this matter.""



    https://mathscholar.org/2021/06/bach-as-mathematician/
    Last edited by pianozach; Sep-17-2021 at 19:21.

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    I think there's a lost Renaissance painting of Bach sitting at the feet of Pythagoras.

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    Senior Member SONNET CLV's Avatar
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    The following info is from OPRAH.COM, a site I visit almost as much as I visit this Talk Classical Forum. Sure I do.


    Dr. Kendra Schmid, an assistant professor of biostatistics, uses the golden ratio and 29 other measurements to study facial sex appeal . These measurements are calculated to determine a person's beauty on a scale of 1 to 10. What does she measure?

    A. First, Dr. Schmid measures the length and width of the face. Then, she divides the length by the width. The ideal result—as defined by the golden ratio—is roughly 1.6, which means a beautiful person's face is about 1 1/2 times longer than it is wide.

    B-aB0a1.jpg

    B. Next, Dr. Schmid measures three segments of the face—from the forehead hairline to a spot between the eyes, from between the eyes to the bottom of the nose, and from the bottom of the nose to the bottom of the chin. If the numbers are equal, a person is considered more beautiful.

    B-aB0a2.jpg


    C. Finally, statisticians measure other facial features to determine symmetry and proportion. On a perfect face, Dr. Schmid says the length of an ear is equal to the length the nose, and the width of an eye is equal to the distance between the eyes.

    B-aB0a3.jpg

    Most people score between 4 and 6, and Dr. Schmid says no one has ever been a perfect 10.


    [Read more: https://www.oprah.com/oprahshow/meas...ixzz76kNwIfVv]



    According to my measurements and Dr. Schmid calculations, I suggest we may have just found our first "perfect 10" in Herr J.S.B.

    B-aB0a.jpg

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    Senior Member jegreenwood's Avatar
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    61x9qySMjzL.jpg

    First thing that came to mind.

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    Quote Originally Posted by jegreenwood View Post
    61x9qySMjzL.jpg

    First thing that came to mind.
    When I see it on my shelf, I keep thinking of "G. Escherbach" which keeps reminding me of Gustav Aschenbach! :-)

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    Senior Member jegreenwood's Avatar
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    Quote Originally Posted by MarkW View Post
    When I see it on my shelf, I keep thinking of "G. Escherbach" which keeps reminding me of Gustav Aschenbach! :-)
    As I head off this evening to hear Colm Toibin read from his new novel, The Magician about Thomas Mann. He will be accompanied by a string quartet playing portions of Beethoven's Op. 132.

    Also there's Richard Powers' The Gold Bug Variations. Well known and celebrated, but frankly I didn't care for it.
    Last edited by jegreenwood; Sep-17-2021 at 23:46.

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    Senior Member Sid James's Avatar
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    Years ago, I went to a performance of Bartok's Music for Strings, Percussion and Celesta, and I remember how the program notes mentioned a connection with the Fibonacci Sequence. Its certainly a fascinating subject, and not only in relation to music, where it is less discernible because of the need to analyse scores. We often associate it with architecture, and there are many examples in visual arts, from Leonardo to Mondrian.

    Its just one example of how science and art have these overlaps. In his Ascent of Man series, Dr. Jacob Bronowski wove together various strands of human endeavour across the ages. In this clip, he begins by saying that "all imagination begins by analysing nature" and describes the process of creativity. He says that it begins with analysis, exploring and observing nature, finding out what is there. It culminates with synthesis, putting the parts together, according to the artist's unique vision and mindset.

    Perhaps his drawing parallels between two sculptors separated by centuries - Michelangelo and Henry Moore - can be done in a musical sense between the likes of Bach and Bartok? Its certainly food for thought.

    Last edited by Sid James; Sep-21-2021 at 23:03.
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    Senior Member hammeredklavier's Avatar
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    https://en.wikipedia.org/wiki/Golden_ratio
    "Ancient Greek mathematicians first studied what we now call the golden ratio, because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons."

    Interestingly, some speculate that Mozart's sonata movements generally tend toward this "golden ratio" of 0.618.
    For example, Mozart's piano sonatas:
    E = number of measures in the exposition
    D = number of measures in the development+recapitulation
    No. 1, K.279 1st movement { E = 38 | D = 62 }
    D/(E+D) = 0.620
    No. 16, K.570 1st movement { E = 79 | D = 130 }
    D/(E+D) = 0.622
    No. 7, K.309 1st movement { E = 59 | D = 97 }
    D/(E+D) = 0.622
    No. 16, K.545 1st movement { E = 28 | D = 45 }
    D/(E+D) = 0.616
    No. 10, K.330 1st movement { E = 57 | D = 92 }
    D/(E+D) = 0.617
    No. 2, K.280 1st movement { E = 52 | D = 88 }
    D/(E+D) = 0.611

    * the entire movement is 100 bars. [ 38 bars of exposition | 62 bars of development + recapitulation ] is the closest ratio you can get with whole numbers to 0.618.

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    Senior Member Captainnumber36's Avatar
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    I think Mozart more naturally found the math behind music when compared to Bach.
    Last edited by Captainnumber36; Sep-25-2021 at 17:15.

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    Senior Member pianozach's Avatar
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    Bach

    Crab Canon



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    Quote Originally Posted by Captainnumber36 View Post
    I think Mozart more naturally found the math behind music when compared to Bach.
    Not to take anything away from Mozart, but I disagree. The amusing thing about the following is Bach writing "Etc" at the bottom of the page. "I could do this all day", as one of the YT commenters pointed out.
    Last edited by dissident; Today at 00:37.

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    Senior Member Captainnumber36's Avatar
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    Quote Originally Posted by dissident View Post
    Not to take anything away from Mozart, but I disagree. The amusing thing about the following is Bach writing "Etc" at the bottom of the page. "I could do this all day", as one of the YT commenters pointed out.
    It's pretty close between the two imo. But I'd take Mozart over Bach.

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    Quote Originally Posted by Captainnumber36 View Post
    It's pretty close between the two imo. But I'd take Mozart over Bach.
    Here's some interesting reading:

    https://www.researchgate.net/publica...OF_FUGUE%27%27

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    Senior Member Sid James's Avatar
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    The classical view of man being the focal point of the four elements might be dated, but we've been trying to figure out unifying principles in all areas since, including music. I think that some music mimics the repetitive patterns of nature (as fractals do in a visual sense), one example being Beethoven's Pastoral, and I opened a short-lived discussion on it way back:
    The origins of "Minimalism"

    Quote Originally Posted by hammeredklavier View Post
    https://en.wikipedia.org/wiki/Golden_ratio
    "Ancient Greek mathematicians first studied what we now call the golden ratio, because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons."
    Particularly in relation to architecture and art, you've also got Leonardo's Vitruvian Man and its modern offshoot, Le Corbusier's Modulor. Despite all the advances in architecture since the modern period, let alone the Renaissance, human scale is still one of the most important considerations in relation to how cities are built. We can now build things which totally dwarf us like the Burj Khalifa. It would be a strange world if that became the norm, even though its unlikely to happen.
    Last edited by Sid James; Today at 11:06.
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