Wednesday, July 31, 2002

Bach’s Minuet in G

Filed under: General — m759 @ 11:29 PM
toys.jpg (17640 bytes)

The Toys

Left to right: June Montiero, Barbara Parritt, and Barbara Harris

From the website http://www.history-of-rock.com/toys.htm

In 1964 they were signed by the Publishing firm Genius, Inc., which teamed them with the songwriting duo Sandy Linzer and Denny Randell…. The writers took a classical finger exercise from Bach and put a Motown bassline to it and “A Lover’s Concerto” was born.

September 1965: “A Lover’s Concerto” on the Dynavoice label went #4 R&B, crossed over to pop charts #2, and also became a #5 hit in England. In 1965 the song sold over a million copies. The Toys began appearing on television shows such as “Shindig!,” “Hullabullo,” and “American Bandstand,”  toured with Gene Pitney, and appeared in the film It’s a Bikini World.

Other sites giving further details on Bach’s Minuet in G:

Search for the sheet music and a rendition of the work at codamusic.com’s Finale Showcase Search Page.

Seeing and hearing the music on this site requires that you download  Coda’s SmartMusic Viewer, and possibly requires that you adjust your browser settings, depending on the operating system you use.

For another look at Bach’s music, along with a midi rendition, you can download Music MasterWorks composing software from the Aspire Software site…


Then download the midi file of the Minuet in G itself,  “Minuet in G,  BWV841” (M.Lombardi), from the website


(To do this, right-click on the minuet link and use the “Save Target As” option, if you, like me, are using Internet Explorer with Windows.)

After you have downloaded the midi file of the minuet, use the “File” and “Open” options in Music MasterWorks to display and play the music.

A comparison of these two versions of Bach is instructive for anyone planning to purchase music composition software.   The MasterWorks creates sheet music from its midi file that is quite sophisticated and rather hard to follow, but this music accurately reflects the superior musical performance in the downloaded midi file versus the rendition in the online Finale Showcase file.   The Showcase file is much simpler and easier to read, as the rendition it describes is also quite simple.

The Gentle Rain

For an even simpler version, those of us who were in our salad days in 1965 can consult our memories of The Toys:

How gentle is the rain
That falls softly on the meadow.
Birds high up in the trees
Serenade the clouds with their melodies.

Oh, see there beyond the hill,
The bright colors of the rainbow.
Some magic from above
Made this day for us just to fall in love.

Those of the younger generation with neither the patience nor the taste to seek out the original by Bach may be content with the following site —

A Lover’s Concerto in Venice

To a more mature audience, the picture of a Venetian sunset at the above site (similar to the photo below, from Shunya’s Italy)

will, together with the lyrics of The Toys, suggest that

The quality of mercy is not strained.
It droppeth as the gentle rain from heaven….

This line, addressed to Shylock in “The Merchant of Venice,” contradicts, to some extent, the statement by Igor Stravinsky in The Poetics of Music (1942, English version 1947) that music does not express anything at all. Stravinsky is buried in Venice.

From  Famous Graves:

Igor Stravinsky,


Tuesday, July 30, 2002

Tuesday July 30, 2002

Filed under: General — Tags: — m759 @ 12:12 AM

Aesthetics of Madness

Admirers of the film "A Beautiful Mind" may be interested in the thoughts of psychotherapist Eric Olson on what he calls the "collage method" of therapy.  The fictional protagonist of "A Beautiful Mind," very loosely based on the real-life mathematician John Nash, displays his madness in a visually striking manner (as required by cinematic art).  He makes enormous collages of published matter in which he believes he has found hidden patterns. 

This fictional character is in some ways more like the real-life therapist Olson than like the real-life schizophrenic Nash.  For an excellent introduction to Olson's world, see the New York Times Magazine article of April 1, 2001, on Olson and on the mysterious death of Olson's father Frank, who worked for the CIA.  Here the plot thickens… the title of the article is "What Did the C.I.A. Do to Eric Olson's Father?

For Olson's own website, see The Frank Olson Legacy Project, which has links to Olson's work on collage therapy.   Viewed in the context of this website, the resemblance of Olson's collages to the collages of "A Beautiful Mind" is, to borrow Freud's expression, uncanny.  Olson's own introduction to his collage method is found on the web page "Theory and therapy."

All of the above resulted from a Google search to see if Arlene Croce's 1993 New Yorker article on Balanchine and Stravinsky, "The Spelling of Agon," could be found online.   I did not find Arlene, but I did find the following, from a collage of quotations assembled by Eric Olson —

"There might be a game in which paper figures were put together to form a story, or at any rate were somehow assembled. The materials might be collected and stored in a scrap-book, full of pictures and anecdotes. The child might then take various bits from the scrap-book to put into the construction; and he might take a considerable picture because it had something in it which he wanted and he might just include the rest because it was there.”

— Ludwig Wittgenstein,
Lectures and Conversations on Aesthetics, Psychology and Religious Belief, 1943/1978

“Not games. Puzzles. Big difference. That’s a whole other matter. All art — symphonies, architecture, novels — it’s all puzzles. The fitting together of notes, the fitting together of words have by their very nature a puzzle aspect. It’s the creation of form out of chaos. And I believe in form.”

Stephen Sondheim
in Stephen Schiff, “Deconstructing Sondheim,”
The New Yorker, March 8, 1993, p. 76.

“God creates, I assemble.”

— George Balenchine [sic]
in Arlene Croce, “The Spelling of Agon,”
The New Yorker, July 12, 1993, p. 91

The aesthetics of collage is, of course, not without its relevance to the creation (or assembly) of weblogs.

Monday, July 29, 2002

Monday July 29, 2002

Filed under: General — m759 @ 8:34 PM

At Random

Today’s birthday: poet Stanley Kunitz 

“I’m Stanley Kunitz. I live in New York City. I published my first book of poems some 70 years ago. Back in 1926, I was roaming through the stacks of the Widener Library at Harvard. While I was walking through the section on English poetry of the 19th century, I just at random lifted my arm and picked a book off the shelf. It was… an author I was not familiar with, Gerard Manley Hopkins. The page that I turned to and began to read was a page devoted to a poem called “God’s Grandeur.” I couldn’t believe what I was reading when I opened this book and started reading that poem. It really shook me, because it was unlike anything else I had ever read before. When I started reading it, suddenly that whole book became alive to me. It was filled with such a lyric passion. It was so fierce and eloquent, wounded and yet radiant, that I knew that it was speaking directly to me and giving me a hint of the kind of poetry that I would be dedicated to for the rest of my life.”

Sunday, July 28, 2002

Sunday July 28, 2002

Filed under: General — m759 @ 3:07 PM

Keats and the Web

From a letter of John Keats on the Web:

“There is the passage in a famous letter of John Keats, 19th February, 1818:

Now it appears to me that almost any Man may like the spider spin from his own inwards his own airy Citadel — the points of leaves and twigs on which the spider begins her work are few, and she fills the air with a beautiful circuiting. Man should be content with as few points to tip with the fine Web of his Soul, and weave a tapestry empyrean full of symbols for his spiritual eye, of softness for his spiritual touch, of space for his wandering, of distinctness for his luxury.”

This seems not unrelated to the observations below on commonplace books and Web logs. 

Sunday July 28, 2002

Filed under: General — m759 @ 2:16 PM

Memories, Dreams, Reflections

Saul Steinberg in The New York Review of Books issue dated August 15, 2002, page 32:

“The idea of reflections came to me in reading an observation by Pascal, cited in a book by W. H. Auden, who wrote an unusual kind of autobiography by collecting all the quotations he had annotated in the course of his life, which is a good way of displaying oneself, as a reflection of these quotations.  Among them this observation by Pascal, which could have been made only by a mathematician….”

Pascal’s observation is that humans, animals, and plants have bilateral symmetry, but in nature at large there is only symmetry about a horizontal axis… reflections in water, nature’s mirror.

This seems related to the puzzling question of why a mirror reverses left and right, but not up and down.

The Steinberg quote is from the book Reflections and Shadows, reviewed here.

Bibliographic data on Auden’s commonplace book:

AUTHOR      Auden, W. H. (Wystan Hugh),              1907-1973. TITLE       A Certain World; a Commonplace Book   
            [selected by] W. H. Auden.
PUBLISHER   New York, Viking Press [1970]
SUBJECT     Commonplace-books.

A couple of websites on commonplace books:

Quotation Collections and

Weblets as Commonplace Books.

A classic:

The Practical Cogitator – The Thinker’s Anthology
by Charles P. Curtis, Jr., and Ferris Greenslet,
Houghton Mifflin Company Boston, Massachusetts
c 1962 Third Edition – Revised and Enlarged

Sunday July 28, 2002

Filed under: General — m759 @ 1:56 PM

A Commonplace Blog

William Safire blogs the word “blog” in his On Language column today.  He specifically mentions xanga.com —

“Blog is a shortening of Web log. It is a Web site belonging to some average but opinionated Joe or Josie who keeps what used to be called a ”commonplace book” — a collection of clippings, musings and other things like journal entries that strike one’s fancy or titillate one’s curiosity…. To set one up (which I have not done because I don’t want anyone to know what I think), you log on to a free service like blogger.com or xanga.com, fill out a form and let it create a Web site for you. Then you follow the instructions….”

Friday, July 26, 2002

Friday July 26, 2002

Filed under: General — Tags: — m759 @ 1:59 PM

Today's birthdays:

Another opening of another show…. Kevin, Kate, and Carl.

Spacey Beckinsale Jung

This is not a big book but a dense book,
as exquisitely rich as a dark fruit cake.

— Jean Brown, review of Jung and the Story of Our Time,
by Laurens van der Post, Hogarth Press 1976, Penguin Books 1978, 1985
(From © For A Change Magazine — December/January 1999 —
http://www.mra.org.uk/fac/de c98/books.html

Thursday, July 25, 2002

Thursday July 25, 2002

Filed under: General — m759 @ 9:18 PM

I’ve been looking for a weblog editor, and Xanga seems like it’s the best.   Too bad they can’t host pre-existing domain names…. I registered the URL log24.com some time ago, and want to use it.  But I also want to use Xanga’s neat entry software.   My solution: Use Xanga for day-to-day entries, with the new URL log24.net (just purchased), and use my log24.com site as an archive.

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , , — m759 @ 10:13 PM

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.


For an animated version, click here.


Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).

The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions


The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator


Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)

Creative Commons License
This work is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivs 2.5 License

Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com


Initial Xanga entry.  Updated Nov. 18, 2006.

Saturday July 20, 2002

Filed under: General — m759 @ 10:04 PM

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