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Tuesday, September 20, 2016

Ode on a Cretan Maze

This one is just for fun.........

It may be that universal history is the history of the different intonations given a handful of metaphors.
– Borges, 'The Fearful Sphere of Pascal'

The Pylos Maze (tablet pre-1200 BCE)*



(a)                                                   (b)
Illustration 1.
Sketch for analysis of the Pylos Maze,
rotated clockwise 90 degrees from photo above
(Prof. Tony Phillips)


Illustration 1 is taken from the article 'Hidden Symmetries of Labyrinths from Antiquity and the Middle Ages' by Tony Phillips appearing in the American Mathematical Society's October 2015 Feature Column: Monthly essays on mathematical topics. As Phillips describes it:
The oldest securely dated labyrinth design appears on the back of an accounting tablet found in the ruins of King Nestor's palace in Pylos, on the western shore of Greece. The tablet (7 X 5.7 cm) was baked when the palace burned in 1200 BC. The design on this tablet is often called the Cretan Maze. It appears on coins minted on Knossos on Crete during the 5th century BC and later, and represented the legendary labyrinth where King Minos kept the Minotaur.
The aim of the article as a whole is 'to point out that the great majority of labyrinth designs share a topological symmetry which, while not obvious, cannot be accidental.' Phillips works through  examples of several labyrinths (the reader is encouraged to go to the entire article), but I'd like to concentrate on the Cretan Maze as another illustration of hidden or 'deep' form – one of the main themes in Essays & Endnotes.

If you stare at the labyrinth on the left as it appears on the tablet (Illustration 1a), you realize that, considered as a 2-dimensional rigid figure, it's asymmetric  – you can't rotate or flip it to get an identical figure. So first, Phillips labels the levels of the labyrinth in red, starting on the outside with 0 and ending at the goal with 8 (Illustration 1b). Now enter the labyrinth at 0 (at the opening on the right) and follow the path. Each time you cross the line of red numbers, record the number you cross. The 'level sequence' you come up with will be 032147658. Phillips reveals the hidden symmetry by first noting that 'if each number n is eight's complement 8–n [in music theory terminology this is "inversion" I8 (mod 9)] the sequence becomes 856741230, which is the original sequence read backwards [retrograde]' – the way to back out of the labyrinth. He then "lifts" the labyrinth into three dimensions by wrapping it around four faces of a cube (see his illustrations in the article) which ultimately reveals its hidden rotational symmetry. 

Using the pictorial representation of permutations that we've used previously (beginning with spiral representations), after some unscrambling of the original labyrinth using Phillips' labelling of the levels, we can see that Illustration 1b is topologically identical to Illustration 2.


Illustration 2.

So the labyrinth path can be analyzed as a concatenation of two identical paths,

(nn+3, n+2, n+1, n+4)                                                         (1)

with n=0,4. Tracing these paths in K9 (mod 9 circle space) with red for n=0 and blue for n=4 (Illustration 3) gives us another view of labyrinth permutation.

Illustration 3.

Now if we let n take the values 0,4,8 we get the obvious extension of the Pylos labyrinth permutation (Illustration 4).


Illustration 4.

And then, working backward, we can translate this into a maze of three 'levels' by surrounding the original two-level maze with another level:
+

=

Illustration 5.

We can indefinitely extend expression (1) that generates the permutation (and the labyrinth) by letting n=4k with k=0,1,2,3,.... Do we then have a labyrinth that has a center but no outside? If we begin at the center and some demon keeps adding to k it will take us an infinite amount of time to get out; that much is clear. But, with a nod to Zeno and Cantor, exactly where could one enter such a labyrinth?




__________________________

* Photograph from Weisstein, Eric W. "Maze." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Maze.html



Thursday, May 5, 2016

Broken Symmetries 3.2


A Tale of Two Tunes

2. Pitch content
in the themes of Messiaen's Île de feu I & II

In the previous post the comparison of the two themes was based on similarity of rhythmic contours. We now consider a pitch set-class comparison which will provide a transformational relationship amongst the two themes and their initial accompaniments. Once again, for reference, here are the two themes:

Example 1
Theme, Idf1 (mm.1-2)

Example 2
Theme, Idf2 (mm.1-7)

We first need to visualize the pitch-class material as nested sets. In Example 3 a black box indicates total pitch-class material in the indicated segment, a grey box indicates the characteristic set for that segment, and a red box indicates a linking tetrachord. (Technically, Example 3b should have a grey box labelled, say, X surrounding the red box and a black box labelled X* around that grey one, but considering that this tetrachord shares all three of those descriptions (J' = X = X*), just the red box is sufficient to show the transformation link we're headed for.)


Example 3.

The object of this game is to draw arrows that connect Examples 3a, 3b, 3c, and 3d. To begin, we note that the total pitch-class content of Idf1's theme, enclosed by the black box in Example 3a, is the set K* = {C#, E, F, F#, G, G#}. (Ignore the parenthetic A# for now. We'll get to it shortly.) Note that the five-note subset of K* marked by the grey box is K = {E, F, F#, G, G#}, SC[01234], a segment of the chromatic collection. To emphasize this structural characteristic we can write

                    K* = {K, C#}                    (1)

Next, the pitch-class content of Idf2's theme (Example 3c) is P* = {E, G, A, A#, C, C#, D}. The characteristic subset of P* is the pentatonic collection P = {E, G, A, C, D}, SC[02479], so to emphasize that structural characteristic we can write a partition of P* as

                   P* = {P, A#, C#}               (2)

We know that the two set classes [01234] and [02479] are homothetically related by M5/M7 (mod12); so there must be some pitch-class transposition such that TnM5/7(K) = P, and indeed:


                     T8M5(K) = P.                   (3)

Now what about the C# and A#? The parenthetic A# doesn't appear in the Idf1 theme at all. But when looking initially at the symmetric transformation relationship between the two themes, it is revealing to include it provisionally as an imaginary element or 'ghost presence' in the Idf1 theme's pc content. So let's posit a set K*+ that includes the A#,

  K*+ = {K*, A#} = {KA#, C#}         (4)

and pretend, for now, that Messiaen didn't find any need to use that A# until Idf2's theme; or, conversely, after its inclusion in the construction of the Idf2 them, he dropped the A# in making Idf1's theme. Now we can summarize the entire transformation relationship between the pitch content of the two themes (Examples 3a and 3c) as either

               T8M5(K*+) = P*,                     (5)

or, going the other direction, its inverse

              M5T4(P*)  =  K*+.                    (6)


This suggests at least one reason why (assuming Messiaen started (pre)compositionally from Idf1), after he applied the M5 transformation to K* (or K*+), he then transposed it by T8 to get the content for Idf2, because that's the only transposition that regains the C# and reifies the ghost element A# after M5 multiplication.

Also note that K*+ and P* share a diminished seventh chord,

   K*+  P* = {A#, C# E, G}.           (7)

In the final post in this thread we will see that the A# marks a 'cut' of the theme that is tempting to call a half cadence. It's at the A# that Messiaen cuts the Idf2 theme into two halves and reverses their order to use as part of connecting material leading into the final Magma Dance. The diminished triad {C#, E, G} is used to create quasi-cadential patterns in both themes.


Next let's consider the initial accompaniment for Idf2's theme in the right hand in mm.1-7 (Example 4).

Example 4.

With the exception of the notes circled in red, the entire pitch-class content of the homorhythmic accompaniment in the right hand in mm.1-7 is the set H = {D#, E#, F#, G#, A#, B, C#}. This belongs to SC[013568t], the 'usual diatonic' set class, which is the set-theoretic complement of Idf2 theme's characteristic set, the pentatonic SC[02479]:

              H = COMPL(P)                    (8)

As noted, what ruins the analytical case for the entire accompaniment H* being a strictly diatonic set is the presence of those four spoilers circled in red: Q' = {G, A, C, D}, SC[0257]. So for the complete accompaniment content H* we have

                  H* = {H, Q'}                     (9)

H* can also be expressed as a string of eleven equally tempered perfect fifths or minor seconds (short the E-natural it would take to make a complete circle), however, focussing on Q' in the accompaniment, which is also found as Q in the theme's characteristic set P, provides the key to unlocking relationships binding the two themes and their initial accompaniments.

Looking at mm1-7 (Example 4) only as a sequence of homorhythmic verticals will miss a salient feature of the accompaniment. First fix the elements of the unordered set Q to create the descending string [D-C-A-G], then transpose up a minor second to form the string q = [D#-C#-A#-G#]. Tracing q and its retrograde and transpositions (Example 7) reveals that it knits together nearly all the pitch-class pairs making up the r.h. accompaniment for Idf2's theme. The pairs untouched by q and its transformations, with the exception of m.7, can be reached by permuting the string [D-C-A-G] into q' = [C-G-A-D] (reading the C as B#). This finds all the notes circled in red in Example 4. Measure 7 is quasi-cadential in the r.h. The voice leading by interval-class 3, F→G# & F#→D#, is broken by Messiaen to create a final decisive hammer stroke – a gesture that lends greater force to the final two notes in the l.h.; keep in mind this motion by ic3, especially as a cadential move.

Example 5
q-strings in Idf2, mm1-7 (rh)
(brought down 2 octaves for easier reading)


We are now nearly finished with a tour around an analytical schema that binds the significant features of the themes and accompaniments for Idf1 and Idf2. Example 6 shows where we've been in this transformation analysis so far, as well as where we are now heading.



Example 6.
Circle of transformations relating Idf1 and Idf2.
To close the circle, we 'flatten' Q' back to a four-note segment of the chromatic:


M5Q' = {A, Bb, B, C} = J'                  (10)

Then returning to Idf1's tetrachord is a simple transposition:

            T8J' = J                                       (11)

While Q is consistently expressed as a melodic cell in Idf2's accompaniment, J' (without  the C-natural that appears only in the 32nds that echo the 16ths up in the theme) is expressed as a tone cluster and treated as percussion in Idf1's drumming accompaniment. (See Example 7 [C-flat was substituted for the published edition's B-natural for clarity].)


Example 7.
Idf1, mm.1-2



 The final installment in this thread, 
 'Broken Symmetries 3.3: Tonality, a.k.a. Strange attractors',  
 is under construction as of 5 May 2016 


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[1] The letter-symbols I'm using here may be confusing but they are not entirely random. Since I am using only the QWERTY keyboard in order to avoid characters that will be unreadable for some browsers/devices, it's necessary to reserve the traditional A-through-G letter names for pitches as well as the now standard transformation symbols, T, I, R and M. In this section (if it will help remember and keep the sets straight) I've chosen 'suggestive' symbols: K is for 'chromatic', P is for 'pentatonic', H is for 'heptatonic', and Q is for the French 'quatre'. I have no idea where I got the J from (maybe from J<K alphabetically).