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Gödel

13 F

[32]

Picc.

Fl.

1 2

Ob.

1 2

Bsn

1 2

a 2

Bsn Bsn 3

Eb Cl. in E♭

Cl. in B♭ 1

Cl. in B♭

2 3

B. Cl. in B♭

C. A. Cl. in E♭

F Sop. Sax. in B♭

Alto Sax. in E♭

Ten. Sax. in B♭

Bari. Sax. in E♭

a 2

Hn in F

1 3

Hn in F

2 4

F

Tpt. in B♭ 1

Tpt. in B♭

2 3

Tbn. 1

a 2 2.

gliss.

Tbn. 2 Tbn 3 3.

gliss.

B. Tbn.

Euph.

Tba.

port.

Db.

F Timp.

Caj.

To Tam.

Caj.

R-toms

To Vib.

Vib.

Sus. Cym.

To Bon. B. Dr.

Pno.

Bon.

BJ Brooks - Symphony #3 - G.E.B. Symphony 3, G.E.B. is meant to be heard with the following programme information in mind.

Preamble

Symphony #3: GÜdel, Escher, Bach is a meditation on human cognition. It was written in the spirit of Douglas Hofstadter’s seminal book of the same name that details, abstracts, and illuminates the mystery of cognition.

In 1979 Dr. Douglas Hofstadter, Indiana University Professor of Cognitive Science and Comparative Literature, published his Pulitzer Prize winning GĂśdel, Escher, Bach: An Eternal Golden Braid, know colloquially as G.E.B. The incompleteness theorems of Kurt GĂśdel, proven with an ingenious use of recursion and self-reference, are illuminated by analogy in prose, the art of M.C. Escher, and in the music of J.S. Bach.

At the beginning of the 20th century the world of mathematics was in crisis. The 19th century had seen an increasing momentum towards mathematical abstraction. Mathematician’s formalization of foundational logical structures were constantly undermined by paradoxes within their constructed rigorous systems. An attempt to avoid the paradoxes and to streamline the logical process through the use of a relatively small number of symbols was written by Alfred North Whitehead and Bertrand Russell and published in 1910. The Principia Mathematica used a specialized grammar that avoided certain paradoxes. It is a work of such logical complexity that it takes the book 362 pages to suggest ⊢:.â?ş,đ?›˝âˆŠ1.⊃:â?şâˆŠđ?›˝=ĘŒ.≥.â?şâˆŞđ?›˝âˆŠ2 after which the authors note, “From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.â€?

The consistency and completeness of Principia Mathematica were all but settled until Logician Kurt GĂśdel published his incompleteness theorems in 1931. The theorems cleverly use a type of recursion and self reference in metamathematical logic to demonstrate that Principia Mathematica can indeed contain paradoxes. Thus GĂśdel proved that 1) for any substantive formal system there are unprovable truths, and 2) a system cannot on its own demonstrate its own consistency. SuďŹƒciently complex formalized systems such as mathematics may either be consistent or complete, but not both.

Programme

Movement 1- GĂśdel

Concept:

The exquisiteness, complexity, and logical journey of GÜdel’s proof is represented throughout movement 1. A chaconne of 6 chords starting with the outline of E-minor, arpeggiated as the pitches G, E, and B in the high winds, begins a series of twenty continuous variations. Self reference, recursion, and sonification are used throughout the movement.

Process:

Self reference appears in numerous ways including the arpeggiated GEB, groupings of notes on staves clustered to appear as the letters “GEBâ€? on the score, harmony created by the untransposed pitches GEB played in unison, serialized melodies using the first movement’s introduction of all twelve chromatic pitches (G, E, B, E-flat, B-flat, D, A, F-sharp, Csharp, F, C, A-flat), sonification (of a type similar to GĂśdel’s arithmetization) of the logical expression in Principia Mathematica that yields 1+1=2, aural and visual representations of Zeno’s paradox, and the use of the chaconne form itself as it insistently reuses the same 6 chords throughout.

Movement 2- Escher

Concept: The M.C. Escher lithograph “Drawing Hands”.

Process: “Drawing Hands” serves as a visual metaphor for the structure of Movement 2- Escher.

The right hand reaches left bringing references from movement 3 to the beginning of the work as the left hand reaches right and brings references from movement 1, though apparently inverted.

Movement 3- Bach

Concept:

Book I of J.S. Bach’s The Well-Tempered Clavier (WTC), BWV 846–893, and the first prelude (p1) in particular serves as the foundation of the third movement.

Harmonies from p1 and melodies from throughout WTC are quoted throughout the movement.

Process:

The collection of notes in each measure of p1 are used in order throughout the duration of the movement. This slow harmonic development is juxtaposed with alterations of the consistent rhythmic texture from p1 realized as an energetic bass line, and as the first melodic motive. As the harmony emerges, other melodic motives from WTC are layered throughout. After the harmonic process completes, the end of the piece concludes with a measure by measure restatement of p1. The order of the melodic motives is derived by serializing p1’s introduction of all twelve chromatic pitch classes (C, E, G, D, A, F, B, F-sharp, B-flat, C-sharp, Gsharp, E-flat). The movement uses 48 melodic quotes in serial order from WTC. The order from WTC is therefore Prelude 1 in C-major, Fugue 1 in C-major, Prelude 2 in Cminor, Fugue 2 in C-minor, Prelude 9 in E-major, Fugue 9 in E-major, Prelude 10 in E-minor, Fugue 10 in E-minor, etc.