Composition based on sonifications of modern mathematics, commented by mixed choir.

5.1 channel audio. Duration ~52 minutes.

Premiere November 2011

COMMENT....................

(SURREAL) NUMBER .....................

DEDEKIND CUTS ............

NUMBER AND NEGATIVITY

SPECTRAL DIGITS .........

PRIME FACTORS ...........

COMPLEX NUMBERS ......

RIEMANN'S ZETA FUNCTION

LINEAR CONGRUENCE / INTERCALATION ...........

INTERLUDE: DISSONANCE / DIVERGENCE ................

CARDINAL NUMBERS .....

א0 \ALEPH_0 .............

ONE (UNITY) ................

REFRAIN NUMBERS NEGATION

COMMENT: Is it possible to perceive changes in a society through changes in its music? Is it possible to understand contemporary music through its implicit relationships with money? Music and economy share a fundamental object, number. Accordingly, it seems, through number, the simultaneity between musical and political-economical development becomes evident: the semitone was accepted not before the Rennaissance, at a time when the merchants appeared; Russolo wrote his "Arte dei Rumori" in 1913; noises found their way into music immediately before the onset of the wars of the 20th century; the unrestricted rise of large orchestras occurred during times of enormous industrial growth. Music serves as a mirror of a time, but it is also a shadow that a changing society projects into the future. And how better could we describe the disciplines of music and economics than through mathematics, their common foundation? In order to understand this relationship better, we draw an acoustic diagram between mathematics, abstraction and affect, the politics of frequency. Through the sonification of a series of mathematical problems, their rendering as an acoustic process in time, we may gain an understanding of the limits of intuition and the character of the correlation between music, economy and mathematics. We pursue the impossible goal of making abstract and descriptive terms congruent. The subversive desire to follow this path of impossibility is the formal core of this idea: a world is not structured by things but by events.

Perhaps, the necessity to calculate has its origin in the beginnings of the abstraction of trade. A simple example: two farmers are trading seventeen geese versus five sheep. The shepherd owns temporally enough geese but is still willing to give away the sheep. In doing so, trust is an important factor. The farmer could still give away the sheep, anticipating that he will receive the seventeen geese whenever he needs them. Modern economics, dealing with consumption, capital, debt and interest as forms of symbolic reproduction, relies on trust, speculation and rhetoric. Thus, many of the foundational economic theories come from moral philosophy rather than from mathematics.

In his book 'Number and Numbers', Alain Badiou sets out how numbers serve, strictly speaking, for everything. They provide a norm for all things. Number governs our conception of the political, of suffrage, of opinion polls, of the majority. What counts, in the sense of what is valued, is that which is counted. Conversely, everything that can be numbered must be valued. Political thought is numerical exegesis. Number governs the quasi-totality of the 'human sciences'. Badiou asks: 'Isn't another idea of number necessary, in order for us to turn thought back against the despotism of number, in order that the subject might be subtracted from it? And has mathematics simply stood silently during the comprehensive social integration of number, over which it formerly had monopoly? In our situation, that of Capital, the reign of number is thus the reign of the unthought slavery of numericality itself. The reverse side of the abundance of capital is the rarity of truth, in every order where truth can be attested to: science, politics, arts, love.

As early as the time of the Pythagoreans, a concept of number for a harmonic relationship between mathematical and musical proportions as constitutional element for an ontology of the cosmos was acknowledged. This idea of music is based on the fact that simple proportions would create the most harmonic intervals from the vibration of a string. In essence, 2000 years of musical history have revolved around the problem of how to minimize numerical paradoxes arising from the desire to make music with multiple fundamentals, or multiple 'Ones'. Around the period of the Pythagoreans, an early second strand in philosophy was formulated by Heraclitus and Lucretius, which we could call the strand of minimal deviation and of continuous change.

Interestingly, it is the problem of the continuum, the dialectic of the discrete and the continuous, which, saturating and subverting the ancient opposition between arithmetic and geometry, compelled mathematicians around the second half of the nineteenth century to rethink the idea of number. The question still remains: is there a concept of number capable of subsuming, under a single type of being and by means of a uniform procedure, at least natural numbers, real numbers and ordinal numbers, whether finite or infinite?

In 1888, Richard Dedekind wrote "Was sind und was sollen die Zahlen?" (What are numbers and what should they be?). A generalization of numbers can be approached by regarding the different number classes as proper parts of sets. Revising Dedekind, Alain Badiou lists three fundamental causes that mark the collapse of Greek thinking about numbers: first, the irruption of the problem of the infinite; second, the problem of the ontology of number, zero, the void; and third, the dislocation of the idea of the One. We find ourselves under the jurisdiction of an epoch that obliges us to hold that being is essentially multiple. Consequently, number cannot proceed from the supposition of a transcendent being of the One.

By discussing the concept of Conway's 'surreal number', he aims to 'limit the glory of number to the important, but not exclusive, glory of being, and thereby demonstrating that what proceeds from an event in terms of truth-fidelity can never be, has never been, counted.' The surreal numbers describe an arithmetic (dis-)continuum containing the real numbers as well as all infinite and infinitesimal numbers respectively larger and smaller than any real numbers. In set theory, surreals are the largest possible ordered field; other ordered fields, such as the rationals, the reals, the rational functions, the superreal and hyperreal numbers, are all subfields of the surreals. Surreals also contain every transfinite ordinal number reachable from within the set theory in which they are constructed. We therefore find the program of unification of the concept of Number (one sole concept which subsumes the natural whole numbers, the negative whole numbers, the rationals, the reals and the ordinals) to be wholly realized, firstly in multiple-being, and then in the operational dimensions. This way, this specific notion of Number mirrors the swirling nature of music.

Exploring the relationship between this world of music and economics becomes evident in the structure of the institution, WDR Studio Akustische Kunst. Where does the world of frequencies end and the world of numbers begin? It was impossible to separate method from result. Here, it could not come down to the production of theory; instead the limits of intuitive understanding are explored. The intelligible and the sensory are inseparably interlaced, but their unity is unattainable.

Credits:

Commentators: Anne Gehrig, Marcus Schmickler

Composition: Marcus Schmickler

Sonification: Julian Rohrhuber

Photo: 'volatile smile' © GEISSLER/SANN

Comissioned by WDR3 Studio Akustische Kunst, 2011

SONIFIED MATHEMATICS:

00 Number (00'00" - 03'40")

Numbers, apparently the simplest of concepts, became increasingly problematic within 19th and early 20th century mathematics. A host of unrelated concepts arose from this new thinking so that it is difficult to capture all the advances in a single term, but perhaps the best candidate would be the surreal numbers, discovered in the 1970s. Here, even numerical operations are conveived of as existing sets instead of as dynamic productions. Surreal numbers represent the largest imaginable ordered field. The sonification shows how even small integers and simple fractions require a complex, multi-layered network of contexts. (see Alain Badiou, "Number and Numbers") (Literature: Badiou, Alain Number and Numbers)

01 Dedekind Cuts (03'40" - 07'20")

An infinite sequence of ratios involving integer numbers may converge upon any point in a continuum. Though this approximation is teasingly accurate, it does not exclude the possibility that there are still other numbers which arise in the gaps between fractions. In the 1870's Dedekind conceived of such irrational numbers as cuts ('Schnitte' in his own language). In an acoustic analogy, they make it possible to divide any body of sound at a single frequency. There is a limiting frequency which really exists. (Lit: Dedekind, Richard Stetigkeit und irrational Zahlen)

02 Number and Negativity (07'40" - 11'00")

Is the presence of a lack any less than its absence? What can gaps add? In Number and Numbers, Badiou shows that, in surreal numbers, the presentation of nothing means a surplus indeed. The presence of emptiness in the form of number by no means signifies any dearth (-). Indeed the presence of an absence can certainly add to the whole (+).

03 Spectral Digits (11'00" - 14'40")

What numbers actually are is not necessarily wholly expressed the way they are written. The idea that a decimal number like 81 could also be written in binary as 01010001 shows how much we forget that we first wrote in base 10. The various radices allow every number to be broken down into a spectrum of numbers, expressed in this part as a pattern of pitches reflecting their divisibility. Whereas prime numbers exhibit regularity on no level, all others give rise to repeated patterns (in the way that 81 is 10000 in ternary). (see Georges Ifrah "Universal History of Numbers", Geroge Lakoff/Rafael Nuñez Where Mathematics Come From)

04 Prime Factors (14'40" - 18'20")

The divisibility of integer numbers might appear to be rather trivial since we all learn about division in school. In fact, though, separating a number into its prime factors alone can soon become a fascinating problem. With large enough numbers, even high-powered computers can be extended to their limits, which is precisely why prime factors are utilized for encryption of data. How similar are the primary factors of two adjacent numbers? Any integer number can be sonified as a frequency spectrum of prime numbers. The common denominator of two numbers is automatically common part of the sound spectrum.

05 Complex Numbers (18'20" - 22'00")

The continuum of the reals might seem like a closed medium of measurability. However, even in solving simple quadratic equations there are instances which have no real number solution. Nevertheless, any quadratic equation can be solved by simply including another, so called "imaginary" dimension of number. Here, however, calculation differs: multiplication is not the same, because both complementary (real and imaginary) parts affect the outcome together. This sonification interprets these complementary aspects as the location and frequency of sound clusters. (Lit.: Krantz, S. G. The Topology of the Complex Plane)

06 Riemann's Zeta Function (22'00" - 25'40")

The Riemann zeta function can be represented as the series of natural numbers to the power of a single complex number. Its rather intricate graph hides a deeper simplicity, and perhaps that is why its importance extends across disciplines, indirectly linking multiple branches of mathematics. Riemann's 1859 postulate that all the roots (zero values) lie on an axis with the real component 0.5 remains unproven. Ken Takusagawa has calculated all the zeta functions on that axis within a range from 2.7 and 29143636.6 to produce a sequence with makes an excellent basis for sonification. The roots become audible for instance in the form of a sequence of impulses.

07 Linear Congruence / Intercalation (25'40" - 29'20")

As already mentioned earlier with Dedekind's Cuts, into every ratio can be interjected another ratio. This creates a series which seems gravitate towards the center of a real number, which is the cut itself. However, from a pair of numbers a chain of numbers can arise by repeated multiplication and integer division. Over a longer or shorter time this chain repeats itself, and so in a sense revolves around an empty, non existing, centre.

08 Interlude: Dissonance / Difference (29'20" - 33'00")

Set theory is a relatively new area in the history of music. One of the first composers who used set theory for his concept of musical scales was Ferruccio Busoni, who in 1907 conceived a theory of 'musical degrees' for a classification of melodic scales. At roughly the same time, Mathias Hauer was investigating the combinatorial properties of all hexachords. Since the 1950s, some speculated properties have insinuated themselves into mathematically motivated music in both Europe and the USA. (see Joseph Schillinger "Mathematical Basis of the Arts", Moreno Areatta, Jean-Michel Bardez, John Rahn (Eds.) Around Set Theory)

09 Divergence (33'00 - 36'40")

In the world of acoustic instruments, as well as that of sound, there are no pure tones. Every sound is a sum of a fundamental tone and overtones, whose arrangement forms the timbre and determines the character of instruments. The correspondence of the overtones plays an important role in the perception of consonance and the selection of audio material and its instrumentation. Yet even a pure sine tone can be thought of as a result of summation. Here a single frequency disintegrates into a pattern of many frequencies. (see Sethares, William A. Tuning, Timbre, Spectrum, Scale)

10 Cardinal Numbers (36'40" - 40'20")

By means of this class of numbers the generalization of natural numbers is used to describe an unordered cardinality of sets. Georg Cantor showed how cardinal numbers can be used to bring different infinite sets into relation with each other. Infinite sets are not necessarily of equal size or cardinality; instead they themselves constantly form new levels of the infinite.

11 The 'smallest' infinite set is known as (Aleph_0) (40'20" - 44'00")

is the cardinality of the set of all natural numbers, with which the large and the small can be defined, thus allowing the possibility of an ordered sequence. Aleph-zero itself plays no part in this order. A psychoacoustic phenomenon may come close to this mixture of order and disorder: a sequence of Shepard tones, whose discrete pitches drift.

12 One (Unity) (44'00" - 47'40")

The one distinguishes between being and the void, which is why philosophical conceptions have pointed to its apparent simplicity since antiquity. Is the world one or do we only consider it as one because it is not conceivable as anything else?

13 Refrain: Numbers Negation (47'40 - 52'00")

The two halves that every surreal number possesses are known as its form and its matter. The form is that which is infinitely perforated and irregular; the matter is the projection surface onto which this incomplete shadow falls. The negation of such a number is like a photographic negative - a number that remains empty wherever the other is filled.

REFERENCES: | |||

2016 | Köln | 09.18 | Photoszene presents Politiken der Frequenz with Greek Choir inside an installation "Volatile Smile" by Geissler/Sann. |

08.28 | CD | new mix CD by Objekt containing a track from "Politiken der Frequenz" on legendary Technolabel Tresor | |

04.09 | Athens | Borderline Festival presents Politiken der Frequenz with choir in Greek language | |

2015 | 03.07 | Los Angeles | Noise and the Possibility for a Future conference at Goethe Institut LA. |

2014 | 04.14 | Release | Marcus Schmickler/ Julian Rohrhuber 'POLITIKEN DER FREQUENZ' (Tochnit Aleph / editions Mego) |

04.10. | Berlin | Politics Of Frequency - Record Release Presentation at KW Kunstwerke for Contemporary Arts | |

2013 | 28.08. | MusikTexte 138 "Sonification in the Context of Composition" by Marcus Schmickler (German) | |

20.04. | Providence | Performance and Talk at Brown University | |

2012 | 14.12. | Radio | ORF presents 'Politiken der Frequenz' as performed at Steirischer Herbst, Graz |

05.10. | Graz | Concert Performance Musikprotokoll im Steirischen Herbst | |

25.05. | Frankfurt | "Politiken der Frequenz" Lecture at Hochschule für Musik und Darstellende Kunst. | |

19.01. | Bielefeld | Kunstverein: Presentation/Talk: Politiken der Frequenz / Concert | |

2011 | 04.11. | Köln | Premiere of Politics of Frequency, WDR3 Radio |

03.11. | MACBA webradio Mix: Interruptions 6 / Ontology of vibration: economics, music and number. |