synesthetics-in-motion

COMPOSING BY NUMBERS – SUMMARY by Dorota Walentynowicz

LECTURE 1 – PYTHAGOREANS

The school: We speak of Pythagoreans rather than of Pythagoras, because little is known of the founder and, therefore it is difficult to assign to any individual the characteristic Pythagorean doctrines. The term „Pythagoreism” refers to the whole of religious, ethic, moral, political and even aesthetic doctrines and concepts formed in VI/V c. BC, assigned by the tradition to Pythagoras himself. Pythagoreism is also a sensu stricto philosophical current formed by the Pythagorean school. And finally we interpret the term „Pythagoreism”, as Plato did, as a certain model of living in a community introduced by the Pythagorean communities, obeying the rules assigned by their master. Pythagorean philosophy came to an end in IV c. BC but Pythagoreism persisted as a form of attitude and model of life until III c. AD.

Mysticism: Pythagoras was believed to be the descendant of Hermes Psychopompos: the Greek god Hermes was responsible for bringing the souls to the underworld and was considered the patron of any secret knowledge (hermetic) – connecting Pythagoras to this particular deity indicates the mystical character of his teaching

Harmony: Aristoteles in “De anima” - „the tradition has passed on still another theory of the soul; they [who are in favor of it] say that the soul is a certain kind of harmony, because harmony is a mixture and synthesis of the adverse things”.
According to Pythagoras the soul was a force, which, taking as an example the harmony of cosmic elements, keeps a similar harmony in the human being. The man is supposed to imitate the order (Greek “cosmos” means „order”) of the universe, thus deserving to be called a microcosm.

Numerical order: Having discovered mathematical ratios in all the phenomena of the world, Pythagoreans begun to worship the number, seeing in it the cause and reason of all order and the ruling power of the universe.
„you shall see the nature of the number and its ruling power not only in the daemonic and devine things, but also always and in all of the doings and words of humans” says Philolaos.
Some of the numbers and geometric values were even identified with certain gods or devoted to the highest deities of Greek pantheon.
Philolaos, a contemporary of Socrates, uses the numbers to define not only the mind but also all other properties and abilities of a man.
„He defined the physiological functions of the body by number 5, animation by number 6, mind, health and light by number 7. Further he claims that love, friendship, wisdom and perception are in number 8.”

Plato's Timeaus and the Regular Polyhedra: Plato's creator god, the Demiurge, gave the basic elements definite geometrical shapes, but just as small and imperceptible as the atoms. The Demiurge chose the first four regular polyhedra for this purpose, because they were the most perfect, the most beautiful, and therefore the best. As earth is most stable and immobile of elements, God chose the cube, with its large base areas, to constitute it. Fire is the least stable and most mobile, so it is made up of regular triangular pyramids (tetrahedra). Water is more stable than air, so it is composed of regular icosahedra. Plato observes (59d) that the bases of icosahedra "give way" more easily than cubes, so that explains why water is able to flow. Perfect octahedra are left to make up the air. Fire, being composed of the smallest solid with the sharpest points and edges, is the most destructive of the elements.


LECTURE 2 – FROM PYTHAGORAS TO J.S.BACH

Consonance:
According to legend, Pythagoras discovered the foundations of music by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. He found out that the explanation was in the weight ratios. He recognized that consonant musical sounds can relate to simple number ratios.
Perfect consonances: unison 1:1, an octave 1:2, a twelfth 3:1, a double octave 4:1, a fifth 3:2, a fourth 4:3.

Pythagorean coma: 12 perfect fifths (the circle of fifths) do not equal any even octave. What is left is called the Pythagorean coma. The resolution of the Pythagorean coma was the major concern for further development of temperament system.
Until 14th century Pythagorean tuning remained popular, because the perfect fifths and perfect fourths were intervals favored by the early medieval ear (example: ” La Messe de Notre Dame” by G. de Machant). Later composers have exploited a rearrangement of the Pythagorean tuning in order to explore possibilities of more blending thirds and sixths.

Temperament: involves small and deliberate deviations from the ideal interval ratios in order to eliminate the Pythagorean coma and allow transpositions
Pythagorean tuning: pure 5ths and 4ths, active 3rds
Meantone temperament: pure 3rds (standard for 16th and 17th century keyboard instruments)
Well-tempered system: tempers all fifths by different amount thus producing a variety of colors, but still permitting all intervals to be playable at any transposition (18th century)
Equal temperament: equally tempers each fifth to disperse the come

Musurgia universalis by Athanasius Kirchner (1650) was one of the most important musical texts of 17th century and a testament to the world’s understanding in his days. Kirchner, combining Pythagoreism with Catholic orthodoxy, held the essentially medieval view that the cosmos was revealed in musical rations and that musical harmony mirrored God’s harmony. Kirchner – also known as inventor of Cat Organ – introduced rhetorical steps of inventio, dispositio and elocutio into musical theory. The “musica poetica”, as it was called, aimed to reveal the meaning of the text in and through the music.
“Music is a hidden arithmetical exercise which counts off subconsciously in the soul” (Leibnitz)
the Christian numerology:
1 – God (unison)
2 – Son (octave)
3 – holy spirit (fifth)
4 – angelic (fourth)
5- human
7- mysterious, holy


J.S. Bach’s use of mathematics: J.S.Bach was reported to be a member of Societat der Musicalischen Wissenschaft, a society devoted to studying Pythagorean philosophy. He was well-known for his use of various self-referential number games within his works.
The use of numbers in Bach’s works:
- Enumeration (simply counting notes, measures, bars, parts, movements etc.)
- Operation (transformations of the numbers – subtractions, additions etc.)
- Translation (use of gemaria, eg. A=1, B=2, C=3 etc.)
- Interpretation (the meaning and symbolism of numbers, usually religious)

Bach’s favorite numbers:
7. stands for perfection, Creator and Creation, or the Beginning and End
12 used for the Church, the disciples, the congregation
14. the sum of the letters in his name: b+a+c+h = 2+1+3+8=14
his last piece, called “the art of fugue” remained unfinished, as – according to his son’s report – the composer dies where the notes B-A-C-H appear in the counterpoint

example:
Symbolum Nicenum of the B-minor Mass:,"Credo in unum Deum" and "Patrem omnipotentem”: 
The total number of measures/bars for these two movements: 129

Apply gematria: C(3)+R(7)+E(5)+D(4)+O(14) = 43.
 The number 129 comes from (43 + 43 + 43) x 3. 3 = Triune God
. The word 'Credo' is sung 7 x 7 = 49 times
. The phrase 'in unum Deum' is sung 7 x 12 = 84 times. 
At the end of the fugue in "Patrem omnipotentem" Bach inserts in his own hand the total number of measures/bars: 84 = (permutation: C=3 x A=1 x (B=2, H=8) the letters BACH rearranged, but this is permissible, as well as combining the B and H into one number as he did here) 12 x 7 = 84 a combination of time and eternity.

Geometry of counterpoint:
To us today seems obvious that high-pitched notes are high, and low-pitched are low. So it comes as a shock to learn that in ancient Greece high-pitched notes weren't heard as high; in fact the highest pitched note of the classical Greek octave was called nete, the nether, or lowest note. it got this name from the fact that the stringed instrument named a kithara was held with the highest pitched string nearest the ground - just how one holds a guitar today. J. Desprez in “huc me sydereo”- sending Jezus down from Mount Olympus in a descending scale of twelve notes puts the seal on the development of the pitch/height relation. Three centuries later J.S. Bach used that notion of direction as basis for various geometric operations on his musical scores.
- imitation
- melodic inversion
- retrograde
- retrograde inversion
- augmentation
- diminuition



LECTURE 3 – XX CENTURY MUSIC

Towards atonality:
Tonal music is that in which a definite sense of key prevails, in which all the notes are related to a central key note (a tonic): changes of key center can be brought up by modulation
J.Cage: “the octave has no more reason to be divided into 5/6/7/9 equal intervals than to be devided into 36/59 parts. It’s just a matter of establishing limits”
C.Debussy was the first to use a scale different from traditional scale: the whole tone scale, obtained by dividing the octave into 6 equal parts. Other possible divisions of the octave include 31 tone scale (M. Marsenne designed a 31 tone keyboard) and 53 tone scale (confirmed musical system in China in early 18th century)

Serialism: method of composing with 12 tones (dodecaphony). Series – twelve notes of chromatic scale arranged in any order by composer’s choice, as long as each note appears only once. A series forms a tone-row which undergoes transpositions (12 possibilities) and transformations (prime, inversion, retrograde, retrograde-inversion), which combined give 48 different possible forms of the same series (= a set complex).
Integers 0 to 11 can be substituted for pitch-class names by assigning these integers to successive notes of ascending semi-tonal scale, whose initial element, pitch-class 0, will be the same as the first note of a given prime form of the set.
A.Schonberg was the first one to use the method of composing with 12 tones related only to each other, thus negating the tonal system and “emancipating the dissonance”
(1907 – Schonberg abandoned the key signature)

Aleatoric music: music in which some element of the composition is left to chance or some primary element of a composed work’s realization is left to the determination of the performer (alea = dice). P.Bulez spread the definition using it to describe works that give the performer certain liberties with regard to the sequencing and repetition of the parts.
Aleatoric music originates from musical dice games popular in 18th century (Musicalische Wurfelspiele) – one such is attributed to W.A.Mozart: these games consisted of a sequence of musical measures with variations, which were selected by tossing dice.
M.Duchamp – “Musical erratum” (1913) : score for three voices
Duchamp wrote this piece of music by placing notes on the staff in the order in which they were randomly drawn from a bag (25 notes from F below middle C to high F); the text was a “ready-made” picked from a dictionary ( J.Cage: “One way to study music - study Duchamp”)

Indetermined music : J.Cage and I-Ching
“As a composer I should give up making choices and devote myself to asking questions; chance determined answers will open my mind to the world around at the same time changing my music” (J.Cage)
For cage time is the most fundamental music category, it exists prior to both pitch and harmony and can contain musical sounds as well as noise and silence.
“Imaginary landscape 4” – written for 12 radio receivers: each radio was played by two players, one controlled the frequency, the other the volume; Cage wrote precise scripts of timing but could not control what will be transmitted at the moment of the performance – Cage sets up an architecture but then allows the internal décor to be subject to chance operation.
“Music of changes” – in this composition Cage uses I-Ching, Book of Changes, the ancient Chinese oracular. I-Ching is organized according to 64 hexagrams, each consisting of 6 lines. The oracle is consulted by tossing 3 coins. Cage used a computer program which generated I-Ching hexagrams as if by throwing coins and later translated the chosen hexagrams into a selection of specific number of alternatives


LECTURE 4 – GAMES AND OTHER STRATEGIES

Many composers of XX century music found numbers and various computational models a useful source of compositional material
- set theory
- game theory
- magic squares
- Fibonacci numbers
- Probabilities
- Physical laws

Ordering structure: how to produce scales on more general level? Mathematic strategies, such as sieves, sets, and probability, can be applied to any characteristic of the sound, provided that it can be ordered. If we take any three values of a given sound characteristic and if we can classify them in such way that one falls in between the two others, then we have an “ordering structure”.

Fibonacci sequence:
I.Xenakis - “Metastasis” pitches based on 12-note rows were assigned a series of durations based on the Fibonacci sequence, along with a range of timbres
K.Stockhausen – “Klaverstuck IX”: a mathematical structure undergirds the gradual dissolution of the piece; the grouping of subsections and choice of tempi are partly dictated by the Fibonacci series [1, 2, 3, 5, 8, 13, 21, 34, 55, . . .]
C.Debussy – “Image, reflection in Water”: the Fibonacci numbers are apparent in the organization of sections; sequence of keys is marked out by the intervals 34, 21, 13, and 8
B.Bartok – “Music for string, percussion and celeste” : the xylophon progression appears in intervals 1:2:3:5:8:5:3:2:1


Probability theory - Stochastic music :
Probability concepts are applied across mathematics when considering random structures. What is probability? – when something repeats itself, but not exactly. When we toss a coin we know that the result will be either heads or tails. When we toss it ten times we can’t say forehead how often it will land on one side or the other. When we repeat this many times then the proportion of heads and tails approaches the value of ½. It is therefore a matter of convergence, meeting the target point. That point is “stochos” (Greek for target : the word “stochastic” was used in17th century in a connection with probability theory).
I.Xenakis “ST” – 9 works composed exploring the stochastic method in order to systematize random events

Sieve/ set theory:
I.Xenakis uses the sieve theory to compose his scales. Sieve theory is a set of general techniques designed to estimate the size of sifted sets of integers. Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer.

Physical laws: control of mass events and recognition of laws that govern nature
(eg. I. Xenakis “Metastasis” changes intensity, register, and density of scoring as the musical analogues of mass and energy)
- kinetic gas theory – particles behaving like billiard balls that will hit each other according to the Newtonian laws of mechanic
- Brown movements – how does a single particle move in a mass of others?
- Markov property – the next move depends only on the current position, and not on the previous state

Graphic structures:
In the works of I. Xenakis, such as “Evryali”, “Mists”, the graphic method of arborescence is used (arbor = tree). The starting point is a graphic structure, drawn on a plane, which undergoes a series of geometric transformations (turn, twist, zoom), based on which the sonic picture is created. CEMAM – graphic electromagnetic system with which you can draw any shape and obtain the corresponding music with the help of computer.
M.Kagel - visually manipulates the forms of traditional notation as a means of developing new music. He creates new musical structures by conceptualizing the visual representation of music. These representations are a combination of classical notation form and geometry. Kagel uses angular relationships to develop new variations utilizing harmonic proportions.



Game theory
Game theory is a branch of applied mathematics that studies decisions that are made in environments where various players interact. It studies the situations where players choose different actions in attempt to maximalize their returns.
(origin of game theory - the so called “marriage contract problem” discussed in the Talmud)

types of games:
- symmetric games: identities of the players can be changed without changing the pay-offs
- zero-sum games: a player benefits only at the cost of the other (one wins exactly as much as the other one looses)
- non-zero-sum games: a gain of one player does not necessary correspond with the loss of the other
- simultaneous games: both players move at the same time, or one is not aware of the choice of the other one when making his
- sequential games: later players have some knowledge about the earlier actions
- games with randomly distributed pay-offs

examples of games:
- prisoner’s dilemma
- chicken
- ultimatum game
- dictator game
- battle of sexes
- deadlock
- matching pennies
- stag hunt


I. Xenakis “Duel”, “Strategy”:
“the game is based on musical events, the tactics, which occur simultaneously, the participants employ a strategy determined by me , which leads to the simultaneity of two particular musical sections out of a number of possibilities. The winner is the one who is more adroit in employing a series of tactics designed to achieve higher final results in accordance with the Matrix of the Game. The winner is the one, who has accumulated the grater number of scores according to the matrix”. (I..Xenakis)


Wolfram tones:
WolframTones is an experiment in applying Wolfram's discoveries to the creation of music. WolframTones works by taking simple programs from Wolfram's computational universe, and using music theory and “Mathematica” algorithms to render them as music. Each program in effect defines a virtual world, with its own special story--and WolframTones captures it as a musical composition. (http://tones.wolfram.com)

posted by Mikela @ 12:25 PM