Stochastic Basics to Mechanism

For Xenakis, Stochastic music is based on developing a mathematical structure that functions on a macrolevel to organize the movement of sound events that are random only within a set of possibilities or densities. Aggregates of sound are thus held within an overarching net of sound movement. [1]

The Diatope
Xenakis Familly Collection

The Diatope, the project by Ianis Xenakis for Paris in 1978. The audience witnessed 1600 strobe lights with eleven loudsppeakers within the space of the canopylike structure. [1]

 

Whatis ?

Stochastic system is a part or parts of which holds a random element associated with it. We could possibly say that anything based on probability is stochastic in a broad sense. It is often used for describing and simulating the subject which is uncertain and heterogeneous with use of random variablesw, notably in contrast to deterministic systemw, input and outcome of which are fixed and certain. In such subject, we only can predict events at mcrolevel but not at microlevel. [1]

Along with Probability theoryw, the science of uncertainty, the system is often applied in various fields of study from economics to nature. It is of interest to note that probability theory arose from the study of gambling in order to analyze so-called games of chance. In modern-day applications, the system is essential for simulating and anticipating non-deterministic events of natural phenomena; stock prices, chemical reactions, population growths, weather, various biological processes like molecular dynamics, artificial intelligence, and even musical compositions. Maestros like John Cage and Karlheinz Stockhausen have incorporated a system as a source for algorithmic compositions since the 1950s. Our research on stochastic systems will be largely based on occurrences in its application to music. On this page, which precedes the topic of musical applications (Stochastic in Musical Compositionspost type), we would like to introduce some basic concepts and usage of the system through mathematics.

To learn more about stochastic processes, please see the Wikipedia entry, Stochasticw, Stochastic processw.

 

Stochastic in Design

Architecture consists of multiple components spatially, structurally, and materially on different scales. There must be an abundant number of strategies as there are a variety of designers for these facets to be well-assembled and well-arranged. The stochastic system is also of help on this specific task, distributing parts in a natural manner, in other words, a “non-deterministic distribution”. In design, the system will not generate a complete picture of your resulting design out of the box, but will rather become a powerful complement to many complex systems, and will work competently with the likelihoods of events that the designer sets, according to the context.

 

Probablity Distribution

Many computer languages have random functions in their libraries. Even Grasshopper has several components to assist you in incorporating randomness in your project.

 

For us to gain better control of a randomly determined organization, we should set up some parameters, one of which could be frequency/the number of all outcomes. To do so, we could borrow a tool from the study of probability. Probability distributionw gives us the means to place to set that frequency, the likelihood of an event, and manage a specific tendency of organizations under randomness.

According to probability theory, probability distribution is “a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.”[1], wherein Random variablesw map outcomes of random processes to numbers, quantifying the outcomes. For example, the distribution of the chance you get a head after flipping a coin 3 times can be represented in probability distribution as P(X=0) = 1/8, P(X=1) = 3/8, P(X=2) = 3/8, P(X=3) = 1/8 where X is the random variable. Probability of X=0, where you get 0 heads, is 1/8, probability of X=1, where you get 1 head, is 3/8, and etc…(Fig. m-2)

 

[Probability Distribution of Coin Toss ]

This simple occurrence of probability distribution called the Binomial Distribution. There are only two possible outcomes, success (a head) and failure (tail), with fixed probabilities summing to one. It consists of a sequence of n identical trials where the function of distribution is P(x) =  \frac{N!}{x!(N-x)!}  \pi^{x}   (1-\pi)^{N-x}. [2] There is a great variation in probability distribution that has already been established in the study of probability. You can view the list of those on wiki List of probability distributionsw. Some of the most common distributions are listed below (Fig. m-2).

  • Binomial Distribution
  • Poisson Distribution
  • Uniform Random Distribution
  • Deterministic Distribution
  • Cumulative Distributions

[List of Probability Distributions]

It would be fun experimentation to implement one of the distributions on the above list in our organization. However, for the specific purpose of design, we need to develop our own customized probability distribution. To begin with, we look at how to implement randomness in our organization both with and without probability distribution through Grasshopper.

 

Algorithm

Shuffling Panels

Randomness applied in our organization both with and without probability distribution through Grasshopper.

Two images presented below are generated by GH definitions that aimed at shuffling panels by replacing their indices randomly. “Shuffling Panels Ver.1” is without distribution and “Shuffling Panels Ver.2” is the one with.

The common set up of our targeted space for both definitions is a 4 x 4 grid along which mesh panels are arranged and color-GHd according to the indices of each grid cell. For the random component, we use one of GH components, Jitter, which allows us to shuffle those list indices of the grid cells with its parameter, shuffling strength and seed. In “Shuffling Panels Ver.2”, in order for us to set frequency of all outcomes as random distribution before shuffling panels, we incorporated GH Graph Mapper and several vb dotNET script components.

 

To see the full description and GH definition, go to Shuffling Panelspost type

Now with “Shuffling Panels Ver.2”, it is possible that frequencies of each index set by the distribution and the location of indices are shuffled randomly. Those two algorithms presented above are meant to be an introduction to the basic usage of the stochastic system.

 

Now we can analyze it and discuss what could be the next. We will do this in the next article. In the meantime, have a look at other available algorithms in Stochastic, their prototypes, and some application in projects at Stochasticpost type

 

Gallery

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