Topophonia: Terrestrial Music
Technical Details

Technical Notes

The way in which Fontcuberta's proposal has been developed, is based on scientific justification of the representation of the profile of a mountain as a melody. The first step has been to obtain a model of the mountain. The geographic data has been kindly granted by the Institut Cartogrāfic de Catalunya. Starting from an elevations model, a triangular grid adapted to the detail of the terrain was obtained. This grid is processed in order to obtain two types of representation: visual and aural.

Given the fact that this particular mountain belongs to the occidental cultural background, the visual representation chosen is perspective viewing and its composition is somehow related to landscape painting tradition. Likewise, the aural representation has been worked around the temperate scale within the eight octaves of the piano and the traditional notion of melody.

The approach used to generate the melodies, is based on the fractal properties of nature. We will not begin an in depth discussion of the matter since it has been widely studied. We will only make an intuitive description and will make reference to those points which have served as foundations for our decisions.

Its is known that shapes produced by nature may be modelled by a certain randomness restricted by gravity, resistance of materials, friction, etc. This does not permit to obtain "totally random" shapes. In other words, we cannot find infinitely high mountains, nor mountains that have the width of a grain of sand. This compromise between randomness and predictability may be modelled by a fractal "noise". That is, by the variations produced over time, the spectral density of which has an order of 1/f. This type of noise is, neither totally random as white noise which has a constant frequency spectrum, nor so correlated as noise with a spectral density of order 1/f².

Similarly, the studies made by Voss in the field of music, have proven that all known melody has fractal properties which correspond to the same 1/f noise. What is very surprising, is the fact that this is true for any type of melody whatsoever, be it from the XI century or contemporary, and more surprising still, it is independent from its origin, be it from the African pigmies, from occidental classics or from the Beatles. All of them obey the compromise between randomness and predictability.

This means that if a noise with 1/f spectrum is generated and from it a melody is written, where the sequence of notes is given by the variation of the noise, the resulting sound would have the necessary characteristics to be recognised as a melody. A melody which cannot be associated to any existing culture, but a melody after all.

We will not generate the melodies from an invented set of data. Instead, we will obtain them from the cartographic data of the shapes of nature: from mountains. This process starts with the mesh of the mountain which was derived from all sorts of geographic measurements: satellite and aerial pictures, field measurements, etc. This mesh represents certain points with world co-ordinates (longitude and latitude) and their height above sea level. Together with the points, the mesh also determines how they are interconnected forming polygons (triangles).

From this mesh, we may obtain the visual representation of the mountain by perspective projection of the polygons and a given point of view of the spectator (chosen by him/herself).

The melody is obtained from the profile viewed by the spectator at any moment. This is done by a process which: (a) calculates the viewed profile, (b) extracts the fractal noise (i.e. the morphologic properties) from the profile and (c) scales the values in a codification of the eight octaves of the piano in the tempered scale. We now briefly detail these steps.

• (a) The profile is computed in real time by sampling it through an adaptation of the Z-buffer algorithm. More specifically, each edge of the grid is perspective projected on the projection plane according to the given point of view. Then we proceed to simulate the drawing of the edge. To do this, we calculate the Y's of the edge (Y_e) for integer X's corresponding to positions of an array of the same length as the profile. These Y_e are then compared with the Y_p that have already been stored as part of the profile. If for a certain X_p of the profile, Y_e is greater than Y_p, then Y_e substitutes Y_p. Therefore we can call this algorithm a Y-buffer. We have tried other algorithms but all of them have a complexity of O( n^a) where n is the number of edges and a <= 2. In contrast, the Y-buffer algorithm has a complexity of O(N) for N=P*n where P is the number of points in the profile and n is the number of edges. The sampling of the profile is done from left to right since it was an arbitrary decision to take one direction or another (figure (a)).
  
  
  figure (a). Sampling of the profile (example profile).
  
  
• (b) Now that we have the profile sampled, we need to obtain the fractal noise within it. For this purpose we subtract one sample from the following one. This gives us the local increment (positive or negative) of the profile, giving a discrete measurement of the fractal noise (figure (b)).
  
  
  figure (b). Fractal noise (Difference of one sample minus the previous one).
  
  
• (c) To be able to convert the measurements of the fractal noise onto notes, we first chose a codification of the white notes of the eight octaves (seven plus the eighth C) of the piano. This codification was as simple as numbering the notes consecutively by positive integers. This gave us a range of numbers (0, 49). The sampled fractal noise was then scanned to find its maximum and minimum giving a range of floating point values (fmin, fmax). This range was then mapped onto the integer range of the codified notes and hence each sample corresponded to a given note (figure (c)).
  
  
  figure (c). Resulting melody by mapping values.