Rotations

Traditional counterpoint techniques, such as inversion, mirroring, and mirrored inversion, as well as various transposition algorithms, could also be applied. As shown below, the triangle as a whole as well as individual rows can be rotated, mirrored, inverted, and further manipulated according to different algorithms.

Readings
Furthermore, the sequences of numbers can, of course, be read in several different ways. This includes the obvious left-to-right, right-to-left, top-to-bottom, and bottom-to-top readings, but many other options like looped sequences, randomness, and changes in direction are also possible. I think of this system as similar to a sequencer module in a modular synthesizer, where different clock pulses are used to advance to the next value according to specified reading patterns.

Applications
Besides being entangled with the other systems presented within this project, the EMP Triangle has been used in several of the patch scenarios. This includes Patch 5: Triangular Progressions and the three Sinew0od transcriptions presented in Patch 6, Patch 7, and Patch 8.

System 3: EMP Triangle

The EMP Triangle is a modular compositional tool for the generation of musical ideas, sonic material, and structure that have been used in my practice for a long time, and hence, intertwined into all the other systems.

 

Construction

It was constructed using the series of 1 - 8 as a first row of numbers and then derive the next row by pairwise summing and taking the digital root of each, i.e., reducing all numbers to single digits (which gives the same result as n mod 9). Continuing this process, a number triangle with symmetrical properties emerge. As seen in the image above, these properties include symmetrical alignment of numbers that sum to nine, the rows all sum up to numbers that can be reduced to nine, and the number nine itself perfectly lined up in the middle of every second row. When i discovered these properties, I thought of it as something magical, and I still find it fascinating, but of course this is just a consequence of the use of the Base-10 number system. However, this triangle has been proven to be useful as a catalyst for new compositions and sonic structures ever since I discovered it. It has laid the foundation for many compositions and is an important part of my composer's toolkit.

 

 


 

Modularity

Over the years, I have developed a modular understanding of the triangle. Instead of applying a unidirectional notion of mapping, this enables a notion of multidimensional patching where digits, rows and symmetries can be used as modules that can be patched both internally and into certain musical dimensions or as containers for musical gestures or parts on a higher structural level. The patch example below shows internal connections between numbers 3, 5, and 7, and their corresponding rows (read from the top). The four nines are patched to 1 and 8, 2 and 7, 3 and 6, and 4 and 5, respectively.


Such patching techniques have been used to generate patterns for note durations, pitches, subdivisions of durations and pitches, chords, various timbral parameters, as a re-mapping filter for musical gestures, and form.

 

A crucial factor for the patching possibilities described here, is a SuperCollider implementation of the EMP Triangle. Working with the triangle in practice thus means that the connections are made in a coding interface. This, in turn, enables a simplified handling of complex patching, e.g., between several instances of the triangle, with different rotations and permutations, and a simplified mapping between musical structure and sonic control.