Hypereuclidean SetsBack to Home PageSummary The idea of hypereuclidean sets (also referred to as "higher-order Euclidean sets") builds upon research by Godfried Touissant, John Clough, Jack Douthett and many others. A hypereuclidean set is created by taking a series of related Euclidean sets and using those with lower n-values as index sets. Elements from higher-order sets are removed recursively by lower-order elements. In this way, the method can generate interrelated structures that have a greater variety of IOIs (inter-onset-intervals) than conventional Euclidean sets. Analytical methods can potentially show relationships among different musical structures by charting shared lineages. The problem of creating, finding and relating hypereuclidean sets lies principally in the sheer quantity of sets that can be derived (see below). Further complications arise in sorting, identifying interesting properties, and relating sets to musical structures. However, solutions to all these challenges exist. The approach is promising because it can be applied either to rhythm or pitch. To better understand the principles, I have written two patches in Max/MSP that vividly illustrate the technique of creating hypereuclidean sets in a simple and visually intuitive way. While Max offers an appealing environment for casual exploration, its scheduler is not designed for computationally intensive tasks. To do the work more efficiently, I wrote a pair of C++ programs that run on the command line. Two C++ programs are available. The first prompts the user for all the relevant information, and generates a single set. This program is intended for targeted searches. The other program only prompts the user for the outer bounds, and generates a family of sets in a way that seeks to achieve some measure of algorithmic optimization. Release Notes & Downloads All the code is available for free on GitHub. You can obtain it here: https://github.com/lauprellim/euclid The Max/MSP patches require the free Bach and Cage libraries. Please be sure they are correctly installed or the patches won't work. You can easily get them through Max's Package Manager. The patches also have three dependencies, namely two JavaScript files and a .json file which stores the presets. Be sure the patcher can access them. The C++ programs are available as source code. After compilation, users can easily write the output of both programs to text files using the command line (pipe tee), and then search for sets using grep. Future plans include organizing the output of the "make-many" C++ program into trees, and other tools to organize the generated material. There are a few small glitches here and there in the patches, and I'm sure I could optimize the C++ source in many ways. I'd be glad to receive whatever feedback the community might offer. This is an ongoing project. Algorithmic Performance Run on a 2.4 GHz Quad-Core Intel Core i5 MacBook Pro (2019!)
Some screen shots Generating Steve Reich's "Clapping Music" as a second-order Euclidean set using the subtract-first method 
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