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This paper is an attempt to create a theoretical foundation for equal division of the octave (EDO... more This paper is an attempt to create a theoretical foundation for equal division of the octave (EDO) tunings. It is specific to 15 notes per octave (15 EDO) tunings, but should be applicable to more broadly. The majority of the non12tone compositions that I am aware of either use a modified version of 12 EDO scales (major, minor, etc.), or simply use trial and error to find interesting melodies and harmonies. I have found a way to define 15 EDO scales according to their own character that should help composers work with any EDO tuning. Section 1. Tuning The tuning I use is based on the way I discovered microtonal non12tone music. I was aware of microtones as used in Indian and midEastern music, (although I only learned about the theory behind them relatively recently). I was not aware of non12tone music when I read about a method of setting 12tone equal temperament notes in the book "e, The Story of a Number" by Eli Maor. In this, an imaginary meeting between Johann Bernoulli and J. S. Bach is portrayed, in which Bach describes the difficulty in determining the frequency of notes in an equal temperament tuning. Bernoulli immediately grasps the problem and presents a mathematical solution, which is to graph a logarithmic spiral and intersect it with 12 lines that are spaced at equal angles around the origin of the XY axis.
This paper is an attempt to create a theoretical foundation for equal division of the octave (EDO... more This paper is an attempt to create a theoretical foundation for equal division of the octave (EDO) tunings. It is specific to 15 notes per octave (15 EDO) tunings, but should be applicable to more broadly. The majority of the non12tone compositions that I am aware of either use a modified version of 12 EDO scales (major, minor, etc.), or simply use trial and error to find interesting melodies and harmonies. I have found a way to define 15 EDO scales according to their own character that should help composers work with any EDO tuning. Section 1. Tuning The tuning I use is based on the way I discovered microtonal non12tone music. I was aware of microtones as used in Indian and midEastern music, (although I only learned about the theory behind them relatively recently). I was not aware of non12tone music when I read about a method of setting 12tone equal temperament notes in the book "e, The Story of a Number" by Eli Maor. In this, an imaginary meeting between Johann Bernoulli and J. S. Bach is portrayed, in which Bach describes the difficulty in determining the frequency of notes in an equal temperament tuning. Bernoulli immediately grasps the problem and presents a mathematical solution, which is to graph a logarithmic spiral and intersect it with 12 lines that are spaced at equal angles around the origin of the XY axis.