a piece from the full-length album, Robert Schneider "Songs for Other Worlds", which will be available on cassette from Cloud Recordings on December 2, 2022

from the cassette liner notes:
The pieces on this album were composed using a mathematically-defined “non-Pythagorean” musical scale tuned in proportion to the logarithms of the whole numbers from 4 to 16. I first introduced the musical scale fifteen years ago on the album New Magnetic Wonder by The Apples in stereo (2007). Five years later, after much experimentation plus completing my B.S. in Mathematics, I summarized my ideas in “A non-Pythagorean musical scale based on logarithms” for the Proceedings of Bridges: Mathematics, Music and Architecture (2012), my first publication. In 2012, I stopped touring in order to go to graduate school in number theory. So this album marks the fifteenth anniversary of my new musical scale, and the tenth anniversary of my first steps as a mathematician. It also marks the thirty-year anniversary of the birth of the Elephant 6 Recording Co., which I co-founded with my friends in 1992.



I thought of the formula that defines the musical scale in 2005 while discussing music theory with Apples co-founder Jim McIntyre, and wondered if it could be used to make music. I generated the tones using a software sine wave generator, and my brother-in-law and collaborator, Craig Morris, converted those into files playable on a MIDI keyboard. The scale is "non-Pythagorean" in that it is constructed using entirely different mathematical rules from those that define the tones in most traditional musical scales, which are attributed to Pythagoras. Instead of dividing the octave into twelve roughly equal half-steps, successive notes in the logarithmic scale get closer and closer together. This produces alien-sounding melodies that are unlike tone sequences in other forms of music.



Logarithms have beautiful mathematical properties, that lead to unusual textures and resonances in non-Pythagorean chords: the beat frequencies between notes in a chord, can themselves be notes in the scale if chosen correctly, giving a second layer of harmonic and melodic information to compose with in the subtle realm of beats. Different chords have distinctive textures, resembling the chiming of bells, or the chirping of crickets, or the hum of machinery. Some compositions began with mathematically deriving nice families of chords, overtones or drones to exploit this property. Beyond that, the pieces were composed by ear, free explorations in a new musical landscape.



The series of compositions represented here was inspired by La Monte Young and Marian Zazeela’s NYC sound and light installation, The Dream House, and by reading Young’s thoughts on sustained tones and beat frequencies, such as in his and Zazeela's Selected Writings (1969). In trying to create non-Pythagorean timbres and overtones, I was heavily influenced by Wendy Carlos’ theory of synthesis summarized on her Secrets of Synthesis album (1987), and by mathematician Thomas Morrill’s paper in the Journal of Mathematics and Music, “Difference tones in ‘non-Pythagorean’ scales based on logarithms” (2019). I was also influenced by new age musician Steven Halpern’s ideas about therapeutic effects of sound, such as those described in his book Tuning the Human Instrument (1978). Most deeply, I am indebted to my teachers, Benjamin Rogers (Ruston, Louisiana) and Raj Pandya (Lexington, Kentucky), for showing me that music theory is limitless.



These compositions were recorded for my friends: to interest them, to soothe them, to express my tender feelings in my own language. I hope my album will interest you, and soothe you, as well.