Math and music, music and math. The relationship between music and mathematics has long been known. When sound waves could be measured, the relationship became even more clear.

Through the years, musicians have experimented with the mathematical relationships between notes and sounds, sometimes with terrifying results, and sometimes with stunningly beautiful results. This edition explores the latter.

The centerpiece of this episode is a brand new release from Glenn Sogge called The Euler Variations. It’s a purely fascinating device with beautiful results. Tune in to hear the music and the story behind it, along with surrounding tracks that also beautifully support the premise.

You know what this means, right? These esoteric, seemingly abstract relationships between numbers and sound waves contribute to the very foundation of our existence.

Music is life!

This is the stuff that lights your host up, in case you couldn’t tell: when the intersection between music and intellect results in such beauty. It’s not always pretty, but it certainly is beautiful.

Fractal Blend – Jack Hertz and Zreen Toyz – Fractal Blend of Primitive Icons and Numbers Stations (2014)

The Number You Have Reached – Different Skies – Imaginary Spaces (2007)

“December 1, 2007 (Little Fractals)” – PalancarAmbient Train Wreck Back Catalog Collection Eleven (2008)

Glenn Sogge – The Euler Variations (2017)

  1. Four (K)nights On Tour
  2. Long Tales
  3. The Whole Story
  4. Turing’s Target

Detailed notes below the playlist.

Geometry of the Skies – Robert RichNumena and Geometry (1997)

Geometry – Thom Brennan – Vista (2005)

Departure / Dark geometry – Joe Frawley – Curious Perspectives (2011)

Notes on the composition of The Euler Variations (2016-2017)

By Glenn Sogge

These pieces are the first ones created using a tool for composition I devised in 2016 based on the concept of the Knight’s Tour.

In essence, given a grid, such as a chessboard, and a single chess knight piece placed on any square, visit all the squares of the board once and only once following the rules of the knight’s move.  If the tour ends with the next move to be to the starting square, it is called a closed tour. Otherwise, it is an open tour. There are many millions of possible tours and the problem has kept amateur mathematicians and beginning computer scientists learning programming occupied for decades.

One of the mathematicians who studied the problem was the giant of the field, Leonhard Euler. One of his solutions for a closed tour is the basis of this tool.

An 8×8 grid is filled in with symbols. On ordered path is then taken through the grid visiting each square once. Since it is a closed tour, one can begin anywhere and be sure that all the squares will be visited. One can also move backward as it is still a closed tour.

For these pieces, the grids were filled with the pitch classes of various scales (notes.) Each piece used a different scale and there were also differences when the scale did not fill or row. Specific details for each piece are provided below. The grids were pages that could be inserted into a page protector sleeve that contained the numbers of the tour.

For these pieces, a mental image of slow, polyphonic chant was driving the style. So, all the events are very long. I like long decays so maximum envelope values were used. As each note ended, the next one began. For a couple of the pieces, rests were used to fill out the grid so long silences got inserted in a part.

The pieces consist of 3 or 4 voices. The same software synth was used for each piece although different (modified) presets might have been used.

In two cases, a ‘parts’ was written for the pieces; in the other two, a dime coin to keep track of the current position on the grid was used while recording. Each piece was recorded one track at a time.

For each voice, a starting location (1-64) and direction (forward/backward) was chosen by random methods. For at least two of the pieces, maybe more, a starting offset from time 0:00 was also determined by random methods for each voice. This was done to further remove any lock-step feeling from sounds that were essentially the same lengths.

The grids (and derived notations) only indicated which pitch class is to be played, not which octave (hints of the serialism of last century.) Octaves were freely chosen while playing the parts.

Sometimes, I listened to the prior voices as the tracks built up; other times, I did not.

Track Specifics

Four (K)nights on Tour

C major scale but each row ends with the starting pitch. By shifting one position left, the remaining rows also create the classical modes. So, it is Ionion, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian, and Ionion again when read as rows (which this method never does.)

Synthesizer: VBox3 from Krakli Plugs

Long Tales

Chromatic scale repeated through the grid with four rests at the end to fill out the grid. Given the method, these pauses appear scattered throughout the piece.

One of things I thought was successful about this one was the complete democratization of the 12 tones (aka the serialism of the last century) but in a process environment that creates harmonies that maybe wouldn’t have been considered.

Synthesizer: M42 Nebula with different factor presets for each voice. Typically, the envelopes had to be modified to get the long tones desired.

The Whole Story

Whole-tone scale based on C#. Each row contains two rests cells as there are only 6 discrete tones in the scale.

Synthesizer: Reaktor Titan using the Icram preset for all parts (only 3 parts in this piece.)

Turing’s Target

The Enigma scale (see the Wikipedia) rooted on C. For those keeping score, Alan Turing’s mission in WWII was to break the Enigma Code of German intelligence.

Synthesizer: Reaktor Prism with 4 very different (and modified) presets.


The same grid structure, etc., could be used to order any type of musical, visual, or physical directions (dance) for 1 or more performers either live or recorded.

These experiments were the working out of the kinks as I could foresee using it and as a way to further the explorations of my on-again, off-again relationship with Western tonality.

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