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Spellspheres: Part 2

Due to COVID-19, and the necessary lockdown, I have had a large number of concerts postponed or cancelled. I decided it was important (for my career and my sanity!) to stay as creative as possible, and so I begun looking for opportunities to have my music played by others. I found the wonderful project One Page Pieces https://www.facebook.com/onepagepieces and decided to write a new Spellsphere for Adam Marks.  (I’ve written about Spellspheres in a previous blog https://www.oca.ac.uk/weareoca/music/spellspheres-1/

Here’s the score and recording: 


I think the piece sounds rather natural and uncontrived; however, its construction is the result of a number of mathematical processes. 

Firstly, I began with a chromatic scale. 


I started with the C#, and selected the note 3 places to the right (E). The, I select the note 5 places to the right. Here, it looks as though we have run out of notes; when this happens, I carry on counting from the beginning so the low A is 5 places to the right of E. I then select the note 1 place to the right of A (which is A#). This pattern continues; 3 places to the right of A# is C#, 5 places to the right of C# is F#, and one place to the right of F# is G. We keep going, counting 3/5/1, circling back to the beginning if we run out of notes. This continues until the sequence of notes repeats; we end up with a pattern of 12 pitches. If we were to keep going, these pitches would then repeat themselves. 


Each of those notes is transformed into a very slow waltz pattern in 3/2. I shift each note down a few octaves (so it functions as a bass note), and each of these is followed by a note that is two octaves and a third higher. This becomes a 12-bar accompaniment, which I repeat three times, resulting in a 36-bar piece. This is, basically, the entire left hand (albeit with two very simple deviations which occur at exactly bar 18 and bar 36). The relentless major thirds imbue the accompaniment with a sense of stability and tonality, however the mathematically constructed 12-bar pattern provides subtle unpredictability. 


36 is what is known as a highly composite number; put simply it possesses many divisors (1, 2, 3, 4, 6, 9, 12, 18, 36). This makes the next bit of the composition process a bit easier!

I divided the 36 bars into four parts. However, rather than opting for 4 equal parts (9 bars in duration) I opted for a pattern that subtly oscillated; 7, 9, 11, 9, and divided the piece into four sections of these durations (the first section is bar 1-7, the second section bar 8-16, the third section 17-27, and the fourth section 28-36). I then composed a small tremolando idea that occurs at the end of each of these four sections. 

As this tremolando idea occurs 4 times, and each iteration lasts 2 bars, a total of 8 bars have been ‘filled’ with music, leaving 28 right hand bars empty. I divided these 28 bars into 7 sections of 4 bars each, and placed a descending gesture on the 3rd and 4th bar of every 4 bars. You can find them if you look at the score and count each bar “1, 2, 3, 4, 1, 2, 3, 4…”, and if you omit the tremolando bars, the descending gestures will always coincide with the “3, 4”.

There should be seven of these descending gestures (on bars 3-4, 9-10, 13-14, 19-20, 23-24, 29-30, and 33-34). However, you might have noticed that the gesture on 29-30 is missing! This is because I decided to replace it with a sequence of four chords (29-32) that acts as a not-particularly-climactic climax. 

The pitches in both the descending gesture and the chords (29-32) have been constructed mathematically as well. I am interested to know if anybody can work out how the descending gestures have been created – as a clue, I’ll include this rather cryptic image below, and see whether anybody can determine the method I used! 


As a starting point, look at the number of each notes in each iteration of the gesture. 9 notes (bars 3-4), 6 notes (bars 9-10), 3 notes, then 6, 9, and finally 12 in bars 33-34). Then look at the intervals contained within each iteration. 

In the final bar, two ideas occur simultaneously. In the right hand we have the tremolando idea, and in the left hand we have the extra chord (which can also be observed in bar 18). Consequently, I decided to reverse the pitches in the tremolando idea; in every other iteration, the tremolando rises in pitch, here the pitch descends, satisfyingly concluding the piece. 

Please, do look at the other recordings at One Page Pieces; my piece was submission number 22, so there are plenty more compositions to delve into!

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Posted by author: Ben Gaunt
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2 thoughts on “Spellspheres: Part 2

  • Very welcome music at the moment. Pleasing to the ear with gentle complexity woven into it. I found a solution to the problem, or thought I did, and took the liberty to send it to you, being one of your students. No point in posting it here. I might spoil everyone else’s fun.

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