Music:

Concrete Jungle

Dates:

2-Feb-2025 (inception), 01-Mar-2025 (finalised), 02-Mar-2025 (released)

Artists:

Music composed, performed, produced by Norman Freund

Artwork by Norman Freund

Artwork photography by Zachary Freund

Duration:

09:44 min:s

Keywords:

Music, Experimental, polymeter, granular, microtonal, soma pipe, soma cosmos, albrecht dürer, magic square, electronic,

Genre:

Electronic

Description:

Put simply, this music was inspired by the sixteen numbers of the Magic Square (Ref. [1]) numbers of Albrecht Dürer's engraving Ref. [2].

Albrecht Dürer's 1514 engraving in the top right hand corner included a four by four matrix of numbers, who's properties are such that all rows add up to 34, all columns add up 34, the lop left to bottom right diagonal adds up to 34 and the top right to bottom left diagonal add up to 34.

The numbers are:

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

After a post on the FaceBook group "Mathematical Tiling and Tessellation" which showed this magic square, but instead of concentrating on the numbers, examined patterns than can be formed by drawing lines between the centre of the tiles. This got me thinking about patterns, mathematics, microtuning in Just Intonation (JI) and poly-meter rhythms.

So here was my music challenge, compose music where Albrech Dürer's magic square numbers must be the centre piece of the music. I enjoy writing poly-meter music, where simultaneously played themes have different time signatures, I also enjoy writing microtonal music. So how could I use these magic square numbers?

At the time I was working on a digital synthesiser that employs an eight note scale. So here was the constraint, use musically the magic square numbers to form a music scale in Just Intonation fashion (frequency ratios of integer numerators and denominators) and form the main theme with polymeters using at least some of those ratios.

There is no unique way of picking out pairs of numbers from the magic square. Odd rows form the numerators, even rows form the denominators; swap around to odd columns form the numerators and even columns form the denominators; or some other concoction. I sketched the ideas down, did some JI ratio to cent calculations and kept an eye out for "nice" numbers like the perfect fifth ratio of 3/2. In the end I picked the second option, with a micro JI tuning scale as listed in Table 1.

Table 1: Eight Note Microtonal JI Scale.

0: 16/3 = 2898c

1: 2/13 = -3241c

2: 5/10 = 1/2 = -1200c

3: 11/8 = 551c

4: 9/6 = 3/2 = 702c

5: 7/12 = -933c

6: 4/15 = -2288c

7: 14/1 = 4569c

where

i: are the scale note indices.

n/d are the frequency ratios, n,d are positive integers.

c = cent, 100c = 1 semi-tone in 12 equal divisions of the octave (12ED).

This tuning is not purely ascending, but no matter, the synthesiser I use has octave up/down buttons for each of the eight voices and besides the unconventional tuning can put you in a new place for inspiration.

I did not stop there, I also wanted to use these frequency ratios as speed factors for clips of rhythmic/melodic ideas. I set a basic 4/4 time signature at 120 bpm in my eight track looper software (GroovieBaby) and played some ideas through the Soma Pipe by sending human voice/articulations to it. Both percussive and melodic. The loops were either 1 bar or 10 bars of 4/4, then time to experiment with the speed at which they were played back at, where these speed factors were chosen from the frequency ratios of Table 1. Further more, all these loops were to be played back simultaneously.

What follows is a break down of the loops used.

Loop 1:

A single drum beat on the first beat of a 1 (4/4) bar played with the Soma Pipe.

Play speed 1/1

Loop 2:

Copy of Loop 1

Play speed 16/3 (the first ratio in the Albrecht Dürer Magic Square matrix derived tuning vector (See Table 1))

Loop 3:

Copy of Loop 1

Play speed 11/8 (4th ratio of the tuning vector)

Loop 4:

Free style drum recording from the Pipe using the drum algorithm sound

10 bars total but used Bar 6 only

Play speed 1/1 of 1 bar (4/4)

Loop 5:

No used, empty

Loop 6:

Human voice (my voice) recorded on the Soma Pipe

10 bars total, all bars used.

Play speed 1/1

Loop 7:

Copy of Loop 6

Play speed 4/13 (One octave up from the 2nd ratio of the tuning vector)

Loop 8:

1 bar (of 4/4 time signature) Pipe recording of an ascending pitch recorded at 1/1 speed.

Play speed 4/15 (7th ratio of the tuning vector)

I like experimenting with distortion, so loops 1 and 2 were feed into an "atan(x,y)" distortion unit. This simply was a lookup table using the inverse of the trigonometry function tan (rise over run).

So how many bars shall I record combining all those seven loops?

The main theme was 10 bars long and the slowest theme (loop 7) 1 bar at 4/13 speed, so these two lengths were used to determine the master length. At 120 bpm of a 4/4 means 1 bar is 2 seconds. So 10 bars is 20 s.

This then gave a basic

20x(13/4) = 65 seconds or 32.5 bars of (4/4)

I guessed that four repeats of this would be enough to capture the other polymeters going on.

So the final master track length was:

4x32.5 = 130 bars (4 minutes)

As it turned out, determined once the music project was completed, the period for all these rhythmic loops to repeat the same combined pattern was 313 billion years (see my FaceBook page for the derivation).

This four minute recording formed the main theme of the composition, which either formed a backing track or was hacked in various ways to form new melodic and rhythmic ideas. One of these loops received improvised speed changes, selecting from the palette of frequency ratios of Table 1.

The composition structure is:

Introduction (00:00 to 0:15 (minutes:seconds))

Transition 1 (0:15 to 0:17)

Movement 1 (0:17 to 5:43)

Transition 2 (5:43 to 5:45)

Movement 2 (5:45 to 9:43)

The first such hacking in the 15 second introduction was granular synthesis with Robert Henke's PitchLoop89, then a portion of this in the 2 second transition 1, using distortion, Ableton Live's Beat Repeat and convolution reverb. This quickly gave way to feeding in the main theme into the Soma Laboratory Cosmos (drifting memory station). The original factory algorithms were used with the Cosmos, where the performer had control of when to drop down and lift up again the recording needle, amounts of drift between the Cosmos memory banks, the blur between them and the algorithm used. This formed the human expression element.

Movement 1 dropped in the main theme recording into Ableton Live, to play this back sending the output through to the Comsos, then feeding it's input back into Live to record the results. Audio monitoring at this stage was through the Cosmos, monitoring in Live turned off to stop and feedback loops (my digital interface has only one stereo input and one stereo output).

Movement 1 also gave the opportunity to record in new material layering ideas in Live, feeding in the Soma Pipe performances using my human voice/articulations as the sound source.

Microtuning has not yet played a role. In movement 2, I played back the main theme using the looper in my custom programmed synthesiser, the CrossFeedFM. The tuning was set to the magic square numbers of Table 1. Time to improvise note choices, feed back loop dials, FM synthesis controls, rhythmic envelope generator parameters and evolving stereo bandpass filter changes. At this stage Live took a back seat, all recording and performance was done with the CrossFeedFM Max(Cycling 74) Standalone synthesiser.

Movement 2 audio recording was then dropped back into Ableton Live for final minor adjustments of volume levels and final render of the whole composition.

The Concrete Jungle cover photograph has it's own interesting story. Some time ago, my son and I watched a video from Benn Jordan Ref. [3] on methods to visualise sound waves in air photographically. Just by accident my son noticed some interesting shadow play in our cat's drinking bowel as the sun peered through the window. What caused those shadows? Air pressure waves, so he recording some video of it and I played one of my flutes very close to the water surface to purposely generate ripples on the water surface. Various frames of this video recording found their way into the cover photograph, plus some colour changes in post. If you look closely, each differently coloured group of tiles, contain the number of sub-tiles from the matrix of sixteen numbers from the Albrech Dürer magic number square.

So 511 years later, Albrech Dürer's magic numbers were used as inspiration for polymeter rhythms, microtuning and tiling art in this work, "Concrete Jungle".

I think, Albrech would be pleased.

References:

[1] "Magic Square", further reading on the concept of magic square number arrays, https://en.wikipedia.org/wiki/Magic_square

[2] Albrech Dürer, 1514, Engraving "Melencolia I", further information: https://en.wikipedia.org/wiki/Melencolia_I

[3] Benn Jordan, 5-Oct-2020, "Seeing Sound Pt. 2 - How To See Actual Sound Waves In The Air", YouTube video, https://youtu.be/oghe1tb2M1Y?si=GaPeUsnyE5Mo9qE9