Lawrence Ball >> Sound Mandalas

SOUND MANDALAS

Lawrence Ball, 22nd February 2001

1. INTRODUCTION
The term Sound Mandala was suggested by Terry Riley, although earlier the (not sure what to call them) were called UFO tones and later "Shapetapes". Apologies if any of this is hard to understand, I will attempt any angle, depth or aspect of clarification, don't hesitate to ask. Apologies also for the slightly pig's breakfast" order of these notes. This work dates from 1983, and I'm still trying to evolve an understanding of this.

Please be aware its for me a bit like trying to describe a huge magic cave of possibilities that goes a long way into the hillside. If any of you can understand it, it would be really great to get some thoughts on where to build the roads from here.

2. ORIGINS/INSPIRATIONS/OVERVIEW
LaMonte Young's work particularly with long held continuous pitches has always inspired me; the late John Whitney (Sr)'s films and moreover his principle of "differential dynamics" led me to apply an extension of such techniques to timbrally-varying drones, (the tamboura also was certainly present as muse), although later its been applied to scores and midi. So initially it began life as a 128 harmonics'-modulating tone/graphics audio-visual sequence where one sees the magical transforms one hears.

2.1 Whitney's Differential Dynamics (see Whitney's out of print book 'Digital Harmony' McGraw Hill (1980?)) A Hundred point "radius": Imagine 100 points (or more, or less) forming the radial line of an invisible circle. Imagine each point, prepared to set off on a circularmotion at a distance from the circle's centre which is equal to its initial distance from it. If moving at the same angular velocity, the points would move in rotation like a stick connected to the circle centre. But- now start off again with each of the 100 points moving with an angular velocity proportional to an index number from 1 to 100 assigned to each point, numbered from one end to the other (it matters not from which end - each option has interesting and beautiful outcomes). This is best timed to last between 10 and 30 minutes (I feel). At first, a winding spiral dominates, but winds itself to 'break-up' point at which stage crystallisations and dissolutionsof order begin to occur.(These are really good).

At the end of the cycle, point 1 ('speed 1 also) will have done one complete circle orbit, point 2 - 2 circles, point 3 - 3 circles etc. upto 100 orbits for point 100. So the cycle at this point would begin again from a line identical to the position of the start. At the 'halfway' stage, all the odd numbered points will have covered n/2 (n divided by 2) laps (where n is the point's index number), which will be 'an integer plus one half' orbits, forming a line halfway around the circle from the start position. The even numbered index points will all be at their start position.

In fact, any fraction up to a denominator of about a fifth or a sixth of the number of points plotted will yield a figure like this halfway shape- but having a number of projecting radial arms equal to the fraction's denominator.

This results in a quite breathtaking display of simple forms emerging from and dissolving into the teeming of moving points.

Another elementary way in which the principle can be demonstrated:

A line of points may be drawn along the leftmost (say), column of a screen. 256 points (say), are each assigned a row of of the screen to move along to the right . Upon reaching the far end of the screen, the points may either reappear at the opposite edge to move again from the left, or , as an alternative system, each one could 'BOUNCE' , i.e. travel backwards, and then eventually forwards again.

Making the points move at speeds proportional to their respective index number, as described earlier gives rise to comparable effects to the circle system, only with vertical instead of radial alignments at the crystallization points.

Maybe this is how an ant would perceive a 128-harmonic tone or a JI chord if it were a) musical and b) its time sense was slowed sufficiently??

3. ORIGINAL EXPERIMENTS
Initially, the question arose- "What happens if I apply Whitney's differential dynamics to the points in a wave table?"

Imagine a sine wave, composed (8 bits- this is 1983!) point 1 moves from its initial position, incredibly slowly, when it reaches the maximum value, its starts downwards again, and bounces back up again when reaching the lowest possible value. It completes a cycle in say 44 minutes. Point 2 completes 2 cycles of movement, and arrives back at the start point in the same 44mins. Point 3 will do 3 cycles &c &c.........and point 256 will have completed 256 cycles.

The sine waveform had index numbers assigned to each point (for speed assignments) from 0 (!) at the left (say) end, to 255 at the right, I found that a sonic journey (through 400,000 timbres) was produced, easily fascinating enough - even on one pitch, which journeyed through complex timbral 'states' formed from many combinations and amplitudes, gradually changing, of 128 harmonics. An 'orchestral', 7-octave sheet of tone-anchored from one, low fundamental.

I ran this timbre-itself-harmonically-transforming on a low B with the 2 speakers fractionally detuned to create a mobile phase loop. Its like applying the laws of sound and harmonics of Helmhlotz's "atmospheric ocean" to the shape itself of the sound waves on that ocean. It quickly became known as "the tone",and "the UFO tone".

The sounds resulting are smoothly varying modulations of timbre that can be very satisfying or stimulating or both, to listen to. I prefer the changes on the slower side.

3.1 Timbral Analysis
A very thin ripple of amplitudes (1 or sometimes 2) harmonics wide runs slowly up and then down the harmonic axis (from 1 to 128 and then back the same way) forming a difference tone at the fundamental resulting from the ascending (ie n+1th) and descending (ie nth) harmonics.

4. BOUNCE MK 2 - MODIFIER SHAPES
A modification of the assignment system of speeds to points was then developed. Rather than always having the linear array of points moving at speeds proportional to their "index number" , it became desirable to assign the speeds in different ways. Such as having the middle points in the sequence travelling fastest, and those at each end slowest. Or vice versa. This gave rise to the concept of a MODIFIER shape. In the elementary system described before, a graphical shape of position in sequence plotted against speed was a ramp form (either ascending or descending). If fastest (or slowest) speeds are assigned to the middle, this can be called a VEE (because of its shape) or TRIANGLE modifier.

In another variant, fastest (or slowest) speeds are assigned to the 1/4 and 3/4 positions along the sequence of movable points. The most successful sound/visual combination (on the video/Sound mandala piece called Triangles') was a triangle wave starting point (this now referred to as the SEED form, with 'two triangles' as a modifier. (Two 'mountains' with maximum speed at 1/4 and 3/4 points. The two halves of the image were set to move in different directions at the start. Many nteresting forms arise, which suggest Inca designs. The timbral and graphic sequences are each characteristic of a particular quality, right through the whole cycle.

4.1 Timbral Analysis
These sound sequences (or tones as they have been called) create on the palette of 128 (or more of course) harmonics constellations or virtual harmonics, gathering, scattering and reforming eg a ripple of amplitude running up and then down the axis of harmonic numbers. If I start with a sine wave and modify it with two humps (like the modulus of a sine wave- values without a minus sign), and starting by moving the left half up and the right down I get a strange bundle (looking at a dynamic FFT) of harmonics 4 or so harmonics wide, travelling up and down.

5. EDIT or PREDESIGN?
We (my then students Michael Tusch, James Larsson and other friends in the 80s, and now Dave Snowdon in the 90s and 00s) decided we like/d to listen to an unedited algorithm, rather than edit fragments a la Shakespeare with portions, we have gone for designing the action into the spec. (in the envisioning of the characteristics and the ramifications) rather than cutting up bits. This I feel is an interesting paradigm approach for all kinds of computer-generated art. More organic? What do you think? John Whitney Sr had a bad reaction to this approach at first, but when I later met him, and discussed/elaborated it, he seemed much more open to what I had been doing.

6. FROM BOUNCE to SEQUENCE: DIFFERENTIAL DYNAMICS TO HARMONIC MATHEMATICS
We quickly generalised Whitney's principle to not only governing motion (controlled by a large 1D array of integers) but also the movement through a table of values (instead of just travelling at constant rate, points would sweep through eg a timbre wave table's values, each at a different integer rate).

I wanted then to make the motions do better than simply go up and down and back to where they started. I conceived of a sequence of values (i.e. positions) through which each point moves, but, again, at speeds related to assigned integer values, which are in turn in whole number ratio (and hence harmonic) to each other. Hitherto, each point moved up, bounced, moved down, bounced, and moved back to starting point.

A harmonically constructed complex timbre form, (eg a timbre with harmonics 1,2,3,4,6,12 with amplitudes gradually decreasing) is used instead to control the movement sequence of each point. This 6-harmonic timbre gives rise to a sequence of intricate, smooth bobbing motions, each point gradually rolling in and out of alignment with others.

The points' motion in this more complex way I call SEQUENCE technique. The crystallization points are spectacular in different ways, the way in which they are approached are also, being not simply an 'overtaking' process, but also having unrolling and twirling motions, implying a 3-dimensional interpretation. Also one gets "magic carpet" effects where layers of points waver and hover.

'Helix' uses a timbre, comprised of those harmonics 1,2,3,4,6 and12, as a sequence. The points start off as a horizontal line (silence), and each point moves through the sequence at a rate proportional to its distance from the left hand end. (i.e. a ramp modifier).

The timbral journey resulting consists of the 6 harmonics "themselves going on a journey" up and down the 7-octave spectrum of that synthesizer, at a speed proportional to the no. of the harmonic.(ie at speeds 1,2,3,4,6,12). This I gleaned from FFTs I ran in the 80s.

Why these particular timbral/harmonic motions ? I don't know, and would like to know, so I can find interesting variant animations of these harmonics' ripples. Doing an FFT is one thing, predicting how to get interesting results is another. Help!

The maths of how and why specific harmonic processes result from the seed and modifier shapes (in BOUNCE technique) and from the modifier and sequence tables (in SEQUENCE technique) still evades me.

Thanks if you got this far.

Let me know any feedback if anyone finds this useful or inspiring I'd be glad to know about it.

Yours mathematically, musically and spiritually
Lawrence Ball
composer, math tutor, director Planet Tree Music Festival
http://www.planettree.org


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SOUND MANDALAS Lawrence Ball, 22nd February 2001 1. INTRODUCTION The term Sound Mandala was suggested by Terry Riley, although earlier the (not sure what to call them) were called UFO tones and later "Shapetapes". Apologies if any of this is hard to understand, I will attempt any angle, depth or aspect of clarification, don't hesitate to ask. Apologies also for the slightly pig's breakfast" order of these notes. This work dates from 1983, and I'm still trying to evolve an understanding of this. Please be aware its for me a bit like trying to describe a huge magic cave of possibilities that goes a long way into the hillside. If any of you can understand it, it would be really great to get some thoughts on where to build the roads from here. 2. ORIGINS/INSPIRATIONS/OVERVIEW LaMonte Young's work particularly with long held continuous pitches has always inspired me; the late John Whitney (Sr)'s films and moreover his principle of "differential dynamics" led me to apply an extension of such techniques to timbrally-varying drones, (the tamboura also was certainly present as muse), although later its been applied to scores and midi. So initially it began life as a 128 harmonics'-modulating tone/graphics audio-visual sequence where one sees the magical transforms one hears. 2.1 Whitney's Differential Dynamics (see Whitney's out of print book 'Digital Harmony' McGraw Hill (1980?)) A Hundred point "radius": Imagine 100 points (or more, or less) forming the radial line of an invisible circle. Imagine each point, prepared to set off on a circularmotion at a distance from the circle's centre which is equal to its initial distance from it. If moving at the same angular velocity, the points would move in rotation like a stick connected to the circle centre. But- now start off again with each of the 100 points moving with an angular velocity proportional to an index number from 1 to 100 assigned to each point, numbered from one end to the other (it matters not from which end - each option has interesting and beautiful outcomes). This is best timed to last between 10 and 30 minutes (I feel). At first, a winding spiral dominates, but winds itself to 'break-up' point at which stage crystallisations and dissolutionsof order begin to occur.(These are really good). At the end of the cycle, point 1 ('speed 1 also) will have done one complete circle orbit, point 2 - 2 circles, point 3 - 3 circles etc. upto 100 orbits for point 100. So the cycle at this point would begin again from a line identical to the position of the start. At the 'halfway' stage, all the odd numbered points will have covered n/2 (n divided by 2) laps (where n is the point's index number), which will be 'an integer plus one half' orbits, forming a line halfway around the circle from the start position. The even numbered index points will all be at their start position. In fact, any fraction up to a denominator of about a fifth or a sixth of the number of points plotted will yield a figure like this halfway shape- but having a number of projecting radial arms equal to the fraction's denominator. This results in a quite breathtaking display of simple forms emerging from and dissolving into the teeming of moving points. Another elementary way in which the principle can be demonstrated: A line of points may be drawn along the leftmost (say), column of a screen. 256 points (say), are each assigned a row of of the screen to move along to the right . Upon reaching the far end of the screen, the points may either reappear at the opposite edge to move again from the left, or , as an alternative system, each one could 'BOUNCE' , i.e. travel backwards, and then eventually forwards again. Making the points move at speeds proportional to their respective index number, as described earlier gives rise to comparable effects to the circle system, only with vertical instead of radial alignments at the crystallization points. Maybe this is how an ant would perceive a 128-harmonic tone or a JI chord if it were a) musical and b) its time sense was slowed sufficiently?? 3. ORIGINAL EXPERIMENTS Initially, the question arose- "What happens if I apply Whitney's differential dynamics to the points in a wave table?" Imagine a sine wave, composed (8 bits- this is 1983!) point 1 moves from its initial position, incredibly slowly, when it reaches the maximum value, its starts downwards again, and bounces back up again when reaching the lowest possible value. It completes a cycle in say 44 minutes. Point 2 completes 2 cycles of movement, and arrives back at the start point in the same 44mins. Point 3 will do 3 cycles &c &c.........and point 256 will have completed 256 cycles. The sine waveform had index numbers assigned to each point (for speed assignments) from 0 (!) at the left (say) end, to 255 at the right, I found that a sonic journey (through 400,000 timbres) was produced, easily fascinating enough - even on one pitch, which journeyed through complex timbral 'states' formed from many combinations and amplitudes, gradually changing, of 128 harmonics. An 'orchestral', 7-octave sheet of tone-anchored from one, low fundamental. I ran this timbre-itself-harmonically-transforming on a low B with the 2 speakers fractionally detuned to create a mobile phase loop. Its like applying the laws of sound and harmonics of Helmhlotz's "atmospheric ocean" to the shape itself of the sound waves on that ocean. It quickly became known as "the tone",and "the UFO tone". The sounds resulting are smoothly varying modulations of timbre that can be very satisfying or stimulating or both, to listen to. I prefer the changes on the slower side. 3.1 Timbral Analysis A very thin ripple of amplitudes (1 or sometimes 2) harmonics wide runs slowly up and then down the harmonic axis (from 1 to 128 and then back the same way) forming a difference tone at the fundamental resulting from the ascending (ie n+1th) and descending (ie nth) harmonics. 4. BOUNCE MK 2 - MODIFIER SHAPES A modification of the assignment system of speeds to points was then developed. Rather than always having the linear array of points moving at speeds proportional to their "index number" , it became desirable to assign the speeds in different ways. Such as having the middle points in the sequence travelling fastest, and those at each end slowest. Or vice versa. This gave rise to the concept of a MODIFIER shape. In the elementary system described before, a graphical shape of position in sequence plotted against speed was a ramp form (either ascending or descending). If fastest (or slowest) speeds are assigned to the middle, this can be called a VEE (because of its shape) or TRIANGLE modifier. In another variant, fastest (or slowest) speeds are assigned to the 1/4 and 3/4 positions along the sequence of movable points. The most successful sound/visual combination (on the video/Sound mandala piece called Triangles') was a triangle wave starting point (this now referred to as the SEED form, with 'two triangles' as a modifier. (Two 'mountains' with maximum speed at 1/4 and 3/4 points. The two halves of the image were set to move in different directions at the start. Many nteresting forms arise, which suggest Inca designs. The timbral and graphic sequences are each characteristic of a particular quality, right through the whole cycle. 4.1 Timbral Analysis These sound sequences (or tones as they have been called) create on the palette of 128 (or more of course) harmonics constellations or virtual harmonics, gathering, scattering and reforming eg a ripple of amplitude running up and then down the axis of harmonic numbers. If I start with a sine wave and modify it with two humps (like the modulus of a sine wave- values without a minus sign), and starting by moving the left half up and the right down I get a strange bundle (looking at a dynamic FFT) of harmonics 4 or so harmonics wide, travelling up and down. 5. EDIT or PREDESIGN? We (my then students Michael Tusch, James Larsson and other friends in the 80s, and now Dave Snowdon in the 90s and 00s) decided we like/d to listen to an unedited algorithm, rather than edit fragments a la Shakespeare with portions, we have gone for designing the action into the spec. (in the envisioning of the characteristics and the ramifications) rather than cutting up bits. This I feel is an interesting paradigm approach for all kinds of computer-generated art. More organic? What do you think? John Whitney Sr had a bad reaction to this approach at first, but when I later met him, and discussed/elaborated it, he seemed much more open to what I had been doing. 6. FROM BOUNCE to SEQUENCE: DIFFERENTIAL DYNAMICS TO HARMONIC MATHEMATICS We quickly generalised Whitney's principle to not only governing motion (controlled by a large 1D array of integers) but also the movement through a table of values (instead of just travelling at constant rate, points would sweep through eg a timbre wave table's values, each at a different integer rate). I wanted then to make the motions do better than simply go up and down and back to where they started. I conceived of a sequence of values (i.e. positions) through which each point moves, but, again, at speeds related to assigned integer values, which are in turn in whole number ratio (and hence harmonic) to each other. Hitherto, each point moved up, bounced, moved down, bounced, and moved back to starting point. A harmonically constructed complex timbre form, (eg a timbre with harmonics 1,2,3,4,6,12 with amplitudes gradually decreasing) is used instead to control the movement sequence of each point. This 6-harmonic timbre gives rise to a sequence of intricate, smooth bobbing motions, each point gradually rolling in and out of alignment with others. The points' motion in this more complex way I call SEQUENCE technique. The crystallization points are spectacular in different ways, the way in which they are approached are also, being not simply an 'overtaking' process, but also having unrolling and twirling motions, implying a 3-dimensional interpretation. Also one gets "magic carpet" effects where layers of points waver and hover. 'Helix' uses a timbre, comprised of those harmonics 1,2,3,4,6 and12, as a sequence. The points start off as a horizontal line (silence), and each point moves through the sequence at a rate proportional to its distance from the left hand end. (i.e. a ramp modifier). The timbral journey resulting consists of the 6 harmonics "themselves going on a journey" up and down the 7-octave spectrum of that synthesizer, at a speed proportional to the no. of the harmonic.(ie at speeds 1,2,3,4,6,12). This I gleaned from FFTs I ran in the 80s. Why these particular timbral/harmonic motions ? I don't know, and would like to know, so I can find interesting variant animations of these harmonics' ripples. Doing an FFT is one thing, predicting how to get interesting results is another. Help! The maths of how and why specific harmonic processes result from the seed and modifier shapes (in BOUNCE technique) and from the modifier and sequence tables (in SEQUENCE technique) still evades me. Thanks if you got this far. Let me know any feedback if anyone finds this useful or inspiring I'd be glad to know about it. Yours mathematically, musically and spiritually Lawrence Ball composer, math tutor, director Planet Tree Music Festival http://www.planettree.org back to top	Introduction Harmonic Mathematics theory & application Sound Mandalas Recording Section Visual Harmony
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