Sound Incubator II.

Algorithm wise and coding oriented

Aa..

I. TMS in Musinum

1) Thue-Morse sequence

Thue-Morse Sequence – Wolfram MathWorld
– TMS was explicitly mention for the first time by Axel Thue in 1906 (used for words combinatorics) and get global attention with M. Morse in 1921 (in use for deferential geometry).
– un-directly  was TMS first described by E. Prouhet in 1851 (applied for number theory).
– it is sequence A010060 in the OEIS.

- while its integration or code looks like unnecessary complication and we (musicians) would be often less poetiacal with statememenst like "amazing sequence can be generated from the substitution system" (link). In fact it is rather unnecessarily simple - just repeat itself - self-similarity:

– altough it is simplistic, it is pretty hard to do in Grid, since it needs store value and DAWs are generally not doing this.. so far I can only imagine Array module can do it, but did not managed it.

     – O  – 01  –  01-10  –  0110-1001  –  0110/1001-1001/0110
                                                                  – resp.: 01101001-10010110
step: 1. – 2.  –  3. (4 nos) – 4. (8 nos)  – 5.  (16 numbers) 
 or :   0    1.       2.                  3.                  4. 

– to say obvious (dumb-proof): similarity here represent sequence of steps starting from 0 and repeat (similarly)) its boolean complement – 1. Following next step is repeating 1. finished step (01) with is inverse counterpart (10) so it gives 0110 and so on so forth..
– of course inverse seq is obtain when first no is 1..

There are easier way to that.

 I. Direct definition

Way the numbers in left column are created
– as one can see u start on 0s and adding successively 1s from left to right (Rpic).
1. step – just add 1 – finished
2. 1 is on second row
        – last row is also changed to 1 – finished
3. 1 is on 3. row
         – last row is also changed to 1
         – next to last row is also changed to 1 while last keep 0
          – ……………………………………………………………………….last keep 1 – finished
And so on so forth.       

Way the numbers are interpreted – right column
– if binary expansion is odd – than it equal 1, if even it equal 0 (very complicated)).
Odd numbers are here also called oddius)) – 1, 2, 4,      7, 8, 11, 13, 14, 16,      19,  21, 22, 25, 26, 28,       31, 32,        35, 37, 38 – odd parity
Even are called evils)) – have even parity    – 0,   3, 5, 6, 9, 10, 12,     15, 17, 18, 20,      23, 24, 27, 29, 30,      33, 34, 36,        39


II. Fast sequence generation

III. Recurrecence relation

IV. L-system

– as u can see in sequence tree u can fallow Lshaped 11 sequences, however it use is so far unknown to me.))

2) TSM in Musinum

For further details and use both Architect and PD enter - here.

– TMS was loosely used in fractal/ self-similarity generator by Lars Kindermann 1999 in Musinum (shortening Music in numbers) soft – regolos.de/musinum

– the soft came w/ explanation page (link above) and also mention, that it is directly based on Number Theory in Science and Communication from Manfred Schroeder – even tough the book is from 1993 (and so one can find it on the net for 3) it get 5th revision in 2009.

- it was then put by M. Brickmann into Pure Data as musinum_generator1 - see/ download at - martin-brinkmann.de/pd-patches.html.
- it was then put by AZZIN into Architect as Musinum - see more infoes/ download at - KVR or here on Architect page.

II. Self-similarity

– ..

–

Koch curve

–

A Koch curve – Wikipedia – has analogically (similarly)) an infinitely repeating self-similarities creating magnified fx:

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–

–

Sieves

– Sieve of Erastothenes – antient algorithm for finding prime numbers.

III. Deterministic dynamic systems

– deterministic dynamic systems are evolving toward equilibrium set by attractor(s) in basin. DDS is described by ODE from its initial position.
– DDS are generally probabilistic in a rather bit philosophical way reflecting real-life – they are exact to its predictability horizon, after then they are chaotic. In summary – probabilistic… 
– are used in self-organization, self-assembly, chaos theory, bifurcation theory,, ..

A. Orbits

1) Planetary motion

– Kepler (sigsaly.xf.cz)
– in PM predictability horizon is about 1 ml. y. However, in some way (I don’t of course understand) it incorporates chaotic elements – trajectory is offset from its equilibrium line set by eigenvalues, according to this offset it can be calculated how will further deviation from equilibrium line unfold.
     – xy, resp. z define position.

B. Cycles

1) Phase spaces – Limit cycles

More on: Phase spaces.

– in typical xy (2D) one value in not defining position, nother characteristic rate of change..

2a) Self-Organization

– Self-organization, a process where some form of overall order arises out of the local interactions between parts of an initially disordered system, was discovered in cybernetics by William Ross Ashby in 1947.

–

–

2b) Self-Assembly

–

The Game of Life (Cellular automataton)

– cellular automaton is a concept from Automaton theory discovered in 40s by S. Ulam and John von Neuman, that get wider attention w/ zero player game (means that game is based only on initial condition and not altering after hitting play) The Game of Life made by British mathematician J. H. Conway in 1970 (Lpic bellow). 

–

– The Game was used in music – there is free The Game of Life Sequencer made for Windows (2009): Game Of Life Music Sequencer – Synthtopia.
   – was also used in M4l – bve.ultomatoncore version 2.0 by ithkaa on maxforlive.com (2012).

The Game of Life (1970).

Puffer train (object puffing stuffs) that leaves other stuff – here gliders.

- for further details and use enter - The Game of Life.

C. Knotted orbits

– knotted orbits are most notably defined by number of crossing its line made, in knots particular – in 1) chapter – it is crossing number, but in knotted curves it differs:
   – chaos attractors in 2) chapter that are more incline to deterministic systems are also torus knots – and so based on p, q – Lorenz Attractor with  ρ 99.96 is T(3,2) .
          – there is also special knot class called Lorenz knots..
          – but Attractors generally are based on Differential eq. and so represent all values rate of change (no value is originally used for positioning)

1) Knots, torus knots and Lissajous curves

- to Main page enter - here.
1. chptr.: Knots

– knots are categorized according to their crossing number into so called prime knots – knots that are not composite knots or composite links: simply unique knots, that can not be created from other knot without untie.

– first tabulations of this kind was made by Peter Guthrie Tait did in 1877 (Lpic). 
–  31 – trefoil knot – has 3 crossing, it is the simplest prime not. Can be also (2, 3) torus knot.
– 41 – figure-eight knot – has 4 crossing, it is simplest non torus knot.

– first number defines number of crossing (crossing number – Rpic), second number is given order  based on how many particular knots are w/ particular crossing number.
– first two 3 and 4 crossing number groups have only one type of knots, 5 crossing have already two knots, 6 crossing 3 and 7 crs. 7 knts.

2. chptr.: Torus knots and Lissajous curves

– Torus knots  – are defined by orbits around its height and length – p, q.
– Lissajous curves – are defined by phase offsets of two x, y sin waves and x, y phase speed ratio. 

– compound knots are are either composite knots or composite links – see (2, 8) torus link on right.
– Hopf link – is the simplest non-trivial link of two circles each passing thru centers of the other.
– Torus knots prime nots ..

(3, 2)-torus knot, also known as the left-handed trefoil knot.

    (3, 2) torus

           (7, 3) torus

The Hopf link is cobordant to the unlink.

(2,8) torus link.

– w/ Lissajous curves one get basic curves – which are further check  in separate – n – based curves,

1: 2 Lissajous curve – Parabola, Besace, Lemniscate of Gerono.

First four Lissajous curves.

2) Chaotic attractors

- to main page enter - here.

– chaotic attractor are now (q1 2024) not Imo used in Pd, but are used in M4L: most notably by Dillon Bastan (Max for Live Users | Facebook) – free Strange mod (gumroad.com) and Divisions.
They are also used in Architect (see main pg).

2nd pic Hamiltonian formalism for the Van der Pol oscillator
3rd pic from L is from Brain Dynamics Toolbox – open source in Matlab for neuro waves – these waves are very similar like wavelets and have some useful math backgounds…

– these trajectories are sensitive to initial position – minor offset of starting values will after while cause deviations. This time is known as Lyapunov time – when amount of predictability is diminishing, behavior start to be chaotic and the differences of trajectories will be continuously more noticeable ..   
Lpics: deviation of 2 trajectories (Blue and Yellow) on x by 10^-5.  

Rpics: deviation of double compound pendulum (R2 in basin) – one of the simplest case of chaotic behavior..

– Pen on double compound pendulum is harmonogram – physically one of the first was made in 1890s – known also as Blackburn pendulum (according Hugh Blackburn) –  curves are usually Lissajous type (see Lissajous knot family in Knot page).

2b) Basins

Basin (mathcurve.com)

Full page – enter Basins (sigsaly.xf.cz).

Full page – enter Basins (sigsaly.xf.cz).

IV. n – based curves (resp. p, q or a/b)

Parametric curves (sigsaly.xf.cz)
– non-parametric n based curves related to knotted curves – Torus knots, Lissajous curves, Lorenz Attractors, and Limit cycles, Van de Paul Oscillations...:
Conical rose: Vivani – n=1:  Fish curve, Besace, Lemniscate of Gerono.
          – Quatrefoil – n = 2: Deltoid, Astroid, – Evolute, Involute, Pedal, Pursuit + Base + Catenary + Beetle, Cissoid, Rational Circular cubic,  curve,  Cycloids (hypo/epi-trochoids/cycloids), conchoid, Cyclic hamonic curves, Cassini Curves, Stelloides, Circles.
          – Trifoil – n=3:
Beyond Conical: Bifolium, Mating profile, Helix, Chebyshev.
Transendental curves: Clelias, Sinusoidal spirals, Butterfly curve, Ribacour curve, Spherical cycloid, Spherical trochoids

V. Probability

1) Markov chain

Markov chain – Wikipedia
– markov chain is making further changes – variation – based on actual state: Rpic show 2 states and its probability to change on to nother.

2) Rubik´s cube group theory

History
Rubik’s Cube – Wikipedia
– RC was invented 1974 by Hungarian sculptor and prof. or architecture Erno Rubik – who sold the patent to UK Pentagle Puzzles in 1978.
Since 1982 – there is each 2 years a World Rubik Cube championship – first was in Budapest (1982) w/ winning time 23 sec (on single solve), last one was in 8. 2023 w/ 5.21 sec average on 5 solves.)).
Competition is in all 4 types of Cubes.
Next one is in Seattle 3-6 Jully. 

 

Intro
Rubik’s Cube group – Wikipedia
– typical RC is has 6 sides x 9 facets – so 54 facets, but middle 6 facets are unchangeable: so it leaves symmetric group S48 w/ 48 facets – that is also 4x 12 or 32 + 16 so in there is space for musical and math/ programming base approach right out of the box.)
–  each move rotates 6 facets (middle one stay as mentioned basically static) +/- 90° or just 180°.
– there are also 4x4x4, 5x5x5, 6x6x6 and 7x7x7 cubes.

Basics (3x3x3)
– in 1979 UK David Breyer Singmaster (died in 1. 2023) published Notes on the “Magic Cube” – Download PDF (vdoc.pub) – that used group theory to solve RC (1980 he made expanded edition)
Further works on the topic: The Cube: The Ultimate Guide to the World’s Bestselling Puzzle

Sum-up
– the cube has 8 corner and 12 edges
– makes 8! x 3pow7 x (12!/2)  

Basic 90°180°-90°
 turns the front clockwise turns the front clockwise twice turns the front counter-clockwise
 turns the back clockwise turns the back clockwise twice turns the back counter-clockwise
 turns the top clockwise turns the top clockwise twice turns the top counter-clockwise
 turns the bottom clockwise turns the bottom clockwise twice turns the bottom counter-clockwise
 turns the left face clockwise turns the left face clockwise twice turns the left face counter-clockwise
 turns the right face clockwise turns the right face clockwise twice turns the right face counter-clockwise

–

– 

– 

3) Group theory

Parametric curves (sigsaly.xf.cz)
– 

VI. Topology

– changing topological structure generally means stretching, twisting, crumpling, and bending without making hole.
– it is also closely related to knots, torus, circle/ figure 8 stuffs (as u could see….

2-sphere

– 2-sphere is 2D sphere (pic bellow) – just its surface in 3D – generally easier and pretty standard way of mapping in topology. 
– hyperbolic volumes and other stuff often use 3-sphere (3-manifold) – see bellow.,.

– 

Figure-eight

– ..

A figure eight in the torus (Rose (topology) – Wikipedia) 
– it is general rule, that technically u cannot morph from violet (longitude) link the red one (since it is 2D), but u can make by nulling (narrow down) of red one the violet one. (see fundamental square (polygon) in torus knots chpt.) – intrinsic topological properties – defining fundamental group.
          – tough u can make red cr. from violet one by morphing (interchange – see further) their set points, but not in space sense – regular topological properties.
Similarly one can make: Sphere → Cube, Ellipsoidal surface..
            Annulus (mezikruží) → cylindrical surface..
            Euclidean plane → spherical surface
But similarly u cannot go: Disk → Sphere    
           Twisted → untwisted band – since the shapes would self-intersect, but not if immersed in 4D (I have absolutely no idea why it is then possible and shapes are not intersecting)      ..

Rose – bouquet of n-circles

– A rose with four petals.

– topological space of petals (circles) glued along a single point.
– beside algebraic topology are related to free groups.
– in topology – rose is wedge sum of circles.
      – WS is one point union of tpl. spaces 

Wedge sum of two circles..

Genus-h

– genus number h defines number of holes in genus shape.

Genus 0: Planar graph
– objects w/ 0 holes: S^1 – Circle (1-manifold)
           – S^2 – 2-Sphere

Genus 1: Toroidal graph
– objects w/ 1 hole: T^2 – Torus.
                         – Torus and mug are homemorphic (Rpic).
             – P^2 – Projective plane
             – K^ – Klein bottle –  R2pic:2D KB immersed in 3D space.

Klein Bottle was first described by Felix Klein in 1882.

Figure8 immersion of Klein btl.
R2-bluepic: Klein bagel cross section, showing a figure eight curve (the lemniscate of Gerono – R1pic).

Genus 2: Double-toroidal graph
– objects w/ 2 holes: 3D Pinched torus/ 4D Mobius tube

Genus 3: Pretzel graph
– objects w/ 3 holes.

- for further details and use enter - here.

VII. Coding in Codebox (M4L)

Particles

–

Attractors

–

VII. Lua 

Meshing

On program Fugu – Fugu (monash.edu) – one can see how few lines of codes can spell pretty decent mayhem in modulation…)
– meshing (or mesh generation etc.) 
     – if elements are 3D, then 2D entities are faces. 

Meshes from image (Pd Project)

– check on: ofBook – Basics of Generating Meshes from an Image (openframeworks.cc)

Lpic is source and Rpic is generated.
– unfortunately OpenFrameWorks were discontinued, so this may be simply practice ..

Fibonacci sequence – Plugdata

– Fibonacci sequence, Pisano period – Wikipedia
– FS also known as Fibonacci numbers (Fn) – sq starting from the pair 0, 1, so the next number is always the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, … (sequence A000045 in the OEIS)
– sq was known to Indian mathematicians 200 BC and introduced to European math in 1200 by Leonardo Pisano – known as Fibonaci.
– numbers grow pretty quickly (with the ratio of successive numbers approaching the Golden ratio), thus for musical use its range is useful to limit particularly w/ modulo m, i.e., just retain their remainders when divided by the given modulus → Pisano period.
Sequences obviously have a finite range and must repeat eventually, but PP have a surprisingly large period even for small values of m.

Add: Modulo – integer wrap. F. e. Time resp clock represent 12 modulo – 9h + 4 h is 1 h.

Pissano period

n = 3 (Lpic)
n = 5
n = 10 (Rpic)
– 0 is also counted (so values are -1 looking).

– each line of the graph is – obviously – displaying sum of 2 last values – previous and current. Starting w/ set 0, 1 displayed as red – actually third value – and then follow rainbow coloring.

Pi(3)
– sq A082115 w/ 8 values in cycle: 0112 0221
– in graph above: First half cycle: 0) 01 – first constant steps pair.
1-third value) red – 0,1 (1), 2) org – 1, 1 (2),
Second half of the cycle: 3) l-green – 1, 2 (3 – 3 is modulo, so it is reset to 0), 4) d-green – 2, 0 (2), 5) tyrkys – 0, 2 (2), 6) l-blue – 2, 2 (4 – 1), ……… repeating cycle part 7) d-blue 2, 1 (0), 8) violet – 1, 0 … 
– so the blue following w/ violet is in multiple cycles actually second possible real start of the sequence – which repeats 0, 1 constant setting.
     – as u can see this sort of duality of the start of cycle is presented in all cycles – red line is starting to make period constructing on two (arbitrary) set up values 0, 1 and u of course u get same these values (0, 1) again when finishing first cycle (resp. starting the second one)..

pi(10)
– sq A003893 w/ 60 value cycle: 0112358-31459437 077415617853819 099875279651673 033695493257291
– it applies modulo 10 – so second 3 after dash obviously should be 13, but since modulo is 10, counting is reset after 10. Next 1 is actually 11 and so on so forth.

– circular arc – connecting opposite corners of F. tiling. (Rpic)


Least common multiple
Lcd of the denominators of two fractions is the “lowest common denominator” ,

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Pissano in Pd-lua

– this sequence is claimed to be not easy to do in Plugdata, but relatively easy in Pd-lua.

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Pissano in Pd-lua

– this sequence is claimed to be not easy to do in Plugdata, but relatively easy in Pd-lua.

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