Lately I ran into this book Into The Cool by Schneider and Sagan. It's on the developing science Non Equilibrium Thermodynamics (NET). It's central theme is the 2nd law of thermodynamics, nicely formulated as "nature abhors gradients". Nature likes to even out differences. Concentrations of heat get dissipated in time. Nature likes disorder. The novelty of the book is that it includes life into the 2nd law. That life is an integral part of the machinery of dissipation of order. The paradox here is that life is a process of building order. While it builds order, it consumes gradients which are larger order.
I am still reading the book, but I could not help commenting on the paradox. It is pretty clear that, while nature abhors gradients, it also loves them. It first builds them, and then rides them. For example, the gradient between the heat of the sun and the cool of the earth is consumed by life. Life builds intermediate structures to even out the temperature gradient. But the sun does not come out of the blue. It rides the gradient of the black hole in the center of the galaxy. And the galaxies are, in turn, processes that ride another invisible gradient, probably a white hole.
Another example is the seperation of the sexes. The male-female gradient is built by gradual evolution, starting from hermaphrodites. And then nature uses this gradient to make offspring.
Wednesday, February 28, 2007
Wednesday, February 14, 2007
An analysis of symmetry in BWV-999
This is my second and more detailed look at BWV-999, which is a lone prelude amongst the lute works by Bach. It seems to be the opening piece of all the lute works, which are 4 suites, prelude-fuge-allegro-998 which is a toughie and also very lovely, and a lone fugue which is also from the violine works. Apparently 999 is specifically marked to be for lute, and it also suits the 6 stringed modern guitar perfectly. It is also regarded to be one of the "kleine" keyboard preludes. It would be easy for the keyboard but not so for the guitar, with it's difficult chord changes and fast arpeggio. I will analyse all of it's 83 bars one by one, by looking at it from harmonic, melodic and fractal aspects, as much as I can with my limited musical knowledge. I hope to learn some more music through this analysis, as the piece is a relatively simple one. The main emphasis will be it's fractal structure. I am currently practising 999, 998 and the first suite.
The most outstanding aspect of a musical piece is it's measure, which is 3/4 here. 3/4 is a very special number with mythical, religious and mathematical significance. 999 is a uniform arpeggio piece with 12 sixteenths for each measure. 12 is also a magical number. Bach opens the keyboard prelude and fuges with an arpeggio prelude as well. A bit like a historical reference to the ancient harp. 999 opens with D-minor and closes with A-major.
Like most good musical pieces, it has {bass,mid-range,treble} symmetry. The mid-range is the spine of the piece, and bass and treble reflect each other and dance around it. I will use letters for notes and numbers for octaves, like D and 2D. Here is the first measure coded;
1. [(D,2D,2F,2A),(2F,2D,2F,2D),(A,2D,F,2D)]
The second measure is the same;
2. [(D,2D,2F,2A),(2F,2D,2F,2D),(A,2D,F,2D)]
The spine of the piece falls in the even numbers, or the alternate beats, if a measure is numbered from 1-12. All alternate beats are the same except the 4th sixteenth. The odd numbered notes make either side of the spine.
The first beat is the bass, and sets the dominant tone of the measure. It also completes the harmony started in the end of the previous measure. This bass linking is typical of most, if not all, musical pieces, and it comes from the linking of the scales. To see this, consider a 2 octave C-major scale;
(C,D,E,F,G,A,B,(2C),2D,2E,2F,2G,2A,2B,3C)
When a 1 octave scale is played, it sounds complete only after adding the octave higher root (2C here). Thus we get 8 notes in a complete sounding octave, although there are only 7 different notes. When 2 octaves are played, it sounds complete with 15 notes. The middle note 2C both ends the previous octave and also starts the next. Therefore 2C has a dual, or linking, role.
The measure is divided into 3 quarters, each with 4 sixteenths. The 1st and 3rd quarters reflect each other around the mediator 2nd quarter. The 1st quarter(D,2D,2F,2A) is a typical 4 note chord with it's bass and 3 trebles (D minor here). The last 3 notes of the 1st q (2D,2F,2A) are reflected by the 1st and 3rd notes of the 3rd q ((A),2D,(F),2D), and the 1st note of the 1st q ((D),2D,2F,2A), in reverse order, half speed and in the previous octave (2D,2F,2A)-(A,X,F,X,D). The 2nd q acts as a mirror. But the symmetry is slightly distorted by the shifted position of the 1st chord. This creates tension and movement. Thus we get two chords reflecting each other and dancing around a spine, in a slightly imbalanced form. When you consider different tones for the chords in the rest of the piece, you get the fireworks.
The genius of Bach here shows itself in the complexity contained in the simplicity.
A measure is the smallest unit that contains all essential properties of a piece. In this sense, it corresponds to atoms of physics and cells of life. This is such a powerful metaphor. Universe and life are essentially musical forms.
To continue...
The most outstanding aspect of a musical piece is it's measure, which is 3/4 here. 3/4 is a very special number with mythical, religious and mathematical significance. 999 is a uniform arpeggio piece with 12 sixteenths for each measure. 12 is also a magical number. Bach opens the keyboard prelude and fuges with an arpeggio prelude as well. A bit like a historical reference to the ancient harp. 999 opens with D-minor and closes with A-major.
Like most good musical pieces, it has {bass,mid-range,treble} symmetry. The mid-range is the spine of the piece, and bass and treble reflect each other and dance around it. I will use letters for notes and numbers for octaves, like D and 2D. Here is the first measure coded;
1. [(D,2D,2F,2A),(2F,2D,2F,2D),(A,2D,F,2D)]
The second measure is the same;
2. [(D,2D,2F,2A),(2F,2D,2F,2D),(A,2D,F,2D)]
The spine of the piece falls in the even numbers, or the alternate beats, if a measure is numbered from 1-12. All alternate beats are the same except the 4th sixteenth. The odd numbered notes make either side of the spine.
The first beat is the bass, and sets the dominant tone of the measure. It also completes the harmony started in the end of the previous measure. This bass linking is typical of most, if not all, musical pieces, and it comes from the linking of the scales. To see this, consider a 2 octave C-major scale;
(C,D,E,F,G,A,B,(2C),2D,2E,2F,2G,2A,2B,3C)
When a 1 octave scale is played, it sounds complete only after adding the octave higher root (2C here). Thus we get 8 notes in a complete sounding octave, although there are only 7 different notes. When 2 octaves are played, it sounds complete with 15 notes. The middle note 2C both ends the previous octave and also starts the next. Therefore 2C has a dual, or linking, role.
The measure is divided into 3 quarters, each with 4 sixteenths. The 1st and 3rd quarters reflect each other around the mediator 2nd quarter. The 1st quarter(D,2D,2F,2A) is a typical 4 note chord with it's bass and 3 trebles (D minor here). The last 3 notes of the 1st q (2D,2F,2A) are reflected by the 1st and 3rd notes of the 3rd q ((A),2D,(F),2D), and the 1st note of the 1st q ((D),2D,2F,2A), in reverse order, half speed and in the previous octave (2D,2F,2A)-(A,X,F,X,D). The 2nd q acts as a mirror. But the symmetry is slightly distorted by the shifted position of the 1st chord. This creates tension and movement. Thus we get two chords reflecting each other and dancing around a spine, in a slightly imbalanced form. When you consider different tones for the chords in the rest of the piece, you get the fireworks.
The genius of Bach here shows itself in the complexity contained in the simplicity.
A measure is the smallest unit that contains all essential properties of a piece. In this sense, it corresponds to atoms of physics and cells of life. This is such a powerful metaphor. Universe and life are essentially musical forms.
To continue...
Saturday, February 10, 2007
The need to touch down
One day I was watching a panaromic rain scene on TV. I observed that some lightnings were followed by sudden bursts of rain. I thought, overly excited localities were dumping the extra energy in the lateral electrical dimension when ionisation potential was reached. This caused a sudden cooling down in the locality, leading to precipitation.
Then I thought, it must had been like this when the universe was first starting out it's evolution. The lightnings were it's 5 fields in some succession, and matter was it's rain. It was it's creative moment. This process is called Spontaneous Symmetry Breaking in physics. Rising on two legs was a creative moment of evolution for humans. Excess symmetry was dumped by freeing the hands. The creative moment works similarly for brains. When we mull over a problem long enough, we get a sudden flash out of the blue and realise a point. We get a touch down. Creative moments can be the sources of most intense pleasures. They are the purest orgasms of our minds.
In our more and more materialised world, we seek quick and definite solutions to our more and more intricate problems. We have come to expect rewarding sex through little tablets. We seek creativity through magic mushrooms and marijuana. We expect happiness in a short and expensive "romantic" getaway. But we soon realise that there is a prize for every quick fix.
When boredom arrives at the door of our inner selves, most ppl take up arms against it. Some drink, some smoke, some inject. Some go for a quick wank. Some abuse themselves. Drown themselves in work, in slavery for some arse hole corporation. Some abuse others and become soldiers and killing machines. We seek external solutions to an essentially inner problem. We try and scare it away. Mask it out. Drown it in alcohol. How can we like boredom? It is not a good feeling. If we can address it with a quick and easy solution, why not do it?
Boredom can also be a prelude to a creative moment. If you allow yourself to be bored for long enough, sometimes a sustainable and pleasurable solution comes out of the blue. After a while you may realize that boredom is a good thing. It heralds inner tension. Like weather before lightning and rain. If you ride it long enough, you finally get a flash inside. You may go for a long walk to nurse it. Take a pencil and paper and write it down. Draw or paint it. Tell a friend. Make it. After a while you seek boredom. You know it is good for you.
Long drowned boredom digs into our soul and covers itself. Makes a knot inside. It creates a tension. The space inside loses a part of it's coherence. Makes us biassed unconsciously. We become the knot that we tied and forgotten. If you keep ignoring your boredom, it finally makes an ensemble of knots inside and finally the person becomes a nut case. An alcoholic. A junkie. A freak. A derelict.
Sometimes you have no choice but ignore yourself. You may be poor. You may not be able to satisfy your most basic needs. You may be forced to bury your innermost desires for survival. You may have unhappy parents. Abusive parents. You may fall for wrong company. You may fall in love with the wrong girl. Sometimes it is just bad luck. You may be born in the wrong country to wrong parents.
Too many of us carry the inner knots to the grave and die unhappy. And maybe spend an eternity trying to find ourselves, life after life. Is the truth within? Is it my own self? Who am I? Just an elaborate concoction of chemicals, or something much deeper?
Then I thought, it must had been like this when the universe was first starting out it's evolution. The lightnings were it's 5 fields in some succession, and matter was it's rain. It was it's creative moment. This process is called Spontaneous Symmetry Breaking in physics. Rising on two legs was a creative moment of evolution for humans. Excess symmetry was dumped by freeing the hands. The creative moment works similarly for brains. When we mull over a problem long enough, we get a sudden flash out of the blue and realise a point. We get a touch down. Creative moments can be the sources of most intense pleasures. They are the purest orgasms of our minds.
In our more and more materialised world, we seek quick and definite solutions to our more and more intricate problems. We have come to expect rewarding sex through little tablets. We seek creativity through magic mushrooms and marijuana. We expect happiness in a short and expensive "romantic" getaway. But we soon realise that there is a prize for every quick fix.
When boredom arrives at the door of our inner selves, most ppl take up arms against it. Some drink, some smoke, some inject. Some go for a quick wank. Some abuse themselves. Drown themselves in work, in slavery for some arse hole corporation. Some abuse others and become soldiers and killing machines. We seek external solutions to an essentially inner problem. We try and scare it away. Mask it out. Drown it in alcohol. How can we like boredom? It is not a good feeling. If we can address it with a quick and easy solution, why not do it?
Boredom can also be a prelude to a creative moment. If you allow yourself to be bored for long enough, sometimes a sustainable and pleasurable solution comes out of the blue. After a while you may realize that boredom is a good thing. It heralds inner tension. Like weather before lightning and rain. If you ride it long enough, you finally get a flash inside. You may go for a long walk to nurse it. Take a pencil and paper and write it down. Draw or paint it. Tell a friend. Make it. After a while you seek boredom. You know it is good for you.
Long drowned boredom digs into our soul and covers itself. Makes a knot inside. It creates a tension. The space inside loses a part of it's coherence. Makes us biassed unconsciously. We become the knot that we tied and forgotten. If you keep ignoring your boredom, it finally makes an ensemble of knots inside and finally the person becomes a nut case. An alcoholic. A junkie. A freak. A derelict.
Sometimes you have no choice but ignore yourself. You may be poor. You may not be able to satisfy your most basic needs. You may be forced to bury your innermost desires for survival. You may have unhappy parents. Abusive parents. You may fall for wrong company. You may fall in love with the wrong girl. Sometimes it is just bad luck. You may be born in the wrong country to wrong parents.
Too many of us carry the inner knots to the grave and die unhappy. And maybe spend an eternity trying to find ourselves, life after life. Is the truth within? Is it my own self? Who am I? Just an elaborate concoction of chemicals, or something much deeper?
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