Saturday, January 15, 2022

Tuning Tangle

The appearance of orderly structure in the world is a fascinating puzzle. Mathematics studies the properties of orderly structures. Are mathematical objects features of the world, or features of our minds? Do the mathematical regularities we see in the world appear just because that's how our minds process sensory data? Aren't our minds part of the world, anyway?

The vision of the world as mathematically structured is traditionally credited to Pythagoras. One of the cornerstones of this vision is the notion of musical consonance as mathematically structured. Music is built from consonant intervals, the relationships between tones that sound good together. Musically consonant intervals correspond to mathematically simple integer frequency ratios. An "A" pitch with frequency 440 Hertz and the "A" pitch an octave higher, with frequency 880 Hertz, have the frequency ratio 2:1. The 440 A relates to the 660 E that is a perfect fifth above it, with a frequency ratio of 3:2.

Musically, a song is a pattern of notes that are related by a variety of such consonant intervals. Of course songs also involve rhythmic patterns etc., but here I am just focusing on harmonic patterns.

Patterns arise in many ways, but generally they are the outcome of some sort of process. For example, tree rings appear from the varying growth rate of the tree through the regular changing of the seasons. Another kind of pattern arises as liquids cool and solidify. A quick cooling will form finer grained crystals; slow cooling allows the crystals to grow larger. Thermodynamic phase transitions, such as freezing and melting, are a rich field for the study of how order can emerge spontaneously. Musical patterns can be generated by thermodynamic simulation; consonant clusters of notes, such as chords, are similar to crystals that emerge from the process of freezing.

The algorithmic composition method I describe here relies on thermodynamic simulation to choose the pitches to be played at each time. The simulation works with a matrix of points at which a pitch is to be played. This matrix defines connections between such points. Pitches to be played at the same time are connected; pitches played at successive times are connected. Musical patterns generally have a structure of repetition and variation. The matrix is constructed with a fixed repetition structure: connections are made between pitches played at the corresponding points in successive cycles of repetitions.

Thermodynamic simulation is driven by temperature as a key control parameter. Degrees of consonance correspond to energetic possibilities. At high temperatures, pitches are chosen relatively freely; only the most dissonant choices are discouraged. At low temperature, only the most consonant choices are allowed between connected points in the matrix. Initially the points in the matrix are assigned random pitches. The simulation begins at a very high temperature, and then gradually the temperature is reduced. The pitches in the matrix are randomly reassigned again and again. Gradually patterns of mutual consonance begin to emerge.

While the temperature is still quite high, very little orderly structure has emerged: 118edo 3x3x3x3x3 1.

A graphical score also shows a lack of structure:

Here the vertical axis is the pitch, and the horizontal axis is time.

A slow cooling process will allow long range order to emerge, so eventually the entire matrix becomes consonant: 118edo 3x3x3x3x3 22

At an intermediate temperature, there can be fluctuations within an overall harmonic framework, a balance of order and variation that approaches musicality: 118edo 3x3x3x3x3 13

The harmonic movement here is quite limited. One avenue that can open up a richer harmonic landscape is the introduction of tempered tuning. The tuning used here divides octaves into 118 equal steps (118edo), instead of the conventional 12 equal steps (12edo) of a piano. Dividing octaves into some moderate number of equal steps is a practical way to organize the set of pitches used in a composition. If the pure rational intervals such as the perfect fifth 3:2 and the major third 5:4 are used, these can be combined in an infinite number of ways. If the number of equal steps per octave is chosen carefully, good approximations for these pure intervals are available: four steps of 12edo is 1.2599, quite close to the pure 1.25. 38 steps of 118edo is a frequency ratio of 1.2501, imperceptably close to the pure 1.25.

Another feature of these tempered tunings is that the infinite number of ways to combine the fundamental consonances will give only a finite number of results, within an overall pitch range. A given interval can be constructed from multiple combinations of fundamental consonances. For example, in 12edo, a major third can be reached by moving four perfect fifths up and then down two octaves. Each tuning has a different pattern of such combinational coincidences. A Tonnetz diagram provides a useful summary:

In this diagram, the octaves are omitted. E.g. all the ways to play a "C" note in various octaves are all represented as just "C". This diagram is for the 118edo tuning, so instead of the usual 12 note names like "C", "C#", etc., the numbers 0 to 117 are used.

The repeating structure in this diagram, e.g. the multiple occurrances of the 0 pitch, are a result of the tempering of the tuning. E.g., moving by 8 perfect fifths and then a major third will result in the same pitch where one started (moving as many octaves as needed). This property of tempered tunings introduces the possibility of loops in a compositional structure. The Tonnetz diagram shows that loops in 118edo need to be quite long: there are no short paths from a 0 pitch to another 0 pitch in the diagram.

The compositional matrix used above was given a repetition/variation structure of a five dimensional torus with circumferences uniformly size 3. This created a large space but where no large loops will easily arise. Another large compositional space is a two dimensional torus with circumferences size 18. The compositional torus can easily accommodate tuning loops as long as 18 measures. This is long enough that several loops in the tuning space can fit.

Starting the thermodynamic simulation from a random pitch assignment and gradually cooling, these sorts of tuning loops will tend to get trapped in the matrix. When the system is cooled to a very low temperature, the tuning loops remain: 118edo 18x18 cold.

The harmonic movement makes even this very orderly pattern somewhat interesting. At a moderately higher temperature, there are short term fluctuations together with long range movement, producing a composition that is even more musical: 118edo 18x18 10.

Friday, January 7, 2022

Science without Progress

There's a notion of science for which progress is essential to science. Science is a process of steadily broadening, deepening, and refining our knowledge about the world. It's a process of steady improvement. This year's science is better than last year's science, and next year's will be better yet. Whether this process converges on some ultimate theory that captures precisely the way things are, that's a bit beside the point. The sequence of integers 1, 2, 3, etc. steadily get bigger, without ever converging on some final largest integer.

For this kind of steady progress to be the way science works, two things must be true. First, we need a way to compare our scientific knowledge at one time to our scientific knowledge at another time. We need a way to tell which state of scientific knowledge is better. Once we have that measuring stick, then we can at least check empirically whether science is constantly improving. We can develop some kind of model of the evolution of scientific knowledge, and check whether at least the model guarantees continual progress into the future.

It's easy to sketch out a model of the evolution of scientific knowledge that implies perpetual progress. Such a model may not be accurate, though! A major question in examining the dynamics of science is its coupling with the world outside science, with social, ecological, and geological systems. Science is a social institution, intimately connected with the rest of society. When sources of funding, materials, equipment, and personnel dry up, science cannot thrive.

One measure of the state of scientific knowledge is the size of the total accumulation of scientific publications. As long as some library somewhere continues to accumulate the mass of literature, as long as scientific literature is not lost, then scientific knowledge will continue to advance, by this measure.

There are two problems with this logic. First, it is unreasonable to expect all scientific literature to be preserved in perpetuity. It's not even clear what exactly should count as scientific literature. Parapsychology, the study of phenomena such as telepathy, is an example of a discipline whose scientific status has been debated. Should raw data accumulated by scientific instruments count as scientific literure? As our boundary that defines scientific literature changes, our measuring stick to detect progress is being updated. We don't have a consistent measure by which to determine whether science progresses consistently.

Even if we maintained a constant definition of what should count as scientific literature, it is not reasonable to expect all such literature to be maintained in perpetuity. There is some expense involved in preserving information. There is additional expense involved in converting old literature to new formats. Not all printed literature is scanned to digital form. Digital formats are steadily changing, and obscure literature will generally be given a low priority for format conversion.

Even if a record of some coherent piece of scientific knowledge has been preserved in a library somewhere, it can easily happen than no one is alive any more who can make any sense of it. The papers involved may easily refer to scientific instruments that no longer exist, for example.

One can slog through endless such details to determine whether scientific progress is inevitable. In the face of impending climate catastrophe and the profound social upheavals that will bring, the idea that science will somehow weather the storm despite all the challenges... perhaps no amount of detailed argument will convince a true believer!

If progress is essential to science, but if progress is not a secure ground on which to build... must science then crumble, too? Can science survive and even thrive without progress? Is progess, after all, essential to science?

It is a vital project to develop a vision of science that does not depend on progress. We in that part of the world that supports science are at grave risk for a major decline in our general level of prosperity. Science will participate fully in the trajectory of decline and collapse. If we can maintain a thriving science despite that decline, our ability to cushion that decline will be significantly enhanced. We will be better able to respond to recurring crises in medicine, agriculture, etc. If the scientific community cannot find a way to dance with circumstances, we will all suffer from that failure.

An analogy should be useful in developing a vision for science that doesn't depend on progress. Darwin's theory of evolution shows how species are constantly adapting themselves to their circumstances. The steady extension and refinement of scientific knowledge is similar to biological evolution. But biological evolution does not imply any kind of progress. Species today are not more advanced or better adapted than were species ten million years ago. Species ten million years ago were reasonably well adapted to their circumstances back then, which were very different than the circumstances of species today. Some of these changes are surely geological, but they are largely due to the interdependence of species, the nature of the ecological web. When one species develops some new characteristic, that changes the circumstances of other species, pushing them to adapt in new ways. There is no fixed measuring stick by which to determine whether one species is more advanced that some other species.

When we dream of some ultimate scientific truth and view science as a path leading to that goal, progress seems to be essential to science. But if we understand science to be a practical approach to engaging with our experience, enabling us to respond more effectively to our circumstances, then it becomes natural that our scientific knowledge must shift and adapt as our circumstances change.

Sunday, December 12, 2021

The Paradox at the End of Modernity

Modernity can be defined as a culture of faith in progress: newer can be better, should be better, is better. The engine of progress is science. Science is a process of refining our understanding of the world. We are constantly learning about the world, correcting our misunderstandings and extending the frontiers of our knowledge. Science doesn't go backwards. Tomorrow's science is better than yesterday's science. We can use our constantly improving scientific understanding to improve conditions in the world around us, to cure diseases, increase crop yields, etc.

This vision of progress based on science was elaborated by Francis Bacon in the early 17th Century, at the beginning of the modern era. The road of progress we can see in front of us remains limitless. Colonizing Mars, autonomous robots, the extension of life expectancy to multiple centuries and beyond... what barrier can we not imagine transcending? And if we can imagine it, step by step we can use the scientific method to resolve whatever problems limit our ability to achieve it.

On the other hand, as science refines our understanding of the world, it reveals some very challenging limits. Of course the way science understands limits on one day may be overturned the next day. Perhaps the rudest limit science has discovered is the speed of light. As the vastness of the universe has been revealed to us, so has its remoteness. Will we figure out some clever way to leap across distances of thousands of light years? This might be the most elementary form of the paradox we are caught in. An irresistable force is contending against an immovable object. What will happen?

Back down on earth, of course, the speed of light doesn't seem like anything worth much worry. We have plenty of technical problems with much more immediate impact. The general problem of global resources: climate change, water, biodiversity; this problem is foremost among our challenges. Pandemics such as covid-19 are to some degree a result of humanity running up against planetary limits, but there is also the problem of pathogens evolving to evade our countermeasures. Our problems can get worse more quickly than our pace of finding new solutions.

At this point, it is not too farfetched to observe that our progress in scientific understanding is revealing more about the limits to our technical progress than it is enabling further technical progress... "technical" meaning our ability to improve our world.

Nowadays it is very easy to find literature championing each of the two poles of this paradox. There are books that show how things have always been improving and will continue to improve. There are books that map out the trajectory of the collapse of modern civilization that we are riding along.

Will the irresistable force succeed in dislodging the immovable object, or will it be defeated?

Paradoxes, like the paradox of progress that we are caught in today, are not generally resolved by the victory of one pole over the other. In general some kind of deeper understanding of the apparent contradiction is required. A good starting point is simply acknowledging that we really are facing a paradox. It would be foolish to dismiss either the vitality of scientific progress or the reality of the planetary limits to growth. To develop a new understanding of our situation that can encompass these two aspects, that is the challenge we face.

I found Kurasawa's film Dreams to be a wonderful vision of how we got here and where might might be heading. The final segment is "The Village of the Watermills," where we see a joyful celebration of a funeral. Death is a reality. However over the top the visions of the colonization of Mars might be, the visions of human immortality make those look very tame. We really do need to grow up and learn, not just to accept our limited situation, but to cherish it. A joyful funeral is one way to do this. But how we age, that is another vital dimension. What can it mean, to be healthy and old? To be healthy and dying? Such a vision might provide a model for our modern civilization as it runs up against planetary limits.

Sunday, October 17, 2021

The Need for Growth

I've heard many times people say that our economic system requires growth in order to function. Usually people explain this by saying that the only way that interest can be paid on debt is if the money supply increases. This is not true, though. In a debt-based money system, the sum of money accounts is always zero. People who owe money need to be able to provide goods and services that people who have lent the money will purchase, but as long as that is true, there is exactly enough money floating around to pay any debts that are floating around. Understanding this, we can see that paying interest doesn't require a growing economy. So, is a growing economy actually required at all?

Here's a different notion of how our economic system requires growth to function. Our system requires inflation to encourage people to spend and invest. In a deflationary environment, it's better just to hold onto money - in which case, the economy freezes up completely. Inflation has two different meanings that are of course related. One meaning is an overall increasing level of prices. Another meaning is an increasing money supply. If the supply of dollars goes up, then the value of each dollar goes down, which means prices go up.

Our economic system needs prices to go steadily up in order to encourage people to spend and invest, which is what keeps the economy functioning. Prices will go up as the money supply goes up. The money supply goes up as debt goes up. This is what a growing economy is. I.e., our economy indeed requires growth in order to function.

Wednesday, August 18, 2021

Digging Down to the Foundations

Lately I'm reading Michael Millerman's book Beginning with Heidegger: Strauss, Rorty, Derrida, Dugin and the Philosophical Constitution of the Political. Dugin is the target of the book, and the main reason I'm reading it. I'm in the middle of the Rorty chapter at the moment. The overall notion seems to be that Dugin is the one who has picked up Heidegger's ball and is running with it. Strauss and Rorty have either misread Heidegger or anyway have refused to pick up his ball, for opposite reasons. Strauss is more fundamentalist than Heidegger, and Rorty is more historicist.

In a curious coincidence, my wife has been reading The Great Bliss Queen by Anne Klein. She tells me that Klein is discussing a debate within feminism between essentialists and constructivists. It sounds pretty much the same as the debate between Strauss and Rorty - or their followers, anyway. These debates are a bit like the conundrum, "Why not tolerate intolerance?" It's like a dog chasing its own tail.

This brings to mind a simple analogy that I use to illustrate the potential for Buddhist thinking to provide a way to escape the deadend represented by these debates. We're trying to investigate the true reality underlying the diverse appearances that we experience in the world. We start digging down through the shifting sands of the surface, looking for the solid bedrock that holds everything up.

The fundamentalist essentialist vision is that indeed, we can cut through the fog and confusion, and whether we land on the Bible or the U. S. Constitution or Feynman's Lectures on Physics, we will find solid ground. The constructivist historicist vision is that we can dig our way straight through to empty space on the other side. The web of appearances is free floating. It might be a fair amount of work to move the whole mess, but it is quite possible, and perhaps a worthy project. We have that freedom.

The middle way of Buddhism, or of the Madhyamaka school of Buddhism at least, is a third alternative. It's not that we find some third sort of thing once we have dug through appearances. The vision is that we can dig and we can keep digging and actually we can just keep on digging endlessly. The investigation of appearances never reaches any kind of point where further investigation isn't possible. Of course we might run out of the resources needed to keep investigating. But we can also relax our desperate search for foundations once we realize that every layer of appearance is supported by yet another layer of appearance. There is no bottom. It's not that the bottom is hollow - that's the constructivist historicist vision. There is no bottom.

What are the practical consequences of this vision, that's hard to say. Mostly it's a matter of avoiding futile and destructive projects. The MAGA crew seems to want to scrape away the shifting sands to return society to whatever solid ground they put their faith in; once they've killed off all the liberals they can start killing each other over transubstantiation versus consubstantiation etc. The progressive crew seem to want to pick up the whole mess and move it to a less strife-filled place; maybe an annual cycle of presentations from the Human Relations department will do a lot, but the inertia of the entire system will assert itself long before we start knocking up against the constancy of the speed of light and the limits it imposes on interstellar colonization.

Buddhist practise seems to be mostly a matter of letting go of grasping. The subtle details come from a deepening perception of how we are grasping. The extremes of eternalism and nihilism are classical mirages at which we grasp. Fundamentalism and constructivism are modern manifestations of these philosophical extremes.

I've become interested in Dugin because he seems to be a major philosophical inspiration behind the right wing movement. Recently I read an observation, that the right wing extreme in the USA is not really philosophically grounded. It's basically a gang of street thugs. There's this character in the movie A Fish Named Wanda, this thug who lies around reading Nietzsche and shooting his pistol. Perhaps this is a good model for someone like Steve Bannon.

Thursday, January 14, 2021


I'm no lawyer, but what we saw happening at the U. S. Capitol building on the 6th sure looked like criminal activity. What should we do about that?

Generally speaking, the justice system is our collective means to respond to crime. We bring criminals to justice. But there are voices, mostly friendly to the criminals, warning us that justice will further division at a time when we especially need unity. Justice, division, unity: these are complicated ideas with many possible meanings.

What is the proper function of justice? Revenge? Punishment? Compensation? A crime has been committed: what should we do about it?

A crime is a breaking of social bonds. The proper function of justice is to heal those bonds. Justice should be social therapy.

Punishment and compensation can work as components of a therapeutic program. No matter what, a path must be provided by which criminals can be reintegrated into society. Justice must exercise discrimination but never promote division. The wisdom of Solomon is indeed required to judge what form justice should best take in any given situation. We mortals are stuck bumbling along the best we can. But if we at least understand what we're trying to accomplish, that ought to improve results.

Unity: to the extent that it opposes diversity, it is a flawed goal. One image of unity is a watertight boundary surrounding perfect uniformity within. This is a kind of death. A vital society is a cohesive society. Cohesion means rich relationships among diverse components. Diversity without cohesion is merely plural unity, just as dead as singular unity. Society is a fabric, a network.

The core work of justice is in repairing and strengthening social relationships.

Tuesday, November 24, 2020

Diaschismic Tuning

I continue my wandering in the vast world of musical tuning...

One of the first tuning calculations I performed was in 1976. I wrote a simple formula to give scores for possible tunings, to look for good ones. My idea was that a simple calculus-based optimization would yield an excellent result. I soon learned that this formula was not a matter of simple calculus. Now I have returned to evaluating tunings, with a simpler scoring forumula and simpler tunings. Still, a simple calculus optimization is not going to work with a graph like:

But here I am just looking at dividing octaves into some whole number of equal parts, so I can just sort the possibilities to see which are the best:

Here the number of steps in an octave is the column on the left, and the score is on the right. The score is based on how well the tuning approximates the exact harmonics 3 and 5.

The tunings 118edo, 87edo, and 53edo are ones I had worked with already. I had never given thought to 34edo, but there it is on the table! So I thought I should give it a try!

Another key feature of tunings like these is the commas that they temper. A Tonnetz diagram should help:

Each square represents a pitch class, e.g. the pitch C in conventional tuning. C can exist in any octave, but if we just ignore the octave, it's plain old C. There are 12 such pitch classes in conventional tuning. Here there are 34, and instead of using letters I use numbers. Since we're ignoring octaves, one microstep up from pitch class 33 is 0.

A key feature of these tunings is that they only have a finite number of pitch classes. Musically one can move up and down by various intervals, such as octave, perfect fifths, major thirds, etc. in an infinite variety. But there are only a finite number of places to land on. So, for any pitch class, there are an infinite number of ways to get there!

The Tonnetz diagram helps show how this works. Moving from any square to the square above it, that is moving by a major third (plus any number of octaves). In 34edo, a major third is 11 microsteps. Similarly, moving to the right one square is moving a perfect fifth, which is 20 steps. The part of the diagram I show isn't enough to see directly that e.g. by continuing to move by perfect fifths one will eventually circle back to the original pitch class. 20 and 34 have a common divisor 2, so 34 has two seperate circles of fifths, each 17 steps long: one circle for the even-numbered pitch classes, and one circle for the odd-numbered pitch classes.

But the Tonnetz diagram makes it clear that that are many other ways to move in harmonic space and end up back where you started. Intervals in 34edo are near approximations to just-tuned intervals: that's what my scoring formula was measuring. If one were to wander in some direction in just-tuned space, one would never return to the same pitch class except by undoing all the steps one had taken, though perhaps in a different order. Just-tuning requires, or provides, an infinite number of pitch classes. But if the 34edo intervals are close to the just-tuned ones, when 34edo reaches the same 34edo pitch class, the matching just-tuned movement must have reached some just-tuned pitch class quite close to 1:1. Such a just-tuned pitch class is known as a comma. A tempered tuning like 34edo is said to temper a comma when the harmonic movement that would result in that comma in just tuning instead, in the tempered tuning, returns to the same pitch class. In the Tonnetz diagram I have given the common names for the main commas tempered by 34edo.

I use algorithmic composition to explore alternate tunings. I can coax the algorithm in various ways to loop through, or pump, one or more commas tempered by a tuning. Here's a diaschisma pump in 34edo.

Studying the diaschisma a bit more, I realized that a nice 12 pitch class subset can be used to tune a conventional piano:

These pitches can be assigned to piano keys:

The main advantage of such a tuning is that some intervals are closer to their precise just tuning values than e.g. conventional 12edo. From the tuning goodness score table at the top of this post, one can see that 34edo has much better major thirds than does 12edo, while the perfect fifths are significantly worse. In that way, 34edo is similar to a meantone tuning like 31edo, also in the table. But then again, tempered commas are actually a compositional resource. Each tuning creates its own opportunities for making music.

That said, it's interesting to hear what a conventional composition sounds like in an unconventional tuning. This is a minuet by Telemann that I had put into software some years back, so it was easy to convert the tuning. I should note: conventional 12edo was not conventional in Telemann's time. I have no idea what he would have used: in those days, folks were quite creative in finding fresh ways to manage the compromises involved in tuning. I'd like to think that Telemann wouldn't object: Telemann minuet in 34edo diaschismic tuning.

Many high-end digital keyboards allow individual control of the tuning of each pitch class. Here is a tuning table for anyone who'd like to try this: