Non-Euclidean Science

Maybe I should call it non-Aristotelean, or non-Platonic, but the exact name isn't the point. Euclid built geometry up from postulates; Aristotle explained motion as objects returning to their natural state; Plato portrayed experience as the shadow play of forms in an ideal realm. In each case, a complex field of phenomena is explained as the outgrowth of some simple essential foundation. Perhaps I should call my proposal non-foundational science. But I am not proposing any kind of freed-floating science. I am proposing a science that is founded on reality, on the vast tangled web of lived experience. Science is an extract, like resin extracted from the sap of a tree. There is a lot more to a tree than such resin. The tree itself is embedded in soil and climate, in an ecological web, flying pollinators and mycorrhizal fungi. The simple essence emerges from the whole, rather than the whole emerging from the simple essence.

Science as a quest for an inner key that explains everything - such science takes us on a quest into ever more remote realms. It distances us from experiencing what is right at hand. Of course, building and launching the James Webb infrared telescope surely involved considerable attention to experiences right at hand - precision torquing of many bolts, etc. Galaxies and quarks are not objects of direct experience, but neither are they disconnected from direct experience. What I am proposing is no neglect of any corner of the world. I am suggesting a shift in how we understand the way all the bits and pieces fit together.

Non-Euclidean geometry provides an excellent analogy. The surface of a sphere, such as the surface of the earth, is a perfect concrete instance. Euclidean geometry is plane geometry, the geometry of a flat surface. At the scale of a few square miles, the earth is extremely close to a flat surface, and can be mapped onto a flat sheet of paper with great precision. But as the area to be mapped increases to include a significant fraction of the earth's surface, inevitably distortions arise. There is no perfect flat map of the earth.

The impossibility of perfection does not mean that we just give up and produce fantastic maps that have lost any connection with the lived experience of moving around on the earth. The value of a map is exactly in how it relates to such lived experience. Whether a map is good or not, that depends on how the map is to be used. A map that is good for navigation will typically not be a good map for estimating agricultural productivity.

Pure science is science that neglects its relationship to its use. Applied science is science that orients itself to its use. The classical scientific attitude is that applied science grows out of pure science. I am proposing that a healthier approach to science is to see pure science growing out of applied science. Applied science connects to the vast complexity of lived experience. Refining our ideas requires chopping out local regions to be precisely mapped. This always involves distortion and omission: the inevitable price of precision. It's like taking a photograph: a fast shutter speed can reduce blurring from movement, but requires opening the aperture which increases blurring from less depth of field.

Our scientific quest for ultimate theories is like the old searches for the alchemical philosopher's stone or the healer's panacea, a medicine to cure all diseases. Good science requires following the clues wherever they lead, but it also requires a perspective on the actual situation so that one doesn't chase clues just for the sake of the chasing. Good science is science that is engaged with the lived reality of an actual situation.

Double Helices

Here is an image of a torus with a square lattice drawn on it. This lattice is formed by the intersections of two helices drawn on the torus. There are many ways to draw such helices on a torus, and the lattice pattern emerges from the combination of two such helices. This kind of double helix square lattice on a torus is a broad family of geometric shapes.

This kind of geometric shape can be used in music several ways. It can serve as a model for the time evolution of a piece of music. It can also serve as a model for the harmonic relationships between the pitches used in music. Since music is, in large part, a relationship between time and pitch, a piece of music can be modeled as a relationship between two different toruses, a torus of time and a torus of pitch. Of course most music won't fit this model very well or at all. But it can serve as a blueprint for creating music.

Musical time as a helical path on a torus... maybe it's because I have been thinking this way for decades, but it seems quite natural. Of course a piece of music often has more of an arc structure, a beginning, a middle, and an end. But often within that large arc, whole stretches are largely repetitive, where the end of each repetition joins smoothly with the beginning of the next. If the repetitions were exactly the same, this would simply be a circle. But perhaps the words of verses change or other details, so each repetition is slightly different than the last repetition. To bring the last repetition close to the first repetition is of course a more arbitrary choice, but not a very wild one. I hope this makes sense of the notion of musical time as a helical path on a torus.

The idea of pitches being related harmonically in a way similar to a helical lattice on a torus... this is hardly a new idea in the world of music theory. There are many ways to use this kind of geometric shape to represent harmonic relationships. The circle of fifths is the most basic. Major thirds are another fundamental relationship between pairs of pitches. These two intervals then create a mesh of relationships that can be laid out on a torus:

Here the green line traces the circle of fifths. There are four red loops, representing the circles of major thirds.

This torus of harmonic relationships can be drawn for alternate tunings. The different topologies generated display the different musical possibilities of these alternate tunings. One important alternate tuning divides octaves into 19 equal steps instead of the usual 12. The torus of harmonic relationships for 19edo looks like:

The green loop here again represents the circle of fifths. What's more interesting is that major thirds no longer divide up the pitches into separate loops. Instead there is one large loop traversed by major thirds.

With these two toruses and their helical lattices, the harmonic structure of a piece of music can now be mapped out. For the most part, one would expect phrases that are closely related in musical time to be closely related harmonically. There might be abrupt transitions, but they make sense in this approach in the context of surrounding smooth relationships.

The simplest non-trivial mapping uses a loop on the pitch torus, some path through the lattice that returns back to the starting point. This loop is then traversed in musical time. It could be that each repetition traverses the loop. Or perhaps the repetitions don't move much internally, but each repetition moves slightly relative to the previous repetition, so the harmonic loop is traversed over the course of the whole piece.

Algebraic topology is the mathematical discipline where these sorts of smooth mappings are enumerated. Mapping a torus onto a torus is a rather elementary problem in algebraic topology... but there is still a rich variety of possibilities to be explored musically!

Another feature of alternate tunings is that additional basic harmonic relationships can be introduced. Exactly what makes pitches sound harmonically close, that is an endless topic of study and debate. But one fundamental notion with a long history is that frequency ratios very close to a simple rational ratio, that's the basis of close harmonic relationships. An octave is a frequency ratio of 2:1. A perfect fifth is a frequency ratio of 3:2. A major third is a frequency ratio of 5:4. In conventional music, these basic intervals are the foundation of harmony.

One natural step in extending music into wider worlds is to introduce yet another basic interval, governed by the frequency ratio 7:4. This extra relationship makes the torus of harmonic relationships much more difficult to draw... it's not anything that could physically exist in our three dimensional world. But of course mathematically it is nothing very complicated to manage. A tuning that can represent this new interval quite accurately, along with the more conventional intervals, is 171edo, the tuning that divides octaves up into 171 equal steps.

An instrument with so many notes would be physically unwieldy. But with a software synthesizer and algorithmic composition, it is not so hard to build a piece of music based on a mapping of this more complex torus onto a torus of musical time:

Double Helices

Sliding Schismas

For some years now I have been exploring music and tuning, through algorithmic composition. I have a computer program that I tweak, to change tuning systems, scales, etc. How much of the tweaking that I do is actually reflected in the output, in any perceptible way? That's a question worth examining!

Here is a set of musical pieces. The only thing I changed in the software from one piece to the next is that I changed the seed for the random number generator. The random numbers it generates control very many choices in the execution of the program, so these pieces will vary quite a bit. But the primary choice in question is the harmonic movement involved, the key changes. Some of the pieces don't have any movement at all. Others have a progression that is six steps long. Some of the pieces move along the progression in the forward order, other move in the reverse order.

So the question here is: can you divide these pieces into three groups, one group with no key changes, another group that moves in one direction, and a final group that moves in the opposite direction. Can I tell the difference? (The names of the pieces are the seeds I used to initialize the random number generator for each piece.)

• 7000
• 7001
• 7002
• 7003
• 7004
• 7005
• 7006
• 7007

These pieces all use the 53edo tuning system, where octaves are divided into 53 equal steps rather than the conventional 12. These pieces all work with a schisma[17] scale, where 17 notes are selected in each octave out of the full set of 53. In the pieces with no key changes, the scale is constant throughout the piece. In the other pieces, the key changes in a regular pattern, shifting every measure. With six key changes, the scale returns to the starting scale.

Each row in this picture shows which notes are in the scale in one of the keys. In the pieces with key changes, from one measure to the next the scale will shift to the next row up or down in the diagram; in some pieces the key changes move up in the diagram, in other pieces the key changes move down. I repeated the sequence three times in the diagram, and also extended the scale a bit beyond an octave, just to make clear that the pattern continues smoothly through time and up and down the pitch space.

One could play the pieces with no key changes on a piano reasonably accurately. There are five pairs of notes that are very close togther, just one step apart of the 53 per octave. These would correpond to spit keys on a deluxe piano, a slightly sharp version of a note and a slightly flat version. Thinking of the split note as just two versions of a single note, then there are twelve coarse notes per octave, very close to a conventional piano.

The sequence of key changes in the other pieces involve two different shifts in the scale. Moving along the sequence in one direction, the scale shifts five times by a minor third, and once by a minor sixth. Moving in the other direction, the shifts are the inversions, i.e. five major sixths and one major third. In the 53edo tuning system, this combination of key changes brings the scale back to its starting point.

When the scale is shifted by a minor third, the new position of the scale include eight of the notes of the scale before the shift. The shift by a minor sixth has a similar amount of overlap. This overlap allows for good continuity of musical phrases across the shift.

Listen to the pieces above: some have key shifts, and some don't. Can you tell the difference?

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.

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.