Cyclic Paths in Tuning

This diagram shows relationships between the sixteen ways to tune a diatonic scale using just intonation. Each arrow in the diagram represents moving from one tuning to another by shifting a single note by a syntonic comma. The arrows point in the direction of raising the pitch of the note. The diagram has a loop: once all seven notes have been raised by a syntonic comma, one has returned to the same tuning structure that one started with, just a tad higher.

I've made diagrams for each of the sixteen tunings, showing the just tuned perfect fifths, major thirds, and minor thirds. In this first tuning, for example, there is no arrow from G to D. In conventional equal tempered tuning, every interval of seven half steps is the same. In just intonation, not all similar intervals can be tuned the same. In this first tuning, the G-D interval is tuned to a 40:27 frequency ratio, and will sound rather harsh.

Here is an example of tuning 1. I used 87edo to create these examples, rather than just intonation, because my algorithmic composition software works mainly with edo. This software uses weighted random choices to decide what pitches to play. The weights are computed based on the consonance or dissonance of intervals between related notes. So with tuning 1 for example, the program will not very often put a G near a D. It will much more often put A and D near each other.

Here is an example of tuning 2.

Here is an example of tuning 3.

Here is an example of tuning 4.

Here is an example of tuning 5.

Here is an example of tuning 6.

Here is an example of tuning 7.

Here is an example of tuning 8.

Here is an example of tuning 9.

Here is an example of tuning 10.

Here is an example of tuning 11.

Here is an example of tuning 12.

Here is an example of tuning 13.

Here is an example of tuning 14.

Here is an example of tuning 15.

Here is an example of tuning 16.

Diatonic Scale in Just Intonation

I am continuing to explore conventional scales, like the 12 notes of a piano or the 7 white notes, tuned with just intonation.

This interval graph is a simple way to tune a piano - just a little bit out of the ordinary. Here is an algorithmic example using this tuning.

Just toying with possibilities, I came up with a tweaked version:

Here is an algorithmic example in this tuning. This network still has a diatonic scale as a connected subgraph, but this subgraph does not appear as any of the seven tuning modes I listed a few days ago. This got me wondering: how many ways are there to just tune a diatonic scale?

It was a pretty simple tweak to the code I wrote that counted 41844 ways to tune all twelve notes of the piano with just intonation. Looking just at the seven white notes, and requiring these seven notes to be all interconnected by simple just ratios - there are 16 ways to tune a diatonic scale! Here is a list.

Non-Diatonic

I hadn't realized how many ways the 12 conventional notes could be tuned in just intonation!There are so many possibilities with temperaments, with scale sizes... but even with this very restricted approach, there is a lot of room for exploration!

Many of the tunings will be oddly shaped with few options for harmonic movement. Many will be based on conventional diatonic tuning, with the usual seven note pattern connected by close harmonic relationships. The above tuning network does not fit the diatonic pattern. The core of the pattern consists of the two short chains of perfect fifths, C#, Ab, Eb, and E, B, F#. Either A or Bb could be added to make a diatonic scale, but both A and Bb are not directly related.

Here is an algorithmic example of this non-diatonic tuning.

This piece was created in 53edo, which is quite close to just intonation. The table above shows just tuning and also the 53edo approximation for this interval network.

It's simple enough to move A and Bb in the network so that diatonic scales are supported. Here E major and Ab minor will be tuned properly:

Here's a piece in this more conventional tuning.

41844 Ways to Tune a Piano

Here's the big list I generated! This gives twelves pitches per octave, as fractions and as cents values. The twelve pitches in the octave all have to be related to each other by just intervals. In each tuning, there is a tree of simple intervals that relates the twelve pitch classes.

One could certainly extend the notion of simple intervals, e.g. to include ratios like 8:7. How exactly these intervals should appear on the piano keyboard, I don't know. With this list of 41844 tunings, the intervals appear on the keyboard in their conventional way: minor thirds are three half-steps, etc.

Just Intonation

Here is a puzzle: how many ways are there to tune a piano using just intonation? The answer will of course depend on the exact rules.

• All octaves are perfect: C5 is twice the frequency of C4, etc.
• A4 is fixed to 440 Hz.
• Each note must be tuned to at least one other note by a just interval, one of
  • an octave 2:1
  • a perfect fifth 3:2
  • a perfect fourth 4:3
  • a major third 5:4
  • a minor third 6:5
  • a major sixth 5:3
  • a minor sixth 8:5
• these interval relationships must correspond to convention. E.g. if E is linked to C by an just interval, that interval must be a major third.
• there must be a path of these just intervals connecting any two notes

I think these rules are enough to define the puzzle.

The diagram above provides a hint that the number of ways to tune a piano with just intonation is likely quite large. Writing a bit of software to enumerate the possibilities shouldn't be too difficult...

Interval Networks

I am continuing to explore the tuning modes I posted about yesterday. The table of fractions I posted then is compact, but difficult to interpret in terms of simple relationships. So I have made graph or network diagrams for a few of the modes.

Above is a diagram for mode 1. As with the diagrams in the post from a couple days ago for diatonic modes, the green arrows represent perfect fifths, the blue arrows are major thirds, and the red arrows are minor thirds.

This is a diagram for mode 7.

This is a diagram for mode 5. If you remove all the black keys, you can see that it corresponds to the 5th diatonic mode that I posted two days ago. There are seven diatonic modes and twelve dodecatonic modes, so they don't line up exactly. But each diatonic mode will appear as a sort of spine inside at least one of the dodecatonic modes, and their orderings are consistent with each other.

There are many ways to use just intonation to tune the twelve notes on a piano, and to step along a path through these options. I chose this particular approach because it is consistent with the diatonic modes I described.

Yesterday I posted algorithmic examples for modes 1 and 7. Here is an example for mode 5. I've been tweaking my code to work better with these examples. Usually I am experimenting with temperament, working to traverse commas that a tuning tempers out. Temperament creates a non-trivial topology for the interval networks. This topology combines with the non-trivial topology of the rhythmic structure to create knots, so the order that emerges from the thermodynamic simulation doesn't collapse into triviality. These just tuned interval graphs have a trivial topology, i.e. there are no cycles, which means there are no knots that prevent collapse. My approach with these tunings is mostly just to keep the temperature higher. With this piece, I gradually lowered the temperature, watching for some pitch class to start to dominate. So this piece is more about order just starting to emerge, which will happen before the phase transition, i.e. at a higher temperature than most of what I post. Anyway, it still sounds fun enough for me!

Twelve-Tone Modes

Yesterday I posted about an approach to diatonic modes through tuning: there's more than one way to tune the seven notes of the scale using just intonation! I got to wondering whether the cycle of syntonic shifts could be extended to just tuning of all twelve notes on the piano keyboard. Turns out to be quite natural!

The first row here, the first mode, matches Kyle Gann's approach to just tuning. Just like the diatonic tuning modes I posted yesterday, the sequence of modes shifts notes one by one up by a syntonic comma. In this table I have shown mode 6 twice: since C is being used as a reference pitch, I shifted the whole tuning back down to keep C at 1/1. The seven modes I posted yesterday are the same as these modes when the tuning for the black piano keys are ignored.

I suppose I should produce twelve sample compositions as examples for these twelve modes, but for now I have just made examples for mode 1 and mode 7.

Modes of / as Tuning

A few years ago I was reading about modes - Dorian, Locrian, etc. - in W. A. Mathieu's book Harmonic Experience. I got the idea there that modes have distinct tuning, though I didn't manage to work out seven different tunings. More recently I have been contemplating the trade-off between tuning freedom and compositional freedom. A dense network of interval relationships constraints tuning, but gives a composer many options for harmonic movement. A sparse network allows more tuning options, e.g. for more precision. But a sparse network gives a composer fewer options. Thinking about sparser tuning networks brought to mind again the question of modes and how they should be tuned.

I can't say that I really understand how modes should work, but I have found a nice cycle of interval networks for diatonic scales: seven different ways that a diatonic scale can be tuned with just intonation. Do these interval networks correspond to the tradition modes, Dorian etc.? Probably not! But the notion of modes can operate in multiple ways. Perhaps what I describe here could be a fruitful alternative approach.

Here the green arrows represent perfect fifth, the blue arrows are major thirds, and the red arrows are minor thirds. This tuning corresponds well to a natural minor mode. Three minor chords, rooted on D, A, and E, can be just tuned with these relationships. I have used my algorithmic composition software to construct examples for each network, in 53edo which is very close to just intonation. Here is an example composition using this first network.

This second network corresponds to a major mode. The D has been raised by a syntonic comma from the previous network. That's how this cycle works - one note at a time gets raised by a syntonic comma, shifting its position in the network, until all the notes have been shifted and the network returns to the starting configuration. Here is an example composition for this second network.

Next the A is raised by a syntonic comma. The slight shift in tuning is not what is important, but rather the way this shift in tuning changes the interval network, which then constrains composition in a different way. Here is an example composition for this third network.

Now the F is shifted up. There is not a single path possible to cycle through interval networks in this way. The path I have chosen looks the most natural to me, but some further exploration could be worthwhile. Here is an example composition for this fourth network.

E is raised next. Here is an example.

Next is C, and an example.

B is raised next, producing this example.

The final note to be raised by a syntonic comma is G, which returns the network back to its starting shape.

Stepping Outside

A challenge was posted on a facebook tuning group: how about 70edo? That's nothing I have ever explored! Let's try!

The purpose of tuning is to provide useful intervals; useful intervals are those corresponding to frequency ratios that are close to simple rational numbers. So the first step in understanding a tuning system is to see which simple ratios it approximates well. The building blocks of rational numbers are the primes. So, a good start is to look at the primes.

Dividing octaves into 70 equal microsteps: this division is fine enough that most any interval will be approximated tolerably well. The table above shows that primes 5 and 7 fall right in the middle beween the microsteps of 70edo, and yet the resulting error is only about 8 cents. In conventional 12edo, the prime 5 is about 14 cents off; despite that, conventional tuning works quite well enough. If we don't want to worry about precision, 70edo provides an adequate tuning palette to approximate any interval we might want. But precision needn't be discarded so casually. In the right harmonic context, tuning intervals to within a cent or two really helps music sound exquisite. Precise tuning also helps the listener to discern the structure of the music. If the structure is already familiar to the listener, precision is not really necessary. But if the structure is unusual, then precision will help guide the listener.

The primes that 70edo approximates with high precision are 3 and 13. Could I make something musical with 3 and 13? It's definitely an unusual palette!

The next step with a new tuning system is to look at the way intervals combine. A Tonnetz diagram is a useful tool for this. In the diagram above, a shift to the right is movement by a perfect fifth, a frequency ratio of 3:2. A shift up is movement by a ratio 13:8, which I see gets called a tridecimal neutral sixth. The tonnetz diagram shows the commas that are tempered out by the tuning. For example, moving by ten neutral sixths bring one back around to the starting point (ignoring octaves). This reflects the fact that 13^10 is very close to 2^37. Another tempered out comma is traversed by moving seven perfect fifths and then three neutral thirds. This corresponds to the comma 2197:2187, which I see has been called the threedie comma.

I looked for a way to make a scale that would support traversing the threedie comma. The 33 microstep interval, corresponding to a frequency ratio of 18:13, works well for this. A scale with 17 notes per octave, generated by this interval, has steps that are almost all equal: fifteen of the scale steps are 4 microsteps of 70edo, the other two are 5 microsteps. This scale is very close to 17edo. The precision of this tuning also constrains composition. For example, in 17edo, every note has the possibility of moving by a neutral third. In the more precise 70edo scale, only some notes have this possibility.

I used my algorithmic composition software to construct a piece in this scale. I tried to coax it to produce a traversal of this threedie comma, but looking at the scores from my various attempts, I don't think I ever succeeded. But here is a piece that at least sounded musical to my ears!

Twelves Notes per Octave

Consonant intervals are fundamental building blocks of music. Octaves, perfect fifths and major thirds are the primary consonant intervals in most music. The diagram above shows these intervals for the modern conventional equal tempered tuning with 12 notes per octave. With this kind of diagram, notes separated by an octave are considered equivalent, so, for example, from any C to any G can be considered a perfect fifth.

This diagram makes it clear that conventional tuning is quite dense with consonant intervals. This density implies many enharmonically equivalent relationships, i.e. there are many paths between each pair of notes. This density also tightly constrains the tuning: the interval relationships in this diagram fix the tuning to the standard equal temperament.

I use algorithmic composition to explore the sounds made possible by these networks of interval relationships. Here is a piece built using 12 tone equal temperament.

The main problem with conventional tuning is that the major thirds are rather sharp. Historically, before 12 tone equal temperament became dominant, various forms of meantone tuning were used. The diagram above shows the perfect fifths and major thirds available in meantone tuning. Just one perfect fifth has been removed, but also four major thirds. The resulting freedom allows a range of choice in tuning, trading off accuracy between perfect fifths and major thirds.

Here is a piece in 55edo, a meantone tuning similar to some historial tunings.

Diaschismic tuning is another approach to organizing the network of interval relationships among a set of 12 notes per octave. Now two perfect fifths have been removed from the circle, breaking it into two halves that are connected by major thirds. This again allows some freedom of choice in tuning, another way of trading off the errors of perfect fifths and major thirds.

Here is a piece in 34edo, a tuning that supports this network of relationships.

Removing a third perfect fifth allows complete flexibility in tuning perfect fifths and major thirds, in particular allowing just intonation, where a perfect fifth is a 3:2 frequency ratio and a major third is a 5:4 frequency ratio. Removing interval relationships from the network gives more freedom for tuning, but less freedom for composing.

Here is a piece in 118edo, a tuning very close to just intonation.

This graph shows the constraints on tuning for the different interval networks. The x-axis is the size of the perfect fifth, in cents. The y-axis is the size of the major third. The green dot shows just intonation, where the perfect fith and major third are perfectly consonant. The red line shows the tuning possibilities for diaschismic tuning; the blue line shows the possibilities for meantone tuning. I have bracketed the useful regimes for each, where decreasing the error for one interval will increase the error for the other. Outside the useful regimes one can adjust the tuning to reduce the errors for both consonant intervals, which will move one toward the useful regime.

This table gives the tunings, in cents, used in these pieces.

Same and Different

Here are pieces in three different tuning systems:

• 12edo
• 34edo
• 36edo

The tuning system 12edo is the standard tuning system that divides octaves into 12 equal steps. The piece in 34edo also uses 12 notes per octave, but spaced unequally, as I described here in an earlier post. I had my algorithmic composition software build a piece with 12 traversals of the diaschisma comma in 34edo; the 34edo piece here is the result. The 12edo piece is just a mapping of this tuning back to conventional tuning. So these first two pieces should sound almost identical. The 34edo piece should sound a bit more consonant. This is a demonstration of what is possible with precise control of tuning. I presented a similar contrast already in an earlier post; this new piece is just the result with the new interval comparison function.

The piece in 36edo is something very different. It has the same rhythmic topology as the pieces in 12edo and 34edo, but the pitch assignments are entirely different. This piece was inspired by the work of Maat DeMeritt, which used the Well-Tuned Piano system of La Monte Young. The 36edo scale I used differs from the system of La Monte Young in several ways:

• My scale is built from an equal tempered tuning, rather than using just intonation.
• My scale uses 11 notes per octave, instead of 12.
• My scale supports traversal of the slendric comma 1029:1024; just intonation does not allow comma traversal.

Kyle Gann's presentation of La Monte Young's system, linked above, very nicely lays out the scale in a tonnetz diagram based on 3:2 and 7:4. Hmmm, my diagram seems to be upside-down compared to his diagram, but anyway the similarity should be clear. Here is the scale in this 36edo piece:

I think La Monte Young used 12 notes per octave because that is how pianos are set up. I used 11 notes because that gives a scale with two sizes of intervals between notes in the scale: 1 and 6 steps of 34edo.

The Radicalism of Modernity

A friend pointed me to this wonderful physics video. I've only watched the first few minutes so far - the whole thing is almost five hours long! It looks like it will be a delightfully informative five hours! Already at the beginning, from 4:00 to 5:00, a fundamental concept of physics is presented. If we want to get to the fundamental, essential laws of nature, we should take as a starting point an isolated, clean, pure state, a vacuum. I have the impression that the video will be showing us a state that is even cleaner and purer than a vacuum! But I want to head in a different direction.

The Copernican revolution shifted the center of the universe, the perspective from which we can access the essential laws of nature, from the earth to the sun. Giordano Bruno was more profoundly revolutionary: he proposed that the universe does not have a center!

I would like to propose a similar scientific revolution. The center being debated by Ptolemy and Copernicus and Bruno is a location in space. The starting point that Richard Behiel is referring to in the video is not a location in space but a state of matter, in particular a state of absence of matter. From a vacuum, the fundamental, essential laws of nature become apparent. Purity reveals essence. I want to argue that purity is not any particular state. It is true that some situations have a kind of purity that allows clearer revelations of natural law. But there are very many such pure situations, each revealing their own particular species of natural law. There is no uniquely pure situation, no uniquely essential natural law.

I first understood this from reading the book Elementary Excitations in Solids by David Pines. The pure situation here is a crystal, a regular arrangement of atoms. In a crystal, the sorts of elementary particles one finds are different from those found in a vacuum. The most basic such particle is a phonon, the quantum unit of a sound wave. There is no sound in a vacuum!

Our starting points for causal analyses are very diverse. If my automobile engine is mis-firing and I want to understand why, to trace the causal chain back to the big bang through the supernovas that created the metal atoms that condensed to form the earth from which the ore was extracted to allow the casting of the engine block that is mis-firing... however accurate this analysis might be, its complexity is not likely to point me to the need to replace the spark plugs! Instead, the pure state that I should start with would be a properly functioning engine. I can then look at how a disruption to that pure state, e.g. fouled spark plugs, can lead to observed effects like mis-firing.

Physics is the cornerstone scientific discipline, and science is the cornerstone discipline of modern times. The idea that an isolated clean state is the purity on which our analyses should be founded, this is the radicalism that becomes translated onto the political plane. The French Revolution is the paradigm case. The calendar and the units of measure were restructured from rational principles, cut off from tradition. The isolated clean starting point is remote from the tangled web of our immediate experience. We have prioritized what is distant over what is near.

This is not a sustainable approach to managing the world. What we neglect inevitably declines. If that decline really matters, we will generally pick up the pain signal, turn our attention to the decline, and take corrective action. But if we have a strong bias, if we are wearing blinkers that restrict our analyses to remote perspectives, our lack of attention can allow the decline to intensify to the point where it becomes much more difficult to correct.

We can think of earth as just one planet among many: this is a perspective that prioritizes the remote. From this perspective, what happens on earth is not very important. If we think of earth as our home, as our life support system, then what happens on earth is not so remote. It becomes important to look for ways to correct any declines we might observe; it becomes important to pay attention to any possible declines.

A physics-based approach to healthcare is also problematic. We can think of human functioning as some kind of swirling bag of chemicals. A human being is very far from the clean pure state of a vacuum! We can try to understand a disease as a pattern of biochemical reactions, but just to understand health as a pattern of biochemical reactions is already a challenge that is beyond our forseeable grasp. But we can shift our perspective to health as itself a pure state, and study the natural laws that are revealed from that perspective. It's not that the biochemical perspective is wrong - my point is that the biochemical perspective is not uniquely right. There are many sorts of pure states, each providing a perspective that can reveal natural laws specific to it.

Looking at Jupiter through a telescope, one can see its moons orbiting around it. Jupiter and its moons form an orbital system. It is natural to take Jupiter as the center of the universe when studying the orbits of its moons. In just this way, the pure system which can be disrupted, the effects of whose disruptions we can observe: what we should see as a pure system will depend on the problems that we are encountering. If we can remain sensitive to problems and able to shift perspectives so we can analyze problems relative to a normal functioning, where that relationship connects to our ability to respond, then our analyses can empower us to steer away from disaster.

Computing Sustainability

What does it mean for an activity to be sustainable? Just that it can be continued for a long time!

There are many facets to be unpacked here. What's a long time? If my guitar has good sustain, it means that I can play a note and hold it for maybe ten seconds. When looking at the sustainability of fossil fuel combustion, the time horizon is more in the zone of a century.

What is going to put a stop to the activity? The vibrating string on my guitar just diminishes gradually and then goes silent. Looking at the nuclear weapons strategy of Mutually Assured Destruction, the main concern about its sustainability is the possibility of global nuclear annihilation. Looking at the whaling industry, the main concern is the extinction of whales of whatever species.

There are two aspects to the cessation of the activity. We might rely on that activity, and so we will suffer when the activity stops. If the activity stops because the activity has caused enough of a disaster that it is no longer possible to continue the activity, we may well suffer directly from that disaster.

What puts a stop to an activity might not be a consequence of the activity. I like to go hiking through the vacant lots on the hillside to the north of our house. Around here the vacant lots are getting developed quite rapidly. In a few years, I will no longer be able to hike through those lots. My hiking activity is not sustainable, but not as any consequence of the hiking itself.

What's the scope of the activity? Lots of people might be doing the same sort of thing. Or other people might be doing something similar. Fossil fuel combustion is an activity with a rich scope. I drive my car a few hours a week to shop, meet friends, etc. This activity of me driving my car, that has negligible impact on the global environment. But around the world, billions of people are similarly driving their cars a few hours a week. The total impact of everybody driving, that is considerable. And then fossil fuel combustion also includes coal and gas burning power plants, ocean freighters, jet airliners, gas powered residential furnaces, oil fueled industrial boilers, etc. When I consider the sustainability of my driving habits, it makes sense to see this activity as an instance of a larger pattern, and to think about the sustainability of the larger pattern. It's not like everyone else is going to stop driving just so I can continue!

To decide how sustainable an activity is, that involves predicting the future. My hiking is not sustainable because those vacant lots will be developed. But that is my prediction of the future! Maybe those lots won't get developed!

Long term sustainability of activities embedded in complex systems: this sort of puzzle is really unsolvable in any definitive way. It might seem clear enough that, between the depletion of fossil fuel resources and the climate consequences of CO2 emissions, our driving habits are not sustainable in the long run. But maybe fusion power will come to the rescue, with electric vehicles taking over, and we can continue our happy motoring lifestyle. Some sort of scenario analysis needs to be brought in. To know what the future will look like is impossible. But we can more practically sketch out some manageable number of scenarios, combinations of gross features of our ways of living. Looking at the sustainability of whatever activity, we can evaluate that against each scenario. The answer will be relative to the scenario.

Digital electronic computing is a vast and diverse activity in the world these days. How sustainable is it? What might put a stop to it? Probably some amount of computing will continue for a long time, if only at a small scale. But, could the dominance of computing in our society be toppled? How could that happen?

One plausible future scenario is that climate change continues as people continue to burn fossil fuels as long as possible. People stop large scale burning of fossil fuels because climate change destroys our industrial capabilities. Could we continue computing if lose the industrial capability for mining coal etc.? An aspect of computing that is not so visible is the way digital electronic microchips are made. This involves massive technological investment. We will only be able to make computers, and hence be able to compute, if our industrial and technological capabilities are maintained at quite a high level. Certainly if there is just a blip for a few years, computers are quite durable so there needn't be an interruption in our computing capability. But if chip manufacture fails for decades, the impact will be massive.

The unsustainability of computing due to the collapse of our industrial capability, this can be like the unsustainability of my hiking because the vacant lots got developed. My hiking is no causal factor in the development of the vacant lots. Similarly, the collapse of industrial capability could be a result of factors entirely different than computing, e.g. fossil fuel combustion and climate change. But it could also be that computing contributes to its own demise.

Computing could cause its own demise quite directly. We are already seeing the pollution of the web from all kinds of computationally created dangers. Spam, misinformation, fraud, viruses, fishing... the list is constantly growing. The web could get to be so dangerous that usage declines dramatically. The economics of chip manufacturing requires huge volume in order to amortize the huge investment in design and process development. If the demand for hardware declines, the unit cost will rise, further reducing the volume. This can become a vicious cycle that could have massive impact.

A less direct causal path, whereby computing contributes to its own demise, is where computing weakens society, and the weakened society can no longer adequately support the computational infrastructure. A very simple example would be how people lose the ability to do mathematics without digital electronic support, and thus lose the capability to debug software. A more complex example would be where the political polarization driven by misinformation in social media etc. leads to the destruction of universities so there are no more engineers to maintain chip manufacturing facilities. A yet more complex example would be where that political polarization prevents effective response to climate change, which leads to the collapse of our industrial capabilities, including chip manufacturing.

Whether any of these rather wild scenarios could come to pass... Yogi Berra had it right: nothing is harder to predict than the future!

Narrowing the Range

Here is a new piece in 53edo. This is another attempt to create a kleisma traversal. Yesterday I posted a first attempt, whose score did not look like a kleisma traversal. I looked back at the code, and it sure looks like the system had been initialized to a kleisma traversal. So the most likely thing would be that I jostled the system at too high a temperature which erased the kleisma traversal, and then as I brought the temperature down a different structure spontaneously emerged.

To test this hypothesis, I used the same rhythmic structure and the same initialization of pitch values, but just set the temperature near the phase transition and jostled the system at that relatively cool temperature.

Here is a score of the piece. The 32 varying repetitions have been folded on top of each other. The vertical axis is the pitch classes, ordered by minor thirds. I.e. each row is the pitch class one minor third above the pitch class below it. This score looks exactly like a kleisma traversal. There is a gradual ramp from the beginning of each 80 second measure, moving up 6 minor thirds, which then wraps over to the beginning of the next measure but a perfect fifth higher. There is a whole band of pitch classes that is absent: a kleisma traversal has no business visiting all the pitch classes of the tuning. It just needs to follow a path to the tempered out comma, in this case the kleisma.

This brings up another facet of the puzzle of yesterday's piece. This piece did cover all the pitch classes. It looked a bit like a schisma traversal, but that shouldn't cover all the pitch classes either. So I suspect the structure that emerged was some kind of compound comma traversal. I have code to initialize a system with a pattern like that... but how to detect it once it has emerged... I don't know quite how to do that!

Here is another score for the piece, but with the rows reordered so now each row is a perfect fifth above the row below it. There is no helical structure here at all: the dense regions don't connect to form any sort of path. This shows that the piece is not any kind of schisma traversal.

Bug or Feature?

Here is a new piece in 53edo. My intent was for this to be a traversal of the kleisma comma, repeated 32 times with variations. I'm not too sure what I actually got!

The idea behind the thermodynamic approach I use in my software is that order can emerge spontaneously. I use unusual tunings; my hope is that this kind of spontaneous order can reveal some of the potential of these tunings. There is one significant challenge though: is the order that can be discovered in the output something that I introduced accidentally, or is is truly spontaneous?

I often do introduce a simple structure into the system I am simulating, and try to preserve that order. In these cases, I just hope that the variations will reveal additional order around that structure. In the piece here, I initialized the system with a traversal of the kleisma comma along one axis of the system. The variations can then emerge in the other dimensions.

I use various types of graphs or scores in trying to see what sort of order might be present in the output. The graph above is a simple score for the piece. Time is on the horizontal axis, in seconds. Pitch class is on the vertical axis: the pitches in the piece are all folded into a single octave, so the vertical axis runs from 0 to 52. The graph looks a bit like a coarsely woven fabric. Since the topology of the system is 32 repetitions of an 80 second measure, folding all the repetitions together might make the order more clear:

It's clear that the measures share some sort of structure, but it's not so clear what the order is. There is a vague sort of staircase structure, so it looks a bit like a comma traversal. It's pretty surprising that almost all the pitch classes are present. A simple comma traversal doesn't need so many pitch classes!

The kleisma comma is dominated by minor thirds. Six minor thirds is very close to a perfect fifth, and in 53edo they are exactly the same (modulo octaves). So I had the idea of shuffling the pitch classes. This graph has the same rows as the previous graph, but instead of ordering the pitch classes in a sort of chromatic way, just climbing microstep by microstep, in these graph each row is a pitch class a minor third above the pitch class below it. Horizontal stripes appear in this graph, with a period of about 6 rows. These are the perfect fifths. But if this was a kleisma traversal, there should just be a few stripes that angle very slightly so they change height by 6 rows from one side to the other; the right side of this graph wraps over to the left side, so the traversal should look like a helix. This graph has a strongly helical shape, but it is very steep. I don't think I ended up with a kleisma traversal! But all those stripes of a perfect fifth...

Here the rows are ordered so each row is a perfect fifth above the row immediately below it. There are about six stripes that gradually rise from left to right, wrapping to the next stripe to form a helix. Ah, this looks like a traversal of the schisma. With a schisma, eight perfect fifths plus a major third bring one back to the starting point (again, modulo octaves).

My intent was to produce a traversal of the kleisma, but I got instead a traversal of the schisma. I need to go back to my code... did I bungle the initialization? Or maybe I jostled too much, the initialization got erased, and the schisma traversal emerged spontaneously. Either way, when I listen to the piece, it sounds pretty good to my ears!