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:

The Conduct of Emptiness

Buddhism and science share the view that much suffering is due to mistaken ideas about the way the world works, and that much suffering can be avoided by replacing those mistaken ideas with ones that better fit the world. The biggest difference between Buddhism and science is their general strategies for avoiding suffering. Shantideva’s analogy captures the difference. Walking barefoot over rocky terrain is painful. The scientific solution, or at least the technological solution, is to pave a smooth road. The Buddhist solution is to wear shoes. The focus of science is outward; the focus of Buddhism is inward. The more accurate understanding of the world provided by science allows engineers to change the world into a less painful configuration. The understanding of mind taught by the Buddha enables Buddhists to train their minds to respond to the world in less painful ways.

The Buddhist view of the nature of reality is not monolithic. In general, the idea is that the coarse objects of our everyday perception and interpretation, the individual persons and things of our everyday world, do not exist in the cleanly defined discrete way that they appear to. For example, careful examination of a cart reveals that it is just an arrangement of parts such as wheels, an axle, etc. As such analysis is pursued ever deeper, is there some natural stopping point, where analysis reveals some elementary discrete objects that cannot be analyzed any further? Over the millennia of the development of Buddhism, a variety of doctrines have emerged. The Madhyamika school, pioneered by Nagarjuna but also followed by Shantideva, holds that no such elementary objects can exist. Whatever object appears in our investigations of the world, that object can be further analyzed to understand how it arises from the interplay of other objects. Emptiness is a term used to refer to the way objects are never discrete, elementary, and unanalyzable. Interdependent origination is a term used to refer to the way objects arise from the interplay of other objects.

The distinction between the two ways to reduce suffering, changing the world versus changing the mind: this distinction is not so clear and discrete either. Effective engineering requires an understanding of the human mind. For example, designing safe and efficient roadways requires an understanding of how the human perceptual and interpretive systems will parse the various road markings. Mind and world are intimately coupled; indeed, each gives rise to the other: they co-emerge. The Buddhist path of mental transformation is not separate from actions in the world. Ethical and compassionate conduct is one of the cornerstones of the path, along with view and meditation. Cultivating the view of emptiness and interdependent origination is a classic meditation exercise. The view comes alive as it is reflected in one’s conduct.

This perspective flows into conduct mainly through the channels of alertness, curiosity, and sensitivity. Our actions arise out of our perceptions and interpretations of a situation. When we know that further analysis would surely alter those perceptions and interpretations, we don’t commit 100% to the interpretation of the moment. We remain open and curious. The new experiences that unfold as we act can inform us and give us fresh understandings so we can adapt and improvise.

There are classical teaching stories that illustrate the need for keeping our interpretations tentative. One story is about a farmer who finds a beautiful horse and the events that ensue. What first seems like good fortune then turns out to be misfortune; what first seems like misfortune turns out to be good fortune. Things are never quite what they seem.

A further channel for living the view of emptiness and interdependent origination is through community, through collective exploration. So much of our common conversation revolves around who is right and who is wrong. More productive discussion can happen when we understand that no one is altogether right and no one is altogether wrong. Each interpretation of a situation provides another perspective. We may need to take action, and so need to resolve a coherent interpretation on which to base that action. But we can work to act in a way that keeps open opportunities to learn more, for our interpretation to evolve, rather than closing down our perceptions in order to stabilize our interpretation, our actions reinforcing our justifications.

The Character of Character

It’s a simple easy idea that there are good people and bad people. Bad people should be locked up. Bad people should not be allowed on the police force. Police should get tough on bad people. Good people should be trusted. Good people should have opportunities to improve their circumstances.

Perhaps it is the core idea of Buddhism that this notion of good people and bad people is dangerously simplistic. Certainly there is some validity to it. But the more one looks closely, the reality is not so simple. Buddhism is known as The Middle Way for a variety of reasons, but one reason is that it can encompass a simple idea along with an understanding of the limits of that idea, ways that an idea can blind us to important details. How can we be blinded by the idea that some people are good and other people are bad?

How somebody behaves is highly situational. A person whose children are starving and whose efforts to feed them by legitimate means have been constantly frustrated, such a person might be reduced to stealing to save the lives of their children. Another person might have a very active physiology and have a hard time sitting quietly for a long time. Put in a situation where they need to sit quietly, they may not be good at following the rules. In a different situation where constant movement is required, their behavior may well be very good.

Whether a particular behavior is good or bad can be strongly dependent on the perspective of the observer. Someone strongly defending one side of a divisive issue can appear good or bad depending on which side of the issue the observer leans toward.

The behavior of a person is generally very complex. In a day or a month or a year, even in an hour, a person can perform many actions, some of which may be very good and others not so good at all. A person experiences many different situations and also what’s going through their head is always shifting. One situation could make a person angry and they carry that anger into the next situation which can open the door to rather bad behavior. A person could be a hard worker on the factory floor but then go home and abuse their family, taking out their workplace resentments and frustrations.

People do change over time. Some of that is just from getting older and slowing down. Age can bring experience and understanding. But age can also bring bitterness and frustration. There is a feedback loop where a person’s behavior leads them into situations which can then steer their behavior. For example, prisons have a reputation for converting petty criminals into hardened criminals. The possibility for such a feedback loop to lead in a positive direction is equally present.

The most potent driver of change is a conscious intention. Parents, teachers, ministers, supervisors, therapists: each of these can work to guide a person to evolve in a more positive direction. It can certainly happen too that one person works to steer another person into some negative behavior pattern, to exploit that person one way or another. To help another person improve is one of the best behaviors possible; to lead another person astray is one of the worst.

To realize that one’s own pattern of response and behavior is malleable and within one’s own power to shape, this is the essence of the Buddhist path. None of us are fixed quantities. We are each a process of constant becoming. To cherish our own positive potential, and that of everyone we meet, is the seed that yields the fruit that nourishes most deeply.

The dangerously simplistic notion of good people and bad people, that seems to be at the foundation of much of our current political strife. Can we reform society effectively by putting all the bad people in prison? Can we reform the police effectively by firing all the bad police? Certainly we need to find ways to suppress bad behavior of all sorts. A good starting point would be to work to understand what sorts of situations promote bad behavior and to keep people out of those situations. Putting violent criminals in prison and firing abusive police officers, these are reasonable and necessary actions. But those are only superficial remedies.

What is Essential

Saving lives or saving the economy? That is a truly bizarre dichotomy! More lives could be lost from the collapse of the economy than from the pandemic: that is a perspective that at least focuses on what matters. Merely staying alive, that is a very low bar indeed, but it is certainly a good place to start.

The natural instinct in a crisis like this is to work to reestablish the routine by which we had been living adequately enough. Of course, to ignore a pandemic is not going to make it go away. Wishing doesn't make it so. There is no way to escape this unscarred, but facing the situation and realistically engaging with it, that is the approach that will minimize the damage. We do have to balance lives lost due to supply chain disruptions etc. against lives lost directly from the pandemic. People need food, clothing, shelter, and medicine. There is enough to go around. A modicum of ingenuity can keep people supplied.

The shock of this pandemic reveals in a variety of ways that our routines have not served our needs so very adequately. This virus is the kind of thing that comes around every decade or so. A healthy economy, a healthy society, is one that can handle such challenges effectively and efficiently. So far, the United States appears to be stumbling.

In the short run, we need extraordinary measures to make sure that people are fed and housed. Insuring adequate medical care seems, sadly, out of reach in this crisis. In the longer run, we need to restructure our routines so that the next pandemic doesn't knock us off our feet. Our way of life has been driving us against many ecological limits. Plague, drought, flood, famine: we can expect ever more frequent crises as long as we mistake recovering our routines as essential.

Pride and Fall

I was discussing interplanetary colonization with an acquaintance recently. I don’t foresee that in any likely near future, while my interlocutor is convinced it’s a near certainty. Opinions do differ! The discussion did degenerate, sadly, into ad hominem characterizations. I was put into the “genteel-poverty crowd”; I called my acquaintance a “technocrat”. He accepted the label gracefully enough.

Predicting the future involves some kind of model. The simplest common model is an exponential function. For example, the world population might be increasing at 2% per year. With that simple model we can forecast the population at any time in the future. Science is largely a matter of developing, testing, and refining models for various facets of the natural world, by comparing the various models’ predictions against real world experience.

How this whole dialectic interplay between theory and experiment actually proceeds, or should proceed, is a topic of endless discussion among philosophers of science. Scientists have got the orbit of the moon around the earth figured out with remarkable precision. Philosophers of science continue to struggle to come to any basic understanding of how that precise figuring has come about.

Immanuel Kant’s philosophy is a major waypoint in understanding how science progresses. His notion of synthetic a priori judgements accounts for the fact that models cannot be inferred directly from the data of experience. It seems clear by now that our thinking is more adaptable than Kant gave us credit for. But still, it often takes a succession of generations of scientists for a new conception to take root. Our ways of seeing the world are quite deeply rooted.

In my recent discussion on interplanetary colonization, my acquaintance made a remarkable declaration: “Technocrats don’t have preconceptions.” I don’t think this is any kind of unique or even unusual attitude. Most scientists dismiss philosophy of science as being irrelevant. Generally they take for granted a kind of direct insight into the nature of phenomena. The task of science looks, from this perspective, a bit like that of a surveyor mapping out a new territory. Probably even real surveying is a bit trickier than this kind of naïve notion would portray it!

Our modern civilization is built on science. Practically every facet of our lives has been explored scientifically. Our lives are, in turn, structured by these scientific models, be they mechanical, chemical, biological, geological, or whatever. It is quite easy to slip into thinking that we have pretty much figured it all out. The stable structures of our lives mesh neatly with the stable structures of our ideas.

It may be just my own preconception, but things do change. The notion of environmental constraints has been talked about at least since Malthus, a couple hundred years ago, but clearly we humans have been very successful in circumventing whatever limits have appeared. How long this run of success will continue… that’s one of the core controversies of our time. What interests me here is not this or that model on which to build a forecast. What interests me is how the shape of the future can appear unquestionable to the cultish wing of the technocratic faith. More precisely, how might this kind of blind faith affect our ability to navigate any turbulence in the coming decades.

There is a curious paradox here. The notions of Malthusian limits, of climate change, of turbulence on the horizon: these are all scientific ideas. Science forecasts change, but science is not really ready for change.

Supposing that we actually do run out of miracles and Malthusian limits do finally get their teeth into us. What’s the impact going to be? Will it just be a matter of maybe replacing our air conditioners with more efficient units? Or are we facing famine and plague and the decimation of the population? These questions have been discussed extensively. A question that gets much less attention, though: what will happen to science? Certainly there are deep visionaries in the scientific community who can foresee the coming turbulence and begin to sketch what a post-collapse laboratory might look like. But a large fraction of the members of the greater scientific-technocratic subculture share my acquaintance’s blind faith, I fear, in the eternal stability of the present system: a steady growth in watts per capita and the rest of it. This kind of blind faith will not help us navigate any sort of coming collapse!

Schismatic Tuning

I am fascinated by musical tuning, from the conventional 12 equal steps per octave to all sorts of wild possibilities. Sometimes when I am out playing on the fringes, I learn something that can bring me back into much more conventional possibilities. So here is a way to tune a conventional keyboard. I'll call it "schismatic tuning" but I don't doubt that it has been explored again and again in the past.

Musical tuning is essentially a branch of mathematics, especially the way I approach it. Often in mathematics and science the focus is on novelty, on fresh discoveries, where fresh means not previously encountered by the human mind. This focus is unnecessary in math and science, is somewhat distracting or misleading, and will likely serve us much less well in the future. Since the time of Kepler and Galileo, math and science have expanded in a stunning fashion. Predicting the future is a fools game, but it seems unlikely that environmental constraints will permit continuous growth in extracting resources and dumping wastes. The modern trend of constant growth seems destined to end sooner rather than later. Math and science will be of great value in any post-growth society. To keep them alive, though, the focus will need to shift away from novelty.

So here is a historical preface to the schismatic tuning I have (re)discovered: Emilio de’ Cavalieri’s mysterious enharmonic passage - a modern rendition of a renaissance recovery of an ancient Greek tuning! Paul Erlich has written a thorough discussion of tuning A Middle Path Between Just Intonation and the Equal Temperaments - I have barely scratched the surface of this paper! I imagine that schismatic tuning is described in there somewhere! I would just like to share my (re)discovery here of this one small facet of the vast universe of tuning. I offer it as an invitation to explore further!

A quick review of fundamentals. A musical interval is the relationship between two pitches, which can be analyzed as the ratio between their frequencies. If pitches are an octave apart, their frequencies are in a 2:1 ratio; a fifth apart, a 3:2 ratio; a major third apart, a 5:4 ratio. These ratios are ideal. Just Intonation is a tuning that uses these ideal ratios. But for a variety of practical reasons, it is often useful to adjust, or temper, these ratios. There is no perfect solution to the puzzle of temperament. Modern keyboard tuning adjusts the fifth to 2^(7/12) ~= 1.4983 and the major third to 2^(4/12) ~= 1.26. The human ear can detect reasonably well the difference between this tempered major third and the ideal of 1.25.

Schismatic tuning is actually a family of tunings. I will present one version, based on dividing an octave into 53 equal steps, rather than the conventional 12. With 53 steps available, a fifth is tempered to ~1.499941 (31 microsteps) and a major third to ~1.248984 (17 microsteps). The fifth is improved, but the conventional tuning was already very good; the main improvement is in the major third.

How can such an improved tuning be adapted to a conventional keyboard? Here is my proposed schismatic tuning:

The top row names the keys on the keyboard. The second row gives the number of microsteps from the low C to the particular key of that column. The bottom row re-expresses that pitch in terms of cents. The conventional tuning would result in pitches of 0, 100, 200, 300, etc. cents. So this last row makes clear the difference in pitch between the schismatic tuning and conventional tuning, e.g. D is 3.774 cents sharper.

Some points to observe:

• Almost all of the fifths are 31 microsteps, i.e. very accurate. From D to A is only 30 microsteps, though.
• Four of the major thirds are the ideal 17 steps: C to E, F to A, G to B, and D to F#. The others are sharp by a microstep, i.e. closer to a pythagorean major third, 81:64.
• The sizes of the chromatic intervals in this tuning are not all the same: 4, 5, 4, 4, 5, 4, 5, 4, 5, 5, 4, 5.
• The syntonic comma is not tempered. E.g. moving by fifths up from C to E, one must cross the "wolf" fifth from D to A. This is a distinctly unconventional tuning.

One can certainly play in any key signature with this tuning - none of the intervals is too far off. But certainly a piece of music will sound different when the key signature is changed. This tuning does allow though a simple dynamic shift as outlined in my post Dynamically Tuned Piano. With perhaps a push of a foot pedal, A can be sharpened by a syntonic comma:

or a different pedal could instead flatten the D by a syntonic comma:

These shifts will move the wolf fifth up or down a fifth, and also rotate which major thirds are pythagorean, etc.

Rotta Model for Bicycle Tires

I recently discovered Andrew Dressel who recently wrote a PhD dissertation Measuring and Modeling the Mechanical Properties of Bicycle Tires. There's a lot in there, of course, but amongst all that I found a description of the Rotta model for tires. This is much the same as what I have been exploring. I polished up my math and wrote a bit of new software to generate some new tables for recommended inflation pressures base on Frank Berto's rule of 15% squish.

The Rotta model is quite simple:

The tire is flat where it contacts the ground, and has a circular cross section where it is not in contact. Each cross section of the tire is treated independently.

Here are tables from which inflation pressure can be computed. The rows correspond to tire widths, in mm. The columns represent rim width, as a ratio to tire width. E.g. the column with 2 at the top is for tire width twice the inner rim width, e.g. a 50 mm wide tire on a 25 mm wide rim.

The tables give the area in square inches of the contact patch. To compute the inflation pressure, divide the load on the wheel by the area of the contact patch. E.g. a 50 mm tire on a 25-559 ETRTO rim will have a 2.06 sq in contact patch when it is squished down 15% of its width. For that contact patch to support a 100 pound load, the inflation pressure should be 100 / 2.06 = 48.5 PSI.

For 622 BSD:

For 559 BSD:

For 406 BSD:

These tables are calculated purely from theory, i.e. no parameters are used to fit them to any experimental data. Do not follow them blindly! They're food for thought & perhaps provide a useful starting point for exploration.

And a 305 BSD table: