Tuesday, March 26, 2024


In yesterday's post I discussed a type of musical instrument that presents a fixed number of keys the player can strike or press etc. to produce pitches. These pitches would be from a section of the chain of fifths. This is a very traditional meantone approach, but it can be generalized to a chain of whatever other interval is of interest. The instruments I described would have some kind of auxiliary control to allow the section to be shifted along the chain. The idea is that by shifting one step along the chain, the pitch of a single key is changed by some relatively small amount, so the pitch order of the keys is not altered. An example of such an instrument is a harp, which has seven strings per octave. Pedals allow the player to sharpen or flatten strings to follow key signature changes.

This only works for sections of suitable length. For example, a section of five pitches along the chain of fifths would be F C G D A. The next pitch on the chain is E. To shift the section along the chain one step, the F must be replace by E. This does not change the pitch order of the keys, so this works.

If one tries to use a section of six pitches, F C G D A E, to move the section one step along the chain of fifths, one would have to replace the F by B. This is a very large change in pitch that severely disrupts the pitch order of the keys of the instrument and so would be be quite unwieldy.

A seven pitch section is again practical. To shift F C G D A E B along the chain of fifths, the F needs to be replaced with F#, which again does not disrupt the pitch order of the keys of the instrument.

Observe that sometimes moving the section up a fifth along the chain involves sharpening the changing pitch, as with the seven pitch section, and sometimes involves flattening the changing pitch, as with the five pitch section.

Which size sections of the chain will preserve the pitch order when shifting along the chain depends on the exact size of the fifth. For the just tuned fifth of Pythagorean tuning, sections of size 7, 12, and 53 shift by sharpening a pitch; sections of size 5, 17, 29, 41, and 94 shift by flattening a pitch. To take 1/5 comma meantone as a contrasting example, sections of size 7, 19, 31, 74, and 117 shift by sharpening a pitch; 5, 12, and 43 shift by flattening a pitch.

These various lengths all correspond to useful ways to divide octaves to form practical tuning systems. The chain of fifths of length n needs to wrap around the octave circle to come back very close to where it started, if shifting the chain is to involve a small pitch change. So n fifths must be very close to m octaves. So m/n needs to be good approximation to the size of the fifth relative to the size of the octave. This means that dividing octaves into n equal parts will provide a good approximation to the fifth, which will be at m steps of this division.

In celebration of the beauty of these mathematical aspects of music, here is an algorithmic composition using a 19 pitch section of the fifths of 43edo: 43edo scale 19.

Monday, March 25, 2024

Shifting along the Chain of Fifths

Musical tuning systems continue to fascinate me. One path of exploration is innovation, to explore fresh territory. Another path is to look at history and to explore the foundations that the present conventions are built on. Meantone is a historically important tuning system that can still offer innovative possibilities.

The conventional 12 tone equal tempered system of today is built on the circle of fifths. The more fundamental meantone system is build on a chain of fifths. Meantone is actually a family of tuning systems. The conventional 12 tone system is just one of these. Some members of the family will close the chain of fifths into circles of different sizes; others will leave the chain unclosed.

More fundamental than the circle of fifths is the circle of octaves. One can start a scale at C, move up through D, E, F, G, A, and B, and then return to C again: a higher C that where the scale started, but the name repeats. It's another C.

The higher C has twice the frequency of the lower C. One C might be at 256 Hz, and the next higher C would be at 512 Hz. Generally, the theory of tuning systems is built on the principle that two pitches will sound consonant together if their frequencies have a simple ratio. 2:1 is about as simple as a ratio can get; that's why pitches an octave apart are so intimately related that we often just ignore their difference. That's what I'll be doing here, largely. I'll just look at pitches within a single octave range.

After 2:1, the next simplest ratio is 3:2. That's the frequency ratio for pitches a perfect fifth apart. That's why the chain of fifths is so important: the chain is like the main highway, the path built by moving from one pitch along to the pitch it is most consonantly related to (again, ignoring octaves).

The picture above shows the conventional names for the pitches along a central section of the chain of fifths. The chain can be extended indefinitely in either direction, just by adding more and more sharp symbols in one direction, or flat symbols in the other. A musical composition generally uses a finite number of pitches, and so will use just a finite section of this infinite chain. Pentatonic music will just use a five pitch section of the chain. A diatonic scale encompasses a seven pitch segment. The note naming convention A, B, C, D, E, F, G is based on the diatonic scale.

Musical compositions can be based on different sections of the chain of fifths, or might be broken into parts that use different sections, or might use more than seven pitches even in a single part. To slide a seven pitch subset up the chain, one adds the next pitch in the direction of motion and removes the trailing pitch. With the seven pitch subset, these two pitches will be nearby. If the chain is built of precise 3:2 perfect fifths, the difference between the new and old pitches will be a ratio of 2187:2048 (folding the octaves together as needed). This small ratio is the basis for the sharp and flat notation: sliding one step down the chain of fifths involves just a small tweak to one pitch of the seven in a diatonic scale.

The tuning system built on a chain of precise 3:2 ratios is called Pythagorean tuning. It runs into trouble. The Pythagorean major third is a ratio of 81:64. This is very close to the simple ratio of 5:4. When two pitches are quite close to a simple ratio like this, but still significantly off that simple ratio, they sound harsh or out of tune.

Meantone tuning is a solution to this problem. The perfect fifths are slightly flattened, in order to reduce the error in the major third. Exactly how much to flatten, that is not fixed. A range of possibilities has been used historically. The conventional 12 tone system is only a little flat, and the error in the major thirds remains significant. The graph above shows the practical range. One can see that, for example, 1/5 comma meantone balances the errors in the perfect fifth and the major third. 2/7 comma meantone balances the errors in the major and minor thirds. These and other choices have been advocated and used historically.

The fundamental problem with meantone tuning is that, in general, it is still working with the infinite chain of fifths. Many musical instruments can only provide a limited choice of pitches to be played. If there are enough choices, composers won't be overly constrained by the requirement that they don't run so far along the chain that they exceed the capacity of the instrument. Another practical possibility is that the size of the perfect fifth is chosen so that the chain is closed into a circle. This is the great virtue of the conventional 12 tone system. 12 pitches per octave is a very practical number. One can wander up and down the chain of fifths, and because it has been closed into a circle, one will never run into a wall. One can use 19 and 31 pitches per octave, along with other choices. The major thirds can be significantly better than those in the conventional 12 tone system, but the extra pitches per octave can be a bit unwieldy.

While conventional piano keyboards have 12 keys per octave, 7 white and 5 black, harpsichords and organs have been built with additional black keys, tucked alongside the conventional 5. Sometimes just one or two black keys are split. The graph above shows the pitches in a range of meantone choices for a fully extended keyboard, where each black key has been split for a total of ten black keys. One can see in this graph that the sharps and flats cross for the 12 tone conventional tuning system. In the exact Pythagorean system C# is higher than Db. For most of the usual meantone choices, such as 1/5 comma meantone, C# is lower than Db.

Another way to manage the chain of fifths, other than walls and circles, is to add shift controls to instruments. Harps are a good example of this approach. A harp has just seven strings per octave, but allows further movement along the chain of fifths by way of pedal controls. Modern electronic instruments could easily support unbounded movement along the chain of fifths. For example, a conventional 12 tone keyboard could work with a full range of the chain of meantone fifths. At any one time, the keyboard would present a range of 12 pitch choices per octave. There would be one "wolf" fifth in the tuning. But a pedal or other control could be provided, to shift the location of this wolf. At one time the keyboard might provide a range along the chain from Eb to G#. If the music requires moving a fifth up from G#, the required D# is not immediately available. But the auxiliary control could shift the Eb to D#. Generally a piece of music is not going to need the Eb at any time close to when it needs the D#, so handling the auxiliary control should not be too burdensome.

The next natural such circulating keyboard would have 19 keys per octave. These could be arranged by splitting the five black keys, and then adding single black keys between B and C, and between E and F. This could work as a simple system with 19 equal steps per octave, which is approximately 1/3 comma meantone. But such a keyboard could also work in arbitrary meantone tunings using an auxiliary shift control. So, for example, in one setting of the auxiliary control, the key between B and C would provide Cb, while in another setting it would provide B#.

Wednesday, March 13, 2024

Finding the Phase Transition

I've been using an algorithmic approach to music composition, based on thermodynamic simulation. One advantage of an algorithmic approach is its generality. One can use the same software to generate music in a wide variety of tuning systems. Other parameters can be adjusted easily, too. Of course, no algorithm is likely to generate music of the quality that a skilled human composer could produce. On the other hand, there can be some value to music that is outside the usual patterns.

These days a different algorithmic approach to composition has gained some traction, an approach that works with a large number of existing compositions, extracts some patterns, and then follows these patterns to generate a new composition that sounds much like the existing compositions. Just to be clear, the approach I am using does not use existing compositions in the execution of the software. I listen to a lot of music, and I listen to the output of my software; I tweak my software in an effort to coax it into producing something as music-like as I can. But this is a long way from the Deep Learning methods of the predominant Artificial Intelligence software.

Thermodynamic simulation is a randomized process that repeatedly adjusts the values of a large number of inter-related variables. The variables are connected in some kind of network, that defines the interactions between the variables. In my music composition software, the variables are the pitches of what is to be played at each particular time by each particular voice. When a voice is to sound a pitch at one time and then a second pitch at a succeeding time, these pitches should be nicely related if the music is to sound good. The pitches should not be too far apart, and should be reasonably consonant. Similarly, if one voice is sounding one pitch, and another voice is sounding another pitch at the same time, these two pitches need to be consonant if the music is to sound good.

If a piece is ten minutes long, each voice might involve a thousand pitch choices. During the simulation, all of these choices have provisional pitch values. Again and again, one or another of these choices is selected at random, and then the pitch selection for that specific time is reevaluated, in the context of the provisional choices in place for what that voice is to sound before and after, etc. The software will choose a new pitch for that voice at that time, preferring pitches that are consonant with the other pitch choices nearby in space and time. Then some other voice and time will have its pitch reevaluated. Over the course of the composition process, each pitch will change hundreds of times. Other related pitches will have changed between one evaluation and the next, so which pitch is most consonant may well change over the course of the simulation.

Thermodynamic simulation is driven by a key parameter, the temperature. At high temperature, the preference for consonant pitches is not very strong. At low temperature, only the most consonant pitches will be assigned. At very high temperature, the simulation will assign pitches essentially at random, so the music will be pure noise. At very low temperature the simulation will strive to maximize consonance. But if the pitches are initially very random and then the simulation is run at very low temperature, very often it will happen that in evaluating the best pitch for a particular voice at a particular time, the related pitches don't pull in a consistent direction. One pitch choice will be consonant with some neighbors but dissonant with others. There will often be no choice that is consonant with all the nearby pitches.

The way to generate pitch assignments that are mutually consonant throughout the network is to start the simulation at a high temperature and then to slowly lower the temperature. Each pitch selection provides some communication between more remote regions of the composition. The entire system can eventually negotiate mutually agreeable pitch choices in this way.

Thermodynamic simulation thus has the capability of generating pure noise at high temperature and pure order at low temperature. Neither of these makes interesting music: either extreme is quite dull! Interesting music happens in the region between total noise and total order.

The fascinating thing about this kind of thermodynamic system, whether simulated or in real physical systems, is that the transition between order and disorder is often not smooth and gradual, but can be quite abrupt. Right at the boundary between the ordered phase and the disordered phase, the system can exhibit fractal fluctuations as it wavers between the behaviors of the different phases. Fractal fluctuations are a characteristic of interesting music. So the approach I generally use for generating interesting music with thermodynamic simulation is to set the temperature to where the phase transition happens and make the pitch choices at that temperature, where consonant choices are preferred but not too strongly.

One challenge with this approach is that the temperature at which the phase transition happens is not something one can calculate in any simple way. One is basically stuck with simulating the system at different temperatures, observing its behaviors, and identifying an abupt shift. That's what the graph at the top of this post illustrates. At each temperature, the system will settle into an overall level of consonance, which corresponds to energy in thermodynamics. A highly consonant system has very low energy; a highly dissonant system has very high energy.

The graph above has a clear enough abrupt shift at a temperature of around 380. There is a sudden drop in the energy with a small change in the temperature. Locating this sudden drop is a bit tricky though, because of the random nature of the simulation. The energy is always bouncing around even at a fixed temperature. What I do to filter out this randomness is to fit a smooth curve through each small family of temperature and energy measurements. Then I look for which such small curve shows the most abrupt change in energy over a small change in temperature. That tells me the temperature of the phase transition.

Once the temperature of the phase transition has been determined, I can set the simulation temperature to that value and let the simulation run so all the pitch choices come to reflect that boundary behavior, to exhibit fractal fluctuations.

Here are two pieces generated using this approach:

Thursday, February 29, 2024

A Tale of Two Unconventional Tunings

Nowadays the notes available on a conventional piano, seven white keys and five black keys per octave, form the building blocks for almost all the music in circulation. And of course these building blocks have been very effective at enabling the crafting of a vast treasure chest of music, diverse, profound, and beautiful. And yet, there is value in exploring other tuning systems.
  • Around the world, there are still many different traditional tuning systems in use.
  • Tuning systems evolved over the centuries in Europe, only settling on the present convention some two centuries ago.
  • Different tuning systems enable different musical structures; they are a rich compositional resource.
  • Conventional tuning can be better understood in perspective, as being one alternative among many.

One could spend a lifetime learning about different tuning systems, their histories and features etc. But sometimes when encountering a large building it can be difficult to find an entrance! Recently I have been exploring the tuning system that divides octaves into fifty equal intervals, rather than the conventional twelve. Dividing octaves into fifty three equal intervals is another useful tuning. The sizes of the intervals in these two tunings is not very different, yet the tunings have quite different strengths. Comparing these two systems could work as a doorway into the world of alternate tunings.

A general foundation for tuning theory is the observation that two pitches sound consonant when the ratio between their frequencies is a simple rational fraction. For example, the A above middle C is conventionally tuned to 440 Hertz. The next higher A, an octave higher, is at 880 Hz, a ratio of 2:1. If one tunes the E in between to 660 Hz, it will sound very nicely consonant with either of the As, with ratios of a perfect fifth, 3:2, or a perfect fourth, 4:3. Tuning the C# to 550 Hz will complete a consonant major triad. The interval from the A of 440 Hz and the C# of 550 Hz is a major third, with a frequency ratio of 5:4. The interval from the C# of 550 Hz to the E of 660 Hz is a minor third, with frequency ratio 6:5.

The pitches involved in a piece of music form a network. Each pitch is related to several other pitches, and these related pitchs then relate to yet other pitches. Pitches are thus related by chains of simple intervals. The whole pitch space forms a kind of network. If the simple relationships are built from the consonant relationships of fifths and thirds described above, the network of pitches will look something like this:

These pitches are all inside a single octave range - the network could be replicated in as many octaves as needed. The network can also be extended arbitrarily in any and all directions.

While one can make perfectly good music with a tuning system like this, with very precisely consonant frequency ratios, it does run into difficulties. As the network is extended, each octave gets broken up more and more finely, without limit. It's hard to build instruments that can play so many notes, hard for players to hit the right notes, and hard for listeners to distinguish among so many notes. Over the centuries, musicians, composers, and instrument builders have developed simpler tuning systems that approximate these ideal intervals while avoiding the infinite division problem. And then music has evolved to take advantage of opportunities these simpler tuning systems provide for harmonic movement. A tuning system is a network of pitches with a particular shape. Music is then a kind of dance that moves around through that shape.

The fine divisions brought about by precise consonance first arise with the 81:80 pitch on the right column of the tuning network above. It is very difficult to distinguish that from the 1:1 in the center. So the first tuning simplification is to adjust the pitches in the network somehow so that 81:80 is changed back to 1:1. This changes the shape of the tuning network from a flat plane to a cylinder. If one travels in a suitable constant direction on the surface of a cylinder, one can end up back where one started.

There are many ways to adjust the pitches in the network so the 81:80 is flattened slightly to become 1:1, but in general this tuning system is known as meantone. The way pitches are named in European music is a reflection of the meantone system:

While this system does allow unbounded movement, that movement needs to flow around the cylinder, along the diagonal strip where the sharps and flats don't get too wild. Old keyboard instruments sometimes have extra black keys to accommodate a wider range of movement, but still, it can be challenging to dance freely when there is an abrupt edge that one must steer away from. So the next step of evolution is to wrap the cylinder into a torus:

If one moves a perfect fifth from G#, one arrives at Eb. The network of pitches has been tweaked somehow so that D# and Eb are the same pitch. There are various ways to do this, but the simplest way is to adjust all the fiths and thirds in the same way, so the system is totally uniform. This is our conventional tuning of today.

To review the development so far: A network of precise consonances splinters the pitch space to an impractical unbounded extent. Adjusting, or tempering, the intervals allow the network to wrap back on itself, so the number of pitches required can be limited.

Fundamentally, a tuning system is a compromise between simplicity and precision. But tuning must serve music. The shape of the tuning network enables some kinds of harmonic movement but prevents other sorts. Music and tuning evolve in response to each other, meeting each other's demands and taking advantage of each other's opportunities.

One can build a tuning system by dividing octaves into equal intervals of any number. A good tuning system will provide intervals that are close approximations of the precise consonances of 3:2 and 5:4. Dividing octaves into 50 or 53 equal parts will provide reasonable approximations:

This table gives the error, in cents, for each tuning system for each consonant interval. One can see that the conventional tuning system has somewhat large errors for several intervals, though it comes quite close for 3:2. The 53 steps per octave system is quite accurate for all the intervals. The 50 step system is not so good for 3:2, but it is at least better than conventional tuning for the thirds 5:4 and 6:5.

It might seem that, since 53 steps per octave is only slightly more than 50 steps per octave, and provides a significant improvement in precision, that the 50 step per octave system is not very useful. But beyond simplicity and precision, one must look at the shape of the tuning network:

The bright blue highlighted cells marked "0" show the way the torus is wrapped back on itself. Those closely spaced "0" cells along a line sloping slightly down to the right, those cells are wrapped in exactly the way that the meantone tuning system is wrapped. What this means is that most any music written for the meantone system will be playable in the 50 step per octave system. The 50 step system will support even triple sharps and triple flats. It would be a rare piece of music that requires more sharps and flats than that!

The 53 step per octave system has a very different shape:

The pattern of repeated cells, the way the tuning torus is wrapped, does not match the meantone system at all. Music written in the meantone system will fail to return or connect back properly if one tries to play it in the 53 step system.

I have been exploring some of the unconventional musical possibilities of these two tuning systems. For each, I picked a subset of the available pitches to work as a scale. In both systems I built the scale to form a path from lower left to upper right, which is, roughly speaking, a chromatic scale. I then used my algorithmic composition software to generate some music that would flow with the shapes of the scales:

Sunday, April 23, 2023

The Disintegration of Science

In the 1990s, the science wars were fought between advocates of science and folks who saw flaws in science. Nowadays, the science wars underway are between folks claiming scientific support for wildly differing claims. Does our global use of fossil fuels for energy have significant impact on the climate? Are covid vaccines safe and effective? Of course scientific progress is driven by debate, so perhaps these disagreements are healthy.

A healthy organism is constantly fighting off infections and other disturbances. The integrity of an organism is constantly under threat. For a while, various homeostatic processes manage to preserve that integrity, but eventually those processes are overwhelmed, and the organism loses its integrity. Sometimes this lack of integrity means the death of the organism, but it can also mean division into multiple separate organisms. Presevation of integrity and subsequent loss of integrity can happen at many scales, from single cells to cell colonies to insect colonies to human societies.

Science nowadays, for the most part, maintains a very healthy level of integrity. A key component of this integrity is the vision of scientific knowledge as a coherent whole. All the bits and pieces of our scientific knowledge fit together somehow, or eventually will. We're always discovering inconsistencies, but our processes of research and mutual critique keep these inconsistencies under sufficient control that the overall integrity of the system is not under threat. The loud arguments over e.g. climate change are a sore point, but they are certainly at a small enough scale not to threaten the entire system.

And yet... these superficial rashes could be symptoms of a larger systemic problem. Is the rough coherence of scientific knowledge something inevitable? What processes maintain this coherence? What could threaten this coherence?

The coherence of science is maintained by a kind of circulatory system. Information circulates: researchers publish papers but also exchange preliminary results, critiques of draft versions of papers, and also text books and other coordinated summaries of scientific knowledge. People circulate: researchers meet to discuss their work, but also visit each other's laboratories to collaborate on research. Students are trained in one research organization and then get hired to work in other research organizations. Equipment and materials circulate: measuring devices can be calibrated to common standards. Experimental samples are exchanged between laboratories.

What would precipitate the disintegration of science would be the breakdown of this circulatory system. Circulation is supported by the larger social context. Freedom of the press allows research results to be published. Freedom of travel allows people to collaborate. Free trade enables the exchange of equipment and materials.

These freedoms are the hallmarks of liberal society. Science and liberal society have emerged together since early modern times. A free market of ideas allows the best ideas to emerge. Basing policy on effective ideas leads to success and growth, to progress. This progress provides a platform for further exploration, leading to better ideas, more effective policies, and further growth. We have been riding this feedback loop for four hundred years. It's not just science that is coherent, but our global society.

The general pattern in biological systems is that growth is followed by decline. Perhaps this time it will be different, but that is a position that requires a lot of faith! Just as science, liberalism, and progress supported each other in a feedback loop of expansion, there are signs that the same feedback loop may be picking up momentum in the direction of decline.

Of course one can pick a measure of prosperity to support whatever argument one wishes to advance. But it really seems like the financial crash of 2008 is one we have not really recovered from. The rise of vehement anti-liberalism is largely driven by the failure of liberalism. We were promised progress but that is not what we are experiencing. The underlying cause for the lack of progress is probably our reaching various ecological limits, but that's not a message that sells. Science and liberalism have built their castles on progress. As progress falters, so will liberalism, and so will science. Liberalism maintained the circulatory system on which scientific coherence depended.

Of course change is the nature of things. How science might best maintain itself in a new dark age, that is one worthy puzzle. It is valuable to step back a bit, to try to think strategically. How things will unfold in the coming decades and centuries, it is impossible to foresee with any accuracy. What is more feasible is to consider a range of possible trajectories, and to prepare responses across some plausible range. Insurance policies, diversified portfolios, hedged bets: these are effective approaches to dealing with uncertainty. We need to bring these approaches into our investments in scientific research programs.

Thursday, March 30, 2023

Heat Pump Efficiency

Thermodynamics is a fundamental branch of physics. It gets a bit subtle: I find myself getting tripped up often enough!

The cornerstone of thermodynamics is the Carnot cycle, an ideal process for converting heat to work. It's a model for what steam engines do, for example. The Carnot cycle sets a limit on how efficient an engine can be: it is not possible to convert all the energy from heat to mechanical work.

A heat pump is simply an engine running backwards. An engine has heat flowing from a hot reservoir to a cold reservoir, converting some of that heat to mechanical work. A heat pump uses mechanical work to push heat from a cold reservoir to a hot reservoir. The amount of heat added to the hot reservoir will be the sum of the energy from the work and the heat energy removed from the cold reservoir.

To heat a home, one can use a natural gas furnace, or one can use a heat pump. The heat pump runs off electricity, much of which is generated from an engine running off natural gas. Energy is lost when the natural gas heat energy is converted to electricity, but then energy is gained when the electricity is used to heat the home. Since the heat pump is just an engine running backwards, these losses and gains are in some sense reflections of each other, and might seem to cancel out. But they don't!

The missing detail is that there are three heat reservoirs involved. The engine at the utility power generation plant has energy flowing from a furnace to the environment, converting some of that to electrical energy. The heat pump has energy flowing from the environment to the interior living space, driving that with electrical energy:

The two efficiency factors have inverse forms, but the numbers involved are different, so they don't cancel each other.

Plugging in some roughly plausible numbers, a graph can be generated for maximum effiency of the overall system as a function of the outside temperature. As the outside temperature warms to near the interior living space temperature, the round trip efficiency increases without bound. At cold temperatures, the utility's power generation engine can run more efficiently, but the reduction in effectiveness of the heat pump is more dramatic, so the overall effiency is reduced.

Friday, March 24, 2023

Aperiodic Tiling

I've been seeing reports of an aperiodic tiling. At first, I couldn't imagine how a tiling could be aperiodic. Now the pendulum has swung to the other extreme, where it seems trivial:

The tile is just a 1x2 rectangle. Mostly they are all placed vertically, but there is a line along which horizontal tiles are placed. One could interpret the pattern of absence or presence of a horizontal tile in the sequence of columns as expressing a fraction in base 2. If the fraction is irrational, the pattern will be aperiodic. Hmmm, even if there was just one horizontal tile in the middle, the pattern would be aperiodic!

There must be some trickier definition in play, of what aperiodic means. But anyway, now it doesn't seem so impossible!