Freedom and Constraint

Interesting things happen in the space where freedom and constraint play with and against each other. In my musical explorations, with algorithmic composition and CSound synthesis as my vehicles, I have several mechanisms for defining this space of play.

Tuning and consonance are fundamental. I can constrain pitch selection to a scale, to a subset of the full set of pitches provided by the tuning system. Vertical relationships can be regulated, requiring chords to conform to some set of shapes. A variety of horizontal relationships, adjacency in a voice but also across longer scale repetition structures, can be guided more or less rigidly to some set of consonant intervals.

The large scale repetition structure of the piece is another aspect of constraint. Low dimensionality means few horizontal relationships, allowing greater freedom. High dimensionality introduces many horizontal relationships, clusters of clusters, which constrain the pitch selections.

The thermodynamic approach of my algorithm provides a temperature parameter. High temperature allows more freedom, low temperature imposes more constraint. There is generally a transition where long range order emerges, with fractal fluctuations at the transition.

In this new piece I don't target the phase transition. I gave the piece a high dimensionality, so it was tending to jump into a very orderly state. To forstall this, I initialized it randomly and then cooled it just enough to let a moderate amount of order emerge... that's another dimension of the freedom-order interplay: how the pitches are initialized, and how long the consonance optimizer is run.

This piece is in 171edo and uses the same chord shape constraint as the piece I posted a few days ago. But this piece has three voices instead of five. This gives the piece more freedom to move harmonically. My idea was that this would reduce the tendency to fall into a highly ordered state... but it didn't seem to work that way! I thought I could get away with increasing the dimension; I did keep the higher dimension, but just reduced the amount of pitch optimization jostling to preserve some of the initial freedom.

Chord Progression

Here is a new piece in 171edo. 171edo, the tuning system that divides octaves into 171 equal steps, provides very precise approximation to the just intervals 3:2 (a perfect fifth), 5:4 (a major third), and 7:4 (an unconventional interval). 3:2 is approximated by 100 steps of 171edo, 5:4 by 55 steps, and 7:4 by 138 steps. If one starts at any pitch, and moves up a perfect fifth, then up five major thirds, and then up again by a 7:4 interval, the total movement will be 100 + 5*55 + 138 = 513 steps, which is exactly three octaves, equivalent to the starting point. This piece moves around this loop 36 times, once per 63 seconds. All 36 cycles are superimposed in this score:

This piece has five voices, which form relatively complex chords. In constructing this piece, the chord shapes have been constrained:

This is a fragment of the Tonnetz diagram for 171edo. It shows the three dimensional network of relationships among the pitch classes. Horizontal neighbors are connected by perfect fifths, vertical neighbors by major thirds, and the third dimension, in and out of the page, shows pitch classes related by 7:4. The green and purple boxes here have that same shape: the purple box is simply shifted to the right. Each box encloses 8 pitch classes. These boxes represent the constraint on chord shape. At any instant in time, the pitch classes assigned to the five voices must be contained in a box of this size and shape. Picking 5 points out of a total set of 8 allows for 56 different chord shapes.

What fascinates me at the moment is the relationship between the chord constraint and the harmonic movement driven by the 63 second cycle. With the five voices often starting and stopping at different times, much of the time the pitch class of just one voice will change at a time. The cube shaped chord constraint used here will allow unbounded harmonic movement even with this kind of overlap. The green box and the purple box in the diagram include four pitch classes in their intersection: 7, 40, 123, and 156. A five note chord might add pitch class 78, which would be allowed because all five pitch classes are in the green box. But then the voice sounding the 78 could switch to pitch class 52, which would be valid because all the pitch classes are in the purple box. The other voices could all move within the purple box to set up another move to the right. The same tactic works for movment in the other directions.

Consonance and Dissonance

Here is a new algorithmic piece in 50edo. 50edo, the tuning system that divides octaves into 50 equal parts instead of the conventional 12 equal parts, is still a quite conservative tuning system. It is very close to 2/7-comma meantone, whose history goes back to the 16th Century. I was inspired to create this piece from some discussion about diminished chords, chords built by stacking minor thirds. In conventional 12edo, four minor thirds add up to an octave: each minor third is 3 steps of 12edo, and 4*3=12, the number of steps in an octave in 12edo. In 50edo, a minor third is 13 steps, so four minor thirds adds up to 52 steps, 2 steps sharper than an octave. In just intonation a minor third is a 6:5 frequency ratio, so four minor third combine to make 1296:625, sharper than an octave by 648:625. In this piece I wanted to explore what kind of rich chord structure is made available by the greater precision of 50edo.

This piece has five voices, enabling quite complex chords. I didn't want the chords to get too wild, so I constrained the structure of the chords. The diagram above has a green template imposed on a Tonnetz diagram for 50edo. Since 50edo is a meantone tuning, the conventional names for notes can be used; but note that with 50edo, e.g. C# and Db are distinct pitches.

The chords in this piece, the combinations of notes that are sounded at the same time, are constrained by the rule that there should be some positioning of the template that covers all the notes in the chord. In the position shown for the template, the notes of the C major chord C-E-G are all covered, so the C major chord is allowed. An Fb major chord Fb-Ab-Cb is not covered by the template in the position shown, but the template can be shifted down two rows to cover the Fb major chord, so a Fb major chord is allowed. The template defines the shapes of the allowed chords. If a shape is allowed, it is allowed however it might be transposed.

An example of a forbidden chord is a two semitone chord such as D#-E-F. There is no way to slide the template to cover these notes together. Some common tetrads are allowed, such as a major seventh and a dominant seventh. A diminished triad is allow, but a diminished tetrad is not. The question that inspired this piece was about diminished tetrads, so their exclusion here is a bit of a disappointment, but I wanted to keep the template reasonably bounded in hopes of creating some coherent music!

Exotic Intervals

Here's an algorithmic composition in 270edo, the tuning system that divides octaves into 270 equal steps. This precision allows exotic intervals to be introduced with great clarity. This piece uses intervals such as 7:4 and 11:8, and more complex combinations such as 11:7 or 11:10 or 9:7.

Any tuning system with discrete steps will temper out commas, i.e. will map multiple just-tuned intervals to the same pitch. 270edo maps both 96:55 and 110:63 to 217 steps. This new composition is based on a pattern where pitch class 0 moves to pitch class 217 by a sequence of intervals corresponding to one of these just ratios, and then returns by a sequence corresponding to the other.

This composition is built mostly from just 15 pitch classes, out of the full set of 270. This diagram shows the pitch classes used, and the fundamental intervals that relate them to each other. More complex ratios such as 6:5 or 7:6 etc. are not shown.

This pieces progresses clockwise around this diagram 16 times, each cycle taking about 177 seconds.

This score superimposes all 16 cycles to show the general pattern.

Compound Traversal

Here's a new algorithmic composition, in the tuning system 34edo, which divides octaves into 34 equal steps rather than the conventional 12 equal steps.

Changing the tuning system like this does two things. A tuning system makes available some collection of intervals, the building blocks of music. Sometimes this collection of intervals is unconventional. But here I am using very much the same basic intervals in 34edo that conventional musics uses in 12edo, most fundamentally the major third and the perfect fifth, along with the octave. This new composition is not something that would work in 12edo, but the difficulty is not with the basic intervals being used.

The other characteristic of a tuning system that matters musically is how the intervals combine to form more complex intervals. With a tempered tuning system, some complex combinations end up being equivalent to a unison. Such a complex combination is known as a comma; when the tuning system makes it equivalent to unison, then the tuning system is said to temper out the comma.

The conventional 12edo system tempers out the syntonic comma: combine four perfect fifths and a minor sixth and, in 12edo, you will end up three octaves above your starting point. 34edo does not temper out the syntonic comma: the same combination of intervals ends up slightly sharper than three octaves.

The most basic commas tempered out by 34edo are the diaschisma, four perfect fifths and two major thirds, and the Würschmidt comma, eight major thirds and a perfect fourth. This new algorithmic composition is built from traversals of these two commas. A comma traversal, sometimes called a comma pump, is simply a sequence of the intervals that compose a comma. When the comma is tempered out by the tuning system, the comma traversal will return to the starting pitch. Thus one can repeat the comma traversal and the pitch of the music will not drift up or down.

Conventional 12edo does not temper out the Würschmidt comma. That's why this piece would not translate to 12edo. The same sequence of intervals could be used, but in 12edo the piece would drift in pitch.

The piece here is predominantly 27 traversals of the diaschisma. But every third traversal, the sequence is shifted by a major third. With 9 such triple traversals, separated by 8 major thirds, the first and last triples are separated by a perfect fourth. The end of the piece ties back to the beginning.

The way the piece is constructed algorithmically, it starts with just this simple sequence of intervals. The piece is almost 27 minutes long, so each traversal of the diaschisma is about a minute in duration. There are six intervals in the diaschisma. So the initial structure of the piece has pitches being repeated over about ten seconds, then shifting by a perfect fifth or a major third, then another ten second repetition, etc.

The algorithm I use is based on statistical mechanics, with temperature as a key parameter. Consonance is treated as low energy, and dissonance as high energy. At low temperature, pitches will be chosen to be maximally consonant. At higher temperature, more dissonance will be allowed. In a system like this, there will typically be a phase transition, a temperature below which the system will have long range consonance, and above which the overall consonance will break down. Right around the phase transition, there are typically fractal variations. The hope is that such fractal variations in consonance and dissonance will provide musical interest.

This is a plot of the temperature and energy computed in the course of the construction of this piece. The thermodynamic simulation is initiated with the simple comma traversal structure and with low temperature. The temperature is gradually raised; this involves reassigning pitch values to moderately less consonant alternatives. A phase transition is characterized by a rapid rise in energy with a small change in temperature. The composition process ends when this rapid rise is detected.

Tempering Commas

My grand project is to cultivate a philosophy of science based on Buddhist principles. The foundation is seeing concepts in general as of limited value, as conventional and pragmatic. A common idea about science is that it is a convergent process, getting closer and closer to some final comprehensive theory. I propose instead that scientific theories are adaptive to particular circumstances, and will shift as circumstances shift. In part this is a reflexive process, where theories drive the changes in circumstances to which they must then respond. Our present ecological crisis is a primary instance of this reflexive instability. In general, clinging to concepts, overvaluing them, is the root of suffering. We seem to be doubling down on the modern technological project of controlling the world. I am hoping that we can somehow avoid learning the hard way what a profound global cultural bankruptcy might look like.

Musical tuning is a very small technical discipline, simple enough yet rich enough to provide a good sandbox for exploring what non-convergent theorizing can look like. The conventional tuning system, 12 equal divisions per octave, or 12edo, is so well established that it can easily seem that the process of convergence is complete and the ultimate theory has been achieved. These are the universally and absolutely true notes or intervals. Any alternative is necessarily a lesser approximation to the truth.

The limits and flaws of the conventional tuning system, 12edo, are sufficiently evident that many people have explored alternate systems. There are two ways to think about this kind of exploration. One can see it as convergent, that somehow a better tuning system than 12edo will be found, and then perhaps an even better system. The other perspective is that the exploration is more about real alternatives. Tuning systems are not particularly better or worse, but simply different. One or another tuning system might be better or worse for some particular purpose, for a particular piece of music or for a particular instrument. But there may be no absolute ranking independent of the details of some particular intended use.

Just intonation is a tuning system that can easily tempt a person to believe in, as an ultimate theory. But looking at the practical use of tuning systems, there is a lot of music that just cannot work with just intonation. With just intonation, intervals will conform to rational frequency ratios involving small primes, e.g. a perfect fifth of 3:2 and a major third of 5:4. But these intervals can easily be combined to form intervals such as 81:80 which are very close to 1:1. These small intervals are known as commas. Tempered tuning systems will provide some more limited collection of pitches, treating as equivalant some pairs of intervals that would differ in just intonation by such commas.

This is a chart of just intonation intervals involving the prime factors of 3 and 5. All intervals have been folded into a single octave range. The numbers are in terms of cents: 1200 times the base 2 logarithm of the ratio. Moving from one cell to the cell on its right is multiplying by 3, which, when folded into a single octave, becomes 3/2 or 3/4. The base 2 logarithm of 3/2 is 0.585. Multiplying this by 1200 gives 701.955 cents. Moving up or down a cell in the chart is moving up or down by major thirds, by 5/4 or 8/5.

I have highlighted in the chart the small intervals, the commas, that are less than 50 cents. This chart could of course be expanded indefinitely. It should be clear that there are very many such small intervals. This is an echo of the one of the earliest crises in the project to build an ultimate theory of reality. The Pythagoreans discovered irrational numbers, which ruined their project to understand the world in terms of rational numbers.

Here I have zoomed in to the central part of the chart, and labeled most of the commas by their conventional names. (I used tonalsoft as a source for these names.) I would like to focus here on three commas, all of which are tempered out by 12edo.

• the syntonic comma is a combination of four perfect fifths and a major sixth. This is the ratio 81:80.
• the diesis is a combination of three major thirds, the ratio 125:128.
• the diaschisma is four perfect fifths and two major thirds, the ratio 2025:2048.

The simplest class of tuning systems is those that divide octaves into some number of equal parts. This table shows many of the most useful such systems. It gives the accuracy of the tuning, as the difference from just intonation. It also highlights which of these three commas are tempered out by the tuning system.

Conventional music in the tradition of e.g. Mozart is based on tempering the syntonic comma. This means that dividing octaves into 19 or 31 steps instead of the conventional 12 will still allow one to play most such music just as it is written. Dividing octaves into 53 equal steps provides intervals very close to those of just intonation, but since none of these simple commas are tempered out... one is forced into rather unconventional music.

This leaves 34edo. It has an attractive degree of tuning accuracy, but also tempers out the diaschisma. It is not as accurate as 53edo, but a bit more conventional.

A few years ago I presented a 12 note per octave subset of 34edo. Here is a new algorithmic composition using this 12 note subset.

With 12 notes per octave, this composition can be mapped straightforwardly into a 12edo version. This provides a good demonstration of what tuning accuracy is about, what difference it makes.

Traversing 65625:65536

Here is a piece of music generated by my software: 171edo inner. My code uses a lot of randomization, but within a definite structure. I'd like to show some of that structure here.

This is a score for the piece, or a graph. The x-axis is time, in seconds. The y-axis is the pitch. This piece uses a tuning system that divides octaves into 171 equal parts. The y-axis labels are in terms of this tuning. In the synthesis process I assigned pitch 0 to 110 Hertz. The total range is from about -200 to +500, or 700 steps from bottom to top. Each octave is 171 steps, so the full range of the piece is about four octaves.

It's easy to see from this graph that the pitches are not totally random. The graph has a texture, maybe a bit like knurling. This texture can be made more clear by folding all the pitches into a single octave.

Now the y-axis runs just from 0 through 170. Pitches in the piece that have values -171, 0, 171, and 342 will all be mapped to pitch class 0 on this graph. The knurled texture is very clear here. Part of the structure I imposed on this piece is a scale: out of the full 171 pitch classes per octave, I only allow 22 to occur. These 22 pitch classes are somewhat evenly spaced, with narrower and wider spaces interleaved in a pattern somewhat reminiscent of the whole and half steps of a conventional diatonic scale.

It's also clear here that there is some pattern that is repeating across time. I set the program up to start with a fixed sequence that was repeated 64 times, and then let the program randomly adjust that pattern to create some interesting variation. The time axis can be folded in the same way that the pitch axis was folded, so all 64 cycles are super-imposed:

This is the basic elementary structure that gets repeated. It looks a little bit like a staircase. The 22 note per octave scale was designed to accommodate this staircase structure. To understand how this structure works, we need to dive into some tuning theory. A Tonnetz diagram is an effective tool to guide this exploration.

The numbers in this matrix are pitch classes in the tuning with 171 steps per octave. In this tuning, the best approximation to the just tuned perfect fifth, a frequency ratio of 3:2, is 100 steps. So, for every cell in the matrix, the next cell to the right is 100 steps higher - possibly folded back into a single octave by subtracting 171.

The best approximation to the just tuned major third, a frequency ratio of 5:4, is 55 steps. So, for each cell, the next cell above it has a value 55 steps higher, again perhaps folded back into the single octave.

So this diagram is a map of how one moves across the available pitch classes using steps of perfect fifths and major thirds. Since there are only 171 possible pitch classes in this tuning, wandering around by these intervals will not continue to result in new pitch classes forever: at some point there will have to be some repetition. It's easy to see in this diagram that moving by 8 perfect fifths and 1 major third, one ends up back at the same pitch class where one started. This would correspond to a just tuned interval of 32805:32768, a very small interval. These small intervals are called commas; this particular comma is known as a schisma. For 171edo, moving by this comma returns one back to the starting pitch class. Because of this, 171edo is said to temper out the schisma.

Conventional music, as, for example, Mozart composed, is based on these fundamental intervals, the perfect fifth and the major third. Another fundamental interval that the science of acoustics presents but that is not so evident in most music, is 7:4. It doesn't match very well any interval on a piano or in a conventional scale, so it doesn't even have a widely used name. There's a significant community among musicians that have been exploring how to use this and related intervals in music. I would not be surprised to learn that their use has a long history in various traditions different from that represented by Mozart. Anyway, my algorithmic composition methods are my way of exploring a broader palette of musical intervals.

One challenge with this broader palette: the Tonnetz diagram needs another dimension! This is not so easy to show on a flat screen or piece of paper! A bit of imagination will be required.

The just tuned interval of 7:4 is best approximated in the 171edo tuning by 138 steps. So one can imagine layers of cells above and below the Tonnetz matrix, where the next cell above a cell has a pitch class 138 steps higher, again potentially folded back into the single octave.

At this point we have the materials in hand to explain the 22 note scale I have engineered. I should say up front: I expect that other folks have used this same scale, and probably across many decades, if not centuries. All this is simple mathematics. People have been exploring music and mathematics for a very long time. But I like to work these things out for myself, and that's what I am doing here!

The 22 colored cells in the Tonnetz matrix represent the 22 pitch classes in the scale. Again these cells repeat in the matrix just because there are only 171 pitch classes so they just have to repeat. If we zoomed out, there would be further repetitions in other directions. But we need to imagine also the vertical dimension. Above the 0 cell is the 138 cell, and below the 138 cell is the 0 cell. One can thus see from the diagram: if one starts at the 138 cell, moves 5 major thirds, through the 22 cell, the 77 cell, the 132 cell, the 16 cell, to the 71 cell, and then from there one can move a perfect fifth to the 0 cell. From the 0 cell one can move by the 7:4 interval to get back to the 138 cell. Altogether this combination of intervals corresponds to the just interval 65625:65536, known as the Horwell comma. 171edo tempers out the Horwell comma.

So this describes the pattern that I set up in my software and repeated 64 times: 5 moves of a major third, one move of a perfect fifth, and one move of 7:4. This set of moves brings us back to where we started, ready to repeat the cycle. The software can work with this pattern without being constrained by a scale, but my hope is that the scale gives a little extra regularity to make it easier to listen to. The scale does help the program too, by preventing the randomization from going too far off track!

Circulation

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.

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#.

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:

• 50edo
• 31edo

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:

• 50 step
• 53 step