Interval Networks

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

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

This is a diagram for mode 7.

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

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

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

Twelve-Tone Modes

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

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

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

Modes of / as Tuning

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

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

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

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

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

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

E is raised next. Here is an example.

Next is C, and an example.

B is raised next, producing this example.

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

Stepping Outside

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

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

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

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

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

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

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

Twelves Notes per Octave

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

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

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

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

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

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

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

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

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

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

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

Same and Different

Here are pieces in three different tuning systems:

• 12edo
• 34edo
• 36edo

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

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

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

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

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

The Radicalism of Modernity

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

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

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

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

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

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

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

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

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

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