Cyclic Paths in Tuning

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

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

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

Here is an example of tuning 2.

Here is an example of tuning 3.

Here is an example of tuning 4.

Here is an example of tuning 5.

Here is an example of tuning 6.

Here is an example of tuning 7.

Here is an example of tuning 8.

Here is an example of tuning 9.

Here is an example of tuning 10.

Here is an example of tuning 11.

Here is an example of tuning 12.

Here is an example of tuning 13.

Here is an example of tuning 14.

Here is an example of tuning 15.

Here is an example of tuning 16.

Diatonic Scale in Just Intonation

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

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

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

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

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

Non-Diatonic

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

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

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

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

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

Here's a piece in this more conventional tuning.

41844 Ways to Tune a Piano

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

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

Just Intonation

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

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

I think these rules are enough to define the puzzle.

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

Interval Networks

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

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

This is a diagram for mode 7.

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

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

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

Twelve-Tone Modes

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

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

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

Modes of / as Tuning

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

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

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

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

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

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

E is raised next. Here is an example.

Next is C, and an example.

B is raised next, producing this example.

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

Stepping Outside

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

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

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

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

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

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

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

Twelves Notes per Octave

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

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

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

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

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

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

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

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

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

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

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