Hanson[34]

Scales generated by a minor third support traversing the kleisma, a comma tempered out by 53edo. Larry Hanson built keyboards that exploited this fact, so the scales are called Hanson scales. Hanson[19] is a common size such scale. Starting from pitch class 0, a sequence of eighteen minor thirds generates a scale with 19 notes per octave. The last pitch class in the series is 40. The whole sequence can be shifted by a minor third, by ommitting the starting pitch class, 0, and adding one more pitch class at the end, pitch class 1. Thus the shift is accomplished by sharpening 0 by a single step of 53edo.

With scale sizes of 7, 11, 15, and 34, shifting the sequence ahead a minor third is accomplished by flattening the 0 pitch class. For a scale size of 7, the 0 pitch class would be flattened to pitch class 45. For a scale size 11, 0 is flattened to 48; for size 15, to 51; for 34, to 52. Hanson[34] is largest such scale, a scale that can be shifted by sharpening or flattening a pitch class to an adjacent pitch class.

Here is an algorithmic piece in Hanson[34].

Yet More Ways to Tune a Piano

I've been exploring the possibilities for just intonation for the twelve notes per octave on a piano. One can pick a subset of the intervals to tune exactly, and then the other intervals will be have even worse errors than with conventionl equal temperament. I counted 41,844 ways to choose which intervals to tune exactly!

But much of the fun and fascination of tuning has to do with temperament. By relaxing the requirement for intervals to be tuned to exact simple frequency ratios, one can increase the number of intervals that sound acceptably. Of course equal temperament pushes this to the limit, but at the cost of thirds sounding rather rough. There are many other possible choices.

Another way to expand the range of choices is to tune unconventional intervals between the piano keys, or to let go entirely of the seven white and five black keys of the piano, to change the layout of the keyboard. Historically, with meantone tuning, keyboards often enough had split black keys, e.g. seperate black keys for G# and Ab.

Still, it is a nice exercise to stick with the twelve piano keys and their conventional intervals, to start with a choice from among the 41,844 just intonation possibilities, and then to introduce temperament to add a few more acceptably tuned intervals.

This is a tonnetz diagram showing the Pythagorean tuning of the twelve piano keys. The piano is tuned to a chain of perfect fifths. There are no just tuned major thirds available in this tuning.

When this tuning is mapped to 53edo, the tuning system that divides the octave into 53 equal steps, the chain of perfect fifths gets flattened slightly, which brings some of the major thirds into a good approximation. These new relationships change the topology of the tuning: instead of a line segment, the tuning has been wrapped into a circular shape, a loop. The schisma is one of the commas that is tempered out by 53edo. This scale supports traversal of the schisma.

Here is an example of this tuning. This piece is built from a traversal of the schisma, looping around 64 times. A scale built from a chain of perfect fifths, and tempering out the schisma: this is a quite conventional way to tune. This piece does not sound too terribly exotic, at least to my ears!

Here is another of the 41,844 just tunings of a piano. This is built from four chains of major thirds. There are not many perfect fifths in this tuning! It's a more more exotic tuning.

When this tuning is mapped to 53edo, an additional major third is added, which forms a loop that traverses the semicomma.

Here is an example of this tuning, built from 64 traversals of the semicomma.

This exotic tuning does not have a very neat structure: for example, there are three sizes of steps between the notes. By adding one more note per octave, and shifting a couple of the other notes, a more neatly structured tuning can be created:

This tuning does not fit well on a piano keyboard. It's not just that there are thirteen notes per octave. This tuning is built from chains of major thirds of length four and five. Conventional tuning does not allow such chains!

Here is an example of this unconventional and exotic tuning, again built from 64 traversals of the semicomma.

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...