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.