Incrementality

Here is a new piece in 53edo. It uses the scale highlighted in the above diagram, with 17 notes per octave. This scale is built from repeated steps of a perfect fifth, just like the conventional diatonic scale. 53edo is not a meantone tuning, however: it does not temper out the syntonic comma. This can be seen in the diagram above: start at C and move four steps to the right, corresponding to four perfect fifths. One arrives at E. But start at C and move up one cell, corresponding to a major third. One arrives at E-. The - mark stands for either the Pythagorean comma or the syntonic comma. Both these commas are represented as 1 step of 53edo.

The note names in the diagram are doubtless very similar to the names others have used, but I just laid them out in a way that makes sense to me without consulting any references. Well, for that matter, I stumbled onto the excellent qualities of 53edo by just sitting with my pocket calculator, without consulting any references! It turns out that 53edo has been explored for centuries. That's math for you!

The circulating quality of this scale is clear from the diagram. To move from the key of C to the key of G, one would remove F#- from the scale and add F+.

There are many ways to explore tuning. One can use intervals that are quite different from those in conventional tuning. One can use chord progressions that traverse commas that are not tempered out by conventional tuning. The most conservative approach is to use a meantone tuning, a tuning that was widely used before the hegemony of 12edo was so well established. The scale here is just one step beyond meantone. The intervals used here are all very similar to conventional intervals. The comma traversed is the schisma, which is also tempered out by conventional tuning. This scale's precision allows distinctions to be made: it reduces ambiguity. It is a subtler language.