Bohlen-Pierce

Here's a new piece in 41edo, using a scale that is a very good approximation to the equal tempered Bohlen-Pierce scale. One key feature of this scale is that it repeats with a period of frequency ratio 3, rather than the usual 2. A frequency ratio of 2 is an octave: most scales repeat with a period of an octave. Not this one!

The other feature of the scale is that factors of 2 are omitted from all the scale ratios, at least in the just-tuned version. Here I am using an equal tempered version. My software uses a consonance score to drive the probability distributions for choosing pitches. I tweaked the score formula here to omit factors of 2.

Naturalness

Here is a new piece in 118edo, using a 17 note per octave scale essentially the same as the scale I discussed here yesterday. 118edo tempers out the schisma, so this scale structure works the same way it does in 53edo. 118edo gives intervals even closer to just intonation, so the harmonies should sound purer.

A 17 note scale built from a sequence of perfect fifths: this is so close to conventional tuning, it is natural to fit it onto a conventional keyboard. We just need to split the black keys, so instead of five black keys there are five pairs of black keys. The diagram above shows how this scale could be put on such a keyboard.

The first thing to notice is that the white keys are all assigned conventional notes. The notes are not tuned exactly the same as conventional 12edo, but they are not too far off. For example, E on a conventional keyboard would be at 400 cents, while here it is at 407 cents.

Since the scale is built from perfect fifths, almost every note has a good perfect fifth available, as indicated in the color coded row. It is just Bb+ that is left wanting. A just tuned perfect fifth is 701.95 cents, for which 118edo gives an excellent approximation at 69 steps, or 701.69 cents. Conventional 12edo has perfect fifths of 700 cents, of course; still very good, but not as precise as 118edo. But there is no note in this scale which is 69 steps from Bb+. From Bb to F is 69 steps, so one could just again use F from Bb+, giving 67 steps. Or one could move up to the next note up, F#-, giving 76 steps. No other note in this scale will move up a fifth to F#-, so maybe Bb+ to F#- is the natural option, but it is not going to be pretty!

Since 118edo tempers out the schisma, a chain of perfect fifths will give rise to intervals of major thirds. But only some of the notes will have good major thirds. The lower color coded row indicates which major thirds are available. A just tuned major third is a ratio 5:4. The peach and mauve colors show which notes have a good approximation for 5:4, which is 38 steps of 118edo. For example, from D to F#- is 38 steps.

A Pythagorean major third is a ratio 81:64, which is 40 steps of 118edo. The difference between a just tuned major third and a Pythagorean major third is the syntonic comma, the ratio 81:80. In a meantone tuning like conventional 12edo, the same tempered interval is used for both the just tuned and the Pythagorean major third. The strength and weakness of a more precise tuning like 118edo is that these major thirds are distinguished. That precision closes off some musical possibilities, as it opens up others. This is really the heart of what I am trying to explore, the relationship between tuning and music.

A Pythagorean major third is built from a chain of four perfect fifths. Since this scale is built from a chain of perfect fifths, it's implicit that many notes will have Pythagorean major thirds available. These notes are indicated by the beige and peach colors in the lowest row of the keyboard diagram. So, for example, C is marked beige, so it has only a Pythagorean major third available. 118edo has a good just tuned major third available, but here we have picked out for our scale only 17 of the 118 pitch classes of the tuning system.

The peach colors indicate that a note has both just tuned and Pythagorean major thirds available. E.g. with D one can use F#- to form a just tuned major third, or F# to form a Pythagorean major third.

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.