42edo first came to my attention a few weeks ago. I was thinking about tuning systems that provide good approximations to the just interval 7:4. Meantone temperament is a class of tuning systems that provide good approximations to the just interval 5:4. Perhaps an approach that mimics meantone could work!
Meantone evolved from Pythagorean tuning, which uses a chain of perfect fifths, of the just interval 3:2. The fundamental problem of temperament is shown in this graph. The marks with note name labels are a chain of just tuned perfect fifths. The green horizontal line is the just interval 5:4. The note E is the just interval 81:64. The gap between this and 5:4 is the syntonic comma. It is small enough that 81:64 tends to sound like a mis-tuned 5:4. Tempered scales tweak intervals to eliminate such annoying slight differences. Meantone tuning uses a slightly flattened perfect fifth, which pulls that E note closer to the 5:4 line.
The red horizontal line is the just interval 7:4. The Pythogorean Bb and A# are both somewhat close to 7:4. One could flatten the perfect fifths to pull that A# down closer to the 7:4 line. To pull the A# down so it lands exactly on the 7:4 line requires perfect fifths about 5.07 cents flat of just. Remarkably, to pull the E down so it lands exactly on the 5:4 line requires almost the same tweak, perfect fifths 5.38 cents flat of just. 31edo has perfect fifths 5.18 cents flat of just, so it accomplishes both goals splendidly. It is a popular alternative to conventional 12edo for very good reason!
So what about 42edo? Well, instead of A#, one could pull Bb down to the 7:4 line, by making perfect fifths about 13.6 cents sharp of just. This is clearly a bit of a wild idea, because it pushes E away from the 5:4 line. But still, it's fun to explore. 42edo has perfect fifths that are about 12.3 cents sharp of just. Hmmm.
Then, a total coincidence as far as I can tell, the XA - Monthly Tunings group on facebook voted to make 42edo the tuning of this month. Well, OK! I guess I do need to give it a whirl! This is the sort of thing my algorithmic approach to composition is for, to make it easy to try out novel tuning systems.
As a first step, I looked at how well 42edo approximates various just intervals. This chart gives the errors in terms of steps of 42edo, for just ratios with numerators along the top and denominators along the left side. For example, a just perfect fifth, 3:2, is 24.568 steps of 42edo, so it will be approximated by 25 steps. The error is 0.432 steps, as shown in the chart at the intersection of the column labeled 3 and the row labeled 1 (factors of 2 can be ignored, since 42edo provides exact octaves). This 0.432 corresponds to the 12.3 cent sharpness of the perfect fifth: each step of 42edo is 1200/42 = 28.57 cents.
In this chart, the boxes are grey when the numerator and denominator are not relatively prime, i.e. the entry is redundant. The boxes are green when the error is small. The green ratios would be good to use as building blocks in music in 42edo. The two simplest green boxes are 7:4 and 5:3. These could form a good foundation for musical structures.
My next step was to build a tonnetz diagram with these two building block intervals. 7:4 is represented by 34 steps of 42edo, and 5:3 by 31 steps. I like to use comma traversal as a large scale structure for composition. Commas tempered out by 42edo appear as repeated cell labels on this tonnetz diagram. If you start, say, at a cell labeled 0 and move 6 cells to the left, a chain of 5:3 intervals, you arrive at a cell labeled 18. You can also arrive at a cell labeled 18 by moving down 3 cells, a chain of 7:4 intervals. This coincidence corresponds to the fact that 42edo tempers out the comma 250047:250000.
The readers of this blog post can be divided roughly into two categories: those who know more about tuning systems than I do, and those who know less. The distribution of tuning expertise is surely a good example of fat tails, which is to say that most people who know more than I do almost all know a lot more, and those who know less know a lot less. As I explore and learn about tuning, I am constantly amazed that I never, for all practical purposes, discover anything new. Those who know a lot less than I might be skeptical. Surely I am out into uncharted territory, at best bordering on nonsense, if not deeply plunged! But notice... this comma 250047:250000 was named the Landscape Comma almost twenty years ago!
Once I spot an interesting comma to traverse, a typical next step is to define a scale that supports the traversal. One fun feature of scales is when they have a period that is a fraction of the octave. With tunings like 31edo that divide octaves into a prime number of equal steps, this kind of scale is impossible. But 42 = 2x3x7 is not prime! I noticed that the path from a 0 cell to a 0 cell has nice stepping stones of 14 and 28. So a scale with a period of 14 looks very nice. I put 5 notes of the the 14 in the scale, as highlighted in green on the tonnetz diagram. These form a nice path for traversal of the Landscape Comma. The music linked at the top of this post uses this scale.
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