293edo: Difference between revisions

293edo does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 20% [[relative interval error|relative error]], and all primes besides 13 have over 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval. As such, it is only [[consistent]] to the [[5-odd-limit]].

Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }}. It also tempers out the [[marvel comma]] in the 7-limit.  

The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.  

When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmonics have +36 and +37 error respectively, which together cancels out to 1.

One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. For example, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.