Ed5/2

The equal division of 5/2 (ed5/2) is a tuning obtained by dividing the classic major tenth (5/2) in a certain number of equal steps.

Properties

Division of 5/2 into equal parts does not necessarily imply directly using this interval as an equivalence. The question of equivalence has not even been posed yet. The utility of 5/2, (or another tenth) as a base though, is apparent by, beside being the base of so much common practice tonal harmony, 5/2 being the best option for “no-threes” harmony excluding the octave. Many, though not all, of these scales have a perceptually important false octave, with various degrees of accuracy.

Incidentally, one way to treat 5/2 as an equivalence is the use of the 2:3:4:(5) chord as the fundamental complete sonority in a very similar way to the 3:4:5:(6) chord in meantone. Whereas in meantone it takes three 4/3 to get to 6/5, here it takes three 3/2 to get to 6/5 (tempering out the comma 3125/3048). So, doing this yields 5-, 7-, and 12-note mos, just like meantone. While the notes are rather closer together, the scheme is exactly identical to meantone. "Macrodiatonic" might be a perfect term for it because it uses a scheme that turns out exactly identical to meantone, though severely stretched. These are also the mos formerly known as Middletown because a tenth base stretches the meantone scheme to the point where it tempers out 64/63.

Another option is to treat ed5/2's as "no-threes" systems (like how edts are usually treated as no-twos), using the 4:5:7:(10) chord as the fundamental complete sonority instead of 4:5:6:(8). Whereas in meantone it takes four 4/3 to get to 6/5, here it takes one 10/7 to get to 7/5 (tempering out the comma 50/49 in the no-threes 7-limit), producing a nonoctave version of jubilic temperament. Doing this yields 5-, 8-, 13-, and 21-note mos.

Individual pages for ed5/2's

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