Ed4/3: Difference between revisions

Incidentally, one way to treat 4/3 as an equivalence is the use of the 12:13:14:(16) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes (an octave-reduced stack of) four [[3/2]] to get to [[5/4]], here it takes (a fourth-reduced stack of) eight [[7/6]] to get to [[13/12]] (tempering out the comma [[5764801/5750784]]). So, doing this yields 13-, 15-, and 28-note [[mos scale]]s for ed4/3's. While the notes are rather closer together, the scheme is uncannily similar to meantone.

One of the key benefits of equal divisions of the perfect fifth (3/2) is that they create scales where the distance between the unison (1/1) and the direct mapping of the minor third (6/5) matches the distance between the direct mapping of the major third (5/4) and the perfect fifth (3/2). This is because the product of (6/5) and (5/4) equals (3/2). As a result, the errors in approximating the minor third and the major third are of the same magnitude but opposite in direction. Similarly, with equal divisions of the perfect fourth (4/3), the distance between the unison (1/1) and the direct mapping of the septimal major second (8/7) is equal to the distance between the direct mapping of the septimal minor third (7/6) and the perfect fourth (4/3), since (8/7) multiplied by (7/6) equals (4/3). Therefore, the errors in approximating the septimal major second and the septimal minor third are also equal in size but opposite in direction. In essence, equal divisions of the perfect fourth (4/3) relate to 7-limit intervals in the same way that equal divisions of the perfect fifth (3/2) relate to 5-limit intervals.