Ed4/3

An equal division of the fourth (ed4/3) is an equal-step tuning in which the perfect fourth (4/3) is justly tuned and is divided in a given number of equal steps. The fourth can be treated as an equave, but it is not necessary and, more importantly, it is not well known whether most listeners can hear it as such.

The expression equal division of the fourth could be interpreted as applying to other intervals in the region of the fourth (see Category: Fourth), such as 15/11. However, these should be named more specifically and be treated on other pages to avoid any confusion.

The utility of the fourth as a base is apparent by being used at the base of so much Neo-Medieval harmony. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy, but which context(s), if any, it is very perceptually important in is as yet an open question.

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 scales 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.

ED4/3 tuning systems that accurately represent the intervals 8/7 and 7/6 include: 13ed4/3 (1.31 cent error), 15ed4/3 (1.25 cent error), and 28ed4/3 (0.06 cent error).

In a way, 13ed4/3, 15ed4/3, and 28ed4/3 are to the division of the fourth what 9ed3/2, 11ed3/2, and 20ed3/2 are to the division of the fifth.

Individual pages for ed4/3s

• 3ed4/3 (aka Cube Root of P4)
• 4ed4/3
• 5ed4/3 (aka Quintilipyth scale^{[citation needed]})
• 6ed4/3 (aka Sextilipyth scale^{[citation needed]})
• 7ed4/3
• 8ed4/3
• 9ed4/3 (aka Noleta scale)

See also

• Square root of 13 over 10 (previously listed here as an "edIV")