The scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, Play (help·info)) to approximate the interval (=6:5, E♭, 315.64 cents, Play (help·info)).
Carlos' ? (alpha) scale arises from...taking a value for the scale degree so that nine of them approximate a 3:2 perfect fifth, five of them approximate a 5:4 major third, and four of them approximate a 6:5 minor third. In order to make the approximation as good as possible we minimize the mean square deviation.
^Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
^ abcBenson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN0-521-85387-7. "This actually differs very slightly from Carlos' figure of 15.385 ?-scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents."