Why is the DC-9-80 ("MD-80") prohibited from taking off with a flap setting between 13 and 15 degrees?
When the FLAP/SLAT handle is in the 0° to 13° range, the slats are in the mid-sealed position. The slats will be in the extended position whenever the FLAP/SLAT is in the 15° to 40° range. The range between 13° and 15° is the DO NOT USE range.[1]

Given the above and the analogue nature of the input (dial-a-flap), between 13 and 15° the slats are very likely not to settle on a position.

(...) with the flaps flap/slat control lever in the cockpit below flaps 14°, the slats extend to their intermediate position (MID); and with flap/slat control lever positions above flaps 14°, the slats are fully extended (EXT).[2]

In other words:
$$0° \le \text{slat}_1 \lt 14°$$
$$14° \lt \text{slat}_2$$
What happens at 14°? Simply &quot;Do Not Use&quot;. The slat is both alive and dead.
Chronology of Douglas / McDonnell Douglas flaps/slats for context:




EIS Year
Plane
T.O. Flaps
Slats




1959
DC-8
fixed detents
slots (with an O);[a] doors opened when flaps extended


1965
DC-9-10
fixed detents
no slats


1967
Rest of the Original DC-9 Family
fixed detents
manual slats; separate lever


1971
DC-10
dial-a-flap (patent)
manual slats; separate lever


1980
MD-80
dial-a-flap
automatic flap-based three-position[b] slats (leading to the switch point at 14°)


1990
MD-11
dial-a-flap
automatic slats having only one extension position; slats stow button


1995
MD-90
dial-a-flap
same as MD-80


1999
717 (MD-95)
dial-a-flap
automatic slats having only one extension position




a: the DC-8 slots failed to produce the desired pitch-down on the DC-9 during development; tests revealed it was due to the lack of engine pylons; further experiments led to the invention of mini-pylons – the vortilons[4]
b: counting the retracted position
How can the narrow-body MD-80 be compared to the much larger tri-jet MD-11? The underlying high-lift system is practically unchanged since the 60s:

The DC-10 models utilize the DC-9 high-lift technology, and the MD-11, being a derivative of the DC-10, is very similar to it. (...) All its airplanes today, the MD-80s, MD-90s, and MD-11s are derivatives of the DC-9 and DC-10 airplanes, and they use their almost-30-year-old high-lift system technology with few changes.[3]


1: MD-80 FCOM
2: Report A-032/2008 – Accident involving a McDonnell Douglas DC-9-82 (MD-82) aircraft
3: Peter, K. C. &quot;High-Lift systems on commercial subsonic airliners.&quot; NASA Contractor Rept 4746 (1996).
4: Shevell, Richard S. &quot;Aerodynamic anomalies-Can CFD prevent or correct them?.&quot; Journal of Aircraft 23.8 (1986): 641-649.