Ermita de Betlem
Yes, it's not in the Tramuntana, but it's a great little climb and you can see the Tramuntana from it! Does that count? Anytime you go near the lovely little town of Arta or are in the Cala Millor area, this scenic 10km detour with little traffic is well worth the extra effort.
Once into the lovely town of Arta (with its big selection of cafés and some nice art galleries ) you need to follow the Ma-333 to start the climb to Ermita de Betlem. The climb is signposted, just look carefully as the road winds its way between the houses before finding the open road.
The gradient is then undulating through some farmland and the gradient only increases towards the latter part of the climb.
There are a number of hairpins and the scenery is excellent. At the top of the climb, the Bay of Alcudia presents itself along with the island of Menorca on a clear day.
As the monastery of Ermita de Betlem is not at the top of the climb, it is worth descending towards it, then turning around and make the short climb back to the top of the hill.
To be aware of
The road is narrow. There are often goats. There are often tourists in rental cars looking at the stunning views, not at you. Occasionally sand on the surface. Slow is better.