Air carrier Captain reported receiving a terrain alert during descent on visual approach. Captain climbed to safe altitude and continued approach.

Narrative

On approach into ROA; while turning base to final for a visual to runway 24; we got a Terrain Terrain aural (in full VMC) while descending out of 4;000 feet to intercept the inbound approach course. I Captain/ pilot flying had briefed in my briefing prior to descent about the company pages note saying not to descend below the lowest ATC cleared altitude until on a published portion of the approach. Due to operational complexity and (in looking back) a pretty fatiguing reserve schedule; I managed to forget to apply that note completely. Tower cleared us for 34; we said we needed 24 for performance reasons. They then cleared us to 5;000 feet and cleared us for the visual. They asked where we were going to turn base for planning for other traffic inbound. We told them we would be turning between HIBAN and PROSE; based on terrain considerations. I should've chosen to intercept outside PROSE based on the company note. I selected 4000 in the alt preselect based on discussion of terrain clearance and NOTAMS with my FO. I set 3700 in the altitude preselect for the inbound intercept and selected 300 fpm for my intercept turn on. Approaching the ridge just prior to course intercept we got Terrain Terrain. I clicked off the autopilot and climbed; FO and I acknowledged that in full VMC we had clearance on the terrain. Turning inbound for the intercept; I asked the FO (PM) to reengage the autopilot. I started down at 800 FPM; and we had to level off momentarily to miss 3 thermalling eagles on final; then I pitched over to get back on profile (acknowledging the change verbally). I touched down on-speed; just past the 1000 footers. I briefed the company notes (terrain considerations; not descending below cleared ATC mins-with care to discuss considerations at HIBAN and WUDAG) on the descent. WUDAG wasn't even on the approach so I added it in as a fix with a 1 mile ring to pay attention to crossing restriction. I did not apply the note correctly in spite of briefing it out loud. FO did not help me remember this either-we were both busy getting newest ATIS; rejecting the initial landing clearance; planning the visual. I was on day 4/4 on my 12th revision to my trip. I was tired; but I'm always tired on day 4; and do not believe I was more tired that normal (still believe that). Think it just got lost in the many changes we were handling on an approach into a new airport for me (FO had been in many times before). I did my math wrong for intercept; and its my responsibility to get that correct. Me thinking that I was cleared for the approach and able to do my own descent planning ran counter to the company note. FO did not contradict me/remember the company notes when I said where I planned to come in for the intercept.It would be helpful to add this to section Y as well; it was section X (and I briefed it); but really its on both of us that it was briefed and not applied correctly.