Steep section of a climbing route requiring a rope
In climbing, a pitch is a section of a climbing route between two belay points (or belay stations), and is most commonly related to the task of lead climbing (going up), but is also related to abseiling (going down). Climbing on routes that require only one pitch is known as single-pitch climbing, and climbing on routes with more than one pitch is known as multi-pitch climbing (where the number is large, it can be big wall or alpine climbing). 
Modern climbing ropes are typically in length, which sets the theoretical maximum length of a 'pitch', however, other factors mean that the average pitch on a multi-pitch route is circa in length. Advanced climbing techniques such as simul climbing can materially reduce—and even completely remove—the need for 'pitches' on a multi-pitch climbing route. The term is also used in caving.
Description.
In lead climbing — either in rock climbing, mountaineering, and ice climbing — the term 'pitch' describes the length of a section of a climbing route between belays of the 'lead climber'. A route that requires a leader to be belayed only once (ignoring any follow-up belay of the 'second climber' by the leader), is called a 'single-pitch climb'; where there are multiple belays of the leader, it is called a 'multi-pitch climb'. The term is also applied to the reverse process of abseiling, where it denotes the number of anchor points needed to complete the abseil (abseil anchor points are often, but not always, lead climbing belay points). 
Most lead climbing — and particularly sport climbing — routes are 'single-pitch'. Single-pitch routes vary from to the full length of the climbing rope, which is typically . Very short single-pitch routes (i.e. under in length) can be classed as highball bouldering routes, and do not require any protection or the need for belaying (i.e. the concept of a 'pitch' is redundant in bouldering).
Multi-pitch climbing introduces greater risk and requires greater skills and additional climbing equipment. In big wall climbing and in alpine climbing, it is possible to have multi-pitch routes with over 30 pitches, with notable examples being the 31-pitch big wall route, "The Nose" (VI, 5.9, C2) on El Capitan, or the +30-pitch alpine climbing route, the "Walker Spur" (ED1, IV, 5c/6a, A1) on the Grandes Jorasses.
Climbing guidebooks will typically have a topo that outlines the key features of each individual 'pitch(s)' on a given climbing route (e.g. grade, length, climbing challenges, availability of climbing protection and belay stations etc.,).
Length.
A single-pitch route can range from to the full length of the climbing rope (by definition, the longest belay of a 'leader' is limited by the length of the rope). In the 1960s to the 1980s, climbing ropes were typically in length, however, modern ropes are typically in length (and can extend to ), which sets the current theoretical maximum length of a 'pitch'. 
On multi-pitch routes, the typical average length of a pitch tends to be in the range, which is well inside the length of the most commonly used modern climbing ropes. This is due to the effect of other factors that also determine the ideal length of pitches (and thus the total number of pitches) on a multi-pitch route, including: 
Linking pitches.
On long but easier routes, the climbing pair may use simul climbing (or a running belay), whereby both climbers "simultaneously" ascend the route. The 'lead climber' acts like on a normal lead climb, however, the 'second' does not remain belaying in a static position, but instead also climbs, removing/unclipping the protection equipment of the 'lead climber'. Both climbers are tied to the rope at all times, and both make sure that there are several points of protection in situ between them. Simul climbing is only performed on terrain both climbers are comfortable on, as any fall is serious; often the stronger climber goes second. In such a scenario, the concept of a 'pitch' is redundant as the pair keeps moving through the belay stations that mark the end of a normal pitch.
Simul climbing techniques, and the linking of pitches, are very common in alpine climbing, where snowfields can be simul-combined as one continuous pitch (although when iced, they have to be belayed in smaller pitches). A classic example is the "1938 Heckmair Route" (ED2, V−, A0, 60° snow) on the Eiger, which has 7-10 sections that most parties will lead climb as individual pitches (e.g. "The Ice Hose", "The Ice Chimney", "The Quartz Crack"), while much of the rest of the route can be simul-climbed as a continuous pitch.
Speed climbing on multi-pitch rock climbing routes can also use simul-climbing techniques for greater efficiency. For example, many speed climbing pairs on the 31-pitch route, "The Nose", use simul-climbing on the easier established pitches, thus reducing the number of actual pitches (i.e. a 'leader' being belayed by the 'second' who is static) that they need, which significantly speeds up their rate of ascent. 
In addition, free solo climbing, which by definition avoids using any form of belaying (and therefore any need for any belay stations), can link all the pitches on a route together, with the free soloist only stopping at rest points (which may or may not be a belay station); the concept of a pitch is, therefore, less relevant in free solo climbing. 
The techniques of simul-climbing and free solo climbing—both of which avoid climbing in 'pitches', as there is no need for a belay—involve significantly greater risk to the climber(s), but by definition make the concept of a 'pitch' redundant.
In caving.
The term 'pitch' is also used by cavers to refer to a very steep or vertical section (called a drop, pit, pot, or a shaft) in a cave that needs ladders or single rope technique to descend and ascend (a drop that can be descended and ascended without equipment is a 'climb'). As caving rope lengths are variable, the length of a 'pitch' is that of the 'drop'. The deepest known pitch is in Vrtiglavica Cave in the Julian Alps, Slovenia.
References.
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