The Auditorio de Tenerife is the white, wave-shaped concert hall on the seafront of Santa Cruz de Tenerife, designed by the Valencian architect Santiago Calatrava and opened in 2003. It is the building on the postcards: a great curving crest of white concrete that rises off a low base and leans out over the building beneath it, apparently unsupported, like a breaking wave held at the moment before it falls.

It is a working concert hall, not a monument. The Tenerife Symphony Orchestra is based here, opera and zarzuela seasons run through the year, and the programme takes in jazz, flamenco, film music and the island's own folk groups. You can see it from the road in thirty seconds, photograph it in five minutes, take a guided tour of the interior, or buy a ticket and hear it working — which is the version we would recommend.

Its full name is the Auditorio de Tenerife Adán Martín, after the island president who pushed the project through. It stands on the Avenida de la Constitución, on a spit of reclaimed land between the port and the sea, next to the Palmetum park and a short walk from the TEA art centre in the old town.

What Calatrava actually built

Strip away the photographs and the structure is fairly simple to describe. There is a base of two performance spaces — a main hall seating around 1,600 and a smaller chamber hall of roughly 400 — wrapped in a curved shell. Over the top of that sits the famous element: a vast crescent of reinforced concrete, a section of a cone, springing from a single base point and curving up and forward until its tip hangs out beyond the building below.

Calatrava calls it the ala, the wing. Everybody else calls it the wave, and the sea directly behind it makes the reading irresistible. It rises to roughly 60 metres and the overhang is the part that makes engineers look twice: the whole thing cantilevers, with the weight carried back into the base by the geometry of the curve rather than by any visible prop. Underneath it sits a smaller dome of tiled concrete, so from the front the building reads as a dome that has grown a crest.

It took the best part of a decade to build, and the concrete work was the hard part: the shell is a doubly curved surface with no repeated sections, so every piece of formwork had to be made specially, and the overrun on time and budget was substantial enough to be a local political row at the time. Nobody mentions that now.

Calatrava is a Valencian architect and engineer whose work — the City of Arts and Sciences in his home city, station roofs, a string of bridges — is built on the same vocabulary of white ribs, bone shapes and improbable cantilevers. The Auditorio is generally reckoned to be one of his best, partly because the site is so good and partly because the brief was small enough to keep the idea clean.

The skin: broken tile, not paint

Get within a few metres of the building and the second surprise arrives. What looks from the road like smooth white render is in fact a mosaic of small broken ceramic pieces, laid by hand across the curving surfaces so that the joints take up the double curvature. The technique is trencadís, the shattered-tile cladding that Antoni Gaudí used all over Barcelona, and using it here is a deliberate salute from one Catalan-trained Mediterranean architect to another.

It is not decoration for its own sake. A compound curve cannot be covered with flat sheets without either cutting them into tiny pieces or accepting visible faceting; small irregular fragments of glazed tile solve the geometry and throw off the Atlantic sun at the same time. Up close the white is not one white but hundreds of slightly different ones, and the surface glitters rather than glares.

The best light is late afternoon, when the sun comes round to the front of the building and the tile picks it up; at midday the white blows out in photographs. The classic angle is from the seaward side by the Palmetum, with the wave against open ocean rather than against the port cranes.

The detail that explains the shape

Here is the thing worth carrying away. The great crest is not structurally necessary. It shelters nothing, it carries nothing but itself, and the concert halls beneath would stand perfectly well without it. It is, in engineering terms, a very large and very expensive gesture.

That was the point. The Auditorio was commissioned in the 1990s by an island whose image abroad was package hotels and a volcano, and which wanted a single photograph that would say something else. It worked with a completeness that town planners elsewhere still study. Within a few years the building was on the cover of the island's tourist material, on posters, on shop-window fridge magnets and in every aerial shot of the city; it became the shorthand image for Santa Cruz in the way the opera house became the shorthand for Sydney. A city that had been a place travellers changed buses in now had a silhouette. Anyone interested in how the islands market themselves will find the whole arc in our account of how tourism built the Canaries.

Hearing something in it

The Sinfónica de Tenerife plays a full season here, and the acoustic of the Sala Sinfónica — a conventional fan-shaped hall inside all that curved concrete — is widely rated. The programme runs from symphonic repertoire and opera to contemporary dance, jazz nights, film-score concerts and recitals in the smaller Sala de Cámara. The Festival de Música de Canarias, the archipelago-wide classical festival held in the depths of winter, uses the building as one of its two main venues.

Tickets for a symphony concert are typically far cheaper than the equivalent in London or Dublin, and there are usually seats available a day or two ahead outside festival weeks. Dress is smart casual rather than formal; islanders do dress up a little for opera. Performances start late by British habit — often around eight — which fits the local rhythm of the day and leaves you eating afterwards, at a perfectly normal Spanish hour.

The programme and ticket prices are published season by season on the Auditorio's own site, and they change every year. We are not going to quote a figure that will be stale by the time you read this — check the current season directly, and book online before you travel if there is something you actually want.

Guided tours

If you cannot get to a concert, the building runs guided visits that take you into the main hall, backstage and out onto the terraces, with an explanation of the structure and the tilework. Tours run on set days and in set languages, numbers are limited, and they are normally booked in advance rather than turning up on spec. Rehearsals and get-ins can close the hall at short notice, so the schedule is not fixed far ahead.

It takes around an hour. If you have any interest in how buildings are put together it is much the better way to see it — the inside of the wing, where the concrete narrows to almost nothing at the tip, explains more than any photograph. Without a tour you can still walk the whole way round the outside, sit on the steps and use the café; the public terraces are open and free.

Getting there and parking

The Auditorio sits at the southern end of the Santa Cruz seafront, about twenty minutes' walk from the Plaza de España along the Avenida de la Constitución, or ten from the cruise terminals. There is underground parking beneath the building and more on the surrounding streets, most of it metered.

Coming from the south of the island, the TF-1 motorway drops you into the city in under an hour from the resorts, and the express buses from Los Cristianos and Playa de las Américas run to the Intercambiador. From the north, the tram from La Laguna gets you to the centre in around half an hour, and the rest is a walk. If you are coming off a ship, the terminal is close enough to reach it on foot without a shuttle or a taxi.

What is next door

Immediately south is the Palmetum, a botanical garden of palms built on top of what was for decades the city's rubbish dump — a rehabilitation project that is remarkable in its own right and gives you the best elevated view back at the Auditorio. It holds one of the largest palm collections in Europe, including species now rare in the wild, and it is quiet even when the seafront is busy. The Palmetum makes an obvious pairing with a morning at the concert hall.

Walking the other way brings you along the marina to the Plaza de España, the market, and the lanes of the old town. A full day in the capital can reasonably take in the Auditorio, the Palmetum, TEA, the museum with the Guanche mummies and a late lunch, and still leave you time to drive out to Las Teresitas for a swim. If carnival is on, the whole area in front of the building becomes a stage — Santa Cruz's carnival is the largest in Europe after Rio and the Auditorio appears in most of the photographs of it.

Before you go

Seeing the Auditorio from outside costs nothing and takes very little time, and every visitor to Tenerife should give it ten minutes at least. Seeing it properly means either a tour or a ticket, and the ticket is the better spend: a building designed around sound is not fully explained by looking at it.

Two practicalities. The seafront here catches the wind off the Atlantic and it can be a good deal cooler after dark than the beach was at four o'clock, so take a layer. And if you are trying to fit the capital into a week based in the south, the drive is longer than the map suggests in rush hour — our island round-up sets out how the main sights space out, and driving here is straightforward once you know where the motorway traffic backs up.