The mathematics of music means piano strings can never be in perfect harmony

Unlike with guitars and violins, pianos’ strings can never be perfectly tuned to one another. The solution? As this short animation from MinutePhysics explains, the instrument’s 88 strings across more than seven octaves means tuning a piano using harmonic intervals will inevitably lead to notes being fractionally off-pitch, with the issue compounding across octaves. So instead of using harmonics, piano-tuners generally keep octaves perfect, while leaving every other interval out of tune by just a tiny fraction. This workaround forsakes the appealing mathematical patterns of harmonics, but makes it possible to keep the kind of uniformity that is so valued in an era of mass production and reproduction of music.

Want to think for yourself? Start with an agonising state of doubt, says Kierkegaard

Influenced by Socrates’ sense of irony, the Danish philosopher Søren Kierkegaard (1813-55) came to believe that a state of doubt – disorienting and horrifying as it could sometimes be – was the cornerstone of a sound philosophical practice. This scepticism of objective truth and ardent belief in thinking for oneself is omnipresent in his pseudonymous works, in which his assumed names sometimes even spar with one another. While amusing, the peculiar literary device also undercuts any sense that the works were written by a voice of authority. In this video from the London Review of Books, the British philosopher and historian Jonathan Rée traces the theme of doubt in Kierkegaard’s life and work using his unfinished, posthumously published novel Johannes Climacus: Or a Life of Doubt as a starting point.

A handful of executives control the ‘attention economy’. Time for attentive resistance

From fitness tracking devices to search engines, it’s easy to think of personalised technologies as convenient shortcuts and useful tools for working towards goals. But, argues James Williams, a doctoral candidate at the Oxford Internet Institute and a former Google employee, the primary aim of personalised tech is to keep users coming back by any means necessary – and often in a way that encourages empty distraction. In this brief animation featuring audio from a 2017 lecture at the RSA (Royal Society for the encouragement of Arts, Manufactures and Commerce) in London, Williams makes the case that the consolidation of the ‘attention economy’ to just a handful of companies is an unprecedented and deeply fraught human experiment – and one that demands active, attentive resistance.

A guided tour of New York’s public art in 1973, in all its charms and contradictions

‘The streets and parks of Manhattan are really a great place to explore what happens to art, and what happens to us, when art steps out from behind the velvet rope … and stands each day in the public eye.’

As evidenced by the increasingly contentious debate over public art, those pieces that a society chooses to exhibit and celebrate in its shared spaces say a lot about its tastes, values and power structures. Released in 1973, the film Art in Public Places is at once a time capsule of its own and a rich window into centuries of New York’s history. Produced by the Metropolitan Museum of Art, the film provides a meditative tour of Manhattan’s eclectic displays of public art, expertly guided by the US painter and writer Russell Connor. Spanning the works of artists famous and forgotten, and pieces both improvised and years-in-the-making, the US director Fred Barzyk captures New York’s public displays in all their eclecticism, charms and contradictions.

The perpetual motion of life and sand on the ‘walking islands’ of the North Sea

The Frisian Islands (or Wadden Islands) off the coast of the Netherlands, Germany and Denmark form the planet’s largest unbroken system of intertidal sand and mudflats. The 300-mile archipelago in the North Sea is notable not only for its scale, but for its continuous eastward drift due to sand erosion. On a geological scale, the islands move at a sprinter’s pace, having forced many a human settlement into the sea over the centuries. In this short documentary, the Dutch filmmaker Paul Klaver chronicles the circle of life within the islands’ rich ecosystem, capturing their flora, fauna and perpetual drift via a combination of observational and time-lapse filmmaking. For more of Klaver’s dazzling nature filmmaking, watch Alaska: The Nutrient Cycle and Winter.