Sunday, October 1, 2017

Mahler 3

Hand-crafted from start to finish: Salonen and the Philharmonia triumph with Mahler 3

*****
From the score and from his letters, we know a lot about Gustav Mahler’s Symphony no. 3 in D minor, so you can do a great deal to prepare yourself for it before going into the concert hall: you can learn the concepts behind each movement (“Pan awakens” for the first, “What the flowers in the meadow tell me” for the second, and so on, you can learn about Mahler’s aspirations for the work, you can learn about Nietzsche’s poetry and all manner of details about how the score is constructed. But in the right hands – and yesterday afternoon at the Royal Festival Hall, Esa-Pekka Salonen and the Philharmonia’s hands were definitely the right hands – you don’t need any of that: just open your ears and your emotions and let the music take you where it will.
Michelle DeYoung, Esa-Pekka Salonen, Philharmonia © David Karlin | Bachtrack Ltd
Michelle DeYoung, Esa-Pekka Salonen, Philharmonia
© David Karlin | Bachtrack Ltd
Whether or not the Third encompasses the whole of creation – as its composer aspired – what you will find is music which resonates with every part of your life: the sad bits, the cheerful bits, the hopeful, the scary, the nostalgic or schmaltzy bits, young love, mature love, grief, world-weariness, the times when you are proud of yourself, the times when you just need to calm down. Salonen proved an indefatigable and sure-footed guide.
Mahler’s big symphonies, of which the Third is up there with the biggest, are so long and complex that most conductors lose their grip on you at least once in the course of the evening: they just can’t keep you fully engaged and at one point or another, your mind wanders away from the music to other things. Salonen proved that, in Ira Gershwin’s words, “it ain’t necessarily so”. The most striking thing about this performance was the unflagging attention to detail: any phrase, however minor, from any section of the orchestra, was hand-crafted to create maximum effect.
The quality started at the very beginning, with the trombones phrasing a solid entry. From there, every instrument seemed to do just that little bit more than usual with their part – the clarion call of trumpets, threat from the cellos, a scream from oboes, powerful drive from the horns, later to be followed by ethereal release from flutes, cheekiness from the clarinets, perfectly timed acceleration from harp glissandi, and then, late in the movement, theatrics from the percussion as three pairs of clash cymbals crashed together. And then, a piece of true virtuosity from principal trombonist Byron Fulcher, morphing a harsh bray into utter lyricism, after which the strings and timpani took the music down to an exquisite calm.
On the podium, Salonen’s movement is spare: he is keeping time, with great clarity and precision even when the tempi are shifting rapidly, but he doesn’t employ the “do three things at once with different body parts” techniques of some conductors. I have to assume that all the hard work of sculpting the music, one bar at a time, has been done in rehearsal, and if that’s correct, this was one of the best rehearsed concerts I’ve ever attended.
Mezzo Michelle DeYoung made a telling contribution: an impressive figure on stage (DeYoung is as tall as Salonen plus height of podium combined), she intoned Nietzsche’s words “O Mensch” from the depths of the ages; her German diction was excellent and her commitment unquestionable. The fifth movement saw real dramatic tension between the grief in her solo role and the clarity and brightness of the Tiffin Boys Choir and Philharmonia Voices.
Over 90 minutes into the work, the intense string start to the last movement was augmented by yet another set of perfectly turned woodwind phrases, with colour and character as just about every phrase had been through all of the preceding music.
Mahler’s Third should be an impossibility – a work of ludicrously overarching ambition, a behemoth of a symphony with a 40-minute first movement hopelessly out of balance with the five that follow, a work that requires gigantic orchestral forces together with a soloist and choir who are only used for a small fraction of its length, music that veers erratically between contrasting styles.
But it works. In the end, it was DeYoung’s face that said it all. After finishing her part, she had been sitting at the front of the orchestra, concentrated on the music. As the texture of the music thickened steadily and wound up into the huge closing march, the pair of timpanists swayed to and fro, pounding their instruments with both sticks, even more powerful than Strauss’ famous opening to Also Sprach Zarathustra. DeYoung’s rapt half-smile turned into a broad grin, her face shining with the excitement that was nearly as evident in most of the audience at a triumphant exposition of Mahler’s immense conception.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Hand-crafted from start to finish: Salonen and the Philharmonia triumph with Mahler 3
*****
By David Karlin, 02 October 2017
From the score and from his letters, we know a lot about Gustav Mahler’s Symphony no. 3 in D minor, so you can do a great deal to prepare yourself for it before going into the concert hall: you can learn the concepts behind each movement (“Pan awakens” for the first, “What the flowers in the meadow tell me” for the second, and so on, you can learn about Mahler’s aspirations for the work, you can learn about Nietzsche’s poetry and all manner of details about how the score is constructed. But in the right hands – and yesterday afternoon at the Royal Festival Hall, Esa-Pekka Salonen and the Philharmonia’s hands were definitely the right hands – you don’t need any of that: just open your ears and your emotions and let the music take you where it will.

Michelle DeYoung, Esa-Pekka Salonen, Philharmonia © David Karlin | Bachtrack Ltd
Michelle DeYoung, Esa-Pekka Salonen, Philharmonia
© David Karlin | Bachtrack Ltd
Whether or not the Third encompasses the whole of creation – as its composer aspired – what you will find is music which resonates with every part of your life: the sad bits, the cheerful bits, the hopeful, the scary, the nostalgic or schmaltzy bits, young love, mature love, grief, world-weariness, the times when you are proud of yourself, the times when you just need to calm down. Salonen proved an indefatigable and sure-footed guide.

Mahler’s big symphonies, of which the Third is up there with the biggest, are so long and complex that most conductors lose their grip on you at least once in the course of the evening: they just can’t keep you fully engaged and at one point or another, your mind wanders away from the music to other things. Salonen proved that, in Ira Gershwin’s words, “it ain’t necessarily so”. The most striking thing about this performance was the unflagging attention to detail: any phrase, however minor, from any section of the orchestra, was hand-crafted to create maximum effect.

The quality started at the very beginning, with the trombones phrasing a solid entry. From there, every instrument seemed to do just that little bit more than usual with their part – the clarion call of trumpets, threat from the cellos, a scream from oboes, powerful drive from the horns, later to be followed by ethereal release from flutes, cheekiness from the clarinets, perfectly timed acceleration from harp glissandi, and then, late in the movement, theatrics from the percussion as three pairs of clash cymbals crashed together. And then, a piece of true virtuosity from principal trombonist Byron Fulcher, morphing a harsh bray into utter lyricism, after which the strings and timpani took the music down to an exquisite calm.

On the podium, Salonen’s movement is spare: he is keeping time, with great clarity and precision even when the tempi are shifting rapidly, but he doesn’t employ the “do three things at once with different body parts” techniques of some conductors. I have to assume that all the hard work of sculpting the music, one bar at a time, has been done in rehearsal, and if that’s correct, this was one of the best rehearsed concerts I’ve ever attended.

Mezzo Michelle DeYoung made a telling contribution: an impressive figure on stage (DeYoung is as tall as Salonen plus height of podium combined), she intoned Nietzsche’s words “O Mensch” from the depths of the ages; her German diction was excellent and her commitment unquestionable. The fifth movement saw real dramatic tension between the grief in her solo role and the clarity and brightness of the Tiffin Boys Choir and Philharmonia Voices.

Over 90 minutes into the work, the intense string start to the last movement was augmented by yet another set of perfectly turned woodwind phrases, with colour and character as just about every phrase had been through all of the preceding music.

Mahler’s Third should be an impossibility – a work of ludicrously overarching ambition, a behemoth of a symphony with a 40-minute first movement hopelessly out of balance with the five that follow, a work that requires gigantic orchestral forces together with a soloist and choir who are only used for a small fraction of its length, music that veers erratically between contrasting styles.
But it works. In the end, it was DeYoung’s face that said it all. After finishing her part, she had been sitting at the front of the orchestra, concentrated on the music. As the texture of the music thickened steadily and wound up into the huge closing march, the pair of timpanists swayed to and fro, pounding their instruments with both sticks, even more powerful than Strauss’ famous opening to Also Sprach Zarathustra. DeYoung’s rapt half-smile turned into a broad grin, her face shining with the excitement that was nearly as evident in most of the audience at a triumphant exposition of Mahler’s immense conception.




Thursday, September 28, 2017

Aida






Instead of posters for English National Opera's new staging of Aida depicting a steep shaft of light over a lone soprano, they could easily have screamed Akhnaten II: Return of the Tomb RaiderPhelim McDermott envisages it as a sister production to his team's mesmeric award-winning production of Philip Glass' Akhnaten of two seasons ago. But would the hypnotic juggling and glacial choreography work for Giuseppe Verdi's Nile epic?
Latonia Moore (Aida) © Tristram Kenton
Latonia Moore (Aida)
© Tristram Kenton
McDermott's Aida, a co-production with Houston Grand Opera and the Grand Théâtre de Genève, is leagues away from the Zandra Rhodes bling which framed ENO's previous effort. Tom Pye's sets have a monolithic feel, dominated by a huge prism standing proud; drawn from part of a hieroglyph, its triangular form finds echoes in the way the drop curtain opens and closes. Bruno Poet's lighting shrouds Memphis in mists, lending a dark, atmospheric feel to the early acts where a strong sense of ceremony and ritual is created, albeit including static choral direction. Much of the movement and spectacle comes via Basil Twist's silk effects, from a billowing purple column to red banners splayed around Eleanor Dennis' lustrous priestess like rays of the sun in the Temple of Vulcan. Act 4's tomb is beautifully cut away, even if a lot of sunlight still manages to seep through to illuminate Radamès and Aida's dying breaths.
Cast and members of Mimbre © Tristram Kenton
Cast and members of Mimbre
© Tristram Kenton
Kevin Pollard's extrovert costume designs clearly echo those in Akhnaten, modern military chic rubbing shoulders with timeless voluminous robes; Michelle DeYoung's viperous Amneris was cocooned in white like a chrysalis at one stage. Headdresses adorned with horns and antlers are worn with evening dress for Act 2, though it's the leopard-skin fez which should clearly be adopted as this year's must-have fashion accessory. The female-led acrobatic team Mimbre entertains Amneris with a silken rhythmic gymnastic display and balance atop each other to form a guard of honour for Radamès' return. McDermott mutes the usual pomp of the Triumphal Scene, coffins draped with flags adding poignancy as the bodies of soldiers are returned to their families.
Michelle DeYoung (Amneris) © Tristram Kenton
Michelle DeYoung (Amneris)
© Tristram Kenton
Leading a strong cast is American soprano Latonia Moore, last heard in London six years ago – also as Aida – at Covent Garden. She looked every inch the African princess, but adorned with blue hair and splashes of tribal face paint. With golden top notes and plush, velvety colouring to her lower register, Moore shaped a commanding “Ritorna vincitor!”. A snatched top C barely detracted from a tender “O patria mia” in Act 3. Her Radamès was ENO stalwart Gwyn Hughes Jones in vibrant voice. “Celeste Aida” was sung ardently with a lovely diminuendo at the end, while his tone in the tomb was beautifully sweet.
Towering over Moore's Aida, DeYoung – making her ENO debut – took time to settle on opening night, sounding cloudy early on, with some weird English vowels. However, she hit form by the Judgement Scene, her mezzo opening up impressively, delivering a powerful curse, even if it didn't quite rattle the Coliseum's foundations. There was a solid UK debut for South African bass-baritone Musa Ngqungwana, who displayed firm upper notes in the Nile Scene. Robert Winslade Anderson's Ramfis was a little too soft-grained for the implacable priest, but Matthew Best, looking disturbingly like Bill Nighy, white coat draped casually over the shoulders of his white suit, sang strongly as Egypt's king.
Musa Ngqungwana (Amonasro) and Latonia Moore (Aida) © Tristram Kenton
Musa Ngqungwana (Amonasro) and Latonia Moore (Aida)
© Tristram Kenton
Keri-Lynn Wilson conducted a lively account of Verdi's score; a few slips between orchestra and ENO's lusty chorus will doubtless be ironed out. There were excellent on-stage trumpets in the Triumphal March, matched for pageantry and grandeur from the pit, while Claire Wickes' perfumed flute solo evocatively set up the Nile Scene. Verdi's music is, of course, often a good deal more animated than Glass' minimalist noodling. Where McDermott slowed the action in Akhnaten to match the music's trance-like torpor, here he sometimes needed to invigorate his singers more, to find a more dramatic pulse with less reliance on “park and bark”. Nevertheless, this credible new production is a considerable success with another fine cast to take over the run at the end of October. 

Aida, English National Opera review - heroine almost saves a dismal day

Phelim McDermott's static and shallow production hosts one glorious performance

Musa Ngqungwana as Amonasro and Latonia Moore as his daughter AidaAll images by Tristram Kenton
If the best is the enemy of the good, then the excellent is also the enemy of the "meh". And if you can stomach Verdi's Aida, go and see English National Opera’s new production for its central performance by Latonia Moore. In what’s become her signature role – the American soprano has sung Aida a hundred times previously – her searingly expressive, silvery tone and complete inhabiting of the character brought the doomed Ethiopian princess in Egyptian enslavement leaping to life. Unfortunately she was the pearl in an oyster that otherwise proved rather resistant to winkling – showing up how wide of the mark much of this production by the usually inspiring Phelim McDermott and his theatre company Improbable has fallen.
Aida and Princess Amneris both love the Egyptian military hero Radames. He loves Aida, so the jealous Amneris decides to destroy them both. Aida’s father, King Amonasro of Ethiopa, captured and hiding his identity, turns her into a honey-trap to get military secrets from Radames, who is tried for treason and ultimately buried alive; Aida hides in his tomb to die with him. The strong simplicity of plot that so endears this opera to its public is also problematic, because if interpretations lack subtlety, in part it’s the fault of Verdi and his librettist Antonio Ghislanzoni: the characters are scarcely more rounded than a pantomime’s. It is down to director and cast to give them some convincing depth.Michelle DeYoung in ENO AidaStill, there’s spectacle, and everyone loves spectacle, except they might love this one a little bit less. It has its moments. The large, static choral scenes match the monolithic shapes of Tom Pye’s sets to some degree, there are some pretty colours, with rippling silk effects created by Basil Twist; and the beautiful, slanted, varied lighting by Bruno Poet is the subtlest thing on stage. But too often the static direction becomes almost ossified. Kevin Pollard’s timeless costumes don’t always help. Elaborate hieroglyph-inspired head-pieces, exotic venerations of Isis (ancient goddess, not so-called Islamic State), and Radames’s rather 19th century military uniform sit awkwardly with otherwise updated imagery and poor old Amneris is trussed into a white contraption that might have looked happier in 1970s science fiction (Michelle DeYoung's Amneris pictured above with attendants).
No elephants. Instead, the triumphal march accompanies the return of some coffins. Arguably, of course, it is a grotesque moment: Aida’s people have been defeated, her father has been taken prisoner, and her enslaver is stealing her lover. So it’s valid – but grim nonetheless. Radames enters at the height of it to flurries of red bits of paper (plus audible pops as they’re released). Redemption could be found in the team of fabulous on-stage trumpeters.Gwyn Hughes Jones as Radames in ENO AidaNone of the visuals would matter if the characters were fully explored and their interactions were vivid, but few were the moments when that happened; and instead of addressing one another, they mostly had to stand metres apart and bark forwards. Some crucial scenes barely featured eye contact.
Michelle DeYoung’s Amneris never lost her regal stance or full tone, but so much more could be made of her inner conflict, terrible jealousy and, above all, the sonic colours of this huge role. Tenor Gwyn Hughes Jones (pictured above) carried the role of Radames well and was strong vocally, with plenty of ping in the high notes, yet even he could not quite transcend the cardboard cut-out direction. Amonasro was the South African bass-baritone Musa Ngqungwana, making his UK debut, with excellent, warm tone, but perhaps without quite the requisite power of presence. The Egyptian King, Matthew Best, again sang creditably, but with scant personality. Eleanor Dennis, an ENO Harewood Artist, was notable as a Priestess with seductively beautiful tone. Despite good individual components, the whole failed to ignite; the final scene, with Aida and Radames united in the tomb and the regretful Amneris above them, nevertheless moved onto another level, taking Verdi on his own terms at last.Scene from ENO AidaThe chorus mostly stood and delivered, which they did extremely well, with the hushed moments especially fine. There was strong playing from the orchestra under the baton of Keri-Lyn Wilson, who took often brisk tempi and provided solid support to her soloists’ voices, mainly without drowning them in orchestral splendour and cavernous Coliseum space.
There is one elephant in the theatre: it’s Verdi in English. Or Verdi in this English. Some composers translate better than others, and Verdi is one of the thorniest of all. Even so, this version is so clunky that it not only disrupts occasional rhythms but also distracts from its purpose, which presumably is to make the drama more immediate. Aida shouldn’t leave you laughing – but when rhyming couplets match “Isis” with “[in]side us”, and “shelter” with “Delta”, splashed loud and clear over the surtitles, you can’t help thinking that ENO’s opera-in-English mandate is past its sell-by date. 
*************************************

English National Opera’s new production of Aida is the work of Phelim McDermott, whose previous stagings for the company include Philip Glass’s operas Satyagraha and Akhnaten, the latter, like Aida, set in ancient Egypt. McDermott’s intention is to preserve the essential air of mystery or otherness that surrounds Verdi’s tragedy while examining its seriousness of purpose and darker implications: the nature of theocracy; the relationship between desire and obsession; and the inevitability of betrayal when war forces its casualties to choose between lovers and family. He does so, however, in ways that don’t always ideally cohere.
His Egypt is a decadent place that half hides its violence behind a veneer of civilised glamour. The columns and monoliths of Tom Pye’s set derive from hieroglyphs from the Egyptian Book of the Dead, but Kevin Pollard’s costumes range backwards and forwards between antiquity and the present day, allowing McDermott to draw continuous parallels between the ancient world and our own. The priests wear greatcoats and totemic animal headdresses. Gwyn Hughes Jones’s Radamès, looking like a Napoleonic general, is subjected to a primitive initiation ritual in the temple of Phta, before leading his troops off to a modern-day conflict. The triumph scene is a solemn procession of flag-draped coffins, observed by a smart-looking crowd in 1930s evening wear. All this leads to an occasionally bewildering first half, and it is not until after the interval that the production begins to settle. The third act has remarkable tautness and intensity, though McDermott weakens the tension of the judgment scene by having Hughes Jones tried in full view of Michelle DeYoung’s Amneris rather than offstage.
There are some musical unevennesses as well, though the evening is blessed by a remarkable central performance from the American soprano Latonia Moore in the title role. This is a remarkable voice, wonderful in its amplitude at full throttle, yet also capable of sustaining beautiful, rapt pianissimos. She’s a superb actor, too, registering every shift of Aida’s torment as the conflicting demands of her father and lover, implacable enemies in war, tear her psychologically in two. This is one of the finest performances of the role to be heard in London for many years.
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Hughes Jones makes a strong Radamès, though it took a few minutes on opening night for his voice to settle. There was, however, some finely shaded singing in his duets with Moore, and a real edge of defiance in his confrontation with DeYoung. Her once beautiful mezzo sadly sounded frayed on opening night, and her singing undercharacterised, lacking the requisite menace and ire, though dramatically she wasn’t helped by a series of unflattering costumes that restricted her movements. 
Musa Ngqungwana’s sonorous-sounding Amonasro might have been a bit more threatening. Matthew Best’s King, presidential in McDermott’s concept, exuded hauteur. Robert Winslade Anderson was the sinister Ramfis. 
Canadian conductor Keri-Lynn Wilson propelled the score forward with considerable passion. There were a couple of awkward moments of stage-pit coordination on opening night, which should sort themselves during the run.

Saturday, September 9, 2017

When Bright Lines We Set Give Bad Guidance


Math has tools for making better sense of gray areas

PHOTO: TOMASZ WALENTA
In trying to make sense of our complicated world, we naturally look for boundaries and cutoff points. They make it easier for us to navigate our lives and make practical decisions. But this need for clear lines often comes with confusing anomalies. Fortunately, math has tools for helping us to make better sense of gray areas.
Consider body-mass index, or BMI, which is one measure doctors use to assess how fat someone is. It is calculated as weight (in kilograms) divided by height (in meters) squared. If you weigh a lot for your height, your BMI will be higher. Medical guidelines say that adults with a BMI of 25 or more are “overweight.” But is there really an important difference in terms of health between someone just under 25 and just over 25?
Or consider the not-unrelated question (for me, anyway) of how much cake to eat. I’m never sure how to indulge my love for cake without becoming a glutton. If I eat just one bite, surely that’s not a problem. But where do I draw the line if each individual bite is not a problem?
The principle of mathematical induction explains why it can be so hard to draw a line. An argument by induction is like climbing a staircase: You only have to know how to climb one step, and then you can climb any staircase, one step at a time. This sort of thinking was a great help to me when I ran the New York City Marathon.
But we’d be in trouble if we applied this logic to every case. It would mean that I could eat any amount of cake, one bite at a time, or safely gain weight so long as it was just a few grams at time. Induction doesn’t allow us to pin down exactly which increment takes us too far over the line—and that can leave us stuffed with cake and unable to fit into our clothes.
A different piece of math, called the intermediate value theorem, gives us a way out. Like many math theorems, this one gives us an overall picture, not an answer to a specific problem. It holds that if a value changes continuously and there is a point where it is negative and a point where it is positive, it must be zero somewhere in between.

MORE IN EVERYDAY MATH

Mathematicians use the intermediate value theorem to show that it’s possible for an equation to have a solution in a given range, in cases where they don’t need to know exactly what the solution is. It allows them not to waste time and effort on producing a more precise answer.
In real-life cases, the theorem tells us that instead of trying to draw an exact line between “safe” and “dangerous,” we should envision a gray area where the line might be. For cake, we might tell ourselves that danger lurks somewhere between eating one slice of cake and eating three. If we want to keep the pounds off, the inductive logic of “one more bite makes no difference” will be a safe guide only if we stay out of the gray zone and have no more than one slice of cake. For BMI, a score of 25 defines a similar gray area: it isn’t good for us to be close to that mark, and going too far beyond it means real risks to our health.
This approach is often useful in making public policy. We consider 18 to be old enough to drive, because we need that bright line as a practical matter, but we also know there is a gray area for younger teens who are ready to drive, so we allow for various kinds of provisional or restricted licenses.
Math gives us tools to approach the same situation in various ways, and it can produce different interpretations that are all logically valid. But it can’t tell us which approach makes sense in practical situations. We have to figure that out ourselves.

Math has tools for making better sense of gray areas

PHOTO: TOMASZ WALENTA
In trying to make sense of our complicated world, we naturally look for boundaries and cutoff points. They make it easier for us to navigate our lives and make practical decisions. But this need for clear lines often comes with confusing anomalies. Fortunately, math has tools for helping us to make better sense of gray areas.
Consider body-mass index, or BMI, which is one measure doctors use to assess how fat someone is. It is calculated as weight (in kilograms) divided by height (in meters) squared. If you weigh a lot for your height, your BMI will be higher. Medical guidelines say that adults with a BMI of 25 or more are “overweight.” But is there really an important difference in terms of health between someone just under 25 and just over 25?
Or consider the not-unrelated question (for me, anyway) of how much cake to eat. I’m never sure how to indulge my love for cake without becoming a glutton. If I eat just one bite, surely that’s not a problem. But where do I draw the line if each individual bite is not a problem?
The principle of mathematical induction explains why it can be so hard to draw a line. An argument by induction is like climbing a staircase: You only have to know how to climb one step, and then you can climb any staircase, one step at a time. This sort of thinking was a great help to me when I ran the New York City Marathon.
But we’d be in trouble if we applied this logic to every case. It would mean that I could eat any amount of cake, one bite at a time, or safely gain weight so long as it was just a few grams at time. Induction doesn’t allow us to pin down exactly which increment takes us too far over the line—and that can leave us stuffed with cake and unable to fit into our clothes.
A different piece of math, called the intermediate value theorem, gives us a way out. Like many math theorems, this one gives us an overall picture, not an answer to a specific problem. It holds that if a value changes continuously and there is a point where it is negative and a point where it is positive, it must be zero somewhere in between.

MORE IN EVERYDAY MATH

Mathematicians use the intermediate value theorem to show that it’s possible for an equation to have a solution in a given range, in cases where they don’t need to know exactly what the solution is. It allows them not to waste time and effort on producing a more precise answer.
In real-life cases, the theorem tells us that instead of trying to draw an exact line between “safe” and “dangerous,” we should envision a gray area where the line might be. For cake, we might tell ourselves that danger lurks somewhere between eating one slice of cake and eating three. If we want to keep the pounds off, the inductive logic of “one more bite makes no difference” will be a safe guide only if we stay out of the gray zone and have no more than one slice of cake. For BMI, a score of 25 defines a similar gray area: it isn’t good for us to be close to that mark, and going too far beyond it means real risks to our health.
This approach is often useful in making public policy. We consider 18 to be old enough to drive, because we need that bright line as a practical matter, but we also know there is a gray area for younger teens who are ready to drive, so we allow for various kinds of provisional or restricted licenses.
Math gives us tools to approach the same situation in various ways, and it can produce different interpretations that are all logically valid. But it can’t tell us which approach makes sense in practical situations. We have to figure that out ourselves.