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Navigatio Britannica: Planar Trigonometry
Chapter 4: Of Trigonometry, Sections I - V
The first half of this chapter is Planar Trigonometry, which I learned in high school and have used on-and-off ever since; the second half of this chapter is Spherical Trigonometry, which I know nothing about. Consequently, I’m dividing this chapter into two parts -- before we let John Barrow attempt to teach me spherical trig (wish me luck!), I want to do a fast recap of what he has to say about planar trig...
Section I: Definitions
Everything is defined geometrically, on the unit circle, via a diagram that I have yet to find in this scan. (Also, even if I do find the page, I don’t have much hope that it will have been scanned correctly, since Google’s scanning machine can turn pages but not unfold them.) Happily, Barrow is pretty good about describing his figures in enough detail that I can reconstruct them as I go, which is the only reason I was able to understand anything in this chapter.
Not many surprises here, although I did learn that cosine, cotangent, and cosecant are the sine, tangent, and secant of the complementary angle, and likewise that the tangent of an angle is called the tangent because its physical instantiation lives on a line tangent to the circle. Also, Barrow defines the Verse-Sine, which was new to me: geometrically, it is the part of the radius that isn’t the cosine. (Algebraically, it is 1 - cosine.) Wikipedia says the versine was important to navigation, so I assume it will come up later.
Section II: Geometrical Constructions of the Tables of Sines, Tangents, Secants, &c.
In which we are instructed to build ourselves a unit circle, mark it off in 1-degree intervals, and construct ourselves a... well, it’s gonna look a bit like a ruler, but it’s going to measure 1 to 90 degrees, on several parallel scales: chords, sines, versed-sines, tangents, etc. To make this thing, you use your compass and measure the length of a chord for a 10-degree angle, then mark it on your chords-scale, and label it “10 deg.” Repeat for the other 89 degrees, and ta-dah, you have a chords-scale! Then do it again for sines, versed-sines, tangents, and so on. When we get to actually solving trig problems, how to use this scale is one of the three standard methods that Barrow is going to teach us.
Section III: Arithmetical Constructions of the Tables of Sines, &c.
First off, Barrow reassures us this is going to be easy-peasy, no need to panic -- which is our first cue that panicking will be required before we’re done.
But true to his word, Barrow starts out easy, using similar triangles to prove all the basic trigonometric identities: tan = sin/cos, sec = 1/cos, etc. All well and good, except it’s all done in proportions and nothing is called out by name, only by referring to various line-segments in his nowhere-to-be-found unit circle diagram. All of which makes it difficult to absorb at a glance, but once you finish decoding everything this is basically just SOHCAHTOA.
Then he proves two variants on the standard trigonometric sum/difference formulas (although he expresses them as proportions and via verbal descriptions, talking about the means of equi-different angles and the differences between them):
cos x = (1/2) (sin y + sin (y + 2x)) / sin (y + x)
sin x = (1/2) (sin (y + 2x) - sin y) / cos (y + x)
You can verify those via the standard trigonometric sum/difference identities if you want. (I did.) But they’re also pretty straightforward geometrically, if you take the time to very carefully reconstruct what his diagram must have been: in the end, it’s all just similar triangles. He then proves several corollaries -- which in hindsight are simple enough (just straightforward algebraic manipulations, multiplying everything by two, or both sides by the denominator), but sadly, I lost MANY HOURS to a rash of typos in them.
Then.
Oh, then.
All hell breaks loose as he endeavours to prove that a semi-circle has an arc-length of pi. I admit to not following this bit: I haven’t seen Newton’s notation for calculus since I was seventeen, when that one weirdo physics professor used it in lectures, and I didn’t really feel like re-teaching it to myself for this. Nor did I really want to get into re-teaching myself binomial expansions. Also, the type-face on all the fractions in the expansions was super-squinchy to read, and you know what, fuck it, I think it’s well-established that a semi-circle has an arc-length of pi, let’s move on.
The point of establishing that a semi-circle has an arc-length of pi is so that we can calculate the arc-length of one minute (simply divide pi by 10,800 minutes, easy-peasy), which we will then use as an approximation of the sine of one minute. ... Which, okay, I suppose if your angle is small enough and your applications are practical enough you can get away with that? But it makes the mathematician in me cry, I’m just saying. (Even as I admit that you really can get away with it for most purposes: according to my handy-dandy TI-84 Plus, pi/10800 differs from sin(1′) in the ninth significant digit. But Dr. Roberts and Dr. Chrestenson would never have let me get away with that shit, never mind that I also have an engineering degree and thus should be okay with this kind of ruthless practicality. In my soul there is a mathematician and an engineer battling to the death over questions like these, you simply don’t know how much shit like this wounds me.)
Anyway, once I finally got over my fit of vapours...
Now that we have an approximation for the sine of one minute, we can calculate the cosine of one minute via the pythagorean trigonometric identity, and then...
And now I want to cry again, because now we get to build our table of sines (and along with it, our table of cosines), minute by freaking goddamn minute, by using the above equations like so:
2 cos (1′) sin (1′) - sin (0′) = sin (2′)
2 cos (2′) sin (2′) - sin (1′) = sin (3′)
2 cos (3′) sin (3′) - sin (2′) = sin (4′)
...
Continue until I cry blood and the seas boil dry.
(At one point Barrow admits that it’s possible to build this table in 5-minute increments and interpolate the intervening minutes when you need them. While this reduces the task to 1/5th of the original, I still want to hug and rock myself and cry.)
Happily, I don’t need to cry, because Barrow includes these tables in the book? But someone cried blood to make those tables, and John Barrow wants us all to know it.
Section IV: Actual Trigonometry Problems, At Long Last!
A ton of sample problems, all worked three ways:
Geometrically: Basically, use a compass and straight edge and your scale-thingie of chords/sines/secants that you made earlier, and draw a triangle of the correct proportions. Then just read/measure your answer right off the actual triangle in question, ta-dah! No abstract math required, just pretty pictures!
Arithmetically: What you learned in high-school, using the tables that someone cried blood over but without calculators (although you can use logs if you want to skip ahead to chapter five for them!) God, it looks miserable and grindingly awful, and I admit I don’t have the strength of character to follow any of these calculations through to the end.
Magic, I mean, Gunter’s Scale: The instructions here are amazingly low-key -- use your chart-dividers (what tumblr calls a pointy-leg-man what it likes to make walk on tippy-toes across charts) to measure off an interval on one scale, and then drop that same interval across the appropriate second scale, and voila! You have found your answer!
Of course I wanted to know what this magical tool is!
Apparently it was a slide rule without the slidey parts -- you used your dividers to accomplish what the slidey bit does on a slide rule -- but with some extra scales especially chosen for the convenience of navigators.
Apparently these things were so common among navigators that they were simply called Gunters, and I WANT ONE SO BAD. Here’s a nifty article about them, complete with pictures, and did I say? I WANT ONE SO BAD. I collect old-school mathematical tools and I WANT ONE SO BAD.
Ahem.
Anyway, Section V is more trig homework, except now we’re no longer dealing with right triangles. I admit it, I skimmed this like fuck.
And ta-dah! That’s Planar Trigonomometry, according to John Barrow in Navigatio Britannica, or, A Complete Guide to Navigation, pub. 1750!
Up next: Section VI - Spherical Trigonometry, what makes William Bush cry. Do I have the fortitude to teach myself spherical trig? PLACE YOUR BETS NOW.
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Des concerts à Paris et autour
Mars
20. Hackedepicciotto – Walrus (gratuit)
20. Les Tambours du Bronx + Acyl – La Machine
21. Hackedepicciotto + Phoenician Drive + City Dragon [Maninkari : ||ANNULÉ||] – Supersonic (gratuit)
21. Egopusher – Centre culturel suisse
21. OD Bongo + Qonicho Ah + Nuage magique – Pointe Lafayette
22>25. Jean-Philippe Renoult & Dinah Bird : musique pour "Absynth" de HeHe (Biennale Nemo) – WIP (gratuit)
22. Goran Bregovic & l'orchestre des mariages et des enterrements – Salle Pleyel
22. Petra pied de biche – Pointe Lafayette
23. Oren Ambarchi (dj) – Chair de poule (gratuit) ||ANNULÉ||
23. PurForm + TRDLX (Biennale Nemo) – Grande Halle de La Villette
23. Pierre Henry (diff.) + Anabelle Playe + John Chantler + Bill Orcutt + Anthony Child (Présences électronique) – Maison de la radio|Studio 104
23. Hey Colossus + Grey Hairs – Espace B
23. Delacave + Keruda Panthers + La Main – Supersonic (gratuit)
23. Vincent Epplay + Art & Technique + Lionel Marchetti & Nicolas Losson + Pascal Battus & Léo Dupleix – Ourcq blanc
23. Zombie Zombie – Le Plan (Ris-Orangis)
23. Paula Temple + Tommy Four Seven + Umfang – Nuits fauves
24. Zombie Zombie (dj) – Smallville records (gratuit)
24. Alva Noto & Anne-James Chaton : "Alphabet" (Biennale Nemo) – Grande Halle de La Villette
24. Else Marie Pade + :such: + Bellows + Phonophani + The Caretaker (Présences électronique) – Maison de la radio|Studio 104
24. Youth Code + Carpenter Brut – Olympia
24. Jessica93 + JC Satàn – La Clef (Saint-Germain-en-Laye)
25. Jacques Lejeune + Chris Corsano + Ben Vida & Marina Rosenfeld + Mads Emil Nielsen + Gravetemple (Présences électronique) – Maison de la radio|Studio 104
25. Alarm + Trashley + Pour X raisons – La Comedia (Montreuil)
28. Caves + Faux départ + Tentaculo + Cookies – Supersonic (gratuit)
29. Chicaloyoh + KosmoSuna – Le Zorba (gratuit)
29. Bleib Modern + La Peste + Nebula Glow – Supersonic (gratuit)
29. Drame + Le Réveil des tropiques – Centre Barbara-FGO
29. Angry Skeletons + Laurence Wasser – Pointe Lafayette
29. Collateral + Le Casquette tour + Grégoire Bruno + Allister Sinclair + Gérard Jugno 106 + Macmann – L'International
29. Laurent Garnier + Scan X – Rex club
30. Orval Carlos Sibelius + Domotic – Le Zorba (gratuit)
30. Méryll Ampe + Carbon Sink – Galerie Planète rouge (gratuit)
30. Luminance + Pure Ground + Bunker Bal (dj) + Haktion (dj) + Dasz (dj) – Le Klub
30. Polar Inertia + Shlømo + Luigi Tozzi + Twin Peaks – Concrete
31. Schlaasss + Petosaure + Enfance de merde – Supersonic (gratuit)
31. Maulwürfe (fest. Artdanthé) – Théâtre de Vanves
31. The Noise Consort (fest. Artdanthé) – Théâtre de Vanves
31. L'émeute philharmonique de SEC + Kouma + Polar Polar Polar Polar + Stratocastors + Joujou – La Parole errante (Montreuil)
31. BIC + Kasper Toeplitz – Mains d'oeuvre (Saint-Ouen)
31. Laura Not + Heap + Tiff + ANDCL – La Station
31. 14anger + AnD + Schwefelgelb + Scalamerya & VSK + Blind Delon + Darzak + SΛRIN – Docks Eiffel
31. Arnaud Rebotini – Nuits fauves
Avril
01. Olden Yolk + Delphine Dora – Espace B
04. Suuns – Elysée Montmartre
05. Michaela Antalova + Amundsen + On lâche les chiens – Le Zorba (gratuit)
05. UUUU + Quentin Rollet & Jérôme Lorichon – Espace B
05. Dance with the Dead + Christine + Mlada Fronta + Confrontational + Midnight Danger – Gibus
05. Eric Chenaux + Bass Clef – Instants chavirés (Montreuil)
06. Le Prince Harry + Whispering Sons + The Guru Guru + It It Anita + Empereur – Supersonic
06. Le singe blanc + Nohaybanda! + Casse gueule – Cirque électrique
06. Litige + Cookies + Fumo Nero – Pointe Lafayette
06. NSI (Tobias & Max Loderbauer) + Sendai Soundsystem (Peter Van Hoesen & Yves De Mey) + ENA + Izabel – Concrete
06. Abdulla Rashim + Vril + LGML + Thomas Evans + Wondernature – Nuits fauves
07. Nozomi Misawa & Marion Bataille (fest. Raccords) – Bibliothèque Françoise-Sagan (gratuit sur résa)
07. Terminal Cheesecake + GuiliGuiliGoulag + Sweet Williams + Futuroscope – Espace B
07. Massicot – théâtre des Amandiers (Nanterre)
10. Jamie Stewart joue Xiu Xiu – Olympic café
10. Structure + Walking Idiots + Oktober Lieber – Le Klub
10. Wolf Eyes Music 2018 + Evil Moisture + Delphine Dora & Sophie Cooper – Instants chavirés (Montreuil)
11. JC Satàn + Cockpit – La Maroquinerie
11. Silent Era + Youth Avoiders + Retsar Bai Naim – La Comedia (Montreuil)
12. Dune Messiah (dj) – Le Sans-Souci
12. The Ex + Anarchist Republic of Bzzz (fest. Banlieues bleues) – La Dynamo (Pantin)
13. Amusement Parks on Fire + Ulster Page + Misty Coast – Supersonic (gratuit)
13. DJ Krush – La Bellevilloise
13. La Secte du futur + Dune Messiah + The Space Padlocks – La Station
13. Èlg, Catherine Hershey, Borja Flames & Sourdure + D.K. + Carval Tarek (dj) – Instants chavirés (Montreuil)
14. Dominique a + My Brightest Diamond – Salle Pierre Boulez|Philharmonie
14. Infecticide + Exo_C + Randy x Marsh + Mauvaise foi + Forge (Monospace fest.) – Petit Bain
14. Techno Thriller – La Station
14. Badbad – 2, rue Paul-Eluard (Montreuil)
14. Function + Shifted – La Machine
14. Lil Louis + Josh Wink + Ellen Allien + Levon Vincent + Acid Arab + Paranoid London + Wlderz + Thomas Delacroix – Paris Event Center
15. Dominique a + Adrian Crowley – Cité de la musique|Philharmonie ||COMPLET||
18. Chrysta Bell – La Maroquinerie
19. Christian Death + Punish Yourself + Volker – La Machine
19. Peter Kernel – Point FMR
19. Ventre de biche + Compas dans l'oeil + Le Mal des ardents + La Coupure – La Station
19. Carole Robinson, Buno Martinez & Charles Curtis : "Naldjorlak I, II et III" d'Éliane Radigue – La Marbrerie (Montreuil)
20. Die Selektion + Gost + Hørd + Koy + Valeskja Valcav – Supersonic (gratuit)
20. Zarkoff + Al Niklaus (dj) + Law & Haktion (dj) + Schwefelgelb (dj) + Philipp Strobel (dj) + Petula Black Sperm – La Station
20. Idles + Lice – Trabendo ||COMPLET||
21. Igorrr + Niveau Zero – Trabendo
21. Yan Wagner + Tristesse contemporaine (fest. Clap Your Hands) – Café de la danse
21. Louisahhh + Jeanne Added + Maud Geffray + Myako + Léonie Pernet + Betty + RAG + Nari Fshr + Sophie Morello + Vikken + Leslie Barbara Butch + Sara Zinger + Catherin + Apollo noir + F/cken Chipotle (10 ans de Wet For Me) – La Machine
22. The Body + Fange – Olympic café
22. Norma Loy + Babel 17 – Supersonic
24>26. Franck Vigroux & Kurt d'Haeseleer – La Pop
25. Nine Eleven + Binaire + Unlogistic + Ayatollah – Le Klub
26. Ought + Foammm – La Maroquinerie
26. Wrekmeister Harmonies – Espace B
26. A Place To Bury Strangers – Trabendo
27. Popsimonova + Sleep Loan Sharks – Le Klub
28. She Past Away + Lebanon Hanover + Selofan – La Machine
28. Rhys Chatham + Krikor Kouchian + Chloé & Vassilena Serafimova – Centre Pompidou
28. Tomaga + Vanishing Twin + Veik – La Maroquinerie
28. Arcade Fire – Bercy Arena
28. Regis + Inigo Kennedy + CTRLS + Anastasia Kristensen – Nuits fauves
30. Koudlam + Bajram Bili + Pointe du lac – La Maroquinerie
30. Iron Fist of The Sun + Am Not + Kevlar + Kontinent – Les Voûtes
30. Seth Troxler & Miss Kittin – Badaboum
Mai
01. Venetian Snares & Daniel Lanois + Joanne Pollock Tryphème – Trabendo
03. Black Leather Jesus + Vomir + Rien – Instants chavirés (Montreuil)
07. Iceage + Pardans – Petit Bain
10. Derya Yildirim & Grup Simsek + Stranded Horse (Le Beau fest.) – La Petite Halle
11. And Also the Trees + Tropic of Cancer + Better Person + En attendant Ana + Magic Island (Le Beau fest.) – Trabendo
12. Deerhoof + Ulrika Spacek + First Hate + Pantin plage + Le Couleur + [Good Morning : ANNULÉ] (Le Beau fest.) – Trabendo
12. I Hate Models + Ancient Methods + Rrose – La Machine
13. God is an Astronaut – Trabendo
14. Bryan's Magic Tears + Le Villejuif Underground + VVVV – La Maroquinerie (gratuit sur résa)
15. Sinivia Alvise + Erwan Keravec & Mats Gustaffson + Orchestre orange & Gëinst (fest. Switch) – Théâtre de Vanves
17>20. DJ AZF + Bundle of Joy (Barnt & Superpitcher) + Xeno & Oaklander + December + B-Ball Joints (aka Low Jack) + La Mverte + Succhiamo... (fest. Ideal Trouble) – La Station
17. Moor Mother + Boy Harsher (fest. Ideal Trouble) – La Station
17. Alessandro Cortini + Fléau – Petit Bain
17. Clément Edouard + Giani Caserotto + Elise Dabrowski & Claudine Simon (fest. Switch) – Théâtre de Vanves
18. Sister Iodine + Ramleh (fest. Ideal Trouble) – La Station
18. Joachim Florent + Nox.3 & Linda Olah + Chassol (fest. Switch) – Théâtre de Vanves
19. Yo La Tengo – Cabaret sauvage
19. Horse Lord + Das Ding (fest. Ideal Trouble) – La Station
19. Deux boules vanille + Mondkopf + The Noise Consort (fest. Switch) – Théâtre de Vanves
20. Biscuit Mouth + Melkbelly + Storm{o} + BadBad – Espace B
20. SNTS (Marvellous Island fest.) – Ile de loisirs (Vaires-Torcy)
22. Zëro + Moodie Black – Petit Bain
23. Otomo Yoshihide + Kaze – Instants chavirés (Montreuil)
24. Otomo Yoshihide & Chris Pitsiokos + Ikuro Takahashi – Instants chavirés (Montreuil)
24. HMLTD + Faire (dj) – Petit Bain
25. Mogwai + Jon Hopkins + James Holden & The Animal Spirits (Villette sonique) – Grande Halle de La Villette
26. Car Seat Headrest + Naked Giants (Villette sonique) – Trabendo
26. Marquis de Sade + Anna Von Hausswolff + Exploded View (Villette sonique) – Grande Halle de La Villette
26/27. Abra + Essaie pas + Flohio + Kelly Lee Owens + Nilüfer Yanya + Smerz + Snail Mail + The Sea & Cake + Tzusing... (Villette sonique) – Prairie du cercle nord et périphérique (gratuit)
28. The Chamelons – Supersonic
29. The Damned – Petit Bain ||COMPLET||
29. Deerhunter + Midnight Sister (Villette sonique) – Cabaret sauvage
30. John Maus + Flat Worms + Kate NV (Villette sonique) – Trabendo
30. The Damned – Petit Bain
30. Igorrr + Ni – Les Cuizines (Chelles)
Juin
02. Penguin Café – Fondation Cartier
02/03. Björk + Beck + Jamie XX + King Krule + Father John Misty + Migos... (fest. We Love Green) – Bois de Vincennes
07. Molly Nilsson – Badaboum ||COMPLET||
08. Molly Nilsson – Badaboum
09. Trisomie 21 + Delacave – La Maroquinerie
09. Waxahatchee – Espace B
11. Preoccupations – La Maroquinerie
12. Damo Suzuki's Network – Espace B
12. Thom York – L'Olympia
13. L7 – La Cigale
14. Ty Segall & The Freedom Band + Mike Donovan – Bataclan
15. Bernard Grancher + Infecticide + Les Trucs – Centre Barbara-FGO
15/16. Ryoji Ikeda : "Formula - c4i - Datamatics" – Centre Pompidou
16. Vitalic (fest. Bains numériques) – Lac d'Enghien-lès-Bains (gratuit)
17. Animal Collective – Le Trianon
22. Modern Life Is War + Cro Mags – Petit Bain
25. Nine Inch Nails – Olympia ||COMPLET||
27. The Jesus & Mary Chain – Le Trianon
29>01.07. Motor City Drum Ensemble + Antal + Golden Dawn Archestra + Tin Man + A Deep Groove + Josey Rebelle + Toshio Matsuura + Cotonete + Zaltan + Lomboy + Ceephax Acid Crew + Nick V + Saint DX + Kamaal Williams + Blake Baxter + Muddy Monk + Eliott Litrowski... (Macki Music fest.) – parc de la mairie (Carrières/Seine)
30. Echo Collective joue "Amnesiac" de Radiohead (fest. Days Off) – Le Studio|Philharmonie
30. Nils Frahm (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
Juillet
03. David Byrne (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
04. MGMT (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
06. Trami Nguyen et Laurent Durupt jouent "Piano Phase" de Steve Reich + Bruce Brubaker + Laake + Fabrizio Rat + Murcof & Vanessa Wagner + Tom Rogerson + Grandbrothers (fest. Days Off) – Cité de la musique|Philharmonie
06. Amelie Lens + Daniel Avery + Floating Points + Folamour + Jeff Mills + Laurent Garnier + Kink b2b Gerd Janson + Not Waving + Solomun... (The Peacock Society) – Parc floral (Vincennes)
07. Richie Hawtin + Tale of Us + Charlotte de Witte + Chloé + Maetrik + Mano Le Tough + Octo Octa + Joy Orbison b2b Kornel Kovacs (The Peacock Society) – Parc floral (Vincennes)
08. Maulwürfe – La Gaîté lyrique
09. Eels – Olympia
13. Ministry + Grave Pleasures – Elysées Montmartre
Août
20. Front Line Assembly + Die Krupps – La Machine
24>26. Dirty Projectors + Idles + The Black Angels + Parcels + Cigarettes after Sex + Bonobo + Carpenter Brut + Justice... (fest. Rock-en-Seine) – Domaine de Saint-Cloud
Septembre
01. Ariel Pink (Paris Inter. Fest. of Psychedelic Music) – La Machine
04. Thee Oh Sees (Paris Inter. Fest. of Psychedelic Music) – La Cigale
22. The Wedding Present – Point FMR
Octobre
04. Ballaké Sissoko & Vincent Segal – La Seine musicale (Boulogne-Billancourt)
20. Tallinn Chamber Orchestra : Fratres, Cantus in memoriam Benjamin Britten, Adam's Lament, Salve Regina et Te Deum d'Arvo Pärt – Salle Pierre-Boulez|Philharmonie
26. Jon Hopkins – Trianon
Novembre
04. Peaches Christ Superstar – Cité de la musique|Philharmonie
18. Ensemble Links : Drumming de Steve Reich – Cité de la musique|Philharmonie
23. Michael Nyman : "War Work: 8 Songs with Film" – Salle Pleyel
Décembre
01. Deux boules vanille (fest. Marathon!) – La Gaîté lyrique
09/10. Moriarty – Cité de la musique|Philharmonie
15. Gaspar Claus – Cité de la musique|Philharmonie
Avril
14. Arnaud Rebotini joue la BO de "120 Battements par minute" – Cité de la musique|Philharmonie
Mai
17. Philip Glass : Études pour piano – Salle Pierre-Boulez|Philharmonie
18. Bruce Brubaker & Max Cooper : Glasstronica – Cité de la musique|Philharmonie
Juin
26. Magma – Salle Pierre-Boulez|Philharmonie
en gras : les derniers ajouts / in bold: the last news
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On Approximating Pi
So, you want to approximate $\pi$ from first principles (ie, no fancy formulas that you don’t know how to prove). One of the most straightforward ways to do this is through the method that Archimedes used thousands of years ago. He did this by looking at the perimeter of regular polygons, which he could compute. The more sides that the polygon had, the closer the perimeter was to $\pi$. In modern notation, his method amounts to the formula
$$ \lim_{x \rightarrow 0} \frac{\sin(\pi x)}{x} = \pi = \lim_{x \rightarrow 0} \frac{\tan(\pi x)}{x} $$
Since we can compute things like $\sin(\pi/2^n)$ without actually knowing what $\pi$ is, we can use this to find an approximation for $\pi$. But just knowing this formula is not good enough. For instance, how do we know how close we are to $\pi$? If we use $x=2^{-100}$ to do this, then exactly how close we are? If we want to get $\pi$ to, say, ten digits, then what power of two do we need in the denominator?
These questions can be answered by going through the proofs and keeping track of how these limits actually work. Through this, we can get a refined look at exactly how to approximate $\pi$. (And gain an appreciation for $\epsilon-\delta$ in limits.)
Our Fundamental Identity
Firstly, we need to show that for any $-\frac{\pi}{2} < x < \frac{\pi}{2}$ we have
$$ |x\cos(x)| \leq |\sin(x)| \leq |x| \leq |\tan(x)| \leq |x\sec(x)|$$
where we have equality only at $x=0$.
This is where the diagram in the picture comes into play. Note that we clearly have equality at $x=0$ and, by symmetry, we just have to check this for $x>0$. In the picture, we have a quarter-circle of radius 1, with a line from the origin to the circle that makes an angle of $\theta$ with the horizontal axis. We can make this to make a triangle as shown. The length of the horizontal leg of this triangle is $\cos(\theta)$ and the length of the vertical leg of this triangle is $\sin(\theta)$. The blue arc in the picture is clearly larger than the vertical leg of the triangle and, since we are in radians, this arc has length $\theta$. This gives us that $\sin(\theta)<\theta$. On the other hand, the pink highlighted arc has radius $\cos(\theta)$ which means that its length is $\theta\cos(\theta)$. Since this arc is less than the vertical leg of the triangle, we get the leftmost inequality. Finally, through similar reasoning with the larger triangle and arc, we get the final two inequalities.
The Limit Lemma
Next we will look at our first limit. We want to show that
$$\lim_{x\rightarrow 0}\cos(x) = 1 = \lim_{x\rightarrow 0} \sec(x) $$
This means working with $\epsilon$s and $\delta$s, and it will be important to keep track of this information, as it tells us how to actually do the approximation.
Let $\epsilon >0$. We have the basic trig identity: $1-\cos(x) = 2\sin^2(x/2)$. This means that, using our above inequality, that $|\cos(x)-1| <x^2/2$. Therefore if $0<|x|<\sqrt{2\epsilon}$, then $|\cos(x)-1|<\epsilon$. This proves the first limit.
We will use this to prove the second limit. We have $\sec(x)-1 = (1-\cos(x))/\cos(x)$. Suppose that $0<|x|<\sqrt{2\epsilon’}$. Using what we just did, we have $1-\cos(x) < \epsilon’$ and $1-\epsilon’ < \cos(x)$. Combining these gives $\sec(x)-1 < \frac{\epsilon’}{1-\epsilon’}$. Choose $\epsilon’=\epsilon/(1+\epsilon)$. Plugging this into $\frac{\epsilon’}{1-\epsilon’}$ simplifies to $\epsilon$. So, if $0<|x|< \frac{\epsilon}{1+\epsilon}$ then $|1-\sec(x)|<\epsilon$. This proves the second limit.
To summarize the important bits of what we just did, since $ \frac{\epsilon}{1+\epsilon} < \epsilon$, it follows that if $0<|x|< \sqrt{\frac{2\epsilon}{1+\epsilon}} $ then
$$ |1-\cos(x)|< \epsilon \;\;\;\;\&\;\;\;\;\; |1-\sec(x)| < \epsilon$$
The Main Limit
We’re now done with lemmas and can prove the main result, which is: For $0<|x|<\frac{\pi}{2}$ we have that
$$ \frac{\sin(x)}{x} < 1 < \frac{\tan(x)}{x}$$
Moreover, we have the limits
$$ \lim_{x\rightarrow 0} \frac{\sin(x)}{x} = 1 = \lim_{x\rightarrow 0}\frac{\tan(x)}{x}$$
Dividing our original inequality through by $x$ gives
$$ \cos(x) < \frac{\sin(x)}{x} < 1 < \frac{\tan(x)}{x} < \sec(x)$$
This holds for all $x$ in our range because we are able to use the evenness of these functions to get rid of the absolute value. We then immediately have the first statement of this, and we just have to do the limit.
For the limits, we will keep track of our $\epsilon$s and $\delta$s. Obviously, using the divided expression, the Squeeze Lemma will give our result, but we want a more refined result, with errors in place. Take the first two inequalities, and rearrange to get
$$ 0 < 1-\frac{\sin(x)}{x} < 1-\cos(x)$$
By subtracting through by $1$ and multiplying by $-1$. Similarly for the latter two inequalities we have:
$$ 0 < \frac{\tan(x)}{x} -1 < \sec(x) -1$$
If we then take $0<|x|< \sqrt{\frac{2\epsilon}{1+\epsilon}}$, we find, using our previous results, that
$$ 0 < 1- \frac{\sin(x)}{x} < \epsilon$$
$$ 0 < \frac{\tan(x)}{x} - 1 < \epsilon$$
This proves the result, and we know how the error works.
Approximating Pi
We will now use this to approximate $\pi$. In our final result, replace $x$ with $\pi x$, multiply our equations through by $\pi$ and replace $\epsilon$ with $\epsilon/\pi$. This results in the statement: If $0<|x|<\frac{1}{\pi} \sqrt{\frac{2\epsilon}{\pi+\epsilon}}$ then
$$ 0 < \pi- \frac{\sin(\pi x)}{x} < \epsilon$$
$$ 0 < \frac{\tan(\pi x)}{x} - \pi < \epsilon$$
This means that the sine inequality gives a lower bound for $\pi$ with error $\epsilon$ and the tangent inequality gives an upper bound for $\pi$ with error $\epsilon$. Explicitly: Whenever $0<|x|<\frac{1}{\pi} \sqrt{\frac{2\epsilon}{\pi+\epsilon}}$ we have
$$\pi\in \left( \frac{\sin(\pi x)}{x}, \frac{\tan(\pi x)}{x} \right) \subseteq (\pi-\epsilon , \pi +\epsilon)$$
Follow this recipe and you’ll be able to approximate $\pi$ to your heart’s content.
Example
We can compute trig functions of angles $\pi/(3\cdot 2^n)$ by hand. If we want to compute $\pi$ to within $k$ digits, then the question we need to answer is: What does $n$ have to be?
Using $\epsilon = 1/10^{k+1}$, $x=3^{-1}2^{-n}$, and the trivial bound $3<\pi$, in our main result, we find that:
$$ \frac{1}{2\cdot 3^n} < \frac{1}{3} \sqrt{\frac{2\cdot 10^{-k-1}}{3+10^{-k-1}}}$$
which simplifies to
$$n> \frac{k+1}{2}\log_2(10) + \frac{1}{2}\log_2(3/2) $$
or
$$n > 1.66097k +1.95345$$
This gives a linear relationship between the number of digits that we get and how many powers of two we have to take.So, if, say, we want to get $k=3$ digits, then we need $n>6.9$, which means that we need $x=\frac{1}{384}$ will do. Archimedes did it for $x=\frac{1}{96}$, which means that $n=5$, which means that we would have to have $k<1.83$, or we can be sure his result is accurate with error $\epsilon = 0.01$. This isn’t too tight of a bound, unfortunately, but it’s what we can prove using elementary trigonometry.
The record is $k=22,459,157,718,361$ digits. If we were to do this using this method, we would have to have $n=3.73\times 10^{13}$, which means $x\approx 3^{-1}\cdot 2^{-10^{13}}$, which would be really tough.
Using Calculus and Taylor Series, we can show that the error is actually much tighter. The question and challenge would then be to figure out if you can tighten this error using elementary methods.
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Solve sin(pi/2+theta) | sin(pi/2+ x) | sin pi/2 + x formula, Find value sin pi by 2 + x
#Solve sin(pi/2+theta)#sin(pi/2+ x)#sin pi/2 + x formula#Find value sin pi by 2 + x#sin#cos#tenerife#co#cot#sec#cosec#math#12th class
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Solve sec(pi/2+theta) | sec(pi/2 +x) | sec pi/2 + x formula, Find Exact value sec pi by 2 + x
#Solve sec(pi/2+theta)#sec(pi/2 +x)#sec pi/2 + x formula#Find Exact value sec pi by 2 + x#sec(pi/2+theta)#se#si#sec#sin#cos#ten#cot#math#nda#cds#pi/2#180#90#1200#classy#gdlmx#gd
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Solve cosec(pi/2+theta) | cosec pi/2 + x formula, csc (pi/2 +x) | Find value cosec pi by 2 + x
#Solve cosec(pi/2+theta)#cosec pi/2 + x formula#csc (pi/2 +x)#Find value cosec pi by 2 + x#math#12th class#formula#sin#cos#ten#cosec#sec#cot
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Des concerts à Paris et autour
Mars
05. Sef III + Muyassar Kurdi + Ronxshka & Jzerhard + Boris Allenou (aka Cathode) – Treize
05. Ritual Extra + Mhönos + Alahuta + Pisitakun – Pointe Lafayette
05. IAMX – La Machine
07. Les Troupes de l'imaginaire (fest. Sonic Protest) – Médiathèque musicale (gratuit)
07. Franck Vigroux & Antoine Schmitt : "Chronostasis" + Alexander Schubert : "Codec Error" – Centre Pompidou ||COMPLET||
07/08. Ryuchi Sakamoto & Shiro Takatani (Dumb Type) : "Dis.Play" – Maison de la Culture du Japon ||COMPLET||
08. Cheval scintillantes + Delphine Dora – Le Zorba (gratuit)
08. Molecule – Elysée Montmartre
08. Regina Demina & Eggplant + Léon Septavaux : "Haters" – Palais de Tokyo
08. Trevor Wishart + Davide Tidoni + Pancrace (fest. Sonic Protest) – Théâtre de Vanves
08. Léonie Pernet – O Gib (Montreuil)
09. Bretzel Göring + Kollektiv Barner 16 + Mohamed Lamouri (fest. Sonic Protest) – Centre Barbara-FGO
09. Elliott Murphy – Mona Bismarck American Center
09. Extrawelt + Ben Men + Präri – Rex Club
09. Adam X + Perc + Pulse One + Parfait – tba
10. ARP TARK (Aymeric Hainaux) – Centre national de la Danse (Pantin) (gratuit)
10. Julien Desprez – Centre national de la Danse (Pantin)
10. Wild Classical Music Ensemble + Dolly Rambo + DNA,AND feat. Ogrob (fest. Sonic Protest) – Centre Barbara-FGO
10. Uriel Barthélémi & Tarek Atoui – Silencio
10. Anaconda + Casio judiciaire + 2035 – Le Jardin d'Alice (Montreuil)
10. Radioactive Man + AZF + Erika b2b Noncompliant + Powder – Concrete
10. Deena Abdelwahed (dj) + Sundae (dj) – Centre national de la Danse (Pantin) (gratuit sur résa) ||COMPLET||
13. Luminous Bodies + Klarinetthor + Enob – Espace B
14. Balmorhea + Martyn Heyne – Point FMR
14. I Hate Model + Marc.Andrea + Sina – Rex Club
15. Arto Lindsay + Seijiro Murayama & Thomas Brinkmann + Masami Kawaguchi (fest. Sonic Protest) – Eglise Sant-Merry
15. Mathias Delplanque + Club Cactus – Espace B
15. Esmerine – Point FMR
15. C.A.R. – Badaboum
15. Regina Demina & Eggplant + Léon Septavaux : "Haters" – Palais de Tokyo
15. Octave Courtin + Emmanuelle Gibello + Frédéric Mathevet + Hélène Singer + Alexandra Spence + Jean-Charles François & Nicolas Sidoroff + Kwangrae Kim – Le Cube (Issy-lès-Moulineaux)
15. Maud Geffray & Lavinia Meijer jouent Philip Glass (fest. Paris Music) – Eglise Saint-Eustache ||COMPLET||
16. The Altered Hours – Supersonic (gratuit)
16. Morton Subotnick, Alec Empire & Lillevan + Kevin Drumm (fest. Sonic Protest) – Eglise Saint-Merry
16. Panico Panico + The Absolute Never + Thharm – Pointe Lafayette
16. Koki Nakano & Vincent Ségal – Le Bal
16. Virginie Despentes & Zëro + Dream Wife + The Pack AD + La Pietà + Léonie Pernet (fest. Les femmes s'en mêlent) – La Machine
16. Minimal Syndicat + Illnurse + Murd + The DJ Producer – Glazart
16. Pär Grindvik + SHXCXCHCXSH + Nozen – Nuits fauves
17. Hélène Breschand & Wilfried Wendling : "Imaginarium" – La Muse en circuit (Alfortville) (gratuit)
17. Vox Low – La Station
17. Arnaud Rebotini + Etienne Jaumet (dj)... (fest. Paris Music) – L'Aérosol
17. Maher Shalal Hash Baz + Mick Harris aka Fret + Russell Haswell + ZB Aids + Paddy Steer + Thomas Tilly + Terrine + Satan (fest. Sonic Protest) – L'Echangeur (Bagnolet)
17. British Murder Boys + Lucy + Dasha Rush + Anetha – Terminal 7
17. Chloé + Curses – Nuits fauves
19. Koban – Supersonic (gratuit)
19. Fever Ray – Olympia
20. Hackedepicciotto – Walrus (gratuit)
20. Les Tambours du Bronx + Acyl – La Machine
21. Hackedepicciotto + Phoenician Drive + Maninkari – Supersonic (gratuit)
21. Egopusher – Centre culturel suisse
22. Jean-Philippe Renoult & Dinah Bird : musique pour "Absynth" de HeHe (Biennale Nemo) – WIP
22. Goran Bregovic & l'orchestre des mariages et des enterrements – Salle Pleyel
22. Petra pied de biche – Pointe Lafayette
23. PurForm + TRDLX (Biennale Nemo) – Grande Halle de La Villette
23. Pierre Henry (diff.) + Anabelle Playe + John Chantler + Bill Orcutt + Anthony Child (Présences électronique) (Présences électronique) – Maison de la radio|Studio 104
23. Hey Colossus + Grey Hairs – Espace B
23. Delacave – Supersonic
23. Zombie Zombie – Le Plan (Ris-Orangis)
23. Paula Temple + Tommy Four Seven + Umfang – Nuits fauves
24. Alva Noto & Anne-James Chaton : "Alphabet" (Biennale Nemo) – Grande Halle de La Villette
24. Else Marie Pade + :such: + Bellows + Phonophani + The Caretaker (Présences électronique) – Maison de la radio|Studio 104
24. Youth Code + Carpenter Brut – Olympia
24. Jessica93 + JC Satàn – La Clef (Saint-Germain-en-Laye)
25. Jacques Lejeune + Chris Corsano + Ben Vida & Marina Rosenfeld + Mads Emil Nielsen + Gravetemple (Présences électronique) – Maison de la radio|Studio 104
29. Chicaloyoh + KosmoSuna – Le Zorba (gratuit)
29. Bleib Modern – Supersonic (gratuit)
29. Drame + Le Réveil des tropiques – Centre Barbara-FGO
29. Angry Skeletons + Laurence Wasser – Pointe Lafayette
29. Laurent Garnier + Scan X – Rex club
30. Orval Carlos Sibelius + Domotic – Le Zorba (gratuit)
30. Polar Inertia + Shlømo + Luigi Tozzi + Twin Peaks – Concrete
31. Schlaasss + Petosaure + Enfance de merde – Supersonic (gratuit)
31. Maulwürfe (fest. Artdanthé) – Théâtre de Vanves
31. The Noise Consort (fest. Artdanthé) – Théâtre de Vanves
31. L'émeute philharmonique de SEC + Kouma + Polar Polar Polar Polar + Stratocastors + Joujou – La Parole errante (Montreuil)
31. 14anger – tba
31. Arnaud Rebotini – Nuits fauves
Avril
04. Suuns – Elysée Montmartre
05. Michaela Antalova + Amundsen + On lâche les chiens – Le Zorba (gratuit)
05. UUUU – Espace B
05. Dance with the Dead + Christine + Mlada Fronta + Confrontational + Midnight Danger – Gibus
06. Le Prince Harry + Whispering Sons + The Guru Guru + It It Anita + Empereur – Supersonic
06. Le singe blanc + Nohaybanda! + Casse gueule – Cirque électrique
06. NSI (Tobias & Max Loderbauer) + Sendai Soundsystem (Peter Van Hoesen & Yves De Mey) + ENA + Izabel – Concrete
07. Nozomi Misawa & Marion Bataille (fest. Raccords) – Bibliothèque Françoise-Sagan (gratuit sur résa)
07. Terminal Cheesecake + GuiliGuiliGoulag + Sweet Williams + Futuroscope – Espace B
10. Jamie Stewart joue Xiu Xiu – Olympic café
10. Structure + Walking Idiots + Oktober Lieber – Le Klub
10. John Olson + Nate Young + Regression + Henry & Hazel Slaughter + Stare Case + Wolf Eyes + Evil Moisture – Instants chavirés (Montreuil)
11. JC Satàn + Cockpit – La Maroquinerie
12. The Ex + Anarchist Republic of Bzzz (fest. Banlieues bleues) – La Dynamo (Pantin)
13. Amusement Parks on Fire + Ulster Page + Misty Coast – Supersonic (gratuit)
13. DJ Krush – La Bellevilloise
13. La secte du futur – La Station
14. Dominique a + My Brightest Diamond – Salle Pierre Boulez|Philharmonie
14. Infecticide + Exo_C + Randy x Marsh + Mauvaise foi + Forge (Monospace fest.) – Petit Bain
14. Techno Thriller – La Station
14. Badbad – 2, rue Paul-Eluard (Montreuil)
14. Function + Shifted – La Machine
14. Lil Louis + Josh Wink + Ellen Allien + Levon Vincent + Acid Arab + Paranoid London + Wlderz + Thomas Delacroix – Paris Event Center
15. Dominique a + Adrian Crowley – Cité de la musique|Philharmonie ||COMPLET||
18. Chrysta Bell – La Maroquinerie
19. Christian Death + Punish Yourself + Volker – La Machine
19. Peter Kernel – Point FMR
19. Carole Robinson, Buno Martinez & Charles Curtis : "Naldjorlak I, II et III" d'Éliane Radigue – La Marbrerie (Montreuil)
20. Idles + Lice – Trabendo
20. Die Selektion – Supersonic
21. Igorrr + Niveau Zero – Trabendo
21. Yan Wagner + Tristesse contemporaine (fest. Clap Your Hands) – Café de la danse
22. The Body + Fange – Olympic café
24>26. Franck Vigroux & Kurt d'Haeseleer – La Pop
26. Ought + Foammm – La Maroquinerie
26. Wrekmeister Harmonies – Espace B
26. A Place To Bury Strangers – Trabendo
27. Popsimonova + Sleep Loan Sharks – Le Klub
28. She Past Away + Lebanon Hanover + Selofan – La Machine
28. Arcade Fire – Bercy Arena
28. Rhys Chatham + Krikor Kouchian + Chloé & Vassilena Serafimova – Centre Pompidou
30. Koudlam + Bajram Bili + Pointe du lac – La Maroquinerie
30. Iron Fist of The Sun + Am Not + Kevlar + Kontinent – Les Voûtes
Mai
07. Iceage + Pardans – Petit Bain
10. Derya Yildirim & Grup Simsek + Stranded Horse (Le Beau fest.) – La Petite Halle
11. And Also the Trees + Tropic of Cancer + Better Person + En attendant Ana + Magic Island (Le Beau fest.) – Trabendo
12. Deerhoof + Ulrika Spacek + First Hate + Good Morning + Pantin plage (Le Beau fest.) – Trabendo
13. God is an Astronaut – Trabendo
14. Bryan's Magic Tears + Le Villejuif Underground + VVVV – La Maroquinerie (gratuit sur résa)
19. Yo La Tengo – Cabaret sauvage
19. Deux boules vanille (fest. Switch) – Théâtre de Vanves
20. Biscuit Mouth + Melkbelly + Storm{o} + BadBad – Espace B
20. SNTS (Marvellous Island fest.) – Ile de loisirs (Vaires-Torcy)
23. Otomo Yoshihide + Kaze – Instants chavirés (Montreuil)
24. Otomo Yoshihide & Chris Pitsiokos + Ikuro Takahashi – Instants chavirés (Montreuil)
24. HMLTD + Faire (dj) – Petit Bain
25. Mogwai + Jon Hopkins + James Holden & The Animal Spirits (Villette sonique) – Grande Halle de La Villette
26. Car Seat Headrest + Naked Giants (Villette sonique) – Trabendo
26. Marquis de Sade + Anna Von Hausswolff + Exploded View (Villette sonique) – Grande Halle de La Villette
29. The Damned – Petit Bain
29. Deerhunter + Midnight Sister (Villette sonique) – Cabaret sauvage
30. John Maus + Flat Worms + Kate NV (Villette sonique) – Trabendo
30. Klimperei – tba
30. Igorrr + Ni – Les Cuizines (Chelles)
Juin
02. Penguin Café – Fondation Cartier
02/03. Björk + Beck + Jamie XX + King Krule + Father John Misty + Migos... (fest. We Love Green) – Bois de Vincennes
09. Trisomie 21 – La Maroquinerie
11. Preoccupations – La Maroquinerie
12. Damo Suzuki's Network – Espace B
13. L7 – La Cigale
14. Ty Segall & The Freedom Band + Mike Donovan – Bataclan
15/16. Ryoji Ikeda : "Formula - c4i - Datamatics" – Centre Pompidou
22. Modern Life Is War + Cro Mags – Petit Bain
25. Nine Inch Nails – Olympia ||COMPLET||
27. The Jesus & Mary Chain – Le Trianon
29>01.07. Motor City Drum Ensemble + Antal + Golden Dawn Archestra + Tin Man + A Deep Groove + Josey Rebelle + Toshio Matsuura + Cotonete + Zaltan + Lomboy + Ceephax Acid Crew + Nick V + Saint DX (Macki Music fest.) – parc de la mairie (Carrières/Seine)
30. Echo Collective joue "Amnesiac" de Radiohead (fest. Days Off) – Le Studio|Philharmonie
30. Nils Frahm (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
Juillet
03. David Byrne (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
04. MGMT (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
06. Trami Nguyen et Laurent Durupt jouent "Piano Phase" de Steve Reich + Bruce Brubaker + Laake + Fabrizio Rat + Murcof & Vanessa Wagner + Tom Rogerson + Grandbrothers (fest. Days Off) – Cité de la musique|Philharmonie
06. Amelie Lens + Daniel Avery + Floating Points + Folamour + Jeff Mills + Laurent Garnier + Kink b2b Gerd Janson + Not Waving + Solomun... (The Peacock Society) – Parc floral (Vincennes)
07. Richie Hawtin + Tale of Us + Charlotte de Witte + Chloé + Maetrik + Mano Le Tough + Octo Octa + Joy Orbison b2b Kornel Kovacs (The Peacock Society) – Parc floral (Vincennes)
08. Maulwürfe – La Gaîté lyrique
09. Eels – Olympia
Septembre
22. The Wedding Present – Point FMR
Novembre
Michael Nyman : "War Work: 8 Songs with Film" – Salle Pleyel
Décembre
01. Deux boules vanille (fest. Marathon!) – La Gaîté lyrique
en gras : les derniers ajouts / in bold: the last news
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Des concerts à Paris et autour
Mars
13. Luminous Bodies + Klarinetthor + Enob – Espace B
14. Balmorhea + Martyn Heyne – Point FMR
14. I Hate Model + Marc.Andrea + Sina – Rex Club
15. Astéréotypie (fest. Sonic Protest) – Médiathèque musicale (gratuit)
15. Arto Lindsay + Seijiro Murayama & Thomas Brinkmann + Masami Kawaguchi (fest. Sonic Protest) – Eglise Sant-Merry
15. Mathias Delplanque + Club Cactus – Espace B
15. Esmerine – Point FMR
15. C.A.R. – Badaboum
15. Regina Demina + Sentimental Rave – Palais de Tokyo
15. Elsa Quintin + Ivan Samokrutkin + Dispoition Matrix – Bar Oued Rhiou
15. Octave Courtin + Emmanuelle Gibello + Frédéric Mathevet + Hélène Singer + Alexandra Spence + Jean-Charles François & Nicolas Sidoroff + Kwangrae Kim – Le Cube (Issy-lès-Moulineaux)
15. Maud Geffray & Lavinia Meijer jouent Philip Glass (fest. Paris Music) – Eglise Saint-Eustache ||COMPLET||
16. The Altered Hours – Supersonic (gratuit)
16. Morton Subotnick, Alec Empire & Lillevan + Kevin Drumm (fest. Sonic Protest) – Eglise Saint-Merry
16. Koki Nakano & Vincent Ségal – Le Bal
16. Virginie Despentes & Zëro + Dream Wife + The Pack AD + La Pietà + Léonie Pernet (fest. Les femmes s'en mêlent) – La Machine
16. Minimal Syndicat + Illnurse + Murd + The DJ Producer – Glazart
16. Pär Grindvik + SHXCXCHCXSH + Nozen – Nuits fauves
17. Dick Voodoo + Bockers + Walking Idiots – Supersonic (gratuit)
17. Hélène Breschand & Wilfried Wendling : "Imaginarium" – La Muse en circuit (Alfortville) (gratuit)
17. Panico Panico + The Absolute Never + Thharm – Pointe Lafayette
17. Vox Low – La Station
17. Arnaud Rebotini + Etienne Jaumet (dj)... (fest. Paris Music) – L'Aérosol
17. Maher Shalal Hash Baz + Mick Harris aka Fret + Russell Haswell + ZB Aids + Paddy Steer + Thomas Tilly + Terrine + Satan (fest. Sonic Protest) – L'Echangeur (Bagnolet)
17. British Murder Boys + Lucy + Dasha Rush + Anetha – Terminal 7
17. Chloé + Curses – Nuits fauves
17. Neil Landstrumm + Fallbeil + ADC 303 + Soul Edifice – Petit Bain
17. Overmono (Truss & Tessela) + 138 + Ontal + Mayeul + JKS + Airod + Ossian + Gëinst + Kuss + Moth + Jusaï + 16H07 + Absurd + The Engineer – tba
19. Koban + IV Horsemen + Night Night – Supersonic (gratuit)
19. Fever Ray – Olympia
20. Hackedepicciotto – Walrus (gratuit)
20. Les Tambours du Bronx + Acyl – La Machine
21. Hackedepicciotto + Phoenician Drive + Maninkari – Supersonic (gratuit)
21. Egopusher – Centre culturel suisse
21. OD Bongo + Qonicho Ah + Nuage magique – Pointe Lafayette
22. Jean-Philippe Renoult & Dinah Bird : musique pour "Absynth" de HeHe (Biennale Nemo) – WIP
22. Goran Bregovic & l'orchestre des mariages et des enterrements – Salle Pleyel
22. Petra pied de biche – Pointe Lafayette
23. Oren Ambarchi (dj) – Chair de poule (gratuit)
23. PurForm + TRDLX (Biennale Nemo) – Grande Halle de La Villette
23. Pierre Henry (diff.) + Anabelle Playe + John Chantler + Bill Orcutt + Anthony Child (Présences électronique) (Présences électronique) – Maison de la radio|Studio 104
23. Hey Colossus + Grey Hairs – Espace B
23. Delacave – Supersonic
23. Zombie Zombie – Le Plan (Ris-Orangis)
23. Paula Temple + Tommy Four Seven + Umfang – Nuits fauves
24. Alva Noto & Anne-James Chaton : "Alphabet" (Biennale Nemo) – Grande Halle de La Villette
24. Else Marie Pade + :such: + Bellows + Phonophani + The Caretaker (Présences électronique) – Maison de la radio|Studio 104
24. Youth Code + Carpenter Brut – Olympia
24. Jessica93 + JC Satàn – La Clef (Saint-Germain-en-Laye)
25. Jacques Lejeune + Chris Corsano + Ben Vida & Marina Rosenfeld + Mads Emil Nielsen + Gravetemple (Présences électronique) – Maison de la radio|Studio 104
25. Alarm + Trashley + Pour X raisons – La Comedia (Montreuil)
29. Chicaloyoh + KosmoSuna – Le Zorba (gratuit)
29. Bleib Modern – Supersonic (gratuit)
29. Drame + Le Réveil des tropiques – Centre Barbara-FGO
29. Angry Skeletons + Laurence Wasser – Pointe Lafayette
29. Collateral + Le Casquette tour + Grégoire Bruno + Allister Sinclair + Gérard Jugno 106 + Macmann – L'International
29. Laurent Garnier + Scan X – Rex club
30. Orval Carlos Sibelius + Domotic – Le Zorba (gratuit)
30. Luminance + Pure Ground + Bunker Bal (dj) + Haktion (dj) + Dasz (dj) – Le Klub
30. Polar Inertia + Shlømo + Luigi Tozzi + Twin Peaks – Concrete
31. Schlaasss + Petosaure + Enfance de merde – Supersonic (gratuit)
31. Maulwürfe (fest. Artdanthé) – Théâtre de Vanves
31. The Noise Consort (fest. Artdanthé) – Théâtre de Vanves
31. L'émeute philharmonique de SEC + Kouma + Polar Polar Polar Polar + Stratocastors + Joujou – La Parole errante (Montreuil)
31. BIC + Kasper Toeplitz – Mains d'oeuvre (Saint-Ouen)
31. 14anger – tba
31. Arnaud Rebotini – Nuits fauves
Avril
01. Olden Yolk + Delphine Dora – Espace B
04. Suuns – Elysée Montmartre
05. Michaela Antalova + Amundsen + On lâche les chiens – Le Zorba (gratuit)
05. UUUU – Espace B
05. Dance with the Dead + Christine + Mlada Fronta + Confrontational + Midnight Danger – Gibus
05. Eric Chenaux + Bass Clef – Instants chavirés (Montreuil)
06. Le Prince Harry + Whispering Sons + The Guru Guru + It It Anita + Empereur – Supersonic
06. Le singe blanc + Nohaybanda! + Casse gueule – Cirque électrique
06. NSI (Tobias & Max Loderbauer) + Sendai Soundsystem (Peter Van Hoesen & Yves De Mey) + ENA + Izabel – Concrete
07. Nozomi Misawa & Marion Bataille (fest. Raccords) – Bibliothèque Françoise-Sagan (gratuit sur résa)
07. Terminal Cheesecake + GuiliGuiliGoulag + Sweet Williams + Futuroscope – Espace B
10. Jamie Stewart joue Xiu Xiu – Olympic café
10. Structure + Walking Idiots + Oktober Lieber – Le Klub
10. Wolf Eyes Music 2018 + Evil Moisture + Delphine Dora & Sophie Cooper – Instants chavirés (Montreuil)
11. JC Satàn + Cockpit – La Maroquinerie
12. The Ex + Anarchist Republic of Bzzz (fest. Banlieues bleues) – La Dynamo (Pantin)
13. Amusement Parks on Fire + Ulster Page + Misty Coast – Supersonic (gratuit)
13. DJ Krush – La Bellevilloise
13. La Secte du futur + Shiny Darkly + The Space Padlocks – La Station
14. Dominique a + My Brightest Diamond – Salle Pierre Boulez|Philharmonie
14. Infecticide + Exo_C + Randy x Marsh + Mauvaise foi + Forge (Monospace fest.) – Petit Bain
14. Techno Thriller – La Station
14. Badbad – 2, rue Paul-Eluard (Montreuil)
14. Function + Shifted – La Machine
14. Lil Louis + Josh Wink + Ellen Allien + Levon Vincent + Acid Arab + Paranoid London + Wlderz + Thomas Delacroix – Paris Event Center
15. Dominique a + Adrian Crowley – Cité de la musique|Philharmonie ||COMPLET||
18. Chrysta Bell – La Maroquinerie
19. Christian Death + Punish Yourself + Volker – La Machine
19. Peter Kernel – Point FMR
19. Carole Robinson, Buno Martinez & Charles Curtis : "Naldjorlak I, II et III" d'Éliane Radigue – La Marbrerie (Montreuil)
20. Idles + Lice – Trabendo
20. Die Selektion + Gost + Hørd + Koy + Valeskja Valcav – Supersonic (gratuit)
21. Igorrr + Niveau Zero – Trabendo
21. Yan Wagner + Tristesse contemporaine (fest. Clap Your Hands) – Café de la danse
22. The Body + Fange – Olympic café
22. Norma Loy + Babel 17 – Supersonic (gratuit)
24>26. Franck Vigroux & Kurt d'Haeseleer – La Pop
25. Nine Eleven + Binaire + Unlogistic + Ayatollah – Le Klub
26. Ought + Foammm – La Maroquinerie
26. Wrekmeister Harmonies – Espace B
26. A Place To Bury Strangers – Trabendo
27. Popsimonova + Sleep Loan Sharks – Le Klub
28. She Past Away + Lebanon Hanover + Selofan – La Machine
28. Rhys Chatham + Krikor Kouchian + Chloé & Vassilena Serafimova – Centre Pompidou
28. Tomaga + Vanishing Twin + Veik – La Maroquinerie
28. Arcade Fire – Bercy Arena
30. Koudlam + Bajram Bili + Pointe du lac – La Maroquinerie
30. Iron Fist of The Sun + Am Not + Kevlar + Kontinent – Les Voûtes
30. Seth Troxler & Miss Kittin – Badaboum
Mai
03. Black Leather Jesus + Vomir + Rien – Instants chavirés (Montreuil)
07. Iceage + Pardans – Petit Bain
10. Derya Yildirim & Grup Simsek + Stranded Horse (Le Beau fest.) – La Petite Halle
11. And Also the Trees + Tropic of Cancer + Better Person + En attendant Ana + Magic Island (Le Beau fest.) – Trabendo
12. Deerhoof + Ulrika Spacek + First Hate + Pantin plage + Le Couleur + [Good Morning : ANNULÉ] (Le Beau fest.) – Trabendo
13. God is an Astronaut – Trabendo
14. Bryan's Magic Tears + Le Villejuif Underground + VVVV – La Maroquinerie (gratuit sur résa)
15. Sinivia Alvise + Erwan Keravec & Mats Gustaffson + Orchestre orange & Gëinst (fest. Switch) – Théâtre de Vanves
17. Clément Edouard + Giani Caserotto + Elise Dabrowski & Claudine Simon (fest. Switch) – Théâtre de Vanves
18. Joachim Florent + Nox.3 & Linda Olah + Chassol (fest. Switch) – Théâtre de Vanves
19. Yo La Tengo – Cabaret sauvage
19. Deux boules vanille + Mondkopf + The Noise Consort (fest. Switch) – Théâtre de Vanves
20. Biscuit Mouth + Melkbelly + Storm{o} + BadBad – Espace B
20. SNTS (Marvellous Island fest.) – Ile de loisirs (Vaires-Torcy)
23. Otomo Yoshihide + Kaze – Instants chavirés (Montreuil)
24. Otomo Yoshihide & Chris Pitsiokos + Ikuro Takahashi – Instants chavirés (Montreuil)
24. HMLTD + Faire (dj) – Petit Bain
25. Mogwai + Jon Hopkins + James Holden & The Animal Spirits (Villette sonique) – Grande Halle de La Villette
26. Car Seat Headrest + Naked Giants (Villette sonique) – Trabendo
26. Marquis de Sade + Anna Von Hausswolff + Exploded View (Villette sonique) – Grande Halle de La Villette
29. The Damned – Petit Bain ||COMPLET||
29. Deerhunter + Midnight Sister (Villette sonique) – Cabaret sauvage
30. John Maus + Flat Worms + Kate NV (Villette sonique) – Trabendo
30. The Damned – Petit Bain
30. Igorrr + Ni – Les Cuizines (Chelles)
Juin
02. Penguin Café – Fondation Cartier
02/03. Björk + Beck + Jamie XX + King Krule + Father John Misty + Migos... (fest. We Love Green) – Bois de Vincennes
07. Molly Nilsson – Badaboum ||COMPLET||
08. Molly Nilsson – Badaboum
09. Trisomie 21 – La Maroquinerie
11. Preoccupations – La Maroquinerie
12. Damo Suzuki's Network – Espace B
13. L7 – La Cigale
14. Ty Segall & The Freedom Band + Mike Donovan – Bataclan
15. Bernard Grancher + Infecticide + Les Trucs – Centre Barbara-FGO
15/16. Ryoji Ikeda : "Formula - c4i - Datamatics" – Centre Pompidou
16. Vitalic (fest. Bains numériques) – Lac d'Enghien-lès-Bains (gratuit)
22. Modern Life Is War + Cro Mags – Petit Bain
25. Nine Inch Nails – Olympia ||COMPLET||
27. The Jesus & Mary Chain – Le Trianon
29>01.07. Motor City Drum Ensemble + Antal + Golden Dawn Archestra + Tin Man + A Deep Groove + Josey Rebelle + Toshio Matsuura + Cotonete + Zaltan + Lomboy + Ceephax Acid Crew + Nick V + Saint DX (Macki Music fest.) – parc de la mairie (Carrières/Seine)
30. Echo Collective joue "Amnesiac" de Radiohead (fest. Days Off) – Le Studio|Philharmonie
30. Nils Frahm (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
Juillet
03. David Byrne (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
04. MGMT (fest. Days Off) – Salle Pierre-Boulez|Philharmonie
06. Trami Nguyen et Laurent Durupt jouent "Piano Phase" de Steve Reich + Bruce Brubaker + Laake + Fabrizio Rat + Murcof & Vanessa Wagner + Tom Rogerson + Grandbrothers (fest. Days Off) – Cité de la musique|Philharmonie
06. Amelie Lens + Daniel Avery + Floating Points + Folamour + Jeff Mills + Laurent Garnier + Kink b2b Gerd Janson + Not Waving + Solomun... (The Peacock Society) – Parc floral (Vincennes)
07. Richie Hawtin + Tale of Us + Charlotte de Witte + Chloé + Maetrik + Mano Le Tough + Octo Octa + Joy Orbison b2b Kornel Kovacs (The Peacock Society) – Parc floral (Vincennes)
08. Maulwürfe – La Gaîté lyrique
09. Eels – Olympia
Septembre
01. Ariel Pink (Paris Inter. Fest. of Psychedelic Music) – La Machine
04. Thee Oh Sees (Paris Inter. Fest. of Psychedelic Music) – La Cigale
22. The Wedding Present – Point FMR
Octobre
20. Tallinn Chamber Orchestra : Fratres, Cantus in memoriam Benjamin Britten, Adam's Lament, Salve Regina et Te Deum d'Arvo Pärt – Salle Pierre-Boulez|Philharmonie
26. Jon Hopkins – Trianon
Novembre
04. Peaches Christ Superstar – Cité de la musique|Philharmonie
18. Ensemble Links : Drumming de Steve Reich – Cité de la musique|Philharmonie
23. Michael Nyman : "War Work: 8 Songs with Film" – Salle Pleyel
Décembre
01. Deux boules vanille (fest. Marathon!) – La Gaîté lyrique
09/10. Moriarty – Cité de la musique|Philharmonie
15. Gaspar Claus – Cité de la musique|Philharmonie
Avril
14. Arnaud Rebotini joue la BO de "120 Battements par minute" – Cité de la musique|Philharmonie
Mai
17. Philip Glass : Études pour piano – Salle Pierre-Boulez|Philharmonie
18. Bruce Brubaker & Max Cooper : Glasstronica – Cité de la musique|Philharmonie
Juin
26. Magma – Salle Pierre-Boulez|Philharmonie
en gras : les derniers ajouts / in bold: the last news
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