This is one of a series of postings of important classical music dates, from the 17th century to the present. Included are the date and location of the birth and death of composers, and the premiere date and location of the first public performance of works. When the premiere date and location is unknown, the date or year of completion of the work is given. Though reasonably comprehensive, this is a subjective list, so the choice of composers and works is mine. If you find any errors, or if you can offer a premiere date and location for a work where only the completion date or year is listed, please post a comment here.
1761 Joseph Haydn (1732-1809) completed Symphony No. 6 in D major, Hob. I:6 “Le matin”
1762 Antonio Sarrier (1725-1762) died in Spain (specific location, unknown)
1764 Joseph Haydn (1732-1809) completed Symphony No. 22 in E-flat Major, Hob. I:22 “The Philosopher”
After October 25 – Wolfgang Amadeus Mozart (1756-1791) wrote Violin Sonata No. 6 in G major, K. 11 (while in London, England)
1765 February 21 – Symphony No. 1 in E♭ major, K. 16 by Wolfgang Amadeus Mozart (1756-1791) was first performed in London, England
1766 January 22 – Symphony No. 5 in B♭ major, K. 22 by Wolfgang Amadeus Mozart (1756-1791) was first performed in The Hague, the Netherlands
March 11 – Gallimathias musicum, K. 32 by Wolfgang Amadeus Mozart (1756-1791) was first performed in The Hague, the Netherlands
1767 December 30 – Symphony No. 6 in F major, K. 43 by Wolfgang Amadeus Mozart (1756-1791) was first performed in Brno, Czech Republic
1768 Wolfgang Amadeus Mozart (1756-1791) completed the one-act singspiel, the comic opera Bastien and Bastienne, K. 50
1769 August 8 – Cassation in B♭ major, K. 99 by Wolfgang Amadeus Mozart (1756-1791) was first performed in Salzburg, Austria
Did you know that both Venus and Neptune have orbits around the Sun that are more circular than the orbit of the Earth? In fact, Venus has the most circular orbit of all the major planets in the solar system. Here are the currentorbital eccentricities of these three planets.
Venus
0.007
Neptune
0.010
Earth
0.017
Perhaps surprisingly, there are currently 1,959 minor planets with orbits more circular than that of Venus. There may be more, but I’ve excluded any minor planets where the 1σ uncertainty in eccentricity is greater than the eccentricity itself. Large uncertainties happen because some minor planets do not yet have well-determined orbits.
To ensure that my restriction of σe < e wouldn’t be too severe, I confirmed that minor planet 232812 has the lowest known orbital eccentricity. The 11 minor planets that nominally have a lower eccentricity are all unnumbered, which means their orbits are not well determined and should thus be excluded.
The most circular orbit of all minor planets with a well-determined orbit is 232812 (2004 RG298). It has not yet received a name. It is a main-belt asteroid and has an amazingly circular eccentricity (e) of 0.0000152412. It orbits at a distance of 2.37 AU from the Sun once every 3.65 years. Its orbit is inclined 2.4° to the ecliptic. Its physical size is unknown
Among the named minor planets, the one with the most circular orbit is 12083 Darone. It is also a main-belt asteroid and has an eccentricity of 0.00177265. It orbits at a distance of 2.13 AU from the Sun once every 3.11 years. Its orbit is inclined 2.6° to the ecliptic. 12083 Darone has a diameter between 3.5 and 3.8 km. The lucky person to have this asteroid named after them is Gregory Darone, who mentored a finalist in the 2019 Regeneron Science Talent Search, a science competition for high school seniors. He teaches chemistry at the Charter School of Wilmington, Wilmington, Delaware.
The two largest trans-Neptunian objects are Pluto and Eris—both with high orbital eccentricities—andamong the major planets, Mercury has the highest orbital eccentricity. Here are their current values.
Eris
0.438
Pluto
0.245
Mercury
0.206
Eris has the highest orbital eccentricity of these three objects by far. Eris orbits between 38.2 and 97.7 AU from the Sun. Its current distance is 95.5 AU and it is very gradually getting closer to the Sun. At times during their respective orbits, Eris can be closer to the Sun than Pluto. Eris orbits the Sun once every 559.9 years at an orbital inclination of 43.9° with respect to the ecliptic.
Pluto orbits between 29.6 and 48.7 AU from the Sun. Its current distance is 35.6 AU and it is gradually getting further from the Sun. During the years 1979 to 1999, Pluto was closer to the Sun that Neptune, and this will next happen again from 2223 to 2243 (approximately). Pluto orbits the Sun once every 244.7 years at an orbital inclination of 17.0° with respect to the ecliptic.
Mercury orbits between 0.31 and 0.47 AU from the Sun. Mercury orbits the Sun once every 88 days at an orbital inclination of 7.0° with respect to the ecliptic.
All three of these objects are smaller than Jupiter’s moon Ganymede!
Incidentally, there is no known minor planet that orbits completelyinside the orbit of Mercury, but (as of this writing) 523 asteroids and 1,608 comets have come closer to the Sun than Mercury ever does. Sometimes a comet comes so close to the Sun that it disintegrates completely. Actual collisions with the Sun do occur.
Getting back to our original topic of nearly-circular orbits, the Earth’s orbital eccentricity has ranged from as circular as 0.000055 to as elliptical as 0.0679 due to gravitational perturbations caused by all the other objects in our solar system. This variation in orbital eccentricity is one of the Milankovitch cycles. Currently, the Earth’s orbital eccentricity is becoming more circular.
The orbits of all objects in our solar system change with time (albeit usually gradually and by a gentle amount). Over millions or billions of years, the gravitational attraction from stars passing close to the solar system can nudge orbits, too. Our solar system is a much more dynamic place than anyone imagined a century ago. Barring the extremely unlikely event of a star (lots of advance warning) or a rogue black hole (much less advance warning) entering our solar system, the Earth’s orbital eccentricity should remain stable for a very long time to come.
I watch very little television—almost entirely documentaries and news—but I must admit that every once in a while I like to dip into happier times through my living-room Atavachron to watch episodes from two 1960s television series1 starring the much-beloved 6 ft. 5 in. actor Fred Gwynne (1926-1993). The first is Car 54, Where Are You? (1961-1963), and the second is The Munsters (1964-1966). I am old enough to remember watching The Munsters during prime time when it originally aired, but I am too young to remember Car 54, Where Are You?, which I only recently discovered. I must admit I’m a fan of this talented actor.
Here are my favorite episodes from Car 54, Where Are You?:
Car 54, Where Are You?
Season 1, Episode 5: I Won’t Go [This is the first time Fred Gwynne and Al Lewis (1923-2006) appear together on screen, beginning a tremendous partnership between these two great actors in both comedy series.]
Season 1, Episode 6: Muldoon’s Star
Season 1, Episode 11: Catch Me on the Paar Show
Season 1, Episode 14: Get Well, Officer Schnauser
Season 1, Episode 15: Christmas at the 53rd
Season 1, Episode 18: Toody & Muldoon Crack Down
Season 1 , Episode 20: How High Is Up?
Season 1, Episode 21: Toody and the Art World
Season 1, Episode 24: Today I Am a Man
Season 1, Episode 26: The Beast Who Walked the Bronx
Season 2, Episode 9: Toody Undercover
Season 2, Episode 10: I Hate Captain Block
Season 2, Episode 13: 142 Tickets on the Aisle
Season 2, Episode 20: Here We Go Again
Season 2, Episode 22: The Biggest Day of the Year
Season 2, Episode 23: Here Comes Charlie
Season 2, Episode 24: See You at the Bar Mitzvah
And, here are my favorite episodes from The Munsters:
The Munsters
Season 1, Episode 3: A Walk on the Mild Side
Season 1, Episode 7: Tin Can Man
Season 1, Episode 33: Lily Munster—Girl Model
Season 1, Episode 35: Herman’s Happy Valley
Season 2, Episode 17: Just Another Pretty Face
Season 2, Episode 24: The Musician
Fred Gwynne 1984 . Credit: Ralph Dominguez/MediaPunch
While we are on the topic of great television series from the 1960s, please see my lists of favorite episodes from the greatest of them all: The Twilight Zone (1959-1964) and Star Trek (1966-1969): The Twilight Zone Star Trek↩︎
As of this writing, three objects have been discovered that have passed through our solar system coming from interstellar space.
1I/’Oumuamua Discovery Date: 2017-10-19 Perihelion Date: 2017-09-09 (40 days after perihelion) Perihelion Distance: 0.26 AU Incoming and Outgoing Speed: 26.4 km/s Dimensions: approximately 115 m × 111 m × 19 m, or 230 m × 35m × 35m Object Type: asteroid?
2I/Borisov Discovery Date: 2019-08-29 Perihelion Date: 2019-12-08 (101 days before perihelion) Perihelion Distance: 2.01 AU Incoming and Outgoing Speed: 32.3 km/s Diameter: 400 m – 1000 m Object Type: comet
3I/Atlas Discovery Date: 2025-07-01 Perihelion Date: 2025-10-29 (120 days before perihelion) Perihelion Distance: 1.36 AU Incoming and Outgoing Speed: 58.0 km/s Diameter: 320 m – 5,600 m (most likely 520 m – 748 m) Object Type: comet
Here are charts showing the direction from which each object entered our solar system, and the direction towards which each object is leaving our solar system.
1I/’Oumuamua came in from the direction of the constellation Lyra1I/’Oumuamua is exiting in a direction inside the Square of Pegasus2I/Borisov came in from the direction of the constellation Cassiopeia2I/Borisov is exiting in the direction of the southern constellation Telescopium3I/Atlas came in from the direction of the constellation Sagittarius3I/Atlas is exiting in the direction of the constellation Gemini
It is perhaps not surprising that 1I/’Oumuamua had the greatest deflection angle of the three interstellar objects. It came quite close to the Sun (0.26 AU, well inside the orbit of Mercury) and had the lowest incoming speed (26.4 km/s).
3I/Atlas, on the other hand, had the greatest incoming speed by far (58.0 km/s), so it was deflected by only 18.7° from a straight-line trajectory (angular separation 180° and deflection angle 0°).
A massive foreground object such as a galaxy or a galaxy cluster (including, of course, dark matter) can brighten and magnify a distant object. This is called gravitational lensing. Light from the distant object radiates out in all directions, but the massive foreground object bends some of these light rays towards the observer that normally would have continued on in a different direction, as shown in the illustration above.
The image we see is brighter because more light rays are directed our way. The image we see is also magnified because the gravitational lens gives the distant object a larger angular size, making it appear to be much closer to us. Again, the illustration above will help you understand why these effects occur.
Of course, unless the foreground object is a star or black hole or some other small spherical object with a reasonably uniform mass distribution, the gravitational lens effect will be complex and distorted, as illustrated below.
Gravitational lenses produce different shaped images depending on the shape of the lensing body. If the lens is spherical then the image appears as an Einstein ring (in other words as a ring of light) (top); if the lens is elongated then the image is an Einstein cross (it appears split into four distinct images) (middle), and if the lens is a galaxy cluster, then arcs and arclets (banana-shaped images) of light are formed (bottom). Credit: European Space Agency
There can be certain locations in a gravitational lens where light from a small region in the background becomes enormously magnified by a factor of up to 10,000 times. These regions are called caustics. Though the concept of a caustic is a bit difficult to describe or illustrate, here is a video of optical caustics caused by a laser pointer shining through a plastic disc with a lumpy surface.
Since moving to Tucson almost four years ago, my biggest complaint has to do with the terrible drivers and lack of enforcement of traffic laws. Here’s what I’ve noticed.
Speeding
Many drivers exceed the posted speed limit by +10 or +15 mph whenever they are not stuck in traffic; exceeding the speed limit by +20 mph is not uncommon.
Many of the drivers who exceed the speed limit constantly change lanes to get ahead; I often catch up to them at the next red light and even if not, how much time are they really saving while putting themselves and everyone around them in greater danger?
I have never seen one of these speeders and lane weavers pulled over by law enforcement.
Other Forms of Aggressive Driving
I’ve frequently experienced
A driver following too close behind a vehicle that is going the speed limit or slightly over, apparently in an effort to bully the law-abiding driver into either going faster or moving over to another lane (if that is possible) to let them through
A driver passing you in a no-passing zone
A driver cutting in front of you too soon after passing so that you have to brake to maintain a safe following distance
At night, lifted trucks and other large personal vehicles with blinding headlights; sometimes their “fog lights” are even brighter than their headlights.
Being honked at for not turning right on red soon enough to suit the driver behind you; sometimes this even occurs with left turns
Again, no enforcement.
Other Unsafe Driving Behaviors
Vehicles accelerating through an intersection while the traffic light is turning red; sometimes they do this after the light has turned red
Drivers running stop signs or only slowing down a little and not stopping before entering the intersection
Drivers not using their turn signals when switching lanes or making a right or left turn
Drivers driving without headlights during twilight and after dark
Drivers driving vehicles with a burned-out headlight; misaligned headlights are common, too
Again, no enforcement
City Life Makes People Crazy?
As a person who has lived most of my life in much smaller towns, I’ve noticed the following here:
A lack of driver courtesy; apparently city life brings out the worst in people when they get behind the wheel
For me, city life has taken all of the enjoyment out of driving.
The more dangerous the traffic conditions (due to traffic volume and/or a complex decision-making environment), the more aggressive drivers often become. Due to the elevated risk of a collision, drivers need to be more cautious in these situations—and many are—but there is often someone (usually a younger male) whose aggressive driving behavior puts themselves and everyone else in danger, including pedestrians and bicyclists (we have a very high pedestrian and bicyclist fatality rate here).
Posted speed limits are about +5 mph too high on many busy arterials with low access control. Most often, the posted speed limit is 40 or 45 mph when it should be 35 mph.
Safety Improvements
I’d like to see the following safety improvements.
At all signalized intersections with dedicated left turn lanes, only allow a left turn on green arrow (these are called protected left-turn lanes). I hate the flashing yellow arrow (turn left with caution) which may or may not be followed by a solid green arrow.
Prohibit right-turn-on-red
Prohibit U-turns at intersections; these are especially dangerous when combined with right-turn-on-red
Left-turn-on-green-arrow-only should be implemented at all signalized intersections with dedicated left turn lanesU-turns should not be allowed at intersections where right-turn-on-red is allowed
Enforcement Improvements
I’d like to see the following enforcement improvements:
Hire more traffic law enforcement officers
Install red light cameras at signalized intersections
Use AI and other modern technologies to automatically detect traffic law violations and issue citations
I’ll conclude by stating that traffic law enforcement in Tucson, Arizona and the surrounding metro area is so lax that violators know they won’t get caught and that only encourages them to violate traffic laws even more. No wonder our auto insurance rates are so high here!
There are three stable (non-radioactive) isotopes of the element oxygen:
16O has 8 protons and 8 neutrons
17O has 8 protons and 9 neutrons
18O has 8 protons and 10 neutrons
All the oxygen in our solar system was forged in stars that existed before the birth of our Sun. The fusion processes that create oxygen from lighter elements require both high temperature and pressure. These conditions exist deep within a star. Different isotopes are created. A nucleus of an atom containing 8 protons identifies it as an oxygen atom, but it is the number of neutrons in the nucleus that determines which isotope it is. Not all isotopes are created in equal abundance.
When the solar system was forming, the oxygen in the “solar nebula” no doubt originally came from various progenitors. A supernova here or there, a planetary nebula somewhere else, and so on. As the solar nebula collapsed to form the Sun and planets, the relative abundance of oxygen to the other elements may or may not have been different in different parts of the solar nebula. Similarly, the relative abundances of the three stable isotopes of oxygen may also have been different in different parts of the solar nebula.
When we measure the relative amounts of the three oxygen isotopes in a terrestrial rock, ocean water, moon rocks, or the solar wind, it may tell us where the oxygen in those materials came from. It may also tell us something about the “life experiences” of the oxygen since the solar system formed. For example, water molecules containing 16O are more likely to evaporate than those water molecules containing the heavier isotopes 17O or 18O. Thus, ground water in the middle of a continent has a higher abundance of 16O than does water in the ocean.
When we look at the solar system today, we find significant differences in the relative abundances of the oxygen isotopes depending on where the material came from. On Earth, 99.75% of the oxygen atoms are of the 16O variety, 0.04% are 17O, and 0.21% are 18O, on average. We see very similar oxygen abundance ratios in moon rocks, indicating perhaps a common origin, but the oxygen abundance ratios in meteorites and solar wind particles are significantly different from this. For example, if you plot the 17O/16O ratio vs. the 18O/16O ratio for a bunch of terrestrial rocks, you get pretty much a straight line. Moon rocks fall along the same line. The calcium-aluminum-rich inclusions (CAI) and iron-magnesium-silicon chondrules in meteorites also form a straight line on this plot, but it has a distinctly different slope.
The solar wind samples collected by the Genesis spacecraft yielded abundances that fall along the same line as the CAIs and chondrules. Mars rocks fall on a line that parallels the Earth-Moon line, but is shifted upwards, indicating that for a given abundance of 18O, the Mars rocks will have a relatively higher abundance of 17O.
If you haven’t yet read Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation by Ismar Volić (Princeton University Press, 2024) I strongly recommend that you do, especially if you live in the United States. It truly is an epiphany, one of the most important books I have ever read. And don’t let “mathematics” in the title scare you. The mathematics in this book is easy and straightforward, and greatly strengthens the validity of the recommendations presented. The book is organized so that a busy person can read just a few pages a day, with ample natural stopping points throughout its 340+ pages. The current edition is hardcover, and can be found for as little as $15. A paperback edition is due out in February.
The author, Ismar Volić, is Professor of Mathematics at Wellesley College and Director of the Institute for Mathematics and Democracy.
What follows here is a high-level introduction to some of the most important topics covered in the book. This is in no way a substitute for reading the book, however. Volić presents many alternatives, the pros and cons of each, and provides many examples, often from recent history. You need to read this book!
Majority vs. Plurality A “majority” means that at least 50% of the voters select the winning candidate. A “plurality” means that whoever gets the most votes wins, no matter how small the percentage. Whenever there is an election with more than two candidates running, the winner often garners less than 50% of the vote in the plurality-based voting used in most U.S. elections. A candidate can win despite the majority having voting against them. This is inherently undemocratic. In this type of voting, we have no idea what a voter’s second choice would have been if their favored candidate does not win. Time and again plurality gives us minority rule.
Why on Earth should candidate A win with only 30% of the vote? This is undemocratic. (p. 30)
Plurality: More disadvantages
Extremely susceptible to external manipulation (e.g. the spoiler effect)
Two-party systems benefit, effectively shutting out other political parties and independents
Ranked Choice Voting (Instant Runoff) A voter is allowed (but not required) to rank two or more candidates so if their first choice is eliminated because that candidate had the fewest votes, their vote then goes to their second-choice candidate, and so on. This process of elimination of the candidate with the fewest votes continues until the winning candidate has received a majority of the votes.1
Supermajority vs. Simple Majority In a two-candidate election, a simple majority (>50%) is the best voting method. A supermajority (some amount greater than a simple majority) is often arbitrary and unmathematical, and it’s inherently undemocratic.
What if you’re electing more than one candidate? Ranked Choice Voting can be generalized if more than one candidate is to be elected. Here’s how it works. Depending only on the number of seats needing to be filled, a threshold percentage is determined. For all those who voted for the candidate with the most votes, only the fraction of your vote needed to meet the threshold is applied to that candidate, and the remainder of your vote goes to your second choice, and so on. For each voter, the fractions always add up to 100%. This method is called Single Transferable Vote.
Important Advantages to Ranked Choice Voting and Single Transferable Vote
Eliminates the “spoiler effect”. You can vote for who you like the best, regardless of their chances of winning, and your vote won’t inadvertently help to elect a candidate you don’t like.
Makes it possible to move away from our current two-party duopoly. Other political parties and independent candidates become viable, thus strengthening our democracy.
Greatly reduces negative campaigning
Greatly reduces political polarization
Increases voter turnout
Eliminates the needs for runoff elections and even primaries, reducing election administration costs and saving taxpayer dollars
Amartya Sen: 1998 Nobel Prize in Economics Individual liberties are incompatible with social needs. A delicate balancing act is needed.
The U.S. House of Representatives
435 seats is completely arbitrary
Until 1913, the size of the U.S. House of Representative has increased with population; since then, it has been frozen at 435 (except temporarily at 437 when Alaska and Hawaii became states)
A bigger House would go a long way toward correcting the multifaceted mess of the 1929 Reapportionment Act
Most of the world’s democracies have a lower chamber size that is close to the cube root of the population; if the U.S. followed that rule, we would have ~700 representatives instead of 435
Increasing the size of the U.S. House would stimulate greater political diversity and would help alleviate the effects of gerrymandering
Apportionment (determining how many House seats each state gets) uses a decent mathematical method known as Huntington-Hill; a slight improvement could be made if we used the Webster method of apportionment
Gerrymandering
Politicians choose the voters rather than the voters choosing the politicians
Independent, non-partisan commissions should determine federal and state congressional districts. After the 2020 census, only four states had independent commissions: Arizona, California, Colorado, and Michigan.
Increasing the size of the House of Representatives and consequently adding more districts and decreasing their size will decrease the ability to gerrymander them
Implementing multi-member districts with proportional representation and single transferable vote will eliminate gerrymandering once and for all; this will necessarily increase the size of the House and usually the size of the districts.
Multi-Member District Considerations
Works best if 3 to 5 representatives are elected for each district
If only 2 representatives for each district are elected, the entry bar is set too high (e.g. no effective challenge to the current two-party entrenched duopoly)
The Electoral College Needs to be abolished. No other democracy in the world has anything like it. An amazing amount and variety of mathematical dysfunctions converge to make the Electoral College wholly unsuitable in a functioning democracy. The President needs to be chosen based solely on the national popular vote.
The Elephant in the Room: Campaign Finance Reform Admittedly, campaign finance reform is beyond the scope of this book, mostly because mathematics has little to offer in support. It is entirely a political problem. Volić’s only comment about this occurs in a footnote on p. 40: “Citizen’s United reversed campaign finance restrictions, giving an upper hand to wealthy candidates.” In my opinion, Citizen’s United was one of the worst U.S. Supreme Court decisions in recent decades, and even calls into question the legitimacy of the Court. A top priority must be campaign finance reform if we are to have a properly functioning democracy. As it is, almost all candidates are bought and paid for by donors and special interests before they even take office. Ideally, all political campaigns should be publicly funded so no candidate has a financial advantage. Level the playing field so that the ideological merits of each candidate, their integrity, and record of public service is used to determine their worthiness, rather than how much propaganda their money can generate.
Concluding Thoughts The democracy reforms needed in the United States are structural in nature. As the world’s oldest still-functioning democracy, the United States is beginning to show its age and is in need of a refresh. Many newer democracies have corrected some of our deficiencies and we could learn a thing or two from them. Thinking that we can effect substantive change only by continuing to feed the “two-headed monster” that is the Democratic and Republican parties is folly.
The ideas presented in this book need to become a regular part of our national conversation. Insist that political candidates publicly support at least once of these reforms, or refuse to vote for them. These reforms are inherently non-partisan. They will improve the quality of life of all citizens, regardless of their political party or persuasion. You can be sure that the most intense opposition to these ideas will come from those who currently hold an inordinate amount of power and wealth, and that don’t want to lose any of their current privileged status. But we vastly outnumber them. If we turn our collective ignorance to knowledge and apathy to action in sufficient numbers, we will succeed.
In very rare cases, RCV instant runoff can result in a plurality rather than a majority election (the candidate wins with less than 50% of the vote). This can only happen when there are a large number of exhausted ballots (i.e. a significant number of voters do not rank candidates, or stop ranking them, say, after two candidates). Worst case scenario: everyone votes for just one person and does not specify a second choice when there are more than two candidates. ↩︎
If you have a neighbor with a dusk-to-dawn insecurity light or a glare bomb that happens to be turned on while you’re engaged in astronomical pursuits, I have come up with a solution.
In the photo above, you will see a Uline 6 × 6′ Welding Screen Shade 8 (H-4610S8). The Shade 8 black vinyl tarp is almost opaque1 and does not have an objectionable odor, so you can store it in the house when not in use. I also ordered the optional set of four swivel casters from Uline (H-5388) so that the frame can easily be moved around by one person. One of these swivel casters is shown in the photo below.
I store the frame outdoors against the house under the patio roof. The canopy weight plates and frame’s minimal wind loading profile ensures that the frame will stay in place even in high winds. Once I roll the frame to the needed location to block the offending light, I use the canopy weight plates to keep the frame from moving while I am observing.
DSG Canopy Weight Plates from Dick’s Sporting Goods come as a set of four, and each one weighs 7.5 lbs. These keep the frame from moving during storage and when in use to block a neighbor’s light.
Since I wanted an easy way to hang the vinyl tarp on the frame and then remove it for storage inside after my observing session, I used four S-hooks from Ace Hardware. I used heavy-duty slip joint pliers to squeeze together the wide part of the S-hook that goes over the top of the frame so that it can’t come off of the frame. See the photo below.
To keep the S-hooks from sliding down off of the top of the frame, I made a couple of stops using Snake Wrap as shown in the photo below.
In addition to using the four S-hooks to hang the tarp, I also use a shoelace through the grommet nearest the top on each side to keep the tarp fully extended during light to moderate winds, as shown below.
In the final two photos, you see how the tarp moves as the wind blows against it. Allowing the lower part of the tarp to move when the wind blows reduces the wind loading on the light-blocking curtain.
Only a tiny amount of light gets through the Uline Shade 8 black vinyl tarp (i.e. you can see where the light is on the other side of the tarp if you look right at it). If the light you are trying to block is extremely bright and you require a completely opaque tarp, if you can’t find a suitable 6 × 6′ tarp with at least four grommets on each side, I’d recommend purchasing a 5′ 6″ × 7′ 6″ Heavy Duty Reflective All-Purpose Weather-Resistant 9 mil Tarp from Harbor Freight. You’ll want to fold the long side at the bottom and hook the flap to the rest of the tarp using cable ties through the grommets so the length doesn’t exceed 6 ft. ↩︎
