Harp strings and design

Having previously blogged about the mathematics of the harp, I thought I might say some more about harp design issues. It’s an interesting question that involves both physics and human factors. For simplicity, I’m going to talk about just one string, one playing the note A at 440 Hz. Of course, a real harp will have between 21 and 46 other strings.

The physics of vibrating strings gives us Mersenne’s laws, which tell us that the frequency of a string of length L is (1 / 2L) √ T / μ , where T is the tension force on the string (in newtons), and μ is the density per unit length of the string (in kg per metre).

The diagram below shows the required tension force (in newtons) for a nylon string of various lengths and diameters to play the note A at 440 Hz (click to zoom). A newton corresponds to roughly the gravitational force on 100 grams.

The first, and most obvious, design factor is that too much tension causes the string to break. Setting a design limit of 90% of the expected breaking strength means that the string must be less than 560 mm in length. Interestingly, this limit is independent of the diameter of the string.

The string must also be playable. A string that is too floppy or too tight cannot be effectively played. A rough guide is that the tension should be at least 35% of the expected breaking strength, which means that the string must be at least 350 mm long. Additional limits, which I’m ignoring here, relate to how much room the string needs to vibrate.

Thirdly, the frame can only take so much. If the frame of a 40-string harp is built to withstand 10,000 newtons (roughly the gravitational force on 1000 kg), then the average string has a limit of 250 newtons. This restricts us to the design space on the diagram outlined in red.

Finally, a harp has levers or pedals which shift the strings to be sharp or flat (the basic harp strings correspond only to the white keys on a piano). Those devices set further limits on the design space for strings.


A modern electric lever harp (photo: Athy)

It is interesting to relate this to my other interest, that of solar cars. They are vehicles, which means that they must hold a driver (and new guidelines on World Solar Challenge driver space have just been announced). They are solar, which means that their upper surface must hold a solar panel of specified size. And they race, which means that their aerodynamic drag must be as low as possible. This necessitates a variety of compromises, just as with the design of a harp. The problem is much more complex however; the space of possible solar car designs has many more dimensions than two.


COVID-19 in the UK #2

The chart above (click to zoom) is an updated view of registered deaths in England and Wales according to the ONS up to 26 June. The difference between the red and black lines (highlighted in yellow) indicates deaths where COVID-19 was mentioned on the death certificate. The red line shows that a spike in non-COVID-19 deaths also took place.

Sombre news, but the COVID-19 peak seems to have passed.

Edit 1: Updated chart for more recent data.

Edit 2: The Telegraph is expressing concern at the spike in non-COVID-19 deaths, which seems to reflect under-treatment of cancer and other serious diseases during the lockdown.


COVID-19 in the UK

The chart above (click to zoom) shows registered deaths in the UK according to the ONS up to 10 April (note that during holiday periods, some deaths may be “carried over” to the next week). The year 2020 is on the way to passing 2018 as the worst year of recent times, with the fortnight to 10 April being particularly bad.

The difference between the red and black lines (highlighted in yellow) indicates deaths where COVID-19 was mentioned on the death certificate (this includes deaths “with” as well as “from” COVID-19, although other data suggests that in most cases COVID-19 would be the actual cause of death). A clear COVID-19 spike is visible.

The jump in the red line is also disturbing, however. The the red line shows deaths excluding deaths where COVID-19 was mentioned on the death certificate. The jump in the red line may indicate:

  • COVID-19 deaths where no test was done (unlikely, because the records show only a slight increase in deaths by non-COVID respiratory illness); or
  • deaths from other causes exacerbated by lack of hospital beds; or
  • deaths due to the current lockdown itself (e.g. suicides).

At present, I have no way of deciding which of those three options are the correct ones. Hopefully both COVID-19 and those other factors will pass soon (the IHME model suggests that COVID-19 deaths in the UK reached their peak on 21 April).

I should note that CNBC has also looked at this dataset, but they’ve compared this year against an average period that excludes 2018 and 2019. I don’t know why they did that.


The Circle of Fifths

I have often tried to visualise the circle of fifths in a way that makes sense both musically and mathematically. Above (click to zoom) is my latest attempt.

There are 12 notes in an octave (7 white piano keys and 5 black piano keys), and the diagram shows these 12 piano keys wrapped into a circle. A fifth is a step of 7 semitones (7 piano keys, counting black ones, e.g. C→D♭→D→E♭→E→F→F♯→G). The coloured spiral in the chart shows the “circle of fifths” resulting from moving up a fifth 12 times (moving left to right, and hence moving anticlockwise).

The reason that this works is that 7 and 12 have no common factor – and therefore the first multiple of 7 that is also a multiple of 12 is 7 × 12. Therefore every time you move up a fifth you get a different note, returning to the starting note only when you have moved up 12 times. In the process, you have hit every other note exactly once.


Topic Analysis on the New Testament

I have been experimenting recently with Latent Dirichlet allocation for automatic determination of topics in documents. This is a popular technique, although it works better for some kinds of document than for others. Above (click to zoom) is a topic matrix for the Greek New Testament (using the stemmed 1904 Nestle text, removing 47 common words before analysis, and specifying 14 as the number of topics in advance). The size of the coloured dots in the matrix shows the degree to which a given topic can be found in a given book. The topics (and the most important words associated with them) are:

A better set of topics can probably be obtained with a bit more experimentation. Alternatively, here (as a simpler form of analysis) are the relative frequencies of some Greek words or sets of words, scaled to the range 0 to 1 for each word set (with the bar chart showing the total number of words in each New Testament book). Not surprisingly, angels appear more frequently in Revelation than anywhere else, while love is particularly frequent in 1 John:


World Solar Challenge 2019: even more charts

Adding to my earlier list of World Solar Challenge distance/speed plots, here are 8 more (mostly circulated previously on Twitter). Night stops and notable events are marked on the bottom of each chart in a highlight colour. Control stops are in black.

Michigan traditionally comes third in the World Solar Challenge. They were third again this year. Their chart shows no drama, just fast, steady racing.

Control stop times for Michigan: Katherine: Sunday 12:29:00, Daly Waters: Sunday 16:08:02, Tennant Creek: Monday 12:13:30, Barrow Creek: Monday 15:14:24, Alice Springs: Tuesday 10:02:07, Kulgera: Tuesday 13:42:10, Coober Pedy: Wednesday 10:25:19, Glendambo: Wednesday 14:21:01, Port Augusta: Thursday 9:14:26, Adelaide: Thursday 14:56:00.

Western Sydney, in their beautiful car Unlimited 3.0, battled electrical issues, motor problems, and a wind gust that finally took them out. They still found time to help out Sonnenwagen Aachen on the road south. The photograph in the chart is mine.

Control stop times for Western Sydney: Katherine: Sunday 12:55:00, Daly Waters: Sunday 16:59:06, Tennant Creek: Tuesday 11:51:31.

There was no such drama for ETS Quebec (Éclipse), just steady consistent driving, finishing as best Canadian team, 2th North American team, and 9th in the world. That’s why they received my consistency gem.

Control stop times for Éclipse: Katherine: Sunday 13:27:04, Daly Waters: Monday 8:55:47, Tennant Creek: Monday 16:08:23, Barrow Creek: Tuesday 11:13:27, Alice Springs: Tuesday 16:10:27, Kulgera: Wednesday 11:59:00, Coober Pedy: Thursday 9:48:25, Glendambo: Thursday 13:56:55, Port Augusta: Friday 9:32:09, Adelaide: Friday 14:21:48.

Swedish team Jönköping University (JU) also had plenty of drama. They were forced to stop under cloudy skies with a flat battery and they needed an overnight repair. But they still finished tenth!

Control stop times for JU: Katherine: Sunday 12:51:56, Daly Waters: Monday 8:07:49, Tennant Creek: Monday 14:31:05, Barrow Creek: Tuesday 9:41:27, Alice Springs: Tuesday 14:13:37, Kulgera: Wednesday 12:39:00, Coober Pedy: Thursday 9:53:47, Glendambo: Thursday 13:51:40, Port Augusta: Friday 10:04:55, Adelaide: Friday 14:44:20.

Antakari had a smooth and largely uneventful race, apart from a couple of stops of a few minutes each. The GPS track shows them hunting around for a good campsite each night. They finished 7th (just ahead of NITech).

Control stop times for Antakari: Katherine: Sunday 13:15:43, Daly Waters: Monday 8:56:38, Tennant Creek: Monday 15:06:40, Barrow Creek: Tuesday 9:55:51, Alice Springs: Tuesday 14:17:59, Kulgera: Wednesday 10:34:05, Coober Pedy: Thursday 8:45:34, Glendambo: Thursday 12:58:06, Port Augusta: Friday 8:33:08, Adelaide: Friday 13:07:11.

Nagoya Institute of Technology (NITech) also had a smooth and largely uneventful race, finishing 8th (just behind Antakari).

Control stop times for NITech: Katherine: Sunday 12:56:50, Daly Waters: Monday 8:06:31, Tennant Creek: Monday 14:42:02, Barrow Creek: Tuesday 9:38:31, Alice Springs: Tuesday 14:40:56, Kulgera: Wednesday 10:22:50, Coober Pedy: Thursday 8:45:20, Glendambo: Thursday 13:01:29, Port Augusta: Friday 8:38:35, Adelaide: Friday 13:24:10.

The team from Durham University crossed Australia on solar power, in spite of minor electrical problems (they are the first UK team to do so for many years). Unfortunately they only managed around 2830 km, not quite reaching Adelaide. In the past, cars have been permitted to drive on Saturday mornings, whereas this year, cars had to cease driving on Friday evening. Judging from the graph, Durham might not have realised this for the first few days.

Control stop times for Durham: Katherine: Sunday 14:26:58, Daly Waters: Monday 10:34:22, Tennant Creek: Tuesday 9:39:42, Barrow Creek: Tuesday 13:45:32, Alice Springs: Wednesday 10:53:29, Kulgera: Wednesday 15:59:45, Coober Pedy: Thursday 14:36:36, Glendambo: Friday 10:01:30, Port Augusta: Friday 14:42:19.

Swedish newcomers Chalmers Solar Team managed two control stops, but were slowed significantly by the hilly terrain in the first part of the route. They therefore trailered at around 735 km.

Control stop times for Chalmers: Katherine: Sunday 14:56:54, Daly Waters: Monday 12:49:32.


World Solar Challenge 2019 Revisited: some additional charts

Revisiting the World Solar Challenge, the chart below shows distance/speed plots for seven WSC teams (for other teams I was either missing some GPS data, or did not have access to explanatory social media). Distances shown are road distances (not geodesic), while speeds are estimated from distance and time information (because speeds were not included in the GPS data that was kindly supplied to me). As a result of data limitations, the compulsory 30-minute stops at Katherine etc. are shown by sharp speed dips, but not necessarily ones that drop all the way to zero.

Vattenfall (3) had a devastating battery fire; Top Dutch (6) were the best new team, finishing 4th; Agoria (8) won the Challenger class; Twente (21) tragically crashed while in the lead; Sonnenwagen Aachen (70) were one of two teams to come back from a serious crash and still finish; Blue Sky (77) were the 11th and last Challenger team to reach Adelaide on solar power; and Kogakuin (88) were the other team to recover from a major crash.

Night stops in the chart above are marked in red. Photos are from their tweet posted minutes before the fire and from a tweet posted shortly afterwards.

Control stop times for Vattenfall: Katherine: Sunday 12:19:21, Daly Waters: Sunday 15:59:21, Tennant Creek: Monday 11:45:23, Barrow Creek: Monday 14:51:01, Alice Springs: Tuesday 9:30:33, Kulgera: Tuesday 12:51:41, Coober Pedy: Wednesday 8:39:31, Glendambo: Wednesday 12:40:58, Port Augusta: Wednesday 16:44:32.

The chart shows Top Dutch’s multiple stops just out of Tennant Creek with battery problems. Top Dutch finished 4th, and won the WSC Excellence in Engineering Award.

Control stop times for Top Dutch: Katherine: Sunday 12:16:27, Daly Waters: Sunday 15:57:03, Tennant Creek: Monday 12:12:55, Barrow Creek: Monday 15:28:35, Alice Springs: Tuesday 10:39:32, Kulgera: Tuesday 14:41:40, Coober Pedy: Wednesday 12:00:43, Glendambo: Wednesday 15:48:50, Port Augusta: Thursday 10:47:39, Adelaide: Thursday 15:30:00.

Night stops in the chart above are marked in blue. Agoria had a virtually perfect race, winning the Challenger class. Visible in the chart after Coober Pedy is the 80 km/h speed limit imposed by the WSC on Wednesday morning after wind gusts caused multiple crashes.

Control stop times for Agoria: Katherine: Sunday 12:17:04, Daly Waters: Sunday 16:05:54, Tennant Creek: Monday 11:55:56, Barrow Creek: Monday 14:56:40, Alice Springs: Tuesday 9:46:28, Kulgera: Tuesday 13:10:50, Coober Pedy: Wednesday 9:22:36, Glendambo: Wednesday 13:05:06, Port Augusta: Wednesday 16:51:59, Adelaide: Thursday 11:52:42.

Twente’s tragic crash (due to a strong wind gust) occurred at about 2165 km from Darwin, just before Coober Pedy. Photos are from a tweet posted the day before the crash and from a tweet posted shortly afterwards. I was one of the people that signed the car after the crash. Twente won the Promotional Award, for their excellent media.

Control stop times for Twente: Katherine: Sunday 12:08:43, Daly Waters: Sunday 15:32:39, Tennant Creek: Monday 11:33:01, Barrow Creek: Monday 14:31:33, Alice Springs: Tuesday 9:17:16, Kulgera: Tuesday 12:40:40.

Sonnenwagen Aachen stopped for five hours to repair their car on Wednesday, just before Coober Pedy, after the car was blown off the road (their first priority was the driver, of course). There was another stop between Glendambo and Port Augusta, due to a broken shock absorber that had been damaged in the crash (Western Sydney Solar Team kindly helped get them back on the road). Sonnenwagen Aachen finished 6th. They also won the Safety Award and and the Spirit of the Event Award (for not giving up).

Control stop times for Sonnenwagen Aachen: Katherine: Sunday 12:27:34, Daly Waters: Sunday 16:09:06, Tennant Creek: Monday 12:17:32, Barrow Creek: Monday 15:12:27, Alice Springs: Tuesday 9:52:06, Kulgera: Tuesday 13:31:00, Coober Pedy: Wednesday 15:08:20, Glendambo: Thursday 9:35:27, Port Augusta: Thursday 14:49:06, Adelaide: Friday 10:03:48.

Blue Sky (Toronto) had several brief stops of a few minutes (including for electrical issues on Monday), but no particularly dramatic events. They were also slowed somewhat by clouds on Wednesday morning. Blue Sky finished 11th (the last Challenger team to reach Adelaide on solar power).

Control stop times for Blue Sky: Katherine: Sunday 12:49:29, Daly Waters: Monday 8:03:38, Tennant Creek: Monday 14:42:34, Barrow Creek: Tuesday 9:54:44, Alice Springs: Tuesday 14:23:40, Kulgera: Wednesday 11:00:30, Coober Pedy: Thursday 10:34:55, Glendambo: Thursday 15:01:18, Port Augusta: Friday 10:53:25, Adelaide: Friday 15:47:10.

Kogakuin was forced to stop with an overheated motor just after Katherine. They also crashed twice due to strong winds. The second, more serious, crash was due to a mini-tornado or willy-willy just before Glendambo (see their report here), and required overnight repair in town on Wednesday night. Kogakuin finished 5th. They won the CSIRO Technical Innovation Award, for their hydropneumatic suspension. Their dramatic after-race video is here.

Control stop times for Kogakuin: Katherine: Sunday 12:18:43, Daly Waters: Monday 8:03:13, Tennant Creek: Monday 13:16:26, Barrow Creek: Monday 16:15:52, Alice Springs: Tuesday 11:03:53, Kulgera: Tuesday 14:32:15, Coober Pedy: Wednesday 12:04:23, Glendambo: Thursday 9:45:18, Port Aug: Thursday 14:18:47, Adelaide: Friday 9:53:00.

For comparison, here is the distance/time chart I did before. In that analysis, higher means slower, and the arrival times (in Darwin time) can be read out on the right:


World Solar Challenge Cruiser Scores Revisited

Above (click to zoom) are the World Solar Challenge 2019 Cruiser practicality scores. Image credits are as per my illustrated list of teams. That list also includes practicality score guesses – but I did a poor job of guessing (a correlation of only 0.75 with actual scores). That reflects the change in practicality scoring this year, with several of the less subjective elemehts removed. Four cars this year had higher practicality scores than when they last appeared:

  • Minnesota (team 35) from 57 to 76.3 (a surprisingly large change, although it is a very nice car)
  • Flinders (team 14) from 46.7 to 63.7 (this team did make many improvements)
  • HK IVE (team 25) from 45.2 to 63.4 (this team did make many improvements)
  • Lodz (team 45) from 79.1 to 82.4 (a surprisingly small change, given the extensive improvements)

Two cars had lower practicality scores than when they last appeared:

  • Bochum (team 11; they fielded this car in 2015) from 80.5 to 75.1 (a surprising score, given the fantastic interior)
  • Sunswift (team 75) from 80 to 69.5 (also surprising, although they did take the rear seats out)

The chart above (click to zoom) shows final Cruiser scores (including practicality) in the style of my previous chart. The score is calculated as a product S = D × H × (1 / E) × P × 0.99l. The chart shows the components of the score on a logarithmic scale, so that multiplying and dividing score components corresponds to adding and subtracting bars. For each team, there are 6 bars (the 6th bar, in a darker colour, is the total score):

  1. The distance travelled in km (D). Teams given credit for completing the entire course score ahead of others.
  2. The weighted average number of humans (H) in the car (so that the product D×H is the number of person-kilometres). A small tick mark above the bar shows the number of seats in the car, which is the maximum possible value of H for that team.
  3. The nominal external energy usage (E) in kWh (initial battery capacity, plus metered charging along the way). This bar is negative, because we are dividing by E.
  4. The practicality score divided by 100. This bar is negative, since the highest possible value is 1.0 (this means that longer bars mean lower values). A tick mark under the column shows the lowest practicality score across all six teams, which is 0.534.
  5. The lateness factor (0.99l), where l is the number of minutes of late arrival, plus the number of demerit points.
  6. The total score (S). The score itself is shown over the bar. It can be seen by inspection that this bar is the sum of the others.

Well done, Eindhoven!


World Solar Challenge Cruiser Scores

The chart above (click to zoom) shows scoring for Cruiser class cars arriving at Adelaide, in a modified version of my “tuning fork” style. The score is calculated as a product S = D × H × (1 / E) × 0.99l. The chart shows the components of the score on a logarithmic scale, so that multiplying and dividing score components corresponds to adding and subtracting bars. For each team, there are 6 bars (the 6th bar, in a darker colour, is the total score):

  1. The distance travelled in km (D). Teams completing the entire course score ahead of others.
  2. The weighted average number of humans (H) in the car (so that the product D×H is the number of person-kilometres). A small tick mark above the bar shows the number of seats in the car, which is the maximum possible value of H for that team.
  3. The nominal external energy usage (E) in kWh (initial battery capacity, plus metered charging along the way). This bar is negative, because we are dividing by E.
  4. The fourth place, labelled P, is reserved for the incorporation of practicality in the final score.
  5. The lateness factor (0.99l), where l is the number of minutes of late arrival, plus the number of demerit points.
  6. The total score (S). The score itself is shown over the bar. It can be seen by inspection that this bar is the sum of the others.

Well done, Eindhoven!

Update 1: I note that Sunswift’s car has been modified to have 2 seats, rather than 4.

Update 2: See here for an updated chart.