Separation: we'll meet again

Submitted by Simon on Sat, 12/12/2020 - 10:18

With the Five o'Clock Handbell Club, we have been trying to ring another quarter of Bristol Maximus. I rang 3-4 in the first one, but now I have switched to 7-8. When I started practising with Mabel, I found it really difficult - so much so that it almost felt like ringing a different method, in comparison with the 3-4 and coursing positions. I started wondering what makes 7-8 so difficult.

The first point is that the bells get a long way apart. I have written previously about calculating the average separation of a pair of bells in a particular method, as a way of assessing difficulty. (The separation is the number of other bells that ring between your pair in a  specified row, and then we can average over the whole plain course). The greater the separation between your bells, the more you have to think of their positions separately, rather than focusing on one bell and thinking about how far away the other bell is (as we do when coursing: ring one bell, let someone else ring, ring the other bell). The previous article focused on surprise major, but here are the tables of average separations for each pair in some of the royal and maximus methods I have been ringing recently.

Royal
Method 1-2 3-4 5-6 7-8 9-0
Cambridge 2.67 2.44 3.38 2.91 1.93
Yorkshire 2.67 2.34 3.78 3.29 1.26
Lincolnshire 2.67 2.41 3.67 2.91 1.68
Bristol 2.67 2.42 3.86 3.39 0.98
Maximus
Method 1-2 3-4 5-6 7-8 9-0 E-T
Cambridge 3.33 2.62 4.17 4.32 3.37 2.19
Yorkshire 3.33 2.49 4.44 4.83 3.65 1.25
Lincolnshire 3.33 2.54 4.36 4.67 3.38 1.71
Bristol 3.33 2.55 4.50 4.91 3.73 0.98

A few observations stand out.

  • In every method, the average separation for a pair corresponds roughly to the distance between that pair in the coursing order. In royal, 5-6 have the largest separation, and in maximus it's 7-8.
  • The separation of the tenors in Bristol is significantly lower than in the other methods. It's true that the tenors follow each other closely in Bristol, but this does show that average separation isn't the only measure of difficulty.
  • The total of all the numbers across each row is constant and independent of the method (depending only on the number of bells). This means that the unusually low separation of the tenors in Bristol must be compensated by higher separations of the other pairs.

However, I don't think average separation is the only statistic worth looking at. When I started ringing 7-8 for Bristol Maximus, my bells seemed often to be moving in opposite directions, and this seemed to be part of the difficulty. This is consistent with the observation that learners often find 3-4 difficult in plain hunting on six. So let's calculate the percentage of changes at which the separation between the bells changes. Why do I think this is interesting? Considering plain hunting, for the tenors, the separation is always 1 except when the bells come together at the front and back. So the separation only changes occasionally, and the percentage of separation changes is small and becomes smaller the more bells we are ringing. This corresponds to the idea that on more bells, we can settle into the general pattern of plain hunting and think less frequently about how the bells "bounce off" the ends of the row. In contrast, when plain hunting in the opposites position, the separation almost always changes, except when the bells are leading and lying and when they come together and cross. So the percentage of separation changes is large and becomes larger on more bells.

Here are the tables of percentage separation changes for the same royal and maximus methods.

Royal
Method 1-2 3-4 5-6 7-8 9-0
Cambridge 52.78 53.89 61.67 55.00 45.56
Yorkshire 52.22 57.22 62.78 61.11 34.44
Lincolnshire 51.67 54.44 62.78 55.56 42.22
Bristol 53.06 58.61 65.28 61.94 39.72
Maximus
Method 1-2 3-4 5-6 7-8 9-0 E-T
Cambridge 52.65 49.62 62.50 62.50 53.79 41.67
Yorkshire 52.27 52.65 66.29 61.74 57.95 30.30
Lincolnshire 51.52 47.73 67.05 65.91 53.03 32.95
Bristol 53.03 51.52 66.67 65.91 62.12 34.09

My hypothesis is that bigger numbers in these tables also correspond to greater difficulty. The observations are a little different from those for average separation:

  • There isn't quite the same correspondence with distance between the pair in the coursing order. For maximus, 5-6 have a higher percentage separation change than 7-8 in most of the methods.
  • The tenors for Yorkshire come out as the easiest pair. Although they have a higher average separation than in Bristol, there are longer periods when they are coursing in a treble bob hunting pattern, and there aren't Bristol's "twiddly bits" when the bells pass the treble and the separation keeps changing in an intricate little pattern.

A final point about 7-8 for maximus is that the working pairs (3-4 and 5-6) can't get into that position in a tenors-together composition. This is because 7-8 are separated by four bells in the coursing order: ... 8 0 T E 9 7 ... but in a five-bell coursing order on 2 3 4 5 6, the maximum distance apart is three bells. If you imagine conversation after a peal, 5-6 might say "I was grateful to have a little rest when I got into the coursing position, and I found the 9-10 position tricky", but 3-4 will never say "Wow, being thrown into the 7-8 position was really difficult". Bearing in mind the current Bristol Maximus project that I mentioned at the beginning, I think I will try to emphasise the lonely and heroic role of 7-8.

Comments

1-2 are more difficult than the tenors, according to either average separation or percentage separation change. In the maximus methods, 1-2 are also more difficult than 3-4, but curiously for the royal methods, 3-4 have slightly higher percentage separation change.

Submitted by Simon on Sat, 12/12/2020 - 17:00