Colour Supplement

Articles by Christians around the world

Sunday 2 September 2007

 

Maps, music, numbers and churches

by Frank Hammond, a member of St. Andrew's Church

 

You can download this article and the diagram as a .pdf file by clicking here.

 

‘Surprised, what possible connection could there be between maps and music?’  Well, if you are unlucky and cannot play by ear then you are obliged to read a picture of music.  There’s the connection, both maps and written music are pictures. ‘What about Churches and other Houses of God?’  Well they are very special and often have their very own symbol on a map.   A church spire is an excellent landmark for hikers.  And, of course, there’s the Church Choir.

 

Finding whole numbers in maps and song is the thread of this story.  Wannabe singers and dancers are often told to count rhythm by numbers: e.g. one, two and three; one, two and three for a waltz; one, two three and four for stirring marching song or any form of four-time.  A whole regiment can keep in step by the rhythm of their marching feet and by counting.  This is not too good, however, when crossing bridges.  The intention here is to turn numbers into music.  Not just the beat but the melody as well.  This will lead us to a solution of a famous conundrum.

 

Please click here to view a figure with the title ‘We will make a home for the Angel Pair 121 on the Baritone Line’. You can toggle between the figure and this article as you read. (Please note - you may need to click on the diagram to see it clearly at full size!)

 

‘Strange figure with an even stranger title’ you might well think.  A small number of readers have been asked what they think it means.  Prior to this study some readers read some words, gave up and never ever look at the figure which contains everything you need to know about the conundrum.  Before reading on it’s worth a guess as to what the picture means.

 

Accountants, in particular, make pictures of numbers. They would recognise it as the height of a column representing the number above it.  They may be foxed by the fact that all the numbers are whole numbers and why aren’t any smaller denominations shown at all.

 

A mathematician would readily recognise it as a sequence of ordinary whole numbers but just two columns are green while the rest are red.  The mathematician would also recognise that all the primes are on the middle line.

 

A cartographer might see that it looks like a section through hilly country or even through a skyscraper skyline.   The cartographer might well deduce that the horizontal lines are contour heights as on an Ordnance Survey Map.

Plus the fact that musical terms are listed at the left, a musician with all the above information would now realise that it’s musical score. In steps, whole notes can be chanted as contour levels.  Level, level, level, down 2, up 2, up again 2, down 2, down 2 and so on.

 

‘But music’ I hear you say ‘is smooth and flowing’.  You have to imagine the smoothness; small and subtle changes have not been included.  In truth, there are more ways to smooth such a figure than there are ways to stroke a cat.  Jazz musicians would readily fill in the detail and do it spontaneously. ‘Remember too, Moses’ staff turned into a snake!’

 

For Ladies’ Barbershop, for instance, the music sheet would be labelled Bass, Baritone, Lead and Tenor as shown.  Lead usually sings the melody. Tenors say ‘they are the icing on the cake’.  Basses provide rhythm and tempo.  Like Tubby the Tuba, Basses occasional want and get the melody but not for long.  These three groups say ‘Baritones are the clever ones’ since they range up from their allotted staff to where Tenors lodge.  This wisdom flavours our story considerably.  All ranges provide Harmony and all is not fixed forever.  If the ladies get it right, they will hear a ring that could break a wineglass, but not necessarily a bridge.

 

Particularly if you are older, you are very likely to know, off by heart, all times tables up to twelve; a very useful asset.  Back to the tables and counting.  Three times table (as indicated) along the top of the music sheet lists odds and evens.  Along the bottom (again as indicated) two times table lists only even numbers.  The number five is a gifted opera singer and can cover the full range (e.g. five, ten, fifteen etc) but only when paired (e.g. multiplied by two or three). Number seven is also gifted and sings like five.  All these gifted singers are in the premier league and line up as lead singers and can be likened to Angels.  They rank along the lead line from smallest to the left by size and carry-on indefinitely to the right growing in size but not in numbers all the time.  It is believed that they play harps.

 

Something odd happens when identical angels pair-up as, for instance, five times five equals twenty-five (25).   Being odd 25 cannot get to the bottom base line.  They cannot together get to the top line since five by five are not divisible by three.  And being twinned they cannot together get back to the melody line.  They haven’t got a proper home: sad to say ‘they are homeless’.  Like Baritones, they are obliged to drift up or down.  ‘Should they settle on the Baritone line near the evens or should they drift up to the Tenor line near one of the odds?’

 

Some philosophers say ‘they can be in two places at once like the number six’.  And they have ‘proved’ it and collected a Nobel Prize as reward.  However, the primes of number six are not identical.  One of them is two with a proper home and the other is three also with a proper home.  Like good friends, one of them can take the other home.  One of the identical twins can be home alone on the Premier Line only when the other is not home.  ‘Are Baritones odd in other ways?’ Well, as every choirmaster will tell you, people with deeper voices are very special, difficult to find and to keep in the choir.  ‘Look what happens to choirboys when they get older!’

 

‘We want proper waves or curves and we shall impose order.  Twenty-five will be given a home on the Baritone line.  A filling of butter-cream if you like.  Forty-nine (seven times seven) will have a home on the Tenor line (marzipan under the icing on the cake) and so on’.

 

‘How has all this knowledge about a heavenly chorus come about?’  Mostly from Art it must be said; and mostly Italian.  They are very good Opera singers too but not such good Soccer players.

 

Knowledge of the heavenly choir has also come about from long years of studying very special maps without bridges; almost a philosophical bridgeless map.  One of these maps is a map of an island and very special since it is known (nay proved) to be a special home for the two times table.  There is another island that is a proven home for the three times table. 

 

There is another sort of island where all the Angels live.  They rarely need to go ‘abroad’.  If invited and escorted by an alien islander the Angels welcome a change of scene.  ‘Perhaps their wings cannot take them far or they fly so high we don’t notice them!’  Angels always welcome visitors from other islands and visitors can settle in the neighbourhood if they wish to stay.

 

For a long time all three islands were thought to have very solid foundations everywhere and so bridges were quite unnecessary.  Also the numbers were not numbers as such but shapes of numbers like triangles, squares, pentagons, hexagons and so on.  Then it was discovered that the foundations were not so solid after all and fault lines were discovered in the 2-times Table Island.  A crack occurs in a square in one of two ways.  Even Angel Island was affected, a pentagon splits apart only one way.  Like a pie, a five-sided polygon can be sliced to the centre five ways.  The fault lines only allow one piece to move leaving the other four as a solid piece.  Now four plus one is five but four minus one is three and so 4-1 is exactly divisible by three. This is not a coincidence.  Fault lines are well known here on Earth and the subject is known as plate tectonics.  Moving plates are responsible for earthquakes.

 

Plates were the way all three islands were related.  Only triangles could not be faulted.  Any polygon exactly divisible by three was either faulted or faultless.  Like a bad apple in a barrel of apples, if a potentially faultless plate has a faulted plate as a neighbour then it too must become faulted.  Now if the island is a Three-Times Table Island (for brevity, a Table3 Island) then the whole island is faultless.  Unlike a Table2 Island which is always faulted.  It was here on Table2 Island that faults were missed since they were masquerading as something of a more innocent looking nature.  In words of a song imitating ‘rings on fingers or bells on toes’.  It was the word ‘ring’ that lacked a clear definition.  Rings were thought to be just ordinary contours.

 

Angel Island is between Table2 Island and Table3 Island. Table2 Island and Table3 Island can be far apart or very close together.  The datum for each Table is already on the music sheet but drawn out as straight staffs.  The number six in the form of a hexagon can live on both Tables: the post-person always knocks (or rings) twice.  A hexagon, for instance, can live both sides because there’s a ‘Bridge of Understanding’ over the staffs.  With a change of perspective the staffs can be likened to a river carrying raw data, music or information.  Alan Turing’s punched paper tape perhaps.  The Bridge can never support any sort of army other than a Salvation Army.

 

A modern computer printer does something that unites peoples of the world.  It can print faster than anyone can read; well almost everyone.  When it prints left to right it prints as Westerners read.  It drops down as Orientals read.  It then prints right to left as Middle Easterners read.  It then repeats the process.

 

‘To visualise the river flowing take the ‘portrait’ picture of music sheet and turn it clockwise to the ‘landscape’ position. Without a lot of effort, a trick that is not too easy to do on a Web Page.  The Track in natural order then becomes; down, down, down, left 2, right 2, right again 2, left 2, left 2 and so on.  Look up, look left and repeat the process.  For convenience, imaginary cuts (fault lines) have been used to fit the pieces on the page or pages.’

 

To see rivers flowing gently and hear birds singing is quite Heavenly. The water eventually reaches the very bottom of the oceans. Someone famously said that he had seen the Promised Land; a land perhaps flowing with milk and honey.

 

Now we have a solution to a well-known puzzle known as the Riemann Hypothesis or ‘Music of the Primes’.  We have built a musical score with whole numbers built with primes.  Rather appropriately for a magazine called ‘Heartbeat’, the Primes are said to have an ‘Irregular Heartbeat’.  Boffins and Techies might get quite excited about it all.  Remember that you heard or read it here first.  ‘And wasn’t it something about most choirboys wanting to leave the choir when their voices break?’  ‘Or was that the wineglass?’

 

Francis D C Hammond 

Previously with the Admiralty’s Hydrographic Office and the Natural Environment Research Council.

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