Columbus, Riemann, Bach and Wagner: what do they have in common?

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I have an indifferent voice, at best, but just this evening I was visiting my brother, and got drawn into singing in the choir for a wedding!  This is a thrill, since we atheists rarely get invited to sing in church choirs.  Luckily for me, this was an amateur choral society, and they were short of male voices.

Everyone has their own way of getting their head in the choir game.  Some folks need to learn the melodies.  Some can simply read off from the notes.  Some have to hear their place in the rest of the choir sound.  Some have to understand the chord for each note they sing.  Me, I have to understand the geography of the piece.

Back in the old days of plainchant, the tune stuck to just a few notes, and once you knew the Home Note (the so-called tonal center, or Tonic), you learned all your notes relative to that one.  They could have called the Home Note 1, and then numbered the remaining notes 2, 3, 4 and so on, and of course note 8 was (essentially) the same as note 1, but an octave higher, so you could call it 1', and the next note 2', and so on.  (You have a problem going downwards.  You could call the note below 1 by 7*, the next lower one 6*, all the way down to 1*, etc, etc.)

As many of us know, the notes were actually called "doh, re, me, fa, soh, la, ti, doh," etc, as charmingly described by Maria in The Sound of Music.  This system is called Tonic Sol-Fa.  Here is an account in Wikipedia.  (Actually, there seems to have been a medieval verse of music of eight lines, the first line starting on the home note, and each successive line starting on the next higher note.  The lines are said to have started with the syllables Ut, re, mi, fa, so, ... , but someone replaced Ut with Do.  Probably got canonized for this innovation.  If anyone can give me a reference to a performance of this verse, I shall be eternally grateful.)

[Added later: Here is a depiction of the original latin hymn to which I referred, and an explanation by Mr Neil Hawes: 

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Centuries passed, and by the time Palestrina was composing (Giovanni da Palestrina -- probably not his real name, but I could be wrong) harmony had been discovered, and people were singing many melodies simultaneously, creating a tapestry of sounds at a single instant.  This meant that a given note could be harmonized in many ways.

As a side-effect, it became possible --and desirable!-- to change the home note temporarily, to create tension, and the feeling of travel.  Many tunes depart from their home note by the second sentence, to coin a phrase.  By the time Bach came around, the skill of changing home notes was not just one of many procedural techniques for composers, but almost their most important stock in trade.

How was music notated, once the home note was changed?  The new Doh could be the old Soh, or practically any note!  To be honest, I don't know how this situation was dealt with in the world of Tonic Sol-Fa, as the Do-re-mi system is called.  But Bach, and even many composers before him had begun to use an absolute notation system that simply showed the notes, and new notes that were required because of moving away from the Home Note were indicated with symbols (so-called accidentals: sharps and flats that were inserted right in front of the note).

So when we're singing along, and a whole lot of accidentals suddenly crop up, of course we know that we're traveling, and incidentally, what note to sing, as well.  But for those of us who are hard to please, we need to know where we are relative to the original key; hence my remark about the geography of the music.

Several months ago, I described a system used by Stephen Malinowski which illustrated relationships between notes.  He actually used several methods, I'm referring to one of them: The Interval Lattice.  Here it is:

Here, the Home Note is represented by 1.  You could think of it as C major, if you like.  Surrounding it are six other notes: counting counter-clockwise from the note immediately to the "east" is 5, or G major; 3, or E major/minor; 6, or A major/minor; 4, or F major; 5#, or 6Flat, G Sharp or A Flat; and finally 3Flat, or E Flat major.

Of course, some folks consider every note to be related to every other note.  This particular scheme gives priority to these 6 notes, but if you take their immediate neighbors, you get all the possible notes included, at (at most) a distance of one (additional) note away.

To help out novices, I should say that some of these 6 notes are closer than others; actually G major, F major, and A minor are the most closely related.  These are most often the first destination when Bach, for instance, leaves home.  Next favorites are D minor and E minor, closely followed by G minor, and C minor.  (By this time, you're seeing that this lattice is not that helpful, really!  Maybe some 3-dimensional lattice might be of help...)  So when I say that I want to know where I am, I don't mean simply which new Home Note; I want to trace the sequence of Home Notes that got me where I am.

Where do Riemann, Columbus and those people come in?

It has to do with maps.  A map, of course, is an abstract representation of a geographic area.  (We've obviously gotten away from thinking of a map as a representation of The World --here be dragons, etc, etc.  Maps of The World are, at best, laughable approximations, because adjacent points are often at opposite ends of the map.)  If we think about it, our system of scales is a map from a set of sounds into the numbers 1, 2, 3, ... , 7.  Mathematicians and geographers soon cottoned onto the fact that any shape other than a plane, such as the Earth, cannot be represented by a single map.

One of the big things-to-do in mathematics is to calculate gradients, such as the pressure gradient, or the temperature gradient, and so on.  A lot of physics is all about gradients, but we math folks own the idea of gradients.  It so happened that everyone knew how to do gradients when using a single map.  Riemann's brilliant contribution to mathematics was to show how to do gradients while using a patchwork of maps!  By and by, Einstein came along, and said that all maps were equally good for physics (though of course some maps were a little more equal than others).

[Added later: As to the mathematics-- A surface such as a cylinder or sphere (or Möbius Strip, for that matter) cannot be represented by a single map.  So you need several maps to represent the whole thing, and these maps must overlap, and there has to be a minimal degree of smoothness in the equations that connect the two maps in the overlap.  These sorts of surfaces, together with all their maps, are called Manifolds, for the simple reason that multiple maps are necessary.  So, all manifolds are locally Cartesian; in other words, a small neighborhood of every point is one at least one of the maps.  Compare this with the remark that Wagner's music is locally diatonic in the following paragraphs.  Later composers have written music that is totally atonal, that is, there is no tonal center at any moment.  In fact, they go out of their way to destroy any feeling of tonality --any feeling that there exists a Home Note, even a temporary one-- in the entire piece.  I must confess that atonal music is not satisfying to me personally, but I might have enjoyed a few seconds of it while my defenses were down.]

Then along came a fellow called Richard Wagner.  I doubt whether he knew anything about Riemann or Einstein, but he was writing music that seemed to completely disregard the home note.  At any given instant in time, one can hear an implied home note.  But unlike Bach, Handel, Mozart and Beethoven, all of whom traveled widely in respect to their home notes, but who always took the time to travel back to the home note, Wagner sometimes set out from the home note, never to go back again in the same piece.  Like Riemann, though, who insisted that every point should be on some regular map that looked more or less like the X-Y grid, Wagner's music is, in a small neighborhood of any moment in the music, like the music of Bach, which is to say, locally diatonic.  This makes Wagner's music easy to apprehend, so long as you did not have a driving urge to "go back home" to the initial Home Note.  [Wagner changed home notes --modulated, which is the technical term for it-- based on the imperatives of the melody he was building.]

Many Wagner tunes are perfectly diatonic.  Some of them, however, are not.  So Wagner is, for music, something like what Riemann (or Einstein) was to mathematics (and Physics).

So, you see, geography is not too far removed from music, conceptually.  To conclude, Merv Griffin's tune used in Final Jeopardy (named by Griffin Think) is an instance of a tune that leaves home in one direction, and arrives home from the opposite direction, having gone completely around the world!

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