increpare games

Below’s the content of a rambling email I sent to mathstuff, thought the description good enough to put up here. Note that there are many other ways to bring covering spaces into music theory – the whole of mazzola’s theory is based around coverings and symmetries in fact, but the below example is probably the simplest.

From xxxxxxx@maths.tcd.ie Sat Jun 4 21:31:22 2005
Date: Sat, 4 Jun 2005 01:42:18 +0100 (IST)
From: Stephen Lavelle
To: mathstuff@maths.tcd.ie
Subject: combinatorics/music ramblings

[description of a specific computational problem, nothing too interesting…snipped]

This just came up when doing some harmony calculations – the set up is
quite pleasant actually ; you have your base-set of notes N, a scale S
which is a subset of N.

You define your consonant intervals (say major and minor thirds and
major fifths, augmented fifths, and tritones), and you form a graph whos vertices are the notes of N, such
that there is an edge between two notes whenever the interval they form is
consonant.

In the case of the major scale, this provides you with a rather amusing
(triangulated surface homeomorphic to a) mobius strip. If you look at
one of Riemann’s harmonic theories and interpret it geometrically, it is
equivalent to fixing an orientation on the strip ;) (of course, Riemann’s
theory makes sense – that it is logically inconsistent is just an artifact
of its verbalization (though I guess the analysis did clarify where
exactly clarification was needed…maybe…)). [But it’s not that cool
when you think about it further – there are much simpler ways to look at
it non-geometrically].

Now, from this graph you can construct the triads (for our specific case
major minor augmented + diminished chords)- that is whenever you
have a triangle (indicating that the three notes are all consonant with
eachother when sounded), the three points involved in their connection
forms a triad.

Ideally you want every note in your scale to be part of *some* triad
(otherwise … well otherwise it’s not really considerable
from a harmonic perspective). If this condition holds, then you can view
your set of triads as being a covering of your scale (that is, every note
is part of some triad, so you can view the triads as hovering in almost
some sublime way above the scale), called, aptly enough, a triadic
covering.

Now, you can construct the *nerve* of this covering (as you can with any
covering), the nerve is a graph associated to a covering, it is a general
topological construction useful in homology + cohomology, and goes as
follows: you take as your set of vertices all the triads, and there is an
edge joining two triads whenever they have some notes in common.

This may seem a bit shite, but it can actually be quite enlightening to
look at (and a practical way of visualising the chord structure on any
particular scale) – though, alas, for many scales the graphs are very high
dimensional (which sucks).

Now, back to something else: you look at all transpositions of N
(which induce transpositions in S, C(N), C(S)). The first thing to
figuring out cadences is to find these minimal sets of notes that
classify a scale up to transposition (that is, when you hear it, you know
for certain where your scale is – scales ambiguous in this way were
well-loved my Messian + co. …this effect also contributes to the
unsettlingness of the wholetone scale), and then, to find the cadence sets
you have to find, for each cadence set, the minimal sets of chords that
cover it (there may be several).

And that’s what I’m trying to program in at the moment (note that though
what I’m doing is a little nice, it’s not a completely accepted theory yet
(and it’s also not my theory, though I’ll try to generalize it a bit some
other time (there are some generalizations of the triadic coverings that
allow one to incorporate some jazz theory (which involve abandoning S
altogether)) :) , but nobody’s generalized the modulatory theory to go
along with it yet ).

Right…so…yeah…that’s my rambling for today.

ciao.

Stephen

So a Croatian walks into a bar, and the bartender says, “We don’t serb
your kind!”