Comparing the
discussion on geometric distances in Julian Hook’s new book Exploring
Musical Spaces (2023) and the Dmitri Tymoczko classic A Geometry of Music
(2011), we can see that there’s been quite the development on geometric distance
research in the twelve-ish years between the two monographs. The main point of
uncertainty still lies, however, in what ways we can measure the distance between
two nodes in any geometric representation of musical space. I will try to
explore the options in this post.
Let us look at the ways
to measure the distance between sets of pitches in OP-space, since this is the
basis of my current research. There should be no difference between using
OP-space or any combination of OPTIC spaces. I have chosen to use OP-space
since it most easily models individual chords without quality equivalence or
inversion equivalence. Therefore, it is possible to model aural perceptions of
tonal and post-tonal words closely, since our understanding of harmony allows octave
equivalence and honestly, permutation equivalence exists to make my life easier,
without thinking about choral inversions. It closely models the Lewinian
transformational system that came before it, and this new model will have some
of the same limitation as transformational models will too. A new post will come
upon the completion of this paper.
Three main ideas are
touched upon in the Hook (2023) and two ideas (overlapping with the Hook) in
the Tymoczko (2011), to measure distance based on either the maximum distance
in one voice, the Euclidian distance between the two nodes, or the ‘taxicab’
distance, the distance that follows the voice leading pathway between the two
nodes.
Distance measurement
ideas (reproduced from Hook 2023):
And the accompanying
figure (Figure 12.4.1):
So, what works the
best? Tymoczko (2011), in the appendix, talks about how not one measurement
works better than another measurement, and it just depends on what the theorist
is measuring for. Therefore, let us use his definition and apply it to this
paper. For this paper, I am measuring distance to determine tensional profiles,
so let us try all three ways of measuring distance and determine which one best
suits our ears in determining the tensional relationships between chords. For
our test, let us move away from a two-voice graph, as we have in the figure
above, and move to a three-voice graph, as shown below (Figure 11.3.4 in the Hook
2023), to hear the relationships between triads, and not dyads:
Since this graph depicts
all three voice triad combinations, related by semitonal voice leading, any combination
of three notes fits onto this graph neatly, and most important to our test, it can
be measured to determine the distance between any two nodes. It is important to
note that this three-dimensional graph is no different than the two-dimensional
graph above in the reproduced Figure 12.4.1, the node (alpha and beta) containing
the three voices of the chord (Xn, Yn, and now with the addition of the third
voice, Zn) are all the exact same in both versions and function the same as
well. This graph just contains every single three voice combination, so we can calculate
chords and collections aside from major and minor chords. This will be useful
in the future, but just not for our upcoming example. When we perform the calculations
based on the three methods above, we do the same in the two voiced version and
the three voiced version.
Analysis of a
real-life example
Let us take an example
from John Adams’ Nixon in China, Act II, Scene 1, from Pat’s Aria “This
is prophetic”. The first six chords are as follows:
Eb minor à B7 dominant à Eb minor à E major à E minor à C major
Let us further skip
over the four-note chord, B7 dominant for now, since cross-types are a whole other
topic, better discussed at length elsewhere. We shall simplify the B7 into a B
major chord, eliminating the seventh.
Then we can calculate
the distances between all the chords in this progression based on our three
methods shown above. First let’s just number our chord relationships for ease:
Eb minor [1] B (major)
[2] Eb minor [3] E major [4] E minor [5] C major
And to keep our
OP-space, we shall accept octave equivalence and permutations of the chord (any
order of voices in our set), so we will seek the smoothest voice leading, as according
to the three-voice graph above. And thus:
distMAX:
[1] 1
[2] 1
[3] 2
[4] 1
[5] 1
distEuc:
[1] 1
[2] 1
[3] sqrt (6)
[4] 1
[5] 1
distVL:
[1] 1
[2] 1
[3] 4
[4] 1
[5] 1
An interesting pattern
is formed! The chords that share two common tones, and thus the ones that can
be modelled with basic Neo-Riemannian transformations, relation [1, 2, 4, 5], agree
in all three systems. They are separated by one semitone. Since these
relationships [1, 2, 4, 5] maps onto transformations [L, L, P, L] is it obvious
that they share two common tones and therefore the distance between them is the
one tone that moves by semitone. Relation three [3], however, is what we’re interested
in. In the voice leading distance, it measures Eb minor to E major as 4
semitones of distance, in the Euclidian, square root of 6 (~2.45) semitones,
and in the ‘maximum’ calculation, 2 semitones. If we put it in Neo-Riemannian
terms, it is a parallel transformation, a SLIDE transformation, and another parallel
transformation, P-SLIDE-P.
So, which distance
calculation is the best (for this analysis)?
What shall we use? Hook
(2023) says that most theorists choose to use the voice leading calculation in
their work, since it most closely models what we hear. I agree with that. It
closely follows the path that the voices need to travel to reach the next chord.
It also models Neo-Riemannian transformations very well, since each movement in
the voice leading space corresponds to a basic Neo-Riemannian transformation,
so it is just like chaining basic transformations together.
If we go the Euclidian
path, it also makes some sense. Since it is a straight line in our perfectly
organized space, the distance is the shortest path to get from one chord to the
next.
The maximum distance
does not seem to work well with what we are looking at. Since it is a
simplification of the voice leading relationships by only looking at the voice
that moves the furthest, it neglects to model situations where the other two
(or more) voices are moving lesser amounts, but not holding common tones. Therefore,
if we have a chord that holds common tones and a chord that does not, the maximum
distance calculation will not make a difference.
The solution is to
combine some aspects of the voice leading calculation, for it seems to model
the way that the notes travel naturally in our ears, while using the Euclidian path
to model distances in the full phrase. In our example above then, the chords
that ‘sound’ next to each other, like Eb minor to B (major), will be analyzed using
the voice leading distance, since they are truly voice leading from one to the
next. But when looking at the distance travelled from Eb minor (first chord) to
C major (last chord), since these two chords are not sounding next to each
other, we will use the Euclidian path, as it is the straight path to measure the
overall distance travelled in a phrase (or whole section). We can then get the
best of both worlds without sacrificing musical logic!
Issues
Certain issues come up with this analysis. First, there is a difference between OP- and O-space. The fact that the chords are permutable, and I can find the most parsimonious path between the chords means that we are not really measuring the voice leading in this section, but rather the perfect world where the voices go very smoothly. In the above example, the chords are actually rather choppy. This will be expanded upon in the future, where I fine tune this thesis to fit the music a bit better.
The second issue was
touched upon very briefly before. How do we do the cross type? How do we move
from three notes to four notes? We will, as I said before, discuss this in a later
post.
The third issue comes when
we get the distance numbers. When graphed, these relationships are just numbers
that rise and fall according to the relationship between the chords. But if we
wanted to see how the whole phrase functions, in what way can we manipulate this
data to give us the ‘mode’ of the rising and falling distance? Should we also
calculate the distance between the first chord and every other chord? Should we
look at how close the phrase is to tonal ideas and map it that way? Many
avenues are still to be explored in the next post.
