Tag: music as math

The Principle of Nested Parentheses

In simplifying algebraic expressions, one starts with the innermost
pair(s) of parentheses, simplifies and reduces it, then works
gradually outwards to the outermost parentheses, i.e. the full

Today, I want to advocate the opposite process for piano
practicing, one that begins, figuratively speaking, with the outermost
parentheses and works inwards until all the details are presented.

A.B. is playing a piece called “Orientale” by Albeniz. We set up our
first, our outermost parentheses, to surround 5, 6 and 7 of the piece.   We left the parentheses empty except for the first chord of measure five
and the first chord of measure seven. Everything in between was omitted. We tried to effect a connection between just these two chord/islands in time, a connection that was crafted to make those two chords in sequence sound musically self sufficient and meaningful. Bear in mind that, as with any good parenthetical statement, the words (notes) inside the curved brackets are of less importance than what lies outside the brackets.

We next subdivided the outermost parenthesis into two two nesting
parentheses. The first nesting parenthesis goes from the beginning of
the fifth measure to the beginning of the sixth measure, the second
from the latter to the beginning of the seventh measure.

Each of the new, nested parentheses is of less importance than the original, surrounding parentheses. Thus the chord at the beginning of measure six is of less importance than either the chord at the beginning of measure five or the chord at the beginning of measure seven. The presence of the sound of the chord at the beginning of measure six should in no way interfere with the way that the chord in measure five connects with the chord in measure seven. This inner chord is not quite trivial, but it is at a different order of magnitude than the other two. This difference in magnitude should be noticeable both in terms of the amount of physical action and exertion used to sound it and in terms of its musical importance.

We continued the process by further dividing each sub-parentheses into
more numerous shorter parentheses. This process continued until, at
the last stage, every note in the original passage is present and
sounding. Gradually all the chords and melody notes appear.* At each
stage the full, or final, picture becomes more and more fleshed
out. The new material added by way of detail is, as in the stage prior
to it, stepped down in terms of the magnitude of physical action and
exertion made to execute it.

In this system the final details, including all the individual notes
in the score which we insert at that last stage are, strangely
enough, the least important. At each stage we discover that we can
make a convincing musical phrase out of just the material constituting
that stage.* Though eventually there will be more notes present, the
notes that are there in each each level sound entire and musical, as
if nothing is being left out – no note or chord missing, each should
note in no way depends for its musical character on any implied notes
we will hear in the future.**

At the final stage, when all the notes are sounding, all the other steps which we have previously enunciated are still “there” in some sense, enriching the overall texture of the passage.

* In one possible stage, we discover, for instance, that playing just the first two of each group of right hand triplet notes, creates its own independent melody without requiring the third note.

** Generally speaking, it is too easy to make connections between two
things that come one right after the other in time. It is their very
proximity that calls our attention to the relation between them. But
who is to say that the current music note in time should not form a
relation with a note that occurs two or more notes later, or later
still. And if there are such medium and long distance relationships,
they are the building blocks of nascently growing organizational
units of the piece until the whole piece is interconnected. As these
units grow longer in time the beginning of the unit is only partially
retained in memory, first as an ‘after image’, and then deeper and
deeper in memory, until they back to mind if they are in some way
reiterated or altered.

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