Here's a new way of illustrating musical form that connects repeating sections in the music with circles:
Seeing Sequences
From melodies to text to DNA, many interesting types of information come in the form of a sequence of symbols. Although it's natural to try to
analyze these sequences graphically, they present a unique challenge for data visualization. A good visualization method should provide a broad
overview of the "shape" of the data--yet with sequences, the only structure ab initio is the completely uniform micro-level connection of each
symbol to its immediate neighbors.
This page describes a technique, the matching diagram, designed to deduce and display macro-level structure in long sequences. (If you want to skip
this description, go directly to these pictures of music.)
Matching diagrams are based on the fact that sequences often represent a hierarchy of ideas. Melodies, for instance, are usually based on combinations
of smaller repeated musical passages; text has repeated words and phrases. A natural way to find macro-level structure, therefore, is to use these
repeated units as signposts.
Unfortunately the human eye is bad at spotting repetition. (Try to find the longest repeated subsequence in 28746391479735648274639137. It's
irritating.) Matching diagrams avoid this difficulty by having the computer find the repetitions and highlighting them explicitly.
www.bewitched.com...
That page goes on to show how they illustrate that ugly number:
So they apply it to some musical pieces to get a sense of all the structure of the piece, from the "micro" level of individual notes repeating, to
larger passages of music repeating in their entirety. Repetition is what gives music a reference point, so that a "structure" based on some kind of
pattern can emerge. That doesn't stop people from writing music that never repeats at all, it's just that most musicians seem to use repetition
anyway.
Mary Had a Little Lamb:
Beethoven's Fur Elise:
The next diagram visualizes Beethoven's Fur Elise. Again, matches are based on equality of pitch; where chords occur we consider only the top
note. Despite this extremely limited definition of musical similarity, the resulting matching diagram reveals an intricate and beautiful structure.
The picture shows how the piece begins and ends with the same passage, while a longer version of that passage repeats throughout at increasing
intervals. You can also see a long stretch in the second half where that passage is not repeated at all and whose structure looks distinctly
different, which corresponds well to what you hear when you listen to the music.
Bizet's Toreador from Carmen:
Not all pieces show as much exact repetition as Fur Elise. For instance, the "Toreador" song from Carmen, diagrammed above, looks completely
different. Instead of a few long passages repeated exactly over and over again, it contains many repeated smaller phrases.
Bach's Minuet in G Major:
As a final example, consider Bach's Minuet in G Major. The diagram shows that the piece divides into two main parts, each made of a long passage
played twice--or what a musician would call an "AABB" structure. It's not surprising to see this in a minuet, which shows that the matching diagram
is picking out structures that correspond to conventional notions of musical form.
The diagram, however, provides much more detailed information than the simple "AABB" notation. For instance, you can see that the A and B passages
are loosely related, as shown by the bundle of thin arcs connecting the two halves of the piece. And the fact that the two main arcs overlap shows
that the end of the A passage is the same as B's beginning.
This is an old-school "AABB" aka "binary" form that consists of 2 parts that are loosely related, if at all, to each other in musical
character/content. You can see the two distinct sections as two big circles in their illustration. And to connect them, since they didn't share
much in common at the transition directly between them, you see how Bach worked in what looks like the structural equivalent of a band-aid to make
them seem more related anyway (not meant to diminish Bach in any way).
I just thought that was interesting and worth sharing, related to forms and structure within music which is basically what I mean when I say
"grammar" in music.
The best thing about music theory is how open to interpretation it has to be by necessity. You can take any "rule" about what something in a piece
of music "is" and turn it on its head a hundred different ways.
I was strapped for time when I posted the OP because I had to run out, but I meant to elaborate on the pentatonic scale.
5 notes.
This scale or very close approximations of it have been found on flutes of indigenous people all over the world, even in ancient artifacts dating back
to the birth of known civilization.
The reasons for these particular notes becomes clear in the mathematics, where you can see that these notes are actually the first overtones produced
by a single note, because they resonate at very harmoniously ratios with the fundamental frequency.
When you pluck a guitar string, for example, if you watch it, it's hard to even comprehend how exactly it's moving because it just looks like a
vague blur. But based on the harmonics the string produces (the actual sound wave it produces, which you can break down into various frequencies on a
computer), it's actually vibrating at all the frequencies such as what you see above simultaneously. The whole string is vibrating up and down, but
while it does so, there are other vibrations of various frequencies vibrating along it as well. And only the frequencies that resonate best with the
vibration of the entire string survive, because all the others die off quickly from attenuation.
(Btw that last sentence exposes one of those truths about reality that make me consider music theory to be possibly the best spiritual study of
all..)
The major pentatonic has an equivalent minor pentatonic scale that makes up basically all blues and classic rock music and a lot of metal, etc., not
to diminish that,either.
But anyway these notes all have a definite relationship with each down to their very vibrating cores (resonate ratios with one another). And we even
speak with inflections and suggestions of these intervals, especially when our speech is laden with emotion. For dramatic effect we'll drop from one
tone to one about a 5th lower, and in music theory the 5th is what typically comes right before the typical last note, the 1st or root note again.
Their relationship with each other is one of those things you can turn on its head an infinite number of ways, but very generally and conventionally
speaking:
The root note (1) of the pentatonic scale is the most resolved and commanding tone in the music. All other notes are basically in reference to that
note, and that carries over into how you actually psychologically interpret the sequence of notes believe it or not. It all goes back to the physics
of the overtone series.
The 5th or
dominant is considered the next most resolving and commanding note of the scale because the other notes of the scale revolve around
it most similarly to the way they revolve around the root note. Another reason for that is because it's also the first natural harmonic to appear
besides the octave. In classical music composers would typically modulate to a key a 5th higher, because the difference was enough to create tension
but not dramatically, since they are naturally so similar anyway.
The 4th degree is somewhat similar to the 5th in relation to the root note but is considered to have a slightly weaker connection to it, and obviously
of a different character, and the 3rd isn't traditionally considered to have much of a connection to the root note at all except to define whether
the scale sounds major or minor. But especially in blues scales the minor 3rd is used all the time to release to the root note, even more than the
5th is.
The interplay of these basic notes with each other, how they are arranged, how they repeat, the patterns they form, is the basic core of what I mean
by musical grammar in a series of notes or melodies. There is a definite pattern in most music that is comparable to what you see in literature and
even the grammatical structure of sentences.
[edit on 31-1-2010 by bsbray11]