Item: The Scale Tree

Item: The Scale Tree is useful in real music as made by real musicians.

Each location on the scale tree shows two pieces of information: a number of
tones in the scale as the sum of the number of members in two scales with fewer
tones, and a number indicating the size of the generating interval of the scale,
relative to an interval of repetition (typically, but not necessarily, an
octave).

A given number of scale tones will appear on the tree in every relatively-prime
set of sums. For example, one finds scales with 12 tones as the sum of 1+11 and
5+7. (The non-relatively-prime sums, 2+10=12, 3+9=12, and 4+8=12, generate
deficient and redundant solutions, so we can ignore them). This is useful
because a given scale's appearance in different places on the tree indicates
different -- and often surprisingly so -- ways of working.

For example, I can make a piece of music based upon a set of five fifths or
fourths in 12tet, and then make a variation based on mapping that same music
onto a series of minor seconds or Major sevenths (that's the M5 transformation
that some twelve-toners get excited about). I won't make the claim that the
relationship between the original piece of music and the variation will be
immediately or ever apparent to a listener (although one could well come up with
a compositional strategy to do so), but simply that the piece and its variation
will share a structure for consuming pitch classes.

Now the numbers associated with each scale positioned on the tree and indicating
the size of the generating interval for that scale can also work if they are
considered to be a bit fuzzy. Don't worry -- I don't know anything about fuzzy
math, but neither does the Resident of the U.S., I just want to indicate that
the many of the properties the scale tree associates with a given generator also
apply to scales with generators very close in size, or even to scales whose
generators are not all of the same size, but average out to a size close to the
generator in the scale tree. This is one way of thinking about "constant
structure" ("CS" among the listfolk), through which I can find efficient ways to
map non-equal tunings onto a keyboard or a notation derived from an equal
temperament.

Now, here are some real-world examples of the scale tree at work: I had a
rehearsal with my (central Javanese) gamelan group on Tuesday and we played a
famous piece of 19th century Javanese music, the composition "Pangkur", a
ladrang (the form) based on a vocal melody in the slendro pathet (roughly: key)
of Sanga. Part of the saron melody goes like this:

. 2 . 1 . 2 . 6 . 2 . 1 . 6 . 5
6 6 . . 5 5 6 i 2 1 5 2 . 1 . 6

For our purposes here, we can think of slendro as 5tet (it's not really, but it
shares a CS with 5tet), with a series of wide (3/5ths octave) fifths going 1 -
5 - 2 - 6 - 3 - 1 (there's no number 4 in the notation, but don't worry about
that). So it's no big deal to imagine a simple transposition, up a tone, to
slendro pathet manyura:

. 3 . 2 . 3 . 1 . 3 . 2 . i . 6
i i . . 6 6 1 2 3 2 6 3 . 2 . 1

That's just like transposition in any other equal temperament -- I've just added
one to each number, or move two places to the right on the circle of slendro
fifths.

But the Javanese also play "Pangkur" in their other tone system, Pelog,
which is also built on a series of fifths, but fifths which average out to be
about 5/9ths of an octave, smaller than the slendro fifths.

So we have this series of pelog fifths:

4 - 1 - 5 - 2 - 6 - 3 - 7

(This doesn't form a closed circle of fifths, but could if two extra tones were
inserted, making a 9-tone tuning).

When I take my Pangkur melody in slendro, I simply map the tones following the
series of fifths:

slendro: 3 - 1 - 5 - 2 - 6 - 3 - 1
pelog: 4 - 1 - 5 - 2 - 6 - 3 - 7

Note that slendro 3 can be mapped to pelog 3 OR 4 and slendro 1 can be mapped
onto either pelog 1 OR 7. In practice the mapping of a piece in slendro manyura
is from sl.1 to pl.7 and sl.3 to pl.3 in pelog patet barang, and a piece in
slendro pathet sanga is to pelog pathet lima or nem with sl.1 to pl.1 and the
choice of 3 or 4 determined by conventions of tessitura and melodic contour.

So, direct mapping from pathet manyura gives me "Pangkur" in pelog pathet
barang:

. 3 . 2 . 3 . 7 . 3 . 2 . 7 . 6
7 7 . . 6 6 7 2 3 2 6 3 . 2 . 7

Or, from pathet sanga, I get "Pangkur" in pelog nem:

. 2 . 1 . 2 . 6 . 2 . 1 . 6 . 5
6 6 . . 5 5 6 i 2 1 5 2 . 1 . 6

What does this have to do with the scale tree? Easy -- I've just gone from
subsets of a five tone scale based on a generating fifth of .6 octaves to
subsets of a nine-tone scale based on a generating fifth of .66_ octave. And I
can guarantee that the relationship between the various versions of "Pangkur" is
not difficult to hear.

The scale tree certainly isn't the only way to work with scales or to describe
how scales may relate to one another, but it sure can be a useful and efficient
way.

Tomorrow's item: "Lattice diagrams can be useful in making real music."

Daniel Wolf