On 2 Jul 2010, at 16:22, Carl Sorensen wrote:
I'm trying to redo auto-beaming so that it matches better with what
I read
in the literature. When I get it done, I hope to have it set up so
that the
default properties do the right thing, and all we have to do is define
exceptions. Working in this framework, I think the LilyPond syntax
needs to
change.
As for meters, there is a need to define those. Currently, LilyPond
seems letting time signatures determining meters. This is fine in CPP.
For other types of music, one would like to depart from that, viewing
the time signature as a derivative of the meter.
I think that the beaming structure is the same for expressing non-
metric accents, only that it applies locally so a segment of music.
There, the current LilyPond syntax does not admit full level accent
level nesting in a convenient way.
And so I'd like some feedback about (a) names of properties, and
(b) the ideas I have for use of properties.
I found it necessary to in part invent a new, more clearly defined
terminology, in the absence of a generally accepted one. So I guess
you will have to likewise.
There is a Germanic-Latin divide in terminology, between Germany and
France going up and down in Europe. On the Latin side, "duple" and
"triple" refers to structure of the music, but not the metric accents.
On the Germanic side, those are 2- and 3-time meters. Best to avoid
this term.
Otherwise, I ended up using mostly Germanic style definitions. There
is a difference between simple and compound, too (see below). It does
not matter what words one is using - I used what was similar in math.
Let me start with a simple classification of meters. In order to
classify
meters (and their beaming) we have three different levels of
structure that
seem to apply: the fundamental unit, the beat, and the beat
combination.
So one generally seems to think of meters built up by fundamental
components rather than divisions from a higher value. This is
complicated by tuplets, which can be metrical, though rare. But adding
it will make the structure the same as for non-metric accents.
The simplest unit is "in one". If it has subdivisions, one beams it as
fully as possible, as there is no subaccent structure to indicate.
Though there are some defaults: 3 is in 3, 6 is in 2. Then "in k",
where k is an integer > 1, means a group where the first subgroup has
a strong accent, and the other subgroups weaker accents. All else is
unspecific, subject to interpretation. Specifically, this is k units
followed by a note value; mathematically a pair (k, n), where n is a
note.
One can define more complicated subgroups by the use of +,
parenthesizes, and "in k" groups.
Each measure of music is divided into a number of rhythmic elements or
beats. For *regular* meters, all beats are the same length. For
irregular
(or complex, or additive) meters, different beats have different
lengths.
Hence, for irregular meters, we can't talk about "the beat length",
because
there isn't a unique beat length value.
Then I ended up defining two concepts: simple and compound meters. The
simple meters are just those "in k", a pair (k, n), which does not use
+ or parenthesizes.
Then one might define a regular meter as one which can be rewritten
into a simple meter by means of a substitution. For example, the
common 6/8 can be rewritten into 2/4 which is the pair (2, 1/4).
Those that are not regular are irregular. I don't think this concept
of regular-irregular helps the beaming question though.
Note the complication of the meter 4/4: there are two interpretations,
as the simple meter (4, 1/4) and as the compound meter (2, 1/4) + (2,
1/4). It think the French Wikipedia indicated both. Strictly speaking,
one has two different meters here, that might require different beaming.
Regular meters can be classified according to the number of beats per
measure. Duple meters have two, triple meters have three, quadruple
meters
have four, and so on. These terms are really not important for
beaming (at
least as far as I can tell) so I won't discuss them more here.
This is the Germanic way (from Scandinavia down to the Balkans). Look
into the ENglish Wikipedia, and you will find another definition.
People seem to be rather emotional when they discover the difference.
Regular meters can also be classified as simple meters or compound
meters.
Simple meters have beats that are regularly divided into two equal
parts;
compound meters have beats that are regularly divided into three equal
parts.
Here it seems you are shifting to a Latin terminology. In Germanic
tradition, 2, 3, 4 are simple, and others, like 6 are compound.
Other divisions are possible, but for purposes of this discussion we
will consider those other divisions to be part of the irregular (or
complex)
meters.
For numbers that are multiples only has factors 2 and 3, the CPP
default is assuming they have this subdivision. Otherwise, one has to
indicate the subdivision using +. For example, 5 is 2+3 or 3+2. One
can divide 6 as 2+2+2; I have an example in Bulgarian music where it
used along with a 7.
There is a complication that one can have "in one". Finale has such an
option, I am told. Beethoven's 5th is normally performed in one,
though written in 2. In Balkan music, one can have 7 = 2+2+1+2 or 9 =
2+2+2+1+2, for example.
For simple meters, the fundamental unit is the same as the beat. So
the
beat in 4/4 time is 1/4, and the fundamental unit in 4/4 time is 1/4.
Hence, it is common to think of the beat as the fundamental unit.
However, for compound meters, the fundamental unit is *not* the same
as the
beat. For example, in 6/8 time, the fundamental unit is 1/8, while
the beat
is 3/8.
If one uses the nesting with +, parenthesizes and simple pairs (k, n)
one comes down to the lowest level of (k, n) pairs. Then then work
upwards first with number of beams until beam groups, possibly with
different dotted bars, and even a metric shift structure.
The trick would be to describe this hierarchical structure.
The concept of a fundamental unit differing from the beat is not
currently
well-supported in LilyPond.
This was my impression, too, but it is a bit more complicated.
In current practice, we would consider the
beatLength of 6/8 music to be 1/8 (the fundamental unit, not the
beat) and
then establish a beatGrouping of (3 3). While functional (it
provides a
data structure that allows the calculation of appropriate
autobeaming), it
differs from the engraving literature. It also causes problems with
beam
subdivision. The practice in the literature is that beams are
divided at
beat boundaries, and subdivided at fundamental unit boundaries. For
the
current LilyPond practice, beams are subdivided at beat boundaries,
and
divided at beat group boundaries.
So here is the connection with metric and other accent structures: I
think they are the same. So focusing on both might be simpler.
Finally, there is another level of metric structure that is not
currently
accommodated in LilyPond. For 4/4 time, beats 1 and 2 or beats 3
and 4 can
be combined to form a unit (but never beats 2 and 3).
So there are really to 4, (using Germanic terminology) the simple 4
and the compound 2+2.
For 3/4 time, beats 1
and 2; or 2 and 3; or 1, 2, and 3 can be combined to form a unit
(but only
if the unit includes *all* of beat 2; if beat 2 is split, 3/4 looks
like
6/8). This level of combination is not supported at all in
LilyPond, and so
we try to make it work with custom beaming rules.
And as for 3, there are also meters: the simple 3, and the "in one"
three. With the notation above, 3/4 can be (3, 1/4) or (1, 3/4).
Users must specify which one the want to have. They may want more to
have in effect more than one subaccent meter, determined by say note
values or other properties.
As I move forward, I think I've settled on the following that I want
to have
as properties:
1) subdivideUnit: a pair (or perhaps moment) that defines the
fundamental
unit of the meter.
2) beatLengths: a list that defines the beat lengths in the meter,
in terms
of the fundamental unit. If the meter is irregular or complex,
beatLengths
will be a list of all of the beats in the measure. I haven't yet
settled on
what happens in a regular meter. Currently, I still use a list of
all the
beats in the measure (which is automatically generated when the time
signature is changed). But I could see that for user overrides it
might be
preferable to only have one entry in beatLengths.
I've got mixed feelings about the following property:
3) beatCombinations: an alist with a key of beam type, and a value
of the
beats that can be combined for that beam type. This is currently
only used
for 3/4 and 4/4 time, and it doesn't really capture the special
rules used
for 3/4 and 4/4 time (although it comes close). I can't decide
whether to
create this general property and use it to define rules for 3/4 and
4/4
time, or whether I should just go ahead and hard-code the special
rules for
3/4 and 4/4 time (so I could get them perfectly).
So I think you need to somehow capture the nested structure of +,
parenthesizes and pairs (k, n), where k is an integer >= 1, and the
note value n is a rational number indicating is tome value. Then to
each level in the tree it defines, one should attach a subbeaming
construct: number of beams, space between groups, various dotted bars
or even metric shifts. The number of beams is standard. The dotted
beam structure must be specified by the user.
Hans
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