Re: Auto-beaming infrastructure redo

[Carl Sorensen]

On 7/2/10 1:38 PM, "Hans Aberg" <address@hidden> wrote:

> 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.

By CPP, do you mean Common Practice Period, i.e. western European Baroque,
Classical, and Romantic?

> For other types of music, one would like to depart from that, viewing
> the time signature as a derivative of the meter.

My current plan is to have LilyPond as a default derive the meter from the
time signature.  However, I hope that one will be able to override those
defaults to arrive at any meter/time-signature combination desired.

> 
> 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.
> 

I don't understand the meaning of this paragraph.  But I'm willing to try.

>> 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.

The terms I use are from music engraving, not music performance.  They
appear to be consistent across Stone, Read, and Ross.

> 
> 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.

How do you propose beaming to indicate "in k"?  I understand "in one"; it
means to connect all the notes to a common beam, i.e. beam the "beat"
together.  In your mind, is "6 in 2" different from "3 in 1" + "3 in 1"?

> 
> One can define more complicated subgroups by the use of +,
> parenthesizes, and "in k" groups.

In your earlier email (which I didn't understand before, but I think I
figured it out now), you proposed a working notation with ' indicating "in
one".  But then every specific proposal you wrote used ', so I wasn't able
to understand the difference between "in one" and "not in one".

> 
>> 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).

I really don't understand this.  How does 6/8 map into 2/4?  The only way I
can see to do this is by the use of triplets.  What am I missing?

> 
> 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.
> 

According to the engraving books, these aren't really two different meters.
They're just two different ways of notating the music.  But that's just the
engraving books.

>> 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.

I can't understand the extra meaning of duple meter (1.5 times the basic
duration) given in the English Wikipedia.

The clearest explanation of meters that I have found is Chapter 10 of Read.

But I suspect that all of this discussion is largely tangential.  What I
most care about is proper beaming patterns.

> 
>> 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.\
> 

No, I'm not shifting.  I'm just using Read's terminology.

>> 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.
> 

I don't think one needs pairs (k,n).  But I'm willing to be convinced.

> 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.
> 

Actually, according to the engraving references, there is no difference in
the metric structure.  It's just that beams having fewer than three stems
can be combined across the 1-2 and 3-4 boundaries (in the case of 4/4), and
that beams of repeating rhythms or equal notes can be comined for all three
beats, or 1 and 2, or 2 and 3 (in the case of 3/4).  According to the
engraving references, it's *not* reflecting a difference in the meter.

>> 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).

Again, what id the difference of the simple 3 (which, as I understand it,
has an accent on beat 1) and the "in one" 3 (which, as I understand it, has
an accent on the first metric unit)?
> 
> 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.
> 
> 
> 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.

One possibilty I've thought of is to specify a context property called (for
the sake of argument) fundamentalMoment.  Then we could define a beat
structure as a nested list.  The list would represent groupings of
fundamentalMoments.  Each group at the top level of the list would be beamed
together, with beams divided between the groups.  If there were a sublist,
the total of all the elements in the sublist would be beamed together, and
then the indvidual elements of the sublist would be subdivided.  This could
go on arbitrarily.

So, for example, we could have a fundamental unit of 1/8, and then for a
time signature of 15/8, we might have a beat structure of

(2 3 (2 (3 2)) 3)

The first two eighth notes would be beamed together, as would the third
through fifth, and the sixth through twelfth, and the thirteenth through
fifteenth.  There would be a subdivision at 16th notes at 7/8, and a
subdivision at 32nd notes at 11/8.  This would try to demonstrate the metric
relationship implicit in the beat structure.

If I were to implement something like this, it would make sense to call the
context property beatStructure.  And actually, that might make sense anyway.
Perhaps the name beatStructure is clearer than anything else I've thought
of.

Thanks,

Carl