spelling

    4/4
  0 w                               | 32
  1 h               h               | 16
  2 q       q       q       q       | 8
  3 e   e   e   e   e   e   e   e   | 4
  4 s s s s s s s s s s s s s s s s | 2
  5 33333333333333333333333333333333| 1
0 0 w-------------------------------| 32
5 3  s.-|                             3
4 2   e.----|                         6
5 3    3|                             1
3 1     q.----------|                 12
5 3      s.-|                         3
4 2       s-|                         2
5 3        3|                         1
2 0         h.----------------------| 24
5 3          s.-|                     3
4 2           e.----|                 6
5 3            3|                     1
3 1             e---|                 4
5 3              s.-|                 3
4 2               s-|                 2
5 3                3|                 1
2 0                 h---------------| 16
5 3                  s.-|             3
4 2                   e.----|         6
5 3                    3|             1
3 1                     q.----------| 12
5 3                      s.-|         3
4 2                       s-|         2
5 3                        3|         1
2 0                         q-------| 8

    3+3/8
  0 h.                      h.
  1 q.          q.          |
  2 e   e   e   e   e   e   |
  3 s s s s s s s s s s s s |
0 0 h.----------------------|
4 3  s.-|
3 2   s-|
4 3    3|
2 1     q-------|
4 3      s.-|
3         s-|
4          3|
2           e---|
4            s.-|
3             s-|
4              3|
1               q.----------|
4                s.-|
3                 s-|
4                  3|
2                   q-------|
4                    s.-|
3                     s-|
4                      3|
2                       e---|
4                        s.-|
3                         s-|
4                          3|

    2+2+2/8
  0 h.
  1 q       q       q       |
  2 e   e   e   e   e   e   |
  3 s s s s s s s s s s s s |
  4 333333333333333333333333|
  0 h.----------------------|
4 2 3s.-|
3 2 2 s-|
4 2 3  3|
2 1     e---|
4 2      s.-|
3 2       s-|
4 2        3|
1 0         h---------------|
4 2          s.-|
3 2           s-|
4 2            3|
2 1             e---|
4 2              s.-|
3 2               s-|
4 2                3|
1 0                 q-------|
4 2                  s.-|
3 2                   s-|
4 2                    3|
2 1                     e---|
4 2                      s.-|
3 2                       s-|
4 2                        3|

  0 h.
  1 q               q               q               |
  2 e       e       e       e       e       e       |
  3 s   s   s   s   s   s   s   s   s   s   s   s   |
  4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 |
  5 666666666666666666666666666666666666666666666666|
0 0 h.----------------------------------------------|
5 3  3.-|
4 2   s.----|
5 3    6|
3 2     s---|
5 2 3    3.-|
4 2 2     3-|
5 2 3      6|
2 1         e-------|

I spent the weekend drawing diagrams like those.  A musician might recognize them as the "optimal" way to spell a note beginning at each point in a measure.  Since my score uses a low level start time + duration concept, I need to figure out how to spell rhythms automatically when I convert to lilypond.  Previously I used a quick and simple technique that worked surprisingly well, but only for duple.  Since it made a 6/8 piece totally ugly I decided it was time to solve the problem for real.  That turns out to be either way more complicated than I thought it would be, or I'm doing it totally wrong!

In any case, it was an exercise in design-by-diagram, since I'm definitely not smart enough to visualize rhythmic notation in my head.

First strategy: try to end up on a major division which isn't over the barline.  So define a "rank" for each possible position in the measure (I only go to 128th notes, so there are 128 per whole note), and pick the duration (possibly including dotted durations) which ends on the lowest rank.  Exploring this with diagrams, it seemed to line up with my expectations:

  0 w                               | 32

  1 h               h               | 16

  2 q       q       q       q       | 8

  3 e   e   e   e   e   e   e   e   | 4

  4 s s s s s s s s s s s s s s s s | 2

  5 33333333333333333333333333333333| 1

5 3  s.-|                             3

  5  w-------------------------------|

  5  h.----------------------|

  5  h---------------|

  5  q.----------|

  5  q-------|

  5  e.----|

  5  e---|

* 3  s.-|

  5  s-|

  4  3|

4 2   e.----|

  >   w-------------------------------|

  4   h.----------------------|

  4   h---------------|

  4   q.----------|

  4   q-------|

* 2   e.----|

  5   e---|

  4   s.-|

  3   s-|

  4   3|

  0 w                               | 32

  1 h               h               | 16

  2 q       q       q       q       | 8

  3 e   e   e   e   e   e   e   e   | 4

  4 s s s s s s s s s s s s s s s s | 2

  5 33333333333333333333333333333333| 1

5 3    3|                             1

  >    w-------------------------------|

  5    h.----------------------|

  5    h---------------|

  5    q.----------|

  5    q-------|

  5    e.----|

  5    e---|

  4    s.-|

  4    s-|

* 3    3|

  0 w                               | 32

  1 h               h               | 16

  2 q       q       q       q       | 8

  3 e   e   e   e   e   e   e   e   | 4

  4 s s s s s s s s s s s s s s s s | 2

  5 33333333333333333333333333333333| 1

3 1     q.----------|                 12

  >     w-------------------------------|

  4     h.----------------------|

  3     h---------------|

* 1     q.----------|

  3     q-------|

  4     e.----|

  2     e---|

  5     s.-|

  4     s-|

  5     3|

  0 h.

  1 q.          q.

  2 e   e   e   e   e   e   |

  3 s s s s s s s s s s s s |

4 3  s.-|

  >  w-------------------------------|

  >  h.----------------------|

  4  h---------------|

  4  q.----------|

  4  q-------|

  4  e.----|

  4  e---|

* 2  s.-|

  4  s-|

  3  3|

Unfortunately this doesn't quite work, but of course I had to implement it and run tests to find out.  Given 6/8, interpreted as 3+3/8, it wil happily put a half-note at the 3rd eighth note, because that ends at the end of the measure, rank 0 by definition.  But that obscures the middle of the measure, which is forbidden:

  0 h.

  1 q.          q.

  2 e   e   e   e   e   e   |

  3 s s s s s s s s s s s s |

0 0 h.----------------------|

            h---------------| <- was

            q.----------|

            q-------|

            e.----|

            e---|             <- should have been

Or rather, it's forbidden, but only for notes which start on rank 2!  The dotted half-note at the beginning of the measure can span it, because it starts at rank 0.  So I switched strategies: a note can extend until it reaches a division it's not allowed to span.  "Not allowed" means one rank lower for complex meters, and two ranks lower for duple ones.  Duple meters fit better with the log2 nature of note durations, so they can be more liberal in their spelling.

Unfortunately, this also has a problem: a 5/4 measure can start off with a whole note.  That means a measure-long note would be spelled c1~c4, which is super ugly.  If I were spelling 5/4, I would need to know whether it is mostly 3+2/4 or 2+3/4 and tie accordingly.  The underlying problem is that first time in the measure is rank 0, so it will happily span the center of the measure.  Normally that's fine, but for irregularly divided measures the spelling needs to be more conservative to make the irregular division clear (and because a single note won't fit).  So I tweaked the definition of irregular measures to give the toplevel major divisions rank 0 instead of just the barline, and never cross rank 0 no matter what.  It winds up doing a few things I wouldn't do, e.g. I would use liberal duple spelling in the duple portion of a compound meter while it doesn't, but tolerance for complex spelling is individual and it doesn't hurt to be a little more conservative.

There remains yet another problem: this spells rests incorrectly.  Since it always tries to find the longest duration that will fit, if I have dotted rests disabled, I wind up with "c4 c2 c4".  Rests are different from notes in that they are all interchangeable, so if there are two rests in a row they should be arranged to align with major beats: "c4 r4 r2".  The solution to this was to bring back the sort-by-rank idea, but only for rests.  If I'm trying to spell a rest, find the maximum duration as above, but then try all of the durations that fit within it, and pick the one that ends on the lowest rank.

But wait!  There's one last wrinkle.  At least it's the last one I'm going to worry about today.  While I normally avoid dotted rests, they look better for the major divisions of a triple meter.  E.g. 3+3/8 should be spelled "r4. r4." rather than "r4 r8 r4 r8".  I wound up allowing dotted rests for triple meters.  I should probably only allow them at the rank that involves the triple division (the denominator's rank), and actually technically in a complex meter like 2+3/4, I should only allow them in the triple fraction of the meter, but it seems like too much bother, so I'm just enabling them universally.

One other practical problem that I'll have to look into is that I have to define where the ranks fall for various meters, which means I have to define each meter before using it.  It seems like it should be possible to figure out everything from just looking at the name of the meter, though I'd have to write it explicitly, e.g. 3+2/4 rather than 5/4.

This is one of those problems where it seems like there might be some simple "best" solution.  But it also might just be a set of heuristics and a notion of prettiness, which is something people are good at but machines aren't.  When I tried to analyze my own process in deciding how to spell a note, I came up with a small set of heuristics, but discovered more and more as the meters become more complicated.

I still feel like if I were smarter I would have been able to identify some optimal heuristic from the beginning, or at least forsee the various problems without bumbling into a problem, patching it up, and then bumbling into another.

This is hopefully the last of the major lilypond changes.  My goal is to typeset a simple piano piece acceptably, and it's getting close.  Recent other lilypond improvements:

• Preserve enharmonics.  That meant extending pitches so they generate not just frequencies, but a symbolic note name, applying whatever diatonic and chromatic transpositions.
• Tuplets.  This was a whole other involved saga.  Since tuplets involve nested chunks of lilypond code, whose composite has a different duration than the sum of its parts I wound up needing to interleave score derivation with lilypond "performance".  I tried to just generate pitches, but when I would up with grace notes nested in a tuplet I realized it had to be more general.
• Grace notes.  Easy once the work was done for tuplets.
• A bunch of simpler things, like various articulations, ottava notation and the like.  I switched from having the lilypond interpret note attributes to having the call directly emit lilypond code if it's being derived for lilypond.  It means lilypond knowledge is scattered throughout the calls instead of consolidated in the lilypond performer, but it makes it really trivial to add a lilypond interpretation to a call, and generally simplifies things, since the call has score context.

Before:

After:

The rests should be dotted but the rest looks much better.