Editing Complex compound time signatures

This snippets implements complex compound time signatures (e.g. (1+3)/8 or (1+3)/8 + 2/4). The function \compoundMeter (defined in the snippet) can be used instead of \time to create those time signatures. The argument is a Scheme list of lists. Each list describes one fraction, with the last entry being the denominator, while the first entries describe the summands in the enumerator. If the time signature consists of just one fraction, the list can be given directly, i.e. not as a list containing a single list. For example, a time signature of (3+1)/8 + 2/4 would be created as <code>\compoundMeter #'((3 1 8) (2 4))</code>, and a time signature of (3+2)/8 as <code>\compoundMeter #'((3 2 8))</code> or shorter <code>\compoundMeter #'(3 2 8)</code>. The beat structure (and thus the automatic beaming) will be derived from the given meter automatically.

<lilypond version="2.24.0" full>
%% http://lsr.di.unimi.it/LSR/Item?id=743
%% see also http://lsr.di.unimi.it/LSR/Item?id=835

\paper { tagline = ##f }

\relative c' {
  \compoundMeter #'(1 2 3 4 8)
  \repeat unfold 10 c8 \repeat unfold 20 c16\break

  \time 3/4
  \repeat unfold 6 c8 \repeat unfold 12 c16\break

  \compoundMeter #'((1 2 3 4 8) (2 4))
  \repeat unfold 14 c8 \repeat unfold 28 c16\break

  \compoundMeter #'((1 2 3 4 8) (2 4) (2 3 8))
  \repeat unfold 19 c8 \repeat unfold 38 c16\break

  \compoundMeter #'(1 2 3 4 8)
  \repeat unfold 10 c8 \repeat unfold 20 c16\break

  \compoundMeter #'((1 8) (3 8))
  \repeat unfold 4 c8 \repeat unfold 8 c16\break

  \compoundMeter #'((3 8) (1 8))
  \repeat unfold 4 c8 \repeat unfold 8 c16\break

  \time 4/4
  \repeat unfold 8 c8 \repeat unfold 16 c16\break

  \bar"|."
}

%{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Formatting of (possibly complex) compound time signatures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

#(define (insert-markups l m)
  (let* ((ll (reverse l)))
    (let join-markups ((markups (list (car ll)))
                       (remaining (cdr ll)))
      (if (pair? remaining)
        (join-markups (cons (car remaining) (cons m markups)) (cdr remaining))
        markups))))

% Use a centered-column inside a left-column, because the centered column
% moves its reference point to the center, which the left-column undoes.
#(define (format-time-fraction time-sig-fraction)
  (let* ((revargs (reverse (map number->string time-sig-fraction)))
         (den (car revargs))
         (nums (reverse (cdr revargs))))
    (make-override-markup '(baseline-skip . 0)
      (make-number-markup 
        (make-left-column-markup (list
          (make-center-column-markup (list
            (make-line-markup (insert-markups nums "+"))
            den))))))))

#(define (format-complex-compound-time time-sig)
  (let* ((sigs (map format-time-fraction time-sig)))
    (make-override-markup '(baseline-skip . 0)
      (make-number-markup
        (make-line-markup
          (insert-markups sigs (make-vcenter-markup "+")))))))

#(define-public (format-compound-time time-sig)
  (cond
    ((not (pair? time-sig)) (null-markup))
    ((pair? (car time-sig)) (format-complex-compound-time time-sig))
    (else (format-time-fraction time-sig))))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Measure length calculation of (possibly complex) compound time signatures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

#(define (calculate-time-fraction time-sig-fraction)
  (let* ((revargs (reverse time-sig-fraction))
         (den (car revargs))
         (num (apply + (cdr revargs))))
    (ly:make-moment num den)))

#(define (calculate-complex-compound-time time-sig)
  (let* ((sigs (map calculate-time-fraction time-sig)))
    (let add-moment ((moment ZERO-MOMENT)
                     (remaining sigs))
      (if (pair? remaining)
        (add-moment (ly:moment-add moment (car remaining)) (cdr remaining))
        moment))))

#(define-public (calculate-compound-measure-length time-sig)
  (cond
    ((not (pair? time-sig)) (ly:make-moment 4/4))
    ((pair? (car time-sig)) (calculate-complex-compound-time time-sig))
    (else (calculate-time-fraction time-sig))))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Base beat lenth: Use the smallest denominator from all fraction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

#(define (calculate-compound-base-beat-full time-sig)
    (apply max (map last time-sig)))

#(define-public (calculate-compound-base-beat time-sig)
  (ly:make-moment 1 (cond
    ((not (pair? time-sig)) 4)
    ((pair? (car time-sig)) (calculate-compound-base-beat-full time-sig))
    (else (calculate-compound-base-beat-full (list time-sig))))))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Beat Grouping
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



#(define (normalize-fraction frac beat)
  (let* ((thisbeat (car (reverse frac)))
         (factor (/ beat thisbeat)))
    (map (lambda (f) (* factor f)) frac)))

#(define (beat-grouping-internal time-sig)
  ; Normalize to given beat, extract the beats and join them to one list
  (let* ((beat (calculate-compound-base-beat-full time-sig))
         (normalized (map (lambda (f) (normalize-fraction f beat)) time-sig))
         (beats (map (lambda (f) (reverse (cdr (reverse f)))) normalized)))
    (apply append beats)))

#(define-public (calculate-compound-beat-grouping time-sig)
  (cond
    ((not (pair? time-sig)) '(2 . 2))
    ((pair? (car time-sig)) (beat-grouping-internal time-sig))
    (else (beat-grouping-internal (list time-sig)))))





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The music function to set the complex time signature
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

compoundMeter =
#(define-music-function (args) (pair?)
"Create compound time signatures. The argument is a Scheme list of "
"lists. Each list describes one fraction, with the last entry being the "
"denominator, while the first entries describe the summands in the "
"enumerator. If the time signature consists of just one fraction, "
"the list can be given directly, i.e. not as a list containing a single list."
"For example, a time signature of (3+1)/8 + 2/4 would be created as "
"@code{\\compoundMeter #'((3 1 8) (2 4))}, and a time signature of (3+2)/8 "
"as @code{\\compoundMeter #'((3 2 8))} or shorter "
"@code{\\compoundMeter #'(3 2 8)}."
  (let* ((mlen (calculate-compound-measure-length args))
         (beat (calculate-compound-base-beat args))
         (beatGrouping (calculate-compound-beat-grouping args))
         (timesig (cons (ly:moment-main-numerator mlen)
                        (ly:moment-main-denominator mlen))))
  #{
    \once \override Staff.TimeSignature.stencil = #ly:text-interface::print
    \once \override Staff.TimeSignature.text = #(format-compound-time args)
    \set Timing.timeSignatureFraction = $timesig
    \set Timing.baseMoment = $beat
    \set Timing.beatStructure = $beatGrouping
    \set Timing.beamExceptions = #'()
    \set Timing.measureLength = $mlen
  #} ))

%}
</lilypond>

[[Category:Rhythms]]
[[Category:Scheme]]