Flat Ties

The function takes the default Tie.stencil as an argument, calculating the result relying on the extents of this default.

Further tweaking is possible by overriding Tie.details.height-limit or with \shape. It's also possible to change the custom-definition on the fly.

\version "2.24.0"

%% http://lsr.di.unimi.it/LSR/Item?id=1031

#(define ((flared-tie coords) grob)
 
  (define (pair-to-list pair)
     (list (car pair) (cdr pair)))
 
  (define (normalize-coords goods x y dir)
    (map
      (lambda (coord)
        ;(coord-scale coord (cons x (* y dir)))
        (cons (* x (car coord)) (* y dir (cdr coord))))
      goods))
 
  (define (my-c-p-s points thick)
    (make-connected-path-stencil
      points
      thick
      1.0
      1.0
      #f
      #f))
 
  ;; outer let to trigger suicide
  (let ((sten (ly:tie::print grob)))
    (if (grob::is-live? grob)
        (let* ((layout (ly:grob-layout grob))
               (line-thickness (ly:output-def-lookup layout 'line-thickness))
               (thickness (ly:grob-property grob 'thickness 0.1))
               (used-thick (* line-thickness thickness))
               (dir (ly:grob-property grob 'direction))
               (xex (ly:stencil-extent sten X))
               (yex (ly:stencil-extent sten Y))
               (lenx (interval-length xex))
               (leny (interval-length yex))
               (xtrans (car xex))
               (ytrans (if (> dir 0)(car yex) (cdr yex)))
               (uplist
                 (map pair-to-list
                      (normalize-coords coords lenx (* leny 2) dir))))
 
   (ly:stencil-translate
       (my-c-p-s uplist used-thick)
     (cons xtrans ytrans)))
   '())))

#(define flare-tie
  (flared-tie '((0 . 0)(0.1 . 0.2) (0.9 . 0.2) (1.0 . 0.0))))

\layout {
  \context {
    \Voice
    \override Tie.stencil = #flare-tie
  }
}

\paper { 
  ragged-right = ##f 
  tagline = ##f
}

\relative c' {
  a4~a
  \override Tie.height-limit = 4
  a'4~a
  a'4~a
  <a,, c e a c e a c e>~ q
  
  \break
  
  a'4~a
  \once \override Tie.details.height-limit = 14
  a4~a
  
  \break
  
  a4~a
  \once \override Tie.details.height-limit = 0.5
  a4~a
  
  \break
  
  a4~a
  \shape #'((0 . 0) (0 . 0.4) (0 . 0.4) (0 . 0)) Tie
  a4~a
  
  \break
  
  a4~a
  \once \override Tie.stencil = 
    #(flared-tie '((0 . 0)(0.1 . 0.4) (0.9 . 0.4) (1.0 . 0.0)))
  a4~a
  
  a4~a
  \once \override Tie.stencil = 
    #(flared-tie '((0 . 0)(0.06 . 0.1) (0.94 . 0.1) (1.0 . 0.0)))
  a4~a
}