Display simple pitches in Just Intonation, optionally specifying fundamental with the note encoding

basic-notation-with-fundamental.ly

\version "2.19.50"

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Basic Notation Example for "Just Intonation with LilyPond" paper
%
% This file is self-contained to be compiled with recent versions
% of LilyPond's 2.19 development version or later. It will not work
% with LilyPond 2.18.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Underlying mathematics

% Convert a ratio to a floating point step representation.
% The integer part is the number of semitones above the fundamental,
% the fractional part is the fraction of a semitone
#(define (ratio->step ratio)
   (* 12 (/ (log ratio) (log 2))))

% Convert a ratio and return a pair with
% - the pitch in semitones
% - the cent deviation above or below (rounded)
% Rounds to the nearest semitone and gives the deviation
% in cents -50 < cent < 49.
#(define (ratio->step/cent ratio)
   (let*
    ((step-cent (ratio->step ratio))
     ;; truncate the floating point number to the nearest integer (scale step)
     (step (inexact->exact (round step-cent)))
     ;; determine the cent deviation by stripping off
     ;; the floating point part of step-ratio
     (cent (* 100 (- step-cent step))))
    ;; construct and return the pair
    (cons step cent)))


% Maintain a current duration to be used when no duration is given,
% initialize to quarter notes (like with regular pitches without duration)
#(define ji-duration (ly:make-duration 2))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Configuration

% Maintain the fundamental, initialize to a default middle c
#(define ji-fundamental (ly:make-pitch 0 0 0))

% Maintain a current duration to be used when no duration is given,
% initialize to quarter notes (like with regular pitches without duration)
#(define ji-duration (ly:make-duration 2))

% Map the semitone returned by ratio->step/cent
% to a LilyPond pitch definition.
% This is based on the middle c and has to be transposed later
#(define (semitones->pitch semitone)
   (let
    ;; two lists defining the 12 steps within the octave
    ;;       c  cis  d  dis  e  f  fis  g  as   a  bes   b
    ((steps '(0  0    1  1    2  3  3    4  4    5  6     6))
     (semis '(0  1/2  0  1/2  0  0  1/2  0  1/2  0  -1/2  0))
     ;; strip semitons of octave
     (index (modulo semitone 12)))
    (ly:make-pitch
     (floor (/ semitone 12))
     (list-ref steps index)
     (list-ref semis index))))

% Local predicate which is necessary to process two optional arguments
% Accepts the first optional argument when it's either a pitch or a duration
#(define (pitch-or-dur? obj)
   (or (ly:pitch? obj)
       (ly:duration? obj)))

% Produce a note in Just Intonation.
% If fund(amental) is given change the fundamental to calculate pitches from
% if dur(ation) is given change the duration.
jiNote =
#(define-music-function (fund dur ratio)
   ((pitch-or-dur?) (ly:duration?) fraction?)
   (if fund
       ;; at least one optional argument has been given,
       ;; but it may be *either* the fundamental pitch *or* the duration
       (if (ly:pitch? fund)
           ;; set new fundamental pitch
           (set! ji-fundamental fund)
           ;; "first" (i.e. only) optional argument is a duration:
           ;; set the new duration
           (set! ji-duration fund)))
   (if dur
       ;; second optional argument is present (implies that the
       ;; first one is there too: set duration
       (set! ji-duration dur))

   (let*
    ;; note as pair of semitone-interval and cent deviation
    ((step/cent (ratio->step/cent (/ (car ratio) (cdr ratio))))
     ;; LilyPond pitch as defined by the ratio
     (pitch-from-ratio (semitones->pitch (car step/cent)))
     ;; LilyPond pitch relative to the current fundamental
     (pitch-effective
      (ly:pitch-transpose
       pitch-from-ratio
       ji-fundamental))
     ;; cent deviation as integer
     (cent (cdr step/cent)))
    ;; finally create the note with generated pitch and markup addition
    (make-music
     'NoteEvent
     'articulations
     (list (make-music
            'TextScriptEvent
            'text (format "(~@f)" (round cent))))
     'pitch pitch-effective
     'duration ji-duration)))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Examples
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Style the display of the cent deviation
% (note that this is completely independent from the
%  calculation and encoding)
\layout {
  \context {
    \Voice
    \override TextScript.direction = #UP
    \override TextScript.font-size = -2.5
    \override TextScript.self-alignment-X = -0.25
  }
}

\markup "Occasionally change fundamental"

{
  \jiNote a, 1 5/1
  \jiNote 7/1
  \jiNote e, 5/1
  \jiNote 7/1
}

\markup "Specify fundamental with each note"

{
  \jiNote c, 1 4/1
  \jiNote bes,, 5/1
  \jiNote a,, 6/1
  \jiNote g,, 7/1
}