Displaying basic Just Intonation pitches

basic-notation.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))


% Map the semitone returned by ratio->step/cent
% to a LilyPond pitch definition.
% This is based on the middle c
#(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 semitones of octave
     (index (modulo semitone 12)))
    ;; generate LilyPond pitch object to be returned
    (ly:make-pitch
     (floor (/ semitone 12))
     (list-ref steps index)
     (list-ref semis index))))

% Produce a note in Just Intonation.
% if dur(ation) is given the active duration is changed.
jiNote =
#(define-music-function (dur ratio)
   ((ly:duration?) rational?)
   (if dur
       ;; optional argument is present: set duration
       (set! ji-duration dur))
   (let*
    ;; note as pair of semitone-interval and cent deviation
    ((step/cent (ratio->step/cent ratio))
     ;; LilyPond pitch as defined by the ratio
     (pitch-from-ratio (semitones->pitch (car step/cent)))
     (cent (cdr step/cent)))
    ;; finally create the note with generated pitch and markup addition
    (make-music
     'NoteEvent
     ;; add text element as articulation
     'articulations
     (list (make-music
            'TextScriptEvent
            ;; round the cent deviation number for display
            'text (format "(~@f)" (round cent))))
     'pitch pitch-from-ratio
     'duration ji-duration)))

%%%%%%%%%%%
% Example %
%%%%%%%%%%%

% 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 "Different ratios over the middle c"

{
  \jiNote 1 #7/4
  \jiNote #3/2
  \jiNote #4/3
  \jiNote #5/4
}