Underlying calculations to convert a frequency ratio in a pair of chromatic scale steps and cent deviation

ratio-to-cent-calculations.ly

\version "2.19.50"

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% Basic Caculation 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.
% But as it only contains function definitions there is no real use
% in compiling it yet.
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% 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)))