Straight-line graph.
This is a demonstration of a straight-line graph. It plots onto the Y-axis the beat in a bar, against on the X-axis the distance of a note measured in cms. from the start of the bar.This
assumes that the music publisher always positions notes within a bar in
proportion to their starting time. This is the normal practice. Let us use
as an example a time signature of 4/4, width of the bar on the sheet
of music of 15 cms, and 4 crotchets printed in the bar. The first
crotchet is placed 0 cms. from the beginning, the second ¼ across at 3.75 cms.,
the 3rd ½ way across at 7.5 cms., and the 4th 3/4 across the bar at 11.25 cms.
The formula used to plot this line is y = cx + d, where
In this example, coefficient c is the number of beats per bar divided by the width of the bar, in other words, the beats per centimetre. When the bar is 15 cms. wide and there are 4 beats per bar, c is 4/15 which is 0.2666. Coefficient d is always 1, because a note at 0 cms. will be beat number 1.
This straight line equation is a sub-set of the general polynomial equation y = ax3 + bx2 + cx + d in which there are no terms for x3 and x2, so coefficients a and b are both zero.
Any computer program which attempts to write musical notation on the screen or on a printer, needs to use a similar calculation in reverse, i.e, given the exact starting time of a note in beats and fractions, it must calculate the horizontal distance of the note from the beginning of the bar.