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.


Straight.htm     02 November 2006         MENU  Theory -> Maths and Physics -> Mathematical equations for music