Logarithmic graphs - Pitch against Frequency

To our ears, the "distance" between each whole tone note of the scale seems equal, and likewise, the distance between any pair of notes one octave apart seems the same. In other words, pitches are represented by an arithmetic scale.

However, the frequency of the higher note is twice that of the note one octave below. and whole tone steps are represented by equal frequency ratios. So frequency works "geometrically".  Equal ratios = equal steps.

In order to plot this relationship on a graph, we need to translate the geometrically increasing frequency into even arithmetic steps for the pitch.  Logarithms provide this facility, because each doubling of a number is represented by an equal addition to its logarithm.

In the demonstration chart, the x-axis is calibrated with frequencies from 10 cycles per second to 1000 cycles per second.  And the Y-axis is calibrated with pitch numbers measured in semitones, relative to a value of 24 for the C  three octaves below middle C.

You can use the cross-hair to read this graph. Try reading frequency 256 from the X-axis. It should correspond to MIDI note number 60 on the Y-axis, this is middle C.  The formula for this graph is   y= a log( bx ) + c,  where a = 17.3,  b= 0.031, and c = 24


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