Just Intonation, part I

The next evolution after Pythagorean Tuning came into common practice sometime around the fifteenth century.  Before the Renaissance, chords other than fifths weren't played much and intervals were thought of as melodic (played in succession) rather than harmonic (played simultaneously).

One of the major artistic changes which occurred during the Renaissance is that music evolved to be polyphonic and harmonic intervals such major and minor thirds and sixths became common.  To understand how the scale evolved, let's go back to the overtone series.  Recall, the first overtone or harmonic is given by a frequency of 2:1 with the fundamental and produces and octave.  The second is 3:1 and produces an octave plus a fifth, which we reduced to 3:2 to keep it within an octave of the fundamental.  These are the only ratios required to complete the Pythagorean Scale.  The next harmonic is 4:1.  Any power of 2 (2, 4, 8, 16, 32, etc.) will produce an octave, so no new information here.




The fifth harmonic, 5:1, however gives us something new.  Reducing this down to be within an octave of the fundamental gives us 5:4, which is a very in tune sounding major third.  The Pythagorean major third has a ratio of 81:64.  For some psycho-acoustic reason, intervals with ratios of smaller numbers sound more harmonious.  A well-known book by Hermann von Helmholtz, On the Sensations of Tone (translated from Die Lehre von den Tonempfindungen), written in 1877, explores this phenomenon extensively.

Before we go on, let's calculate the difference between a Just major third and a Pythagorean major third.  Remember, differences are multiplicative, so we divide:

Calculating the difference between a Pythagorean major third and a Just major third

This difference is an important ratio, called a Syntonic Comma.  When the word comma is used without qualification, it means Syntonic Comma and not Pythagorean Comma.

Let's consider the ratios formed by the major triad (1-3-5, C-E-G or do mi sol). 

C is 1:1, E is 5:4 and G is 3:2.  If we stack all three and use the common denominator 4, we get the ratio

4:5:6

This is a much simpler ratio than the Pythagorean version of a major triad and sounds much sweeter.

Take a listen to the two triads.  Listen for the harshness or beating of the Pythagorean chord and then compare it to the smoothness of the Just chord.  The 5:4 third evens out the roughness of the 81:64 third.

Pythagorean Major Triad


Just Major Triad

There are three major triads in a major scale: one starting from the root, one from the third fourth and one from the fifth.  In the key of C we get

C-E-G,  F-A-C  and  G-B-D

 To form the Just Tempered scale, we make these three triads have the relationship 4:5:6, and keep the Pythagorean fifth (G) and the Pythagorean fourth (F).

The new scale is

More to come.