Musical Tuning Systems And Temperament

A musical temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. "Tempering is the process of altering the size of an interval by making it narrower or wider than pure. A temperament is any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds."[1] Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance.

Just Intonation

Just Intonation is a system of tuning with distances of pitches that are based on the harmonic series.  A pretty simple explanation that is quite difficult to completely understand.  David B. Doty, in his introduction to the Just Intonation Primer, says,

Technically, Just Intonation is any system of tuning in which all of the intervals can be represented by whole-number frequency ratios, with a strongly implied preference for the simplest ratios compatible with a given musical purpose”.

Earlier we looked at a quote from Graham Breed in relation to adding intervals together to come up with the frequencies.  The scale below is pulled from the relationships of harmonics from the harmonic series.  David Cartright puts it this way;

“One way to understand Just Intonation is in terms of the harmonic series; every interval used in Just Intonation can be found somewhere in the harmonic series. By definition, the harmonic series is that sequence of frequencies which is all whole-number multiples of any particular fundamental frequency. Thus, since any just interval is expressible as a frequency ratio of two whole numbers, that interval is also the interval between those same two harmonics. For example, the ratio 7/5 is the interval from the fifth harmonic to the seventh. So by becoming familiar with the harmonic series as a musical scale, one also comes to know all the just intervals included. (To put it in strictly numerical terms, a familiarity with the whole numbers also includes a familiarity with the proportions of whole numbers, i.e., the rational numbers.)” 

If we look at the harmonic series and the difference in intervals of the series from the chart earlier, and listen to the intervals from the monochord, we can begin to see the practical basis for this.  Following is a chart of the ratios for the chromatic scale.

Note Ratio Interval
C Root 1
C# 16/15 Minor 2nd
D 9/8 Major 2nd
D# 6/5 Minor 3rd
E 5/4 Major 3rd
F 4/3 Perfect 4th
F# 45/32 Augmented 4th
G 3/2 Perfect 5th
Ab 8/5 Minor 6th
A 5/3 Major 6th
Bb 9/5 Minor 7th
B 15/8 Major 7th
C 2/1 Octave

The above chart shows the ratios for intervals in relation to the fundamental. By careful examination and by listening to different notes you will see that there are many other possibilities for these same intervals in the harmonic series.  For instance the major second purely from the harmonic series is slightly different from the 2nd to the 3rd tones then from the 2nd to the 1st tones.  There are a number of other differences as such.  What this then gives the musician or composer is a number of different ways to play similar intervals.  This is how the Indian and Middle Eastern systems of music have what is referred to in Western Music Theory as ‘microtones’.  These musicians (as well as some Western musicians) will pick the size of the intervals based on the type of ‘scale’ they are playing.  Next lets look at ‘equal temperament’, currently the predominant temperament in modern Western Music.

Equal Temperament

Equal temperament is the system that has been use for the last two hundred years or so.  In equal temperament all minor second intervals are equal distantces from each.  None of these pitches are in tune with the just intonation equivalents and not tuned to the harmonic series.

Note Ratio Interval
C Root 1
C# 7893/7450 Minor 2nd
D 5252/4679 Major 2nd
D# 10754/9043 Minor 3rd
E 6064/4813 Major 3rd
F 6793/5089 Perfect 4th
F# 11482/8119 Augmented 4th
G 10178/6793 Perfect 5th
Ab 4813/3032 Minor 6th
A 9043/5377 Major 6th
Bb 17189/9647 Minor 7th
B 17843/9452 Major 7th
C 2/1 Octave

Web Resources

Tuning Figures - Lissajous Figures
Just Intonation & Equal Temperament - Sound Examples