Just temperament (or pure intonation) is the system of tuning typically used in modern symphonic ensembles – even if the players are unaware that they are using this system! This temperament (system of tuning) differs from equal temperament, which is used for fixed instruments, such as pianos, and digital MIDI instruments.
Just Temperament is based on ratios that occur naturally in music. These are found in the harmonic series (overtone). This is what allows brass instruments to change partials, flutes to overblow, and strings to play harmonics. In addition to allowing players to access additional higher notes, these overtones naturally occur on any note on a musical instrument (but not pure sine waves). The varying strengths of these overtones give each instrument its characteristic sound. This is how a violin and a flute can play the exact same pitch, but a listener is able to distinguish both instruments.
The overtone series can be notated on a staff. Note the following:
- As we ascend, the distance between notes continues to decrease
- Partials 8 – 16 can approximate a diatonic scale.
- The numbers above the notes indicate the difference (in cents) from equal temperament. Notice that E (the major third) is 14 cents flat while G (the fifth) is two cents sharp!
- These continue beyond the 16th partial
This is where it gets complicated: when relating various notes to the fundamental, ratios are used. The ratio of 2:1 indicates that an the higher note in an octave is half the wavelength of the lower note. Notice that in the ½ wavelength the cycle of the wave is able to complete itself exactly two times before first wavelength is able to complete its cycle.
The first two wavelengths on the left represent two tones an octave apart. Certain ratios of the fundamental wavelength will represent the intervals found in music. They are:
5:4 Major Third
6:5 Minor Third
Equal Temperament divides the twelve notes of the the scale into twelve equally-spaced tones. The best visual illustration is to imagine a clock face. Each number 1-12 would represent a different note, and is equally spaced apart from each other. However, the elegance of this division is lost in math. Every half step would be:
However, without equal temperament, we would be severely limited to which keys of music we could perform in. I have never encountered any Baroque music in the key of B major or D-flat major, despite my extensive study of the era!
Therefore, we compromise. In tuning, we base the root and fifth of a chord off of equal temperament, but will adjust the third to follow just temperament. While this sounds overly complex, it merely means that we engage our ears to place the note where we find it to be pleasing to our ears. It is also why tuners are NOT reliable for tuning in ensembles!
Not all pitches are that far off from Just temperament. Octaves, Fourths, and Fifths have little difference, but you will see a noticeable change in thirds. This chart indicates the differences in the pitch. ET = Equal Temperament. JT = Just Temperament, with C = Tonic.
For more information on the history behind temperament, read Temperament by Stuart Isacoff. Music has gone through drastic changes over the past centuries and we have discarded the notion that music itself follows divine laws. I believe this has been a significant loss to music – especially in worship! How much more impact does music have in worship when viewed it as God-given?