Tuning pitch brackets: part 1
Harmony began developing in Europe around the 14th century. Prior to this, instruments were tuned using the Pythagorean method which employed pitch ratios of 2:1 and 3:2. As musicians began exploring new harmonic possibilities in the 14th and 15th century they found that the Pythagorean scale was problematic. While the interval of a 5th and 4th sounded pure, the major and minor third intervals were dissonant and harsh sounding. Eventually this idea of pure intervals was codified and labelled “just intonation”. All of the pitch intervals in just intonation are ratios of small integers:
- 2:1 octave
- 3:2 perfect fifth
- 4:3 perfect fourth
- 5:4 major third
- 6:5 minor third
To accomplish just intonation, theoreticians adjusted the Pythagorean scales so that major thirds were restored to their pure 5:4 ratio. This worked but introduced new undesirable, dissonant intervals. Also, the number of interval types increased: two kinds of tones, four kinds of semitones. This was particularly a problem for keyboard instruments which, with 31 or more notes per octave, became impractical to construct and difficult to play.
Thus a compromise was made. The two kinds of tones in just intonation were merged into one kind and the four semitones merged into two kind. This was accomplished by making more tweaks to the Pythagorean scale while still preserving the pure 3rd interval which, after all, was most problematic of the original Pythagorean scale. The result was called “meantone” temperament because it averages out the difference between the two tones of just intonation.
As tonality developed through the following centuries, it became useful to transpose music and modulate to different keys. At this time, Bb was not the same as A#. This was a result of the way meantone temperament adjusted the Pythagorean scale. Transposition and modulation could only be done easily if the two kinds of semitones, flats and sharps, were equated. By dividing the octave evenly in to 12 chromatic steps, without any regard to the optimal just-intonation ratios, we end up with the equal tempered system still in use today. We have become accustomed to, for example, the dissonance of the equal tempered major third. This was not the case for those first generation equal temperament listeners.
In the following post, I will show how just-intonation, meantone temperament, and most other tuning systems can be accomplished in pitch bracket notation.
P.S. Thanks to Mr. Khramov for guiding me to the excellent book, Enharmonic: Instruments and Music 1470-1900 by Patrizio Barbieri.