Equal Temperament

An equal temperament, abbreviated ET, is a musical tuning which consists of equal divisions, usually of the octave. (An exception, the Bohlen-Pierce scale, equally divides the tritave, or an octave plus a fifth). Such tunings and scales are said to be equally tempered.

Why equal temperament at all?

Since this site advocates for a particular equal temperament, 31-ET, it is important to ask the question of why one would want to use equal temperament at all. Some of the main benefits of equally-tempered tunings (over tunings based on just intonation) are:

• Flexibility in changing key - Equal temperaments allow for an unlimited amount of key changing and cycling through different intervals, while remaining in a well-defined system with a finite number of intervals. In tunings based on just intonation, as one changes keys or cycles through intervals, an infinite array of possible relationships unfold. This puts a rather strict limit on the degree to which and ways in which you can change keys or cycle through intervals in just intonation: in theory one must retrace ones steps in order to reach the original key. In equal temperament, there is no such constraint.
• Fewer intervals to learn - In harmonically complex music, there are fewer total musical intervals to comprehend when equal temperaments are used, relative to just intonation. This is because many of the smallest intervals, and the distinctions between musical intervals close in size, are tempered out. Some of these intervals are so small that they are beyond the perceptual limits of human hearing. In equal temperaments, on the other hand, even ones with a much greater number of divisions than 31, the fundamental step size is well within the perceptual limits of a typical untrained ear.
• Little lost by approximations - Some equal temperaments are so good at matching certain musical intervals that little is lost by using them. 31-ET is one such system, when it comes to the 3rd and 5th harmonics (corresponding to fifths and major/minor thirds). 12-ET, the dominant tuning in western culture, provides an outstanding match to fifths.
• Practical instrument construction - For instruments such as keyboards, fixed-fret guitars (most guitars), and keyed percussion instruments like marimba and other xylophones, using just intonation is prohibitive and unrealistic, since the instrument must be built ahead of time, and then, only a limited selection of possible notes are available. Equal temperaments allow such instruments to be built so that they can play an exhaustive set of musical relationships within that tuning, something not possible with just intonation. Keyed woodwinds and to a degree valved brass instruments can still benefit from equal temperamet in this way, even though these instruments have more flexibility for the player to bend the pitch of each note.

Personal reasons for preferring equal temperament

On a personal note, I would like to share why I love equal temperaments, and why, compared to equal temperament, I find just intonation to be both difficult to work with and boring:

• I love music that changes keys frequently, such as the fugues of J.S. Bach, especially his later works, and complex jazz, such as Thelonious Monk, Charlie Parker, and other artists from the Be-bop era. I think the use of key changes and unusual harmonic relationships in this music sounds beautiful, emotionally moving, and mentally stimulating. This sort of music is my favorite sort of music, and it is also the sort of music that is best served by equal temperament. The music best served by just intonation, by contrast, I find more boring.
• I am interested in forming an exhaustive or complete understanding of a harmonic system or tuning system. In just intonation, this is not possible, due to the infinite number of relationships. This contributes to just intonation being difficult.
• I personally feel greater freedom when writing or playing music in equally tempered tunings. I find just intonation requires a great deal of restraint, and places great constraints on composers: rather than being free to explore any and all intervals, and travel down all musical roads, following musical intervals and key changes as feel natural to me, I feel like I have to stay close to the original tonal center and always return to it. In equal temperament, I feel totally free, able to go wherever I want without having to look back or worry about how I'm going to get back to where I came from. Then, when I return to older notes, it is by choice, or by accident, rather than by being forced into careful design out of necessity.