Linear Temperaments

A rank two temperament is a regular temperament with two generators. If it is possible for one of the generators to be an octave, such a temperament is called a linear temperament (in the strict sense.) However, rank two temperaments in general are often called linear. The most common choice for generators is for one generator to be an octave, or some nth part of an octave for some integer n; in this case this generator is called the period and the other the generator.

A rank two temperament may be uniquely defined in various ways; one is by means of the wedgie, another by way of the comma sequence, and still another by means of the mapping, or icon, for a reduced set of generators. Here we are calling a pair of generators reduced if one generator is the period, and the other is the unique generator greater than one and less than the square root of the period (or less than half the period in logarithmic terms) which together with the period gives a generator pair for the temperament. This last definition depends on the exact tuning, and hence in theory may not be uniquely determined; in practice this seldom matters but for this and other reasons when working with temperaments using computer programs the wedgie is preferable as a means of defining the temperament. The icon, or tuning map, for the reduced generator pair, listing the period first and the generator second, we may call the standard icon.

Here is a list of seven-limit linear temperaments.

Listed below are some of the important families of linear temperaments.