Tuning system
A musical tuning system or interval system (commonly referred to as a tuning or system) is a set of rules or algorithms – including enumeration – that gives all the notes or intervals theoretically available to a composer. The qualification theoretically is important, as such systems are typically defined (or definable) in the abstract, not taking into account the practical constraints of human hearing or the ability to actually produce all the pitches of the system, though it could involve the decision of what fundamental frequencies the notes of instruments will be "tuned to."
Most musicians in the western world are familiar with only one system, called 12-tone equal temperament, where the interval of the octave is divided into twelve equally spaced notes. There are, however, an infinite number of tuning systems, each resulting in different musical possibilities and characteristics. Xenharmony in general deals with tuning systems of all kinds, provided they are distinct from 12-tone equal temperament.
There are many schools of thought regarding what tuning systems are most useful and how they should be generated. With each of those perspectives comes a different method to generate them. Some of the most common systems are just intonation, regular temperaments, circulating temperaments, and equal divisions. Another source of tunings is those used historically and by various cultures throughout the world. Some of those tunings include Indonesian Pelog and Slendro, the Indian Shruti, Middle Eastern maqamat, and historical meantone tunings.
Open and closed systems
Gene Ward Smith considered the most basic of distinctions among tuning systems to be between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the MIDI tuning standard. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.
Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.
Among open systems, the most important kinds are periodic scales and group systems. The latter refers to "groups" in the mathematical sense of abelian groups, and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them. Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given harmonic limit; all intervals in a just intonation subgroup; and all intervals in a regular temperament.