Just intonation

Just intonation is the set of intervals corresponding to frequency ratios between whole numbers, and the approach to musical tuning which utilizes exclusively such intervals. Just intonation can be described in terms of the harmonic series (which is the set of tones at integer multiples of a fundamental frequency), where all just intervals can be found between notes in the harmonic series. Particularly low-complexity just intervals tend to be perceived as consonant, serving as significant tonal anchors for the building of scales. These intervals include 2/1, 3/1, 3/2, 4/3, and 5/3. Additionally, much of xenharmonic tuning theory tends to be built around just intonation intervals as tuning targets, as they are often easy to recognize by ear and tune to.

In older materials, "just intonation" referred in particular to 5-limit just intonation - that is, just intonation where no frequency ratio could have a prime factor exceeding 5. (Thus, this included 3/2 and 25/24, but not, for instance, 9/7. High-complexity Pythagorean intervals were generally considered separately.)

By convention in modern xenharmonic theory, stacking-based just intonation is seen as an interval space, wherein the prime harmonics (2/1, 3/1, 5/1, 7/1, etc) are seen as the "building blocks" of all intervals. Note that interval stacking corresponds to multiplication.

Just intonation in arbitrary limits has a number of complexities. These include the appearance of commas (small intervals between distantly related notes) and wolf intervals (dissonant intervals resulting from simple targets altered by commas), which often result in chord progressions drifting up or down in pitch over time. These can be taken as features of any given JI system, or as problems to solve (for instance, for composers wanting a fixed set of notes or unlimited modulation) - in the latter case, approaches to solve the problems include temperament.

Approaches

There are many approaches to JI. The following approaches are not necessarily mutually exclusive:

• Stacking-based JI: conceived as taking place on a JI subgroup or lattice
• Free JI: not fixed to a particular set of notes. The dominant approach in Ben Johnston's JI string quartets.
• Linear JI: harmonic series focused, including nejis, which are subsets of harmonic series modes approximating a given scale. Includes primodality, Zhea Erose's JI approach.

JI scale constructions and properties

• Generator sequences: stacking a sequence of JI generators cyclically
• Combination product sets
• Detempering an equal temperament
• Constant structure: An often desired property for JI scales