Diatonic semitone
A diatonic semitone, minor second or limma is the small step of a diatonic scale.
In just intonation, an interval may be classified as a diatonic semitone if it is reasonably mapped to 1\7 and 2\24 (precisely one step of the diatonic scale and one step of the chromatic scale). The use of 24edo's 2\24 as the mapping criteria here rather than 12edo's 1\12 better captures the characteristics of many intervals in the 11- and 13-limit.
Examples
- 256/243, the Pythagorean diatonic semitone (3-limit)
- 16/15, the classic diatonic semitone (5-limit)
- 128/121, the Axirabian diatonic semitone (11-limit; specifically belonging to the 2.3.11 subgroup)
Notation
The number of steps a limma is mapped to in an EDO is referred to as its limmanosity, or penta-sharpness.
Etymology
According to the OED, the earliest English use of limma and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical[1] meaning, is in 1694 in A Treatise of the Natural Grounds and Principles of Harmony[2] by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).