Subset notation
Subset notation is the practice of applying a musical notation system designed for a tuning system to a subset tuning system. It is mostly used with dual-fifth tunings, for which the native fifth notation may be ambiguous or counterintuitive.
The trivial case of subset notation is 12edo subset notation for 1edo, 2edo, 3edo, 4edo and 6edo. Since these edos only include intervals which are also in 12edo, it is easy to apply standard notation to these tuning systems.
The most commonly used subset notations include every other element of the original notation, or one element every three, etc., in a regular fashion. This method allows notating edos which don't have a good approximation of the perfect fifth (3/2) by using the standard notation of a superset edo which has a good approximation of the perfect fifth. For example, 13edo can be notated using 26edo subset notation.
Properties
Subset notation allows multiple tuning systems that have common intervals to share the same notation for these intervals. For example, a piece in 11edo will be more easily read in 22edo for anyone already familiar with 22edo notation.
Subset notation also helps avoiding issues that may arise when trying to apply the native fifth notation to certain tuning systems, such as negatively mapped accidentals. In particular, there is no consensus on the signification of sharps and flats in tunings with a fifth narrower than 4\7, between the regular or harmonic mapping (sharps down, flats up) and the melodic mapping (sharps up, flats down), with both options featuring important drawbacks about which one must be careful.
On the other hand, subset notation allows multiple ways to name notes, depending on the choice of the root note. For example, in 11edo as a subset of 22edo, one option excludes C, D and E natural, while the other excludes F, G, A and B natural. Without a consensus on which of the two subsets to use, one has to become fluent in both subsets.
Edos suitable for subset notation
| Tuning system | Notated as a subset of... |
|---|---|
| 1edo | 12edo |
| 2edo | 12edo |
| 3edo | 12edo |
| 4edo | 12edo |
| 6edo | 12edo |
| 8edo | 24edo (or 32edo, 40edo, 48edo, 56edo) |
| 9edo | 36edo (or 27edo) |
| 11edo | 22edo (or 33edo, 55edo) |
| 13edo | 26edo (or 39edo) |
| 16edo | 48edo (or 32edo) |
| 18edo | 36edo |
| 23edo | 46edo |
| 25edo | 50edo |
| 28edo | 56edo |
| 30edo | 60edo |
| 35edo | 70edo |