Reduced mapping

A reduced mapping of a temperament is any nonstandard, simplified form of its mapping.

The most basic reduced mapping is sometimes called generator-chain mapping, which is simply the mapping without the first row. For instance, the mapping of magic is [⟨1 0 2 -1], ⟨0 5 1 12]], and its generator-chain mapping is ⟨0 5 1 12]. For full-octave temperaments, it preserves the information to find the octave-reduced prime harmonics, though the full mapping cannot be restored without some common sense regarding how many periods should be added to each octave-reduced prime harmonic. It represents a mathematical object (covector) that can be used to compute various related data, e.g. inner product with a monzo to find its number of generator steps in the temperament.

A recent development specifically for rank-2 temperaments observes that the first entry of the generator-chain mapping is always zero, so it replaces the entry with the first entry of the first row of the mapping, which represents the number of periods per octave. It then separates the entry from the rest with a semicolon. For instance, the reduced mapping of magic is ⟨1; 5 1 12]. This is the form of reduced mapping used in Template: Infobox Regtemp on this wiki. Like above, the full mapping cannot be restored without some common sense, but the addition of the number of periods per octave can be helpful in the case of fractional-octave temperaments. To use it as a mathematical object, the first entry needs to be set back to zero.