Reduced mapping
A reduced mapping of a temperament is any nonstandard, simplified form of its mapping. Along with information about the size of the generator, types of reduced mappings can be a way to uniquely identify reasonable temperaments and understand basic properties about them.
Generator-chain 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.
Standard reduced mapping
To make a mapping even more concise and readable, as well as unique, we can make the following reductions, which for the sake of example are done on the temperament "Midnatssol," which is
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 442/441, 561/560, 676/675
Mapping: [⟨2 0 1 3 7 -1 5], ⟨0 2 1 1 -2 4 2], ⟨0 0 2 1 3 2 0]]
| Generator | 1st | 2nd | 3rd |
|---|---|---|---|
| preimage | 140/99 | 26/15 | 10/7 |
| WE tuning | 600.112 | 951.191 | 617.610 |
We start with the mapping above, which must be in Hermite normal form. First we combine the nth element in each row so that we can see the data for each prime rather than for each generator: ⟨2,0,0 0,2,0 1,1,2 3,1,3 7,-2,3 -1,4,2 5,2,0]
Then we mark the number of times the first generator is stacked modulo the number of periods per most reasonable equave with apostrophes or numbers in parentheses, then throw the rest of the data for the first generator away. We can also delete the first 2,0,0 because it always becomes 0, replacing it with "2 |" to indicate that there are two periods per 2/1. If this number were 1, we would leave out the number. (If we were using an equave other than 2/1, like 3/1, for example, we would have used "2<3/1> |".) Thus we have ⟨2 | 2,0 '1,2 '1,3 '-2,3 '4,2 '2,0]
Then we reduce the generator sizes to become less than the period, then take the period-complement of every generator except the first if it is larger than half the period. At this point we now note that the sizes of the second and third generators in cents have became 249c and 18c. This gives us ⟨2 | -2,0 '-1,2 -1,3 2,3 '-4,2 '-2,0]
Then we mark the just intervals in the subgroup after prime 13 or after the first prime we skipped. We also replace our ⟨ ] notation with standard [ ] brackets. Our final reduced mapping is [2 | -2,0 '-1,2 -1,3 2,3 '-4,2 17:'-2,0]