Reduced mapping: Difference between revisions
Restore the old reduced mapping, which has some recognition in the community and is notably used in Template: Infobox Regtemp. The new "standard reduced mapping" isn't standard until accepted by the community; note the author as a means of distinction for now. |
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== Generator-chain mapping == | == 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 {{mapping| 1 0 2 -1 | 0 5 1 12 }}, and its generator-chain mapping is {{val| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many | 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 {{mapping| 1 0 2 -1 | 0 5 1 12 }}, and its generator-chain mapping is {{val| 0 5 1 12 }}. For full-octave temperaments, it preserves the information to find the [[octave reduction|octave-reduced]] [[prime harmonic]]s, though the full mapping cannot be restored without some common sense regarding how many [[period]]s 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 {{val| 1; 5 1 12 }}. This is the form of reduced mapping used in [[Template: Infobox | 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 {{val| 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. | ||
== | == HKM's 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 midnatssol temperament, which is | 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 midnatssol temperament, which is | ||