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Mapping: Difference between revisions

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Turn the mappings into tables. This allows vertical alignment
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At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:
At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:


{{val| a b c }} – period
{| class="right-all left-5 left-6"
 
|-
{{val| d e f }} – generator
| [⟨ || ''a'' || ''b'' || ''c'' || ] || – period
|-
| ⟨ || ''d'' || ''e'' || ''f'' || ]] || – generator
|}


The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
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When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at
When mapping a prime in JI onto a rank 2 temperament, one must think about how many steps of each type of generator it takes to reach the final tempered prime interval. For an example, we'll look at meantone temperament, and we'll start by mapping 2/1. We'll assume that the period is 2/1, and the generator is 3/2. 2/1 maps to one period and zero generators, and hence we arrive at


{{val| 1 _ _ }}
{| class="right-all left-5"
 
|-
{{val| 0 _ _ }}
| [⟨ || 1 || _ || _ || ]
|-
| ⟨ || 0 || _ || _ || ]]
|}


3/1 is slightly more complicated – it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:
3/1 is slightly more complicated – it requires one step along the 2/1 period chain, plus one step along the 3/2 generator chain, to get to 3/1. This is represented by the following mapping:


{{val| 1 1 _ }}
{| class="right-all left-5"
 
|-
{{val| 0 1 _ }}
| [⟨ || 1 || 1 || _ || ]
|-
| ⟨ || 0 || 1 || _ || ]]
|}


5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:
5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:


{{val| 1 1 0 }}
{| class="right-all left-5"
 
|-
{{val| 0 1 4 }}
| [⟨ || 1 || 1 || 0 || ]
|-
| ⟨ || 0 || 1 || 4 || ]]
|}


This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.
This is, in fact, the mapping matrix for meantone temperament, which is what we wanted.
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If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:
If we wanted to lay meantone out as having generators of 2/1 and 4/3, we arrive at the following list of vals:


{{val| 1 2 4 }}
{| class="right-all left-5"
 
|-
{{val| 0 -1 -4 }}
| [⟨ || 1 || 2 || 4 || ]
|-
| ⟨ || 0 || -1 || -4 || ]]
|}


This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
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If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):
If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):


{{val| 1 0 -4 }}
{| class="right-all left-5"
 
|-
{{val| 0 1 4 }}
| [⟨ || 1 || 0 || -4 || ]
|-
| ⟨ || 0 || 1 || 4 || ]]
|}


The [[normal lists #Normal val list|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.
The [[normal lists #Normal val list|normal val list]] is a normalized form among the variety of writing the mapping matrices, and it is what appears in temperament pages on this wiki.
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