Projection: Difference between revisions
Cmloegcmluin (talk | contribs) mNo edit summary |
Cmloegcmluin (talk | contribs) →Generator information types: add diagrams |
||
| Line 1,126: | Line 1,126: | ||
==Generator information types== | ==Generator information types== | ||
[[File:Info types - flow chart.png|thumb|602x602px|A diagram showing how the three information types (approximation, embedding, and form) break down across the tuning map, projection, mapping, generator embedding, multimap, and form matrices. ]] | |||
One way to think about what's happening in this vicinity of RTT is that we have three different generator information types: | One way to think about what's happening in this vicinity of RTT is that we have three different generator information types: | ||
#approximation | #approximation | ||
| Line 1,136: | Line 1,136: | ||
One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see: [[https://en.xen.wiki/w/Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#Pure_representation_of_temperament_information]] | One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see: [[https://en.xen.wiki/w/Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#Pure_representation_of_temperament_information]] | ||
In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the | In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the diagram to the right, with type (1) red, type (2) blue, type (3) green. | ||
So, when you compress the multi-row projection matrix into a single-row tuning map by multiplying it by the just tuning map <math>𝒋</math>, the two types of information are still there, but blended together such that they are unrecoverable, or in other words, it's now ambiguous how we arrived at this <math>𝒕</math> and could have arrived to it from a different combination of <math>M</math> and <math>G</math>. | So, when you compress the multi-row projection matrix into a single-row tuning map by multiplying it by the just tuning map <math>𝒋</math>, the two types of information are still there, but blended together such that they are unrecoverable, or in other words, it's now ambiguous how we arrived at this <math>𝒕</math> and could have arrived to it from a different combination of <math>M</math> and <math>G</math>. | ||
| Line 1,148: | Line 1,146: | ||
And here's a series of tables that show various parts of the tempering process color-coded according to the above diagram: | And here's a series of tables that show various parts of the tempering process color-coded according to the above diagram: | ||
[[File:Info types - checkers.png|frameless|800x800px]] | |||
==Mapping projected intervals== | ==Mapping projected intervals== | ||