Projection: Difference between revisions

# represent regular temperaments,  

# transform [[JI]] intervals, and  

Perhaps the simplest way to explain the difference would be to say that a mapping outputs intervals in the form of [[generator-count vector]]s, whereas a projection outputs intervals that are also in the form of prime-count vectors.  

So, if an entire matrix is mapped by a temperament's mapping matrix to an identity matrix, then that is a very special case; it tells us that each of this matrix's columns can be thought of as a prime-count vector that maps to a different one of each of that same temperament's generators. It is, in other words, a [[generator preimage transversal]].

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A '''generator embedding'''  is an object that represents the ''embedding'' of a [[regular temperament]] from the tempered lattice back into tuning space. It could be thought of as representing the "tuning" information of a temperament, if one leaves out the actual "sizing" part of that (the conversion of prime factors to their logarithmic pitch size). It has one column for each of the temperament's [[generators]]. Each of these columns represents its generator's tuning in the form of a prime-count vector.

A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map|1.000 1.585 2.232}}·{{ket|{{map|1 1 0}} {{map|0 0 0}} {{map|0 1/4 1}}}} = {{map|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map|log₂2 log₂3 log₂5}} ≈ {{map|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map|1.000 1.585 2.232}}·{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{map|1.000 0.580}}, or in cents instead of octaves, that's {{map|1200.000 696.578}}.  

As just mentioned, projection matrices represent specific tunings of abstract temperaments, being the matrix product of a generator embedding which provides the embedding information, and a mapping which provides the approximation information. Notably, not only does the projection matrix represent the tuning of a temperament, it does so ''uniquely''. We can say that mappings and generator embeddings contain not only approximation and embedding information, but also ''generator form'' information, and it is this generator form information which causes them to be non-unique; however, when they are combined into a projection matrix, their generator form information ''cancels out'', and so no matter which combination of matching mapping and generator embedding we choose for a given temperament tuning, we will end up with the same exact projection.<ref>The present author is not sure if any combination of mapping and generator embedding should lead to a projection matrix, and what the conditions on this would be. If anyone can figure this out, please add it to the article.</ref>

To be clear, a mapping matrix does ''not'' uniquely represent approximation information. Multiple mappings can be found that describe the same temperament, in the sense that the same set of [[commas]] are [[Tempering_out|vanished]]. This non-uniqueness is the reason why a [[canonical form]] for mappings was developed, which can be understood as a function which takes any equivalent mapping and converts it to the same exact mapping.  

A <math>(d, d)</math>-shaped projection matrix represents both all of the [[comma]]s of a temperament as well as all of the [[unchanged-interval]]s of the tuning. It does so via <math>d</math> '''unrotated vectors''', which is to say, this projection only ''scales'' the vector. Each one of these unrotated vectors, then, is paired with a corresponding '''scaling factor''', which tells us by how much the projection scales it. Each of these scaling factors — which we'll represent with the Greek letter lambda <math>λ</math> — is equal to either 1 or 0, and no other value is possible (more on this in the next section).

For readers familiar with the linear algebra concepts of [https://mathworld.wolfram.com/Eigenvector.html eigenvector]s and eigenvalues, this all may sound familiar: eigenvector is technical math speak for an "unrotated vector", and eigenvalue is technical math speak for its "scaling factor"; the prefix "eigen-" comes from the German word for "own", and this refers to the fact that the vector is a member of its own [[span]], or in other words, it falls along the infinite line through space we get if we extend the vector in both directions forever (after projecting it). For those readers unfamiliar with these ideas, we recognize that this may be a lot to process at once.  

===Idempotence===

So we can see why the only possible values a projection could scale its eigenvectors by would be 1 and 0, because these are the only values one can repeatedly scale things by without changing them past the first scaling.<ref>Here is a helpful proof, giving another way to look at this fact: https://math.stackexchange.com/questions/1393656/show-that-if-%ce%bb-is-an-eigenvalue-of-a-projection-matrix-p-then-%ce%bb-1-or-%ce%bb-0/1393661#1393661</ref>  

A good way to understand the idempotence of projections is geometrically. This also agrees with our natural language intuitions about projections, such as projecting shadows of objects onto a wall. A projection is any transformation that reduces the dimensionality. What we've done is taken information in <math>d</math>-dimensional space and projected it down by one dimension, into <math>(d-1)</math>-dimensional space. For example we might take some 3D object and project it down to its silhouette. If we take the projection we just made and try to project it again, nothing is left to change.  

  TODO: diagram, at least one diagram here would be really valuable; perhaps this will help https://app.asana.com/0/1200134195404323/1201659347245292. also plane always flat at bottom projection diagram - might be easier to discern how least-squares points straight up and others are at an angle - as opposed to having the x/y/z axes always in the same position and then try to figure out what angle is this plane oriented through the space. this could help too: https://stackoverflow.com/questions/42893410/how-to-get-the-tilt-angle-of-a-plane-defined-by-three-coordinates-in-python

It's rather tricky to visualize planes tilted within volumes, as it turns out. But perhaps a simpler example will be welcome. Consider the following 2D lattice, for 3-limit JI:

caption: All you need is purple to get anything on here (a purple bit, or one blue bit and one red bit). Meaning that if you have any of these:

then you can get everything else using the methods described below.

To obtain the comma basis from <math>P</math>, simply take the nullspace as you would take it of the mapping (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: exploring temperaments#Nullspace]] for more information).  

  <nowiki>In:  P = {{1,1,0},{0,0,0},{0,1/4,1}}

By the way, the general method to find scaling factors is to solve the "characteristic equation" <math>\det(P - λI) = 0</math> for <math>λ</math>; that's lamdba times an identity matrix, or in other words, subtract lambda from each entry along <math>P</math>'s diagonal. But this shouldn't be necessary if one follows the other suggestions provided here.

This section assumes you've reviewed the immediately previous section.

One way to think about what's happening in this vicinity of RTT is that we have three different generator information types:  

Mappings combine types (1) and (3). Generator embeddings combine types (2) and (3). Projections combine types (1) and (2). So each possible subset of two of these pieces of information is accounted for by these three objects.

Projections and generator bases comes up in some methods for calculating tunings according to commonly-used schemes:

==See also==

Note that projection matrices do not have anything deeply to do with [[projective tuning space]] or [[projective tone space]], other than the fact that they both use the mathematical operation of projecting something from a higher dimension to a lower one.