Projection: Difference between revisions

The key reason for this difference is that mappings represent temperaments in the abstract, that is, without any specific tuning information; to find the cents value<ref>Any logarithmic pitch unit — cents, octaves, millioctaves, etc. — may be used, but this article has chosen to consistently use cents.</ref> of a ''mapped'' interval — one that has been mapped by a mapping — one must further map it by a [[generator tuning map]]. On the other hand, a ''projected'' interval — one that has been mapped by a projection, or "projected" — already includes the tuning information, and so their cents value can be obtained by mapping them with the generic [[just tuning map]] for the primes. In other words, the projection has applied tuning to the mapped intervals in a particular way, by embedding them back into the original JI space, where the tuning is known, so all we're really doing at that point is sizing the interval.

While a projection maps one prime-count vector to another prime-count vector, the output prime-count vector is usually quite different from the input prime-count vector. Most notably, the input interval is justly intoned, and therefore the entries of its prime-count vector are integers, while the output interval is tempered, and therefore the entries of its prime-count vector may be non-integers. Some temperament tunings are chosen so that certain JI intervals remain unchanged by the temperament; in such cases, if the input interval is one of the unchanged intervals, then its output will exactly match the input.  

A projection matrix may be found as the combination of a mapping with a generator embedding, through matrix multiplication. The mapping represents the temperament information in the abstract, i.e. abstracted from any specific tuning, while the generator embedding specifies such a tuning. And so together, the projection matrix represents the specific tuning of the given temperament, de-abstracting it.

A '''generator embedding'''  is an object that represents the ''tuning'' of a [[regular temperament]]. 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 information of a temperament than as a generator ''basis'' is as a [[generator tuning map|generator ''tuning map'']]. A 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 tuning information, and a mapping which provides the temperament 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 bases contain not only tuning and temperament information, but also ''generator-size'' information, and it is this generator-size information which causes them to be non-unique; however, when they are combined into a projection matrix, their generator-size 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 temperament 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.  

which technically vanishes the [[meantone comma]] <math>\frac{81}{80}</math> — the main requirement of being a meantone temperament — but it has a period of 76 ¢ and generator of 41 ¢, which is pretty strange indeed.  

What we can say is that generator-size information differentiates these forms of the meantone mapping.<ref>There is a way to represent temperament information without generator-size information, which therefore means a data structure which inherently uniquely represents temperaments, and that is a [[multimap]] (at this time, however, this author is not aware if there is an equivalent structure for uniquely representing tuning information independent of generator-size information, or of representing generator-size information independent of either tuning or temperament information).</ref>

===Matching generator bases===

Supposing one desires to transform from a pair of <math>M_1</math> and <math>G_1</math> to another pair of <math>M_2</math> and <math>G_2</math> where both pairs multiply to the same <math>P</math>, or — said another way — you wish to keep your <math>M</math> and <math>G</math> ''in sync'', the simplest approach would be to — for each elementary row operation you apply to <math>M</math> you must apply the opposite elementary column operation to <math>G</math>, e.g. if you add three times the second row to the first row of <math>M</math>, then you must ''subtract'' three times the second ''column'' from the first ''column'' of <math>G</math>. This is along the same lines as the explanations provided for manipulating generator-size by changing forms of <math>M</math>, which you can find here: [[Generator form manipulation]].

For example, if we have <math>M_1</math> = {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} and <math>G_1</math> = {{bra|{{ket|1 0 0}} {{ket|0 0 <math>\frac14</math>}}}}, then <math>M_1</math> and <math>G_1</math> are in sync because they're both in the form where <math>g_1</math> is ~2 and <math>g_2</math> is ~3/2. Or if we have <math>M_2</math> = {{ket|{{bra|1 0 -4}} {{bra|0 1 4}}}} and <math>G_2</math> = {{bra|{{ket|1 0 0}} {{ket|1 0 <math>\frac14</math>}}}} then they're still in sync because they're both <math>g_1</math> ~2 and <math>g_2</math> ~3 here. But if we mismatched those, they'd be out of sync. Those are both <math>M</math>'s for meantone, and both <math>G</math>'s that can work for quarter-comma meantone, but if you mismatch them with respect to the generator-size information, you won't find the same <math>P</math> by matrix multiplication <math>GM</math>.  

(This notion of "sync" is the same idea pointed out in the diagram at the start of the "Obtaining objects from the projection" section below, with the note on <math>G</math> reading "(the one matching M)". And for more information on generator-size information, see the "Information types" below.)

We note in particular that putting <math>M</math> and <math>G</math> into their canonical forms independently is not a guarantee that they will remain in sync; canonicalization will not necessarily arrive at the same generator-size information in <math>G</math> as it does in <math>M</math>.

In other words, the service it provides is rather to ''give us a way to speak of the generator form of a temperament with respect to a particular mapping basis and a particular generator embedding.'' In other words, if we possess a scheme for unambiguously determining a "home base" <math>M</math> — such as a [[canonical form]], which we ''do'' have — and we possess a scheme for unambiguously determining a particular embedding — such as any tuning scheme which gives tunings expressible as embeddings (e.g. minimean, miniRMS) — then we now also have a way to describe with cold, hard matrices (i.e. not fuzzier instructions like "octave-fifth") what exact form we wish our generators to be in. We can speak of a pair of a linked bases — the generator embedding treated as a basis, and the mapping treated as a mapping-row basis — as the (generator) '''form''' of a temperament. And so a tuning system can be more fully specified than it could previously, leveraging this conceit.

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).

* When a scaling factor is equal to 1, this essentially means "no scaling", since multiplying something by 1 has no effect; so, these unrotated vectors are ''also'' unscaled vectors, which means they are completely unchanged. In other words, these are the unchanged intervals of this tuning of this temperament.

In general, it is an <math>r × d</math> matrix where <math>r = d</math>, or in other words, a <math>d × d</math> matrix. Whereas a rank-2 temperament in 5-limit still boasts three columns (those correspond to the 3 basis primes of the 5-limit), but only has two rows, corresponding to how it has reduced the three generators of pure JI down to two generators approximating JI:

Here's one way to think about it. When we look at a mapping matrix like meantone's {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} above, we're basically plotting vectors in a brand-new 2D space, that is, one completely distinct from the 3D space we started with in JI. So while the <math>x</math>, <math>y</math>, and <math>z</math> axes for JI corresponded to our basis primes <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math>, our <math>x</math> and <math>y</math> axes (no <math>z</math> here!) of our new meantone space correspond to its generators <math>\text{g}_1</math> and <math>\text{p}_2</math>. That's how ''its'' 2D plane exists. However, as for a tuning of meantone's projection matrix, such as quarter-comma's, it remains in the original 3D space with the <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math> axes. What this means is that everything in that space gets projected — or smooshed down, you could think of it — into a single plane. So this plane is 2D, certainly, but importantly, it still requires three coordinates to describe ''because it exists titled at some angle through the original JI space''. It is a 2D object occupying 3D space, just like any 2D document you might have in your real-life 3D physical space sitting up on a bookshelf at some slight angle.  

Any projection will occupy some line/plane/space/etc. that is perpendicular to all of the temperaments commas' vectors, no matter what the tuning happens to be. What differentiates one tuning from another is what path all the other intervals take onto the projection. Remember how we described projections earlier as distortion fields with curvy vortices something like how we might see warm and cold fronts on a weather map, and among those waves of distortion one will occasionally find paths or spots where the distortion works out perfectly straight or perfectly still. So any tuning of a temperament will have the same such straight paths for the commas mapping to the origin (the point with all zeros). But the tunings will all be set aside from each other by which unchanged intervals they have, that is, the "eyes of the storm" if you will, the points in space that don't budge at all by the distortion. And whatever these are will be part of the complicated distortion field that leads to all the other non-eigenvector intervals landing somewhere or another on that set projection line/plane/space (its dimensionality depends on the rank of the temperament).

Remember, the mapping represents the temperament, and the projection represents a particular tuning of this temperament, so no matter which projection we use, while they will each have their own unchanged intervals, they will share the same commas: the commas of the temperament.

An alternative method for finding <math>\textrm{C}</math> is discussed in the "Alternative method for the comma and unchanged interval bases" section below.

===The unchanged interval basis===

Obtaining the unchanged interval basis <math>\textrm{U}</math> means to obtain the unchanged intervals <math>\textbf{u}_1, \textbf{u}_2, …</math> of <math>P</math> or in other words any <math>\textbf{u}_i</math> for which <math>P\textbf{u}_i = \textbf{u}_i</math>. There are many ways to find these, but one way stands out for its clarity. We can rewrite this equation by subtracting <math>\textbf{u}_i</math> from both sides to get:

An alternative method for finding <math>\textrm{U}</math> is discussed in the "Alternative method for the comma and unchanged interval bases" section below.

To obtain (some form of) a generator embedding for a projection, find the unchanged interval basis per the above, and then use <math>G = \textrm{U}(M\textrm{U})^{-1}</math>. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Solving for the generators]].

The answer comes in the form of a tuple. The first element is a list of our scaling factors (eigenvalues), and the second element is a list of our unrotated vectors (eigenvectors). So, this result is telling us that for scaling factors 1, 1, and 0, respectively, we have corresponding unrotated vectors {{vector|0 0 1}}, {{vector|1 0 0}}, and {{vector|4 -4 1}}. The set of eigenvectors with eigenvalue 0 constitute a comma basis, while the set of eigenvectors with eigenvalue 1 constitute an unchanged interval basis (so, quarter-comma meantone is characterized by an unchanged <math>\frac21</math> and <math>\frac51</math>).

===Alternative method for the comma and unchanged interval bases===

On the left, we've highlighted the diagonalized scaling factors with green if they are for unchanged intervals, and with red if they are for vanished commas. Then on the right, we've colored the entire corresponding vector columns, and placed a vertical line between the two green columns corresponding to the two green scaling factors of 1 and the one red column corresponding to the one red scaling factor of 0. And so we can see that the green-colored part of <math>\mathrm{V}</math> is <math>\textrm{U}</math>, and the red-colored part of <math>V</math> is <math>\textrm{C}</math>.

==Information types==

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

Mappings combine types (1) and (3). Generator bases 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.

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 temperament info (1) from the generator-size info (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-size info goes away from both, and you're left with just pure temperament and tuning information. We've used color to help convey this idea in the following diagram, with type (1) red, type (2) blue, type (3) green:

  TODO: diagram (but add a j to the arrow on the right, and generally label these arrows because they each mean different things, and italicize the t, and double-struck the m, and rename generator-size information probably), and also show the green stuff splitting out of the G and M into F in the middle

So again, while the mapping represents temperament information abstracted from any tuning, and the generator embedding represents tuning information that could be applied to any suitable temperament, they are each impure in the sense that they bind their respective informations to a particular size of generator. This is the nature of how they must match to multiply together to give a certain projection. But no matter which two <math>G</math> and <math>M</math> you choose, as long as they do match, then their generator-size information cancels out and we end up with just the temperament and tuning infos.  

(The top-left object is something no one has ever spoken about, as far as we know, and we see no use for it. We can't even say what "pure tuning information" would mean, independent of a temperament, or what it would mean to explore that space, in the way theorists have explored multimap space using temperament addition, etc. So we can see that the "multituning", perhaps we could call it, of quarter-comma meantone, is {{multivector|0 ¼ 0}})

# tunings based on unchanged intervals: see [[Dave Keenan & Douglas Blumeyer's guide to RTT: tuning computation#Unchanged interval method]].