Douglas Blumeyer's RTT How-To: Difference between revisions

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* <math>\frac{28}{19} = 1.474 ≈ \frac{\log(5)}{\log(3)} = 1.465</math>

You may be more familiar with seeing the base specified for a logarithm, but in this case the base is irrelevant as long as you use the same base for both numbers. If you don't see why, try experimenting with different bases and see that the ratio comes out the same<ref>[~~https~~:~~//en.wikipedia.org/wiki/~~List_of_logarithmic_identities This list of logarithmic identities] has been an excellent resource for me in getting my head around logarithmic thinking. As you can see there, <math>\frac{log_{10}{a}}{log_{10}{b}} = log_{b}{a}</math>, so the base doesn't matter; you could put anything in there — 10, 2, e — and it still reduces to <math>log_{b}{a}</math>.</ref>.

You may be more familiar with seeing the base specified for a logarithm, but in this case the base is irrelevant as long as you use the same base for both numbers. If you don't see why, try experimenting with different bases and see that the ratio comes out the same<ref>[[Wikipedia:List_of_logarithmic_identities|This list of logarithmic identities]] has been an excellent resource for me in getting my head around logarithmic thinking. As you can see there, <math>\frac{log_{10}{a}}{log_{10}{b}} = log_{b}{a}</math>, so the base doesn't matter; you could put anything in there — 10, 2, e — and it still reduces to <math>log_{b}{a}</math>.</ref>.

But why take the logarithm at all? Because a) 2, 3, and 5 are not exponents, b) 12, 19, and 28 are exponents, and c) logarithms give exponents.

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The truth about distances between related ETs on the PTS diagram is actually slightly even more complicated than that, though; as we mentioned, the scaled axes are only the first difference from our initial guess. In addition to the effect of the scaling of the axes, there is another effect, which is like a perspective effect. Basically, as ETs get more complex, you can think of them as getting farther and farther away; to suggest this, they are printed smaller and smaller on the page, and the distances between them appear smaller and smaller too.

Remember that 5-limit JI is 3D, but we’re viewing it on a 2D page. It’s not the case that its axes are flat on the page. They’re not literally occupying the same plane, 120° apart from each other. That’s just not how axes typically work, and it’s not how they work here either! The 5-axis is perpendicular to the 2-axis and 3-axis just like typical Cartesian space. Again, we’re looking only at the positive coordinates, which is to say that this is only the [~~https~~:~~//en.wikipedia.org/wiki/~~Octant_(solid_geometry) ~~+++~~ octant] of space, which comes to a point at the origin (0,0,0) like the corner of a cube. So you should think of this diagram as showing that cubic octant sticking its corner straight out of the page at us, like a triangular pyramid. So we’re like a tiny little bug, situated right at the tip of that corner, pointing straight down the octant’s interior diagonal, or in other words the line equidistant from three axes, the line which we understand represents theoretically pure JI. So we see that in the center of the page, represented as a red hexagram, and then toward the edges of the page is our peripheral vision. ''(See Figure 3h.)''

Remember that 5-limit JI is 3D, but we’re viewing it on a 2D page. It’s not the case that its axes are flat on the page. They’re not literally occupying the same plane, 120° apart from each other. That’s just not how axes typically work, and it’s not how they work here either! The 5-axis is perpendicular to the 2-axis and 3-axis just like typical Cartesian space. Again, we’re looking only at the positive coordinates, which is to say that this is only the +++ [[Wikipedia:Octant_(solid_geometry)|octant]] of space, which comes to a point at the origin (0,0,0) like the corner of a cube. So you should think of this diagram as showing that cubic octant sticking its corner straight out of the page at us, like a triangular pyramid. So we’re like a tiny little bug, situated right at the tip of that corner, pointing straight down the octant’s interior diagonal, or in other words the line equidistant from three axes, the line which we understand represents theoretically pure JI. So we see that in the center of the page, represented as a red hexagram, and then toward the edges of the page is our peripheral vision. ''(See Figure 3h.)''

[[File:Understanding projection.png|600px|thumb|left|'''Figure 3h.''' Visualization of the projection process. (In real life, the cube is infinite in size. I made it smaller here to help make the shape clearer.)]]

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In between where the colored lines touch the maps themselves and the page, we see a cluster of more maps, each of which starts with 6. In other words, these maps are about twice as far away from us as the others. Let’s consider {{map|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{map|3 5 7}}. In effect, {{map|6 10 14}} is redundant with {{map|3 5 7}}. If you imagine doing a mapping calculation or two, you can easily convince yourself that you’ll get the same answer as if you’d just done it with {{map|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{map|3 5 7}} in terms of the relationships between the intervals it maps, the only difference being that it needlessly includes twice as many steps to do so, never using every other one. So we don’t really care about {{map|6 10 14}}. Which is great, because it’s hidden exactly behind {{map|3 5 7}} from where we’re looking.

The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps.<ref>Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [~~https~~:~~//en.wikipedia.org/wiki/~~Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.

The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps.<ref>Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [[Wikipedia:Projection_(mathematics)|projected]] onto the page as a single point. That’s why the diagram is called "projective" tuning space.

Now, to find a 6-ET with anything new to bring to the table, we’ll need to find one whose terms don’t share a common factor. That’s not hard. We’ll just take one of the ones halfway between the ones we just looked at. How about {{map|6 11 14}}, which is halfway between {{map|6 10 14}} and {{map|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{map|6 10 15}} is halfway between {{map|6 10 14}} and {{map|6 10 16}}, and its yellow line appears halfway between the red and green lines on the page. What this is demonstrating is that halfway between any pair of n-ETs on the diagram, whether this pair is separated along the 3-axis or 5-axis, you will find a 2n-ET. We can’t really demonstrate this with 3-ET and 6-ET on the diagram, because those ETs are too inaccurate; they’ve been cropped off. But if we return to our 40-ET example, that will work just fine.

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The mediant of two fractions <math>\frac ab</math> and <math>\frac cd</math> is <math>\frac{a+c}{b+d}</math>. It’s sometimes called the freshman’s sum because it’s an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we’re leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of <math>\frac 35</math> and <math>\frac 23</math> is <math>\frac 58</math>, and it’s easy to see in decimal form that 0.625 is between 0.6 and 0.666.

The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions <math>\frac 01</math> and <math>\frac 11</math>. Taking the mediant of these two gives our first node: <math>\frac 12</math>. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So <math>\frac 01</math> and <math>\frac 12</math> become <math>\frac 13</math>, and <math>\frac 12</math> and <math>\frac 11</math> become <math>\frac 23</math>. In the next tier, <math>\frac 01</math> and <math>\frac 13</math> become <math>\frac 14</math>, <math>\frac 13</math> and <math>\frac 12</math> become <math>\frac 25</math>, <math>\frac 12</math> and <math>\frac 23</math> become <math>\frac 35</math>, and <math>\frac 23</math> and <math>\frac 11</math> become <math>\frac 34</math>.<ref>Each tier of the Stern-Brocot tree is the next [~~https~~:~~//en.wikipedia.org/wiki/~~Farey_sequence Farey sequence].</ref> The tree continues forever.

The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions <math>\frac 01</math> and <math>\frac 11</math>. Taking the mediant of these two gives our first node: <math>\frac 12</math>. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So <math>\frac 01</math> and <math>\frac 12</math> become <math>\frac 13</math>, and <math>\frac 12</math> and <math>\frac 11</math> become <math>\frac 23</math>. In the next tier, <math>\frac 01</math> and <math>\frac 13</math> become <math>\frac 14</math>, <math>\frac 13</math> and <math>\frac 12</math> become <math>\frac 25</math>, <math>\frac 12</math> and <math>\frac 23</math> become <math>\frac 35</math>, and <math>\frac 23</math> and <math>\frac 11</math> become <math>\frac 34</math>.<ref>Each tier of the Stern-Brocot tree is the next [[Wikipedia:Farey_sequence|Farey sequence]].</ref> The tree continues forever.

So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there’s more to 7-ET than simply the number 7. The 7 is just a fraction’s denominator. The numerator in this case is 3. So imagine a <math>\frac 37</math> floating on top of the 7-ET there. And there’s more to 5-ET than simply the number 5, in that case, the fraction is the <math>\frac 25</math>. So the mediant of <math>\frac 25</math> and <math>\frac 37</math> is <math>\frac{5}{12}</math>. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: <math>\frac{5}{12}</math> is between <math>\frac 25</math> and <math>\frac 37</math> on the meantone line. You may verify yourself that the mediant of <math>\frac{5}{12}</math> and <math>\frac 37</math>, <math>\frac{8}{19}</math>, is between them in size, as well as <math>\frac{7}{17}</math> being between <math>\frac 25</math> and <math>\frac{5}{12}</math> in size.

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[[File:Interval of repetition vs interval of equivalence.png|200px|thumb|'''Figure 4d.''' Basically, the octave is usually the interval of equivalence, except in cases like tritave-based systems such as [[Bohlen-Pierce]]. And the period is always the interval of repetition. So when the period is the octave, such as with linear temperaments, then it is both the interval of repetition and of equivalence.]]

As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. Or an octave and an [~~https~~:~~//en.wikipedia.org/wiki/~~Augmented_unison augmented unison]. It could even be generated by cycling a fourth against a fifth. And so on.

As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. Or an octave and an [[Wikipedia:Augmented_unison|augmented unison]]. It could even be generated by cycling a fourth against a fifth. And so on.

And so it’s good to have a standard form for the generators of a linear temperament. One excellent standard is to set the period to an octave and the generator set to anything less than half the size of the period, as we did earlier.

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This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting (integer multiples of) these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{vector|-4 4 -1}} + {{vector|10 1 -5}} = {{vector|6 5 -6}}, which is the same thing as <math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!

Try one. How about the hanson comma, {{vector|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{vector|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this are technically called [~~https~~:~~//en.wikipedia.org/wiki/~~Elementary_matrix|elementary column operations]. Feel free to work through any other examples yourself.

Try one. How about the hanson comma, {{vector|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{vector|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this are technically called [[Wikipedia:Elementary_matrix|elementary column operations]]. Feel free to work through any other examples yourself.

A good way to explain why we don’t need three of these commas is that if you had three of them, you could use any two of them to create the third, and then subtract the result from the third, turning that comma into a zero vector, or a vector with only zeroes, which is pretty useless, so we could just discard it.

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[[File:Different nestings.png|400px|thumb|left|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]

We can extend our angle bracket notation (technically called [~~https~~:~~//en.wikipedia.org/wiki/~~Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

We can extend our angle bracket notation (technically called [[Wikipedia:Bra%E2%80%93ket_notation|bra-ket notation, or Dirac notation]]<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).

Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra|{{vector|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket|{{map|19 30 44}}}}.

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 * <math>\frac{28}{19} = 1.474 ≈ \frac{\log(5)}{\log(3)} = 1.465</math>
-You may be more familiar with seeing the base specified for a logarithm, but in this case the base is irrelevant as long as you use the same base for both numbers. If you don't see why, try experimenting with different bases and see that the ratio comes out the same<ref>[https://en.wikipedia.org/wiki/List_of_logarithmic_identities This list of logarithmic identities] has been an excellent resource for me in getting my head around logarithmic thinking. As you can see there, <math>\frac{log_{10}{a}}{log_{10}{b}} = log_{b}{a}</math>, so the base doesn't matter; you could put anything in there — 10, 2, e — and it still reduces to <math>log_{b}{a}</math>.</ref>.
+You may be more familiar with seeing the base specified for a logarithm, but in this case the base is irrelevant as long as you use the same base for both numbers. If you don't see why, try experimenting with different bases and see that the ratio comes out the same<ref>[[Wikipedia:List_of_logarithmic_identities|This list of logarithmic identities]] has been an excellent resource for me in getting my head around logarithmic thinking. As you can see there, <math>\frac{log_{10}{a}}{log_{10}{b}} = log_{b}{a}</math>, so the base doesn't matter; you could put anything in there — 10, 2, e — and it still reduces to <math>log_{b}{a}</math>.</ref>.
 But why take the logarithm at all? Because a) 2, 3, and 5 are not exponents, b) 12, 19, and 28 are exponents, and c) logarithms give exponents.
@@ Line 248: / Line 248: @@
 The truth about distances between related ETs on the PTS diagram is actually slightly even more complicated than that, though; as we mentioned, the scaled axes are only the first difference from our initial guess. In addition to the effect of the scaling of the axes, there is another effect, which is like a perspective effect. Basically, as ETs get more complex, you can think of them as getting farther and farther away; to suggest this, they are printed smaller and smaller on the page, and the distances between them appear smaller and smaller too.
-Remember that 5-limit JI is 3D, but we’re viewing it on a 2D page. It’s not the case that its axes are flat on the page. They’re not literally occupying the same plane, 120° apart from each other. That’s just not how axes typically work, and it’s not how they work here either! The 5-axis is perpendicular to the 2-axis and 3-axis just like typical Cartesian space. Again, we’re looking only at the positive coordinates, which is to say that this is only the [https://en.wikipedia.org/wiki/Octant_(solid_geometry) +++ octant] of space, which comes to a point at the origin (0,0,0) like the corner of a cube. So you should think of this diagram as showing that cubic octant sticking its corner straight out of the page at us, like a triangular pyramid. So we’re like a tiny little bug, situated right at the tip of that corner, pointing straight down the octant’s interior diagonal, or in other words the line equidistant from three axes, the line which we understand represents theoretically pure JI. So we see that in the center of the page, represented as a red hexagram, and then toward the edges of the page is our peripheral vision. ''(See Figure 3h.)''
+Remember that 5-limit JI is 3D, but we’re viewing it on a 2D page. It’s not the case that its axes are flat on the page. They’re not literally occupying the same plane, 120° apart from each other. That’s just not how axes typically work, and it’s not how they work here either! The 5-axis is perpendicular to the 2-axis and 3-axis just like typical Cartesian space. Again, we’re looking only at the positive coordinates, which is to say that this is only the +++ [[Wikipedia:Octant_(solid_geometry)|octant]] of space, which comes to a point at the origin (0,0,0) like the corner of a cube. So you should think of this diagram as showing that cubic octant sticking its corner straight out of the page at us, like a triangular pyramid. So we’re like a tiny little bug, situated right at the tip of that corner, pointing straight down the octant’s interior diagonal, or in other words the line equidistant from three axes, the line which we understand represents theoretically pure JI. So we see that in the center of the page, represented as a red hexagram, and then toward the edges of the page is our peripheral vision. ''(See Figure 3h.)''
 [[File:Understanding projection.png|600px|thumb|left|'''Figure 3h.''' Visualization of the projection process. (In real life, the cube is infinite in size. I made it smaller here to help make the shape clearer.)]]
@@ Line 266: / Line 266: @@
 In between where the colored lines touch the maps themselves and the page, we see a cluster of more maps, each of which starts with 6. In other words, these maps are about twice as far away from us as the others. Let’s consider {{map|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{map|3 5 7}}. In effect, {{map|6 10 14}} is redundant with {{map|3 5 7}}. If you imagine doing a mapping calculation or two, you can easily convince yourself that you’ll get the same answer as if you’d just done it with {{map|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{map|3 5 7}} in terms of the relationships between the intervals it maps, the only difference being that it needlessly includes twice as many steps to do so, never using every other one. So we don’t really care about {{map|6 10 14}}. Which is great, because it’s hidden exactly behind {{map|3 5 7}} from where we’re looking.
-The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps.<ref>Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [https://en.wikipedia.org/wiki/Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.
+The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps.<ref>Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [[Wikipedia:Projection_(mathematics)|projected]] onto the page as a single point. That’s why the diagram is called "projective" tuning space.
 Now, to find a 6-ET with anything new to bring to the table, we’ll need to find one whose terms don’t share a common factor. That’s not hard. We’ll just take one of the ones halfway between the ones we just looked at. How about {{map|6 11 14}}, which is halfway between {{map|6 10 14}} and {{map|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{map|6 10 15}} is halfway between {{map|6 10 14}} and {{map|6 10 16}}, and its yellow line appears halfway between the red and green lines on the page. What this is demonstrating is that halfway between any pair of n-ETs on the diagram, whether this pair is separated along the 3-axis or 5-axis, you will find a 2n-ET. We can’t really demonstrate this with 3-ET and 6-ET on the diagram, because those ETs are too inaccurate; they’ve been cropped off. But if we return to our 40-ET example, that will work just fine.
@@ Line 364: / Line 364: @@
 The mediant of two fractions <math>\frac ab</math> and <math>\frac cd</math> is <math>\frac{a+c}{b+d}</math>. It’s sometimes called the freshman’s sum because it’s an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we’re leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of <math>\frac 35</math> and <math>\frac 23</math> is <math>\frac 58</math>, and it’s easy to see in decimal form that 0.625 is between 0.6 and 0.666.
-The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions <math>\frac 01</math> and <math>\frac 11</math>. Taking the mediant of these two gives our first node: <math>\frac 12</math>. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So <math>\frac 01</math> and <math>\frac 12</math> become <math>\frac 13</math>, and <math>\frac 12</math> and <math>\frac 11</math> become <math>\frac 23</math>. In the next tier, <math>\frac 01</math> and <math>\frac 13</math> become <math>\frac 14</math>, <math>\frac 13</math> and <math>\frac 12</math> become <math>\frac 25</math>, <math>\frac 12</math> and <math>\frac 23</math> become <math>\frac 35</math>, and <math>\frac 23</math> and <math>\frac 11</math> become <math>\frac 34</math>.<ref>Each tier of the Stern-Brocot tree is the next [https://en.wikipedia.org/wiki/Farey_sequence Farey sequence].</ref> The tree continues forever.
+The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions <math>\frac 01</math> and <math>\frac 11</math>. Taking the mediant of these two gives our first node: <math>\frac 12</math>. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So <math>\frac 01</math> and <math>\frac 12</math> become <math>\frac 13</math>, and <math>\frac 12</math> and <math>\frac 11</math> become <math>\frac 23</math>. In the next tier, <math>\frac 01</math> and <math>\frac 13</math> become <math>\frac 14</math>, <math>\frac 13</math> and <math>\frac 12</math> become <math>\frac 25</math>, <math>\frac 12</math> and <math>\frac 23</math> become <math>\frac 35</math>, and <math>\frac 23</math> and <math>\frac 11</math> become <math>\frac 34</math>.<ref>Each tier of the Stern-Brocot tree is the next [[Wikipedia:Farey_sequence|Farey sequence]].</ref> The tree continues forever.
 So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there’s more to 7-ET than simply the number 7. The 7 is just a fraction’s denominator. The numerator in this case is 3. So imagine a <math>\frac 37</math> floating on top of the 7-ET there. And there’s more to 5-ET than simply the number 5, in that case, the fraction is the <math>\frac 25</math>. So the mediant of <math>\frac 25</math> and <math>\frac 37</math> is <math>\frac{5}{12}</math>. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: <math>\frac{5}{12}</math> is between <math>\frac 25</math> and <math>\frac 37</math> on the meantone line. You may verify yourself that the mediant of <math>\frac{5}{12}</math> and <math>\frac 37</math>, <math>\frac{8}{19}</math>, is between them in size, as well as <math>\frac{7}{17}</math> being between <math>\frac 25</math> and <math>\frac{5}{12}</math> in size.
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 [[File:Interval of repetition vs interval of equivalence.png|200px|thumb|'''Figure 4d.''' Basically, the octave is usually the interval of equivalence, except in cases like tritave-based systems such as [[Bohlen-Pierce]]. And the period is always the interval of repetition. So when the period is the octave, such as with linear temperaments, then it is both the interval of repetition and of equivalence.]]
-As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. Or an octave and an [https://en.wikipedia.org/wiki/Augmented_unison augmented unison]. It could even be generated by cycling a fourth against a fifth. And so on.
+As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. Or an octave and an [[Wikipedia:Augmented_unison|augmented unison]]. It could even be generated by cycling a fourth against a fifth. And so on.
 And so it’s good to have a standard form for the generators of a linear temperament. One excellent standard is to set the period to an octave and the generator set to anything less than half the size of the period, as we did earlier.
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 This was a bit tricky for me to get my head around, so let me hammer this point home: when you say "the null-space", you're referring to ''the entire infinite set of all commas that a mapping tempers out'', ''not only'' the two commas you see in any given basis for it. Think of the comma basis as one of many valid sets of instructions to find every possible comma, by adding or subtracting (integer multiples of) these two commas from each other<ref>To be clear, because what you are adding and subtracting in interval vectors are exponents (as you know), the commas are actually being multiplied by each other; e.g. {{vector|-4 4 -1}} + {{vector|10 1 -5}} = {{vector|6 5 -6}}, which is the same thing as <math>\frac{81}{80} × \frac{3072}{3125} = \frac{15552}{15625}</math></ref>. The math term for adding and subtracting vectors like this, which you will certainly see plenty of as you explore RTT, is "linear combination". It should be visually clear from the PTS diagram that this 19-ET comma basis couldn't be listing every single comma 19-ET tempers out, because we can see there are at least four temperament lines that pass through it (there are actually infinity of them!). But so it turns out that picking two commas is perfectly enough; every other comma that 19-ET tempers out could be expressed in terms of these two!
-Try one. How about the hanson comma, {{vector|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{vector|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this are technically called [https://en.wikipedia.org/wiki/Elementary_matrix|elementary column operations]. Feel free to work through any other examples yourself.
+Try one. How about the hanson comma, {{vector|6 5 -6}}. Well that one’s too easy! Clearly if you go down by one magic comma to {{vector|10 1 -5}} and then up by one meantone comma you get one hanson comma. What you’re doing when you’re adding and subtracting multiples of commas from each other like this are technically called [[Wikipedia:Elementary_matrix|elementary column operations]]. Feel free to work through any other examples yourself.
 A good way to explain why we don’t need three of these commas is that if you had three of them, you could use any two of them to create the third, and then subtract the result from the third, turning that comma into a zero vector, or a vector with only zeroes, which is pretty useless, so we could just discard it.
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 [[File:Different nestings.png|400px|thumb|left|'''Figure 5a.''' How to write matrices in terms of either columns/vectors/commas or rows/covectors/maps.]]
-We can extend our angle bracket notation (technically called [https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation bra-ket notation, or Dirac notation]<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).
+We can extend our angle bracket notation (technically called [[Wikipedia:Bra%E2%80%93ket_notation|bra-ket notation, or Dirac notation]]<ref>Bra-ket notation comes to RTT from quantum mechanics, not algebra.</ref>) to handle matrices by nesting rows inside columns, or columns inside rows ''(see Figure 5a)''. For example, we could have written our comma basis like this: {{bra|{{vector|-4 4 -1}} {{vector|-10 -1 5}}}}. Starting from the outside, the {{map|}} tells us to think in terms of a row. It's just that this row isn't a row of numbers, like the ones we've gotten used to by now, but rather a row of ''columns of'' numbers. So this row houses two such columns. Alternatively, we could have written this same matrix like {{ket|{{map|-4 -10}} {{map|4 -1}} {{map|-1 5}}}}, but that would obscure the fact that it is the combination of two familiar commas (but that notation ''would'' be useful for expressing a matrix built out of multiple maps, as we will soon see).
 Sometimes a comma basis may have only a single comma. That’s okay. A single vector can become a matrix. To disambiguate this situation, you could put the vector inside row brackets, like this: {{bra|{{vector|-4 4 -1}}}}. Similarly, a single covector can become a matrix, by nesting inside column brackets, like this: {{ket|{{map|19 30 44}}}}.