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

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Until stated otherwise, this material will assume the [[5-limit|5 prime-limit]].

If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{vector|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. If we need to distinguish these vectors from other kinds of vectors used in RTT, we call these prime count ~~vectors~~ or PC-vectors.

If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{vector|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. If we need to distinguish these vectors from other kinds of vectors used in RTT, we call these [[prime-count vector]]s or PC-vectors.

And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{map|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach 2/1 (the [[octave]]), 19 steps to reach 3/1 (the [[tritave]]), and 28 steps to each 5/1 (the [[pentave]]). Any or all of those intervals may be approximate.

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If you’ve worked with 5-limit JI before, you’re probably aware that it is three-dimensional. You’ve probably reasoned about it as a 3D lattice, where one axis is for the factors of prime 2, one axis is for the factors of prime 3, and one axis is for the factors of prime 5. This way, you can use vectors, such as {{vector|-4 4 -1}} or {{vector|1 -2 1}}, just like coordinates.

PTS can be thought of as a projection of 5-limit JI map space, which similarly has one axis each for 2, 3, and 5. But it is no JI pitch lattice. In fact, in a sense, it is the opposite! This is because the coordinates in map space aren’t prime count lists, but maps, such as {{map|12 19 28}}. That particular map is seen here as the biggish, slightly tilted numeral 12 just to the left of the center point.

PTS can be thought of as a projection of 5-limit JI map space, which similarly has one axis each for 2, 3, and 5. But it is no JI pitch lattice. In fact, in a sense, it is the opposite! This is because the coordinates in map space aren’t prime-count lists, but maps, such as {{map|12 19 28}}. That particular map is seen here as the biggish, slightly tilted numeral 12 just to the left of the center point.

[[File:PTS with axes.png|300px|left|thumb|'''Figure 3c.''' PTS, with axes]]

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|}

And there’s our {{bra|{{vector|4 -4 1}}}}<ref>Interestingly, the other two columns in the bottom half of this matrix are valuable too. They tell you prime count vectors that would work for your generators. In this case, the two vectors are {{vector|1 0 0}} and {{vector|-8 5 0}}, so that tells you that the octave and the [[diesis]] could generate meantone. They're not necessarily the ''best'' generators, though. You can find other generators from these by adding or subtracting temperament commas, because of course — being tempered out — don't change anything.</ref>. Feel free to try reversing the operation by working out the mapping-row-basis from this if you like. And/or you could try working out that {{bra|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

And there’s our {{bra|{{vector|4 -4 1}}}}<ref>Interestingly, the other two columns in the bottom half of this matrix are valuable too. They tell you prime-count vectors that would work for your generators. In this case, the two vectors are {{vector|1 0 0}} and {{vector|-8 5 0}}, so that tells you that the octave and the [[diesis]] could generate meantone. They're not necessarily the ''best'' generators, though. You can find other generators from these by adding or subtracting temperament commas, because of course — being tempered out — don't change anything.</ref>. Feel free to try reversing the operation by working out the mapping-row-basis from this if you like. And/or you could try working out that {{bra|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.

It’s worth noting that, just as 2 commas were exactly enough to define a rank-1 temperament, though there were an infinitude of equivalent pairs of commas we could choose to fill that role, there’s a similar thing happening here, where 2 maps are exactly enough to define a rank-2 temperament, but an infinitude of equivalent pairs of them. We can even see that we can convert between these maps using Gauss-Jordan addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{map|12 19 28}} is exactly what you get from summing the terms of 5-ET {{map|5 8 12}} and 7-ET {{map|7 11 16}}: {{map|5+7 8+11 12+16}} = {{map|12 19 28}}. Cool!

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So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{vector|1 1}}.

If the familiar usage of vectors has been as prime count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator count vectors or GC-vectors. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.

If the familiar usage of vectors has been as prime-count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator count vectors or GC-vectors. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.

We haven’t specified the size of either of these generators, but that’s not important here. These mappings are just like a set of requirements for any pair of generators that might implement this temperament. This is as good a time as any to emphasize the fact that temperaments are abstract; they are not ready-to-go tunings, but more like instructions for a tuning to follow. This can sometimes feel frustrating or hard to understand, but ultimately it’s a big part of the power of temperament theory. In this case, a common method for optimizing tunings from temperaments would give these two generators as 73.756¢ and 118.945¢, respectively, which gives the tempered 10/9 as 73.756¢ + 118.945¢ = 192.701¢, which is about 10¢ sharp from its JI cents value of 182.404¢. We’ll learn more about tuning methods later.

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|map

|list of maps

|list of prime count lists

|list of prime-count lists

|prime count list

|prime-count list

|-

!linear algebra structure

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|<math>\textbf{i}</math>

|<code>i</code>

|[[Prime count vector|interval]]

|[[Prime-count vector|interval]]

|

|<math>\text{p}</math>

@@ Line 45: / Line 45: @@
 Until stated otherwise, this material will assume the [[5-limit|5 prime-limit]].
-If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{vector|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. If we need to distinguish these vectors from other kinds of vectors used in RTT, we call these prime count vectors or PC-vectors.
+If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{vector|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. If we need to distinguish these vectors from other kinds of vectors used in RTT, we call these [[prime-count vector]]s or PC-vectors.
 And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{map|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach 2/1 (the [[octave]]), 19 steps to reach 3/1 (the [[tritave]]), and 28 steps to each 5/1 (the [[pentave]]). Any or all of those intervals may be approximate.
@@ Line 178: / Line 178: @@
 If you’ve worked with 5-limit JI before, you’re probably aware that it is three-dimensional. You’ve probably reasoned about it as a 3D lattice, where one axis is for the factors of prime 2, one axis is for the factors of prime 3, and one axis is for the factors of prime 5. This way, you can use vectors, such as {{vector|-4 4 -1}} or {{vector|1 -2 1}}, just like coordinates.
-PTS can be thought of as a projection of 5-limit JI map space, which similarly has one axis each for 2, 3, and 5. But it is no JI pitch lattice. In fact, in a sense, it is the opposite! This is because the coordinates in map space aren’t prime count lists, but maps, such as {{map|12 19 28}}. That particular map is seen here as the biggish, slightly tilted numeral 12 just to the left of the center point.
+PTS can be thought of as a projection of 5-limit JI map space, which similarly has one axis each for 2, 3, and 5. But it is no JI pitch lattice. In fact, in a sense, it is the opposite! This is because the coordinates in map space aren’t prime-count lists, but maps, such as {{map|12 19 28}}. That particular map is seen here as the biggish, slightly tilted numeral 12 just to the left of the center point.
 [[File:PTS with axes.png|300px|left|thumb|'''Figure 3c.''' PTS, with axes]]
@@ Line 806: / Line 806: @@
 |}
-And there’s our {{bra|{{vector|4 -4 1}}}}<ref>Interestingly, the other two columns in the bottom half of this matrix are valuable too. They tell you prime count vectors that would work for your generators. In this case, the two vectors are {{vector|1 0 0}} and {{vector|-8 5 0}}, so that tells you that the octave and the [[diesis]] could generate meantone. They're not necessarily the ''best'' generators, though. You can find other generators from these by adding or subtracting temperament commas, because of course — being tempered out — don't change anything.</ref>. Feel free to try reversing the operation by working out the mapping-row-basis from this if you like. And/or you could try working out that {{bra|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.
+And there’s our {{bra|{{vector|4 -4 1}}}}<ref>Interestingly, the other two columns in the bottom half of this matrix are valuable too. They tell you prime-count vectors that would work for your generators. In this case, the two vectors are {{vector|1 0 0}} and {{vector|-8 5 0}}, so that tells you that the octave and the [[diesis]] could generate meantone. They're not necessarily the ''best'' generators, though. You can find other generators from these by adding or subtracting temperament commas, because of course — being tempered out — don't change anything.</ref>. Feel free to try reversing the operation by working out the mapping-row-basis from this if you like. And/or you could try working out that {{bra|{{vector|4 -4 1}}}} is a basis for the null-space of any other combination of ETs we found that could specify meantone, such as 7&12, or 12&19.
 It’s worth noting that, just as 2 commas were exactly enough to define a rank-1 temperament, though there were an infinitude of equivalent pairs of commas we could choose to fill that role, there’s a similar thing happening here, where 2 maps are exactly enough to define a rank-2 temperament, but an infinitude of equivalent pairs of them. We can even see that we can convert between these maps using Gauss-Jordan addition and subtraction, just like we could manipulate commas to get from one to the other. For example, the map for 12-ET {{map|12 19 28}} is exactly what you get from summing the terms of 5-ET {{map|5 8 12}} and 7-ET {{map|7 11 16}}: {{map|5+7 8+11 12+16}} = {{map|12 19 28}}. Cool!
@@ Line 860: / Line 860: @@
 So in this meantone mapping, the best approximation of the JI interval 10/9 is found by moving 1 step in each generator. We could write this in vector form as {{vector|1 1}}.
-If the familiar usage of vectors has been as prime count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator count vectors or GC-vectors. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.
+If the familiar usage of vectors has been as prime-count vectors or PC-vectors, we can now generalize that definition to things like this {{vector|1 1}}: generator count vectors or GC-vectors. Since interval vectors are often called monzos, you’ll often see these called tempered monzos or [[Tmonzos_and_Tvals|tmonzos]] for short. There’s very little difference. We can use these vectors as coordinates in a lattice just the same as before. The main difference is that the nodes we visit on this lattice aren’t pure JI; they’re a tempered lattice.
 We haven’t specified the size of either of these generators, but that’s not important here. These mappings are just like a set of requirements for any pair of generators that might implement this temperament. This is as good a time as any to emphasize the fact that temperaments are abstract; they are not ready-to-go tunings, but more like instructions for a tuning to follow. This can sometimes feel frustrating or hard to understand, but ultimately it’s a big part of the power of temperament theory. In this case, a common method for optimizing tunings from temperaments would give these two generators as 73.756¢ and 118.945¢, respectively, which gives the tempered 10/9 as 73.756¢ + 118.945¢ = 192.701¢, which is about 10¢ sharp from its JI cents value of 182.404¢. We’ll learn more about tuning methods later.
@@ Line 1,137: / Line 1,137: @@
 |map
 |list of maps
-|list of prime count lists
+|list of prime-count lists
-|prime count list
+|prime-count list
 |-
 !linear algebra structure
@@ Line 1,184: / Line 1,184: @@
 |<math>\textbf{i}</math>
 |<code>i</code>
-|[[Prime count vector|interval]]
+|[[Prime-count vector|interval]]
 |
 |<math>\text{p}</math>