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

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“[[Rank]]” has a slightly different meaning than dimension, but that’s not important yet. We’ll define rank, and discuss what exactly a rank-2 or -3 temperament means later. For now, it’s enough to know that each temperament line on this 5-limit PTS diagram is defined by tempering out a comma which has the same name. For now, we’re still focusing on visually how to navigate PTS. So the natural thing to wonder next, then, is what’s up with the slopes of all these temperament lines?

[[File:Diagrams to understand PTS for RTT.png|thumb|left|400px|'''Figure 3a.''' How the tempered comma affects slope on PTS. A temperament defined by a comma with a 0 for a prime will be perpendicular to that prime's axis, because the tuning of that prime does not affect whether or not the comma is tempered out. Therefore the prime corresponding to the 0 in the comma is represented by x, which can be anything, while the proportion between the other two primes must remain fixed.]]

Let’s begin with a simple example: the perfectly horizontal line that runs through just about the middle of the page, through the numeral 12, labelled “[[compton]]”. What’s happening along this line? Well, as we know, moving to the left means tuning 5 sharper, and moving to the right means tuning 5 flatter. But what about 2 and 3? Well, they are changing as well: 2 is sharp in the bottom right, and 3 is sharp in the top right, so when we move exactly rightward, 2 and 3 are both getting sharper (though not as directly as 5 is getting flatter). But the critical thing to observe here is that 2 and 3 are sharpening at the exact same rate. Therefore the approximations of primes 2 and 3 are in a constant ratio with each other along horizontal lines like this. Said another way, if you look at the 2 and 3 terms for any ET’s map on this line, the ratio between its term for 2 and 3 will be identical.

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There are even temperaments whose comma includes only 3’s and 5’s, such as “[[bug]]” temperament, which tempers out [[27/25]], or {{monzo|0 3 -2}}. If you look on this PTS diagram, however, you won’t find bug. Paul chose not to draw it. There are infinite temperaments possible here, so he had to set a threshold somewhere on which temperaments to show, and bug just didn’t make the cut in terms of how much it distorts harmony from JI. If he had drawn it, it would have been way out on the left edge of the diagram, completely outside the concentric hexagons. It would run parallel to the 2-axis, or from top-left to bottom-right, and it would connect the 5-ET (the huge numeral which is cut off the left edge of the diagram so that we can only see a sliver of it) to the [[9edo|9-ET]] in the bottom left, running through the 19-ET and [[14edo|14-ET]] in-between. Indeed, these ET maps — {{val|9 14 21}}, {{val|5 8 12}}, {{val|19 30 45}}, and {{val|14 22 33}} — lock the ratio between their 3-terms and 5-terms, in this case to 2:3.

Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes. But all the other temperament lines follow a similar principle. Their slopes are a manifestation of the prime factors in their defining comma. If having zero 5’s means you are perfectly horizontal, then having only one 5 means your slope will be close to horizontal, such as meantone {{monzo|-4 4 -1}} or [[helmholtz]] {{monzo|-15 8 1}}. Similarly, magic {{monzo|-10 -1 5}} and [[würschmidt]] {{monzo|17 1 -8}}, having only one 3 apiece, are close to parallel with the 3-axis, while porcupine {{monzo|1 -5 3}} and [[ripple]] {{monzo|-1 8 -5}}, having only one 2 apiece, are close to parallel with the 2-axis.

Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes ''(see Figure 3a)''. But all the other temperament lines follow a similar principle. Their slopes are a manifestation of the prime factors in their defining comma. If having zero 5’s means you are perfectly horizontal, then having only one 5 means your slope will be close to horizontal, such as meantone {{monzo|-4 4 -1}} or [[helmholtz]] {{monzo|-15 8 1}}. Similarly, magic {{monzo|-10 -1 5}} and [[würschmidt]] {{monzo|17 1 -8}}, having only one 3 apiece, are close to parallel with the 3-axis, while porcupine {{monzo|1 -5 3}} and [[ripple]] {{monzo|-1 8 -5}}, having only one 2 apiece, are close to parallel with the 2-axis.

Think of it like this: for meantone, a change to the mapping of 5 doesn’t make near as much of a difference to the outcome as does a change to the mapping of 2 or 3, therefore, changes along the 5-axis don’t have near as much of an effect on that line, so it ends up roughly parallel to it.

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We mentioned that the generator value changes continuously as we move along a temperament line. So just to either side of 12-ET along the meantone line, the tuning of 2, 3, and 5 supports a generator size which in turn supports meantone, but it wouldn’t support augmented. And just to either side of 12-ET along the augmented line, the tuning of 2, 3, and 5 supports a generator which still supports augmented, but not meantone. 12-ET, we could say, is a convergence point between the meantone generator and the augmented generator. But it is not a convergence point because the two generators become identical in 12-ET, but rather because they can both be achieved in terms of 12-ET’s generator. In other words, 5\12 ≠ 4\12, but they are both multiples of 1\12.

Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 3a)''.

Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 3b)''.

[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 3a.''' Generator sizes of linear temperaments in PTS]]

[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 3b.''' Generator sizes of linear temperaments in PTS]]

And note that I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because those temperaments all have non-octave periods.

@@ Line 322: / Line 322: @@
 “[[Rank]]” has a slightly different meaning than dimension, but that’s not important yet. We’ll define rank, and discuss what exactly a rank-2 or -3 temperament means later. For now, it’s enough to know that each temperament line on this 5-limit PTS diagram is defined by tempering out a comma which has the same name. For now, we’re still focusing on visually how to navigate PTS. So the natural thing to wonder next, then, is what’s up with the slopes of all these temperament lines?
+[[File:Diagrams to understand PTS for RTT.png|thumb|left|400px|'''Figure 3a.''' How the tempered comma affects slope on PTS. A temperament defined by a comma with a 0 for a prime will be perpendicular to that prime's axis, because the tuning of that prime does not affect whether or not the comma is tempered out. Therefore the prime corresponding to the 0 in the comma is represented by x, which can be anything, while the proportion between the other two primes must remain fixed.]]
 Let’s begin with a simple example: the perfectly horizontal line that runs through just about the middle of the page, through the numeral 12, labelled “[[compton]]”. What’s happening along this line? Well, as we know, moving to the left means tuning 5 sharper, and moving to the right means tuning 5 flatter. But what about 2 and 3? Well, they are changing as well: 2 is sharp in the bottom right, and 3 is sharp in the top right, so when we move exactly rightward, 2 and 3 are both getting sharper (though not as directly as 5 is getting flatter). But the critical thing to observe here is that 2 and 3 are sharpening at the exact same rate. Therefore the approximations of primes 2 and 3 are in a constant ratio with each other along horizontal lines like this. Said another way, if you look at the 2 and 3 terms for any ET’s map on this line, the ratio between its term for 2 and 3 will be identical.
@@ Line 335: / Line 337: @@
 There are even temperaments whose comma includes only 3’s and 5’s, such as “[[bug]]” temperament, which tempers out [[27/25]], or {{monzo|0 3 -2}}. If you look on this PTS diagram, however, you won’t find bug. Paul chose not to draw it. There are infinite temperaments possible here, so he had to set a threshold somewhere on which temperaments to show, and bug just didn’t make the cut in terms of how much it distorts harmony from JI. If he had drawn it, it would have been way out on the left edge of the diagram, completely outside the concentric hexagons. It would run parallel to the 2-axis, or from top-left to bottom-right, and it would connect the 5-ET (the huge numeral which is cut off the left edge of the diagram so that we can only see a sliver of it) to the [[9edo|9-ET]] in the bottom left, running through the 19-ET and [[14edo|14-ET]] in-between. Indeed, these ET maps — {{val|9 14 21}}, {{val|5 8 12}}, {{val|19 30 45}}, and {{val|14 22 33}} — lock the ratio between their 3-terms and 5-terms, in this case to 2:3.
-Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes. But all the other temperament lines follow a similar principle. Their slopes are a manifestation of the prime factors in their defining comma. If having zero 5’s means you are perfectly horizontal, then having only one 5 means your slope will be close to horizontal, such as meantone {{monzo|-4 4 -1}} or [[helmholtz]] {{monzo|-15 8 1}}. Similarly, magic {{monzo|-10 -1 5}} and [[würschmidt]] {{monzo|17 1 -8}}, having only one 3 apiece, are close to parallel with the 3-axis, while porcupine {{monzo|1 -5 3}} and [[ripple]] {{monzo|-1 8 -5}}, having only one 2 apiece, are close to parallel with the 2-axis.
+Those are the three simplest slopes to consider, i.e. the ones which are exactly parallel to the axes ''(see Figure 3a)''. But all the other temperament lines follow a similar principle. Their slopes are a manifestation of the prime factors in their defining comma. If having zero 5’s means you are perfectly horizontal, then having only one 5 means your slope will be close to horizontal, such as meantone {{monzo|-4 4 -1}} or [[helmholtz]] {{monzo|-15 8 1}}. Similarly, magic {{monzo|-10 -1 5}} and [[würschmidt]] {{monzo|17 1 -8}}, having only one 3 apiece, are close to parallel with the 3-axis, while porcupine {{monzo|1 -5 3}} and [[ripple]] {{monzo|-1 8 -5}}, having only one 2 apiece, are close to parallel with the 2-axis.
 Think of it like this: for meantone, a change to the mapping of 5 doesn’t make near as much of a difference to the outcome as does a change to the mapping of 2 or 3, therefore, changes along the 5-axis don’t have near as much of an effect on that line, so it ends up roughly parallel to it.
@@ Line 384: / Line 386: @@
 We mentioned that the generator value changes continuously as we move along a temperament line. So just to either side of 12-ET along the meantone line, the tuning of 2, 3, and 5 supports a generator size which in turn supports meantone, but it wouldn’t support augmented. And just to either side of 12-ET along the augmented line, the tuning of 2, 3, and 5 supports a generator which still supports augmented, but not meantone. 12-ET, we could say, is a convergence point between the meantone generator and the augmented generator. But it is not a convergence point because the two generators become identical in 12-ET, but rather because they can both be achieved in terms of 12-ET’s generator. In other words, 5\12 ≠ 4\12, but they are both multiples of 1\12.
-Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 3a)''.
+Here’s a diagram that shows how the generator size changes gradually across each line in PTS. It may seem weird how the same generator size appears in multiple different places across the space. But keep in mind that pretty much any generator is possible pretty much anywhere here. This is simply the generator size pattern you get when you lock the period to exactly 1200 cents, to establish a common basis for comparison. These are called [[Tour_of_Regular_Temperaments#Rank-2_temperaments|linear temperaments]]. This is what enables us to produce maps of temperaments such as the one found at [[Map_of_linear_temperaments|this Xen wiki page]], or this chart here ''(see Figure 3b)''.
-[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 3a.''' Generator sizes of linear temperaments in PTS]]
+[[File:Generator sizes in PTS.png|800px|thumb|'''Figure 3b.''' Generator sizes of linear temperaments in PTS]]
 And note that I didn’t break down what’s happening along the blackwood, compton, augmented, dimipent, and some other lines which are labelled on the original PTS diagram. In some cases, it’s just because I got lazy and didn’t want to deal with fitting more numbers on this thing. But in the case of all those that I just listed, it’s because those temperaments all have non-octave periods.