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

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=== beyond the 5-limit ===

So far we’ve only been dealing with RTT in terms of prime limits, which is by far the most common and simplest way to use it. But nothing is stopping you from using other JI ~~subgroups~~. For example, you could create a 3D tuning space out of primes 2, 3, and 7 instead, skipping prime 5. ~~Instead of calling this “7-limit”, you~~ would call it “the 2.3.7 subgroup”. ~~For consistency,~~ you could call the ~~5-limit “the~~ 2.3.~~5 subgroup” if you wanted~~. ~~Subgroups are~~ even ~~more flexible than this: you can use~~ combinations of primes, such as the 2.5/3.7 ~~subgroup~~. ~~You~~ can ~~also~~ use ~~non-~~primes~~, like~~ the 2.3.25 ~~subgroup~~. ~~You~~ can even use irrationals, like the 2.π.5.7 ~~subgroup~~! The sky is the limit. Whatever you choose, though, this core structural rule <span><math>d - n = r</math></span> holds strong ''(see Figure 4d)''.

So far we’ve only been dealing with RTT in terms of prime limits, which is by far the most common and simplest way to use it. But nothing is stopping you from using other JI groups. What is a JI group? Well, I'll explain in terms of what we already know: prime limits. Prime limits are basically the simplest type of JI group. A prime limit is shorthand for the JI group consisting of all the primes up to that prime which is your limit; for example, the 7-limit is the same thing as the JI group "2.3.5.7". So JI groups are just sets of harmonics, and they are notated by separating the selected harmonics with dots.

Sometimes you may want to use a JI [[https://en.xen.wiki/w/Just_intonation_subgroup|subgroup]]. For example, you could create a 3D tuning space out of primes 2, 3, and 7 instead, skipping prime 5. You would call it “the 2.3.7 subgroup”. Or you could just call it "the 2.3.7 group", really. Nobody really cares that it's a subgroup of another group.

You could even choose a JI group with combinations of primes, such as the 2.5/3.7 group. Here, we still care about approximating primes 2, 3, 5, and 7, however there's something special about 3 and 5: we don't specifically care about approximating 3 or 5 individually, but only about approximating their combination. Note that this is different yet from the 2.15.7 group, where the combinations of 3 and 5 we care about approximating are when they're on the same side of the fraction bar.

As you can see from the 2.15.7 example, you don't even have to use primes. Simple and common examples of this situation are the 2.9.5 or the 2.3.25 groups, where you're targeting multiples of the same prime, rather than combinations of different primes.

And these are no longer ''JI'' groups, of course, but you can even use irrationals, like the 2.π.5.7 group! The sky is the limit. Whatever you choose, though, this core structural rule <span><math>d - n = r</math></span> holds strong ''(see Figure 4d)''.

The order you list the pitches you're approximating with your temperament is not standardized; generally you increase them in size from left to right, though as you can see from the 2.9.5 and 2.15.7 examples above it can often be less surprising to list the numbers in prime limit order instead. Whatever order you choose, the important thing is that you stay consistent about it, because that's the only way any of your vectors and covectors are going to match up correctly!

[[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]]

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On the other side of duality, septimal meantone’s mapping has two rows, corresponding to its two generators. We don’t have PTS for 7-limit JI handy, but because septimal meantone includes, or extends plain meantone, we can still refer to 5-limit PTS, and pick ETs from the meantone line there. The difference is that this time we need to include their 7-term. So the union of {{val|12 19 28 34}} and {{val|19 30 44 53}} would work. But so would {{val|19 30 44 53}} and {{val|31 49 72 87}}. We have an infinitude of options on this side of duality too, but here it’s not because our nullity is greater than 1, but because our rank is greater than 1.

=== normal form ===

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 === beyond the 5-limit ===
-So far we’ve only been dealing with RTT in terms of prime limits, which is by far the most common and simplest way to use it. But nothing is stopping you from using other JI subgroups. For example, you could create a 3D tuning space out of primes 2, 3, and 7 instead, skipping prime 5. Instead of calling this “7-limit”, you would call it “the 2.3.7 subgroup”. For consistency, you could call the 5-limit “the 2.3.5 subgroup” if you wanted. Subgroups are even more flexible than this: you can use combinations of primes, such as the 2.5/3.7 subgroup. You can also use non-primes, like the 2.3.25 subgroup. You can even use irrationals, like the 2.π.5.7 subgroup! The sky is the limit. Whatever you choose, though, this core structural rule <span><math>d - n = r</math></span> holds strong ''(see Figure 4d)''.
+So far we’ve only been dealing with RTT in terms of prime limits, which is by far the most common and simplest way to use it. But nothing is stopping you from using other JI groups. What is a JI group? Well, I'll explain in terms of what we already know: prime limits. Prime limits are basically the simplest type of JI group. A prime limit is shorthand for the JI group consisting of all the primes up to that prime which is your limit; for example, the 7-limit is the same thing as the JI group "2.3.5.7". So JI groups are just sets of harmonics, and they are notated by separating the selected harmonics with dots.
+Sometimes you may want to use a JI [[https://en.xen.wiki/w/Just_intonation_subgroup|subgroup]]. For example, you could create a 3D tuning space out of primes 2, 3, and 7 instead, skipping prime 5. You would call it “the 2.3.7 subgroup”. Or you could just call it "the 2.3.7 group", really. Nobody really cares that it's a subgroup of another group.
+You could even choose a JI group with combinations of primes, such as the 2.5/3.7 group. Here, we still care about approximating primes 2, 3, 5, and 7, however there's something special about 3 and 5: we don't specifically care about approximating 3 or 5 individually, but only about approximating their combination. Note that this is different yet from the 2.15.7 group, where the combinations of 3 and 5 we care about approximating are when they're on the same side of the fraction bar.
+As you can see from the 2.15.7 example, you don't even have to use primes. Simple and common examples of this situation are the 2.9.5 or the 2.3.25 groups, where you're targeting multiples of the same prime, rather than combinations of different primes.
+And these are no longer ''JI'' groups, of course, but you can even use irrationals, like the 2.π.5.7 group! The sky is the limit. Whatever you choose, though, this core structural rule <span><math>d - n = r</math></span> holds strong ''(see Figure 4d)''.
+The order you list the pitches you're approximating with your temperament is not standardized; generally you increase them in size from left to right, though as you can see from the 2.9.5 and 2.15.7 examples above it can often be less surprising to list the numbers in prime limit order instead. Whatever order you choose, the important thing is that you stay consistent about it, because that's the only way any of your vectors and covectors are going to match up correctly!
 [[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]]
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 On the other side of duality, septimal meantone’s mapping has two rows, corresponding to its two generators. We don’t have PTS for 7-limit JI handy, but because septimal meantone includes, or extends plain meantone, we can still refer to 5-limit PTS, and pick ETs from the meantone line there. The difference is that this time we need to include their 7-term. So the union of {{val|12 19 28 34}} and {{val|19 30 44 53}} would work. But so would {{val|19 30 44 53}} and {{val|31 49 72 87}}. We have an infinitude of options on this side of duality too, but here it’s not because our nullity is greater than 1, but because our rank is greater than 1.
 === normal form ===