Douglas Blumeyer's RTT How-To: Difference between revisions
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[[File:Mapping and comma basis dnr.png|400px|thumb|right|'''Figure 4c.''' The relationship between dimensionality d, rank r, and nullity n]] | [[File:Mapping and comma basis dnr.png|400px|thumb|right|'''Figure 4c.''' The relationship between dimensionality d, rank r, and nullity n]] | ||
=== 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 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)''. | ||
[[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]] | [[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]] | ||
Alright, here’s where things start to get pretty fun. 7-limit JI is 4D. We can no longer refer to our 5-limit PTS diagram for help. Maps and vectors here will have four terms; the new fourth term being for prime 7. So the map for 12-ET here is {{val|12 19 28 34}}. | Alright, here’s where things start to get pretty fun. 7-limit JI is 4D. We can no longer refer to our 5-limit PTS diagram for help. Maps and vectors here will have four terms; the new fourth term being for prime 7. So the map for 12-ET here is {{val|12 19 28 34}}. | ||
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Finding the HNF is not all too different from the other matrix transformations we’ve practiced so far. Basically you just perform Gaussian elimination until you reach your target. The target in this case requires that the biggest square matrix subset of your matrix you can fit in the top-left corner is an identity matrix. In other words, top-left corner is 1, and you have all 1’s along the main diagonal, and 0’s between any of these 1’s and the top or left of the matrix. If you can’t get a 1 along the main diagonal, shoot for a number whose absolute value is as low as possible, and lower or equal to any further numbers down the diagonal. | Finding the HNF is not all too different from the other matrix transformations we’ve practiced so far. Basically you just perform Gaussian elimination until you reach your target. The target in this case requires that the biggest square matrix subset of your matrix you can fit in the top-left corner is an identity matrix. In other words, top-left corner is 1, and you have all 1’s along the main diagonal, and 0’s between any of these 1’s and the top or left of the matrix. If you can’t get a 1 along the main diagonal, shoot for a number whose absolute value is as low as possible, and lower or equal to any further numbers down the diagonal. | ||
== multi(co)vectors == | == multi(co)vectors == | ||