Nick Vuci's Fundamentals of Xen: Difference between revisions
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=== Lattices === | === Lattices === | ||
[[File:Wilsonic lattice.mp4|thumb|Wilsonic lattice|none]] | Lattices are another way to visualize JI: [[File:Wilsonic lattice.mp4|thumb|Wilsonic lattice|none]] | ||
=== Conclusion === | === Conclusion === | ||
In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit. | In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit. | ||
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[[File:7-Limit Tonality Diamond.png|none|thumb|7-Limit Tonality Diamond]] | [[File:7-Limit Tonality Diamond.png|none|thumb|7-Limit Tonality Diamond]] | ||
We can remove the 6 (i.e. the 3) and we get this: | We can remove the 6 (i.e. the 3) and we get this: | ||
[[File:2.5.7 Subgroup.png|none|thumb|2.5.7 Subgroup Tonality Diamond]] | |||
[[ | |||
And if we input A = 1, B = 1, C = 7, you get these intervals. | And if we input A = 1, B = 1, C = 7, you get these intervals. | ||
[[File:2.5.7 Subgroup Triadic Diamond.png|none|thumb|2.5.7 Subgroup Triadic Diamond]] | |||
[[ | |||
Same shape, but different harmonic structure: 2.5.7 subgroup. | Same shape, but different harmonic structure: 2.5.7 subgroup. | ||
Now let’s go back to the infinite JI lattice and look at the small interval 50/49. I said any interval that goes thru that line before, but that was just to evoke mental imagery. Really, it’d go through every other interval on the line the same distance apart, e.g.: | Now let’s go back to the infinite JI lattice and look at the small interval 50/49. I said any interval that goes thru that line before, but that was just to evoke mental imagery. Really, it’d go through every other interval on the line the same distance apart, e.g.: | ||
[[File:Infinite JI Lattice 50-49 comma.png|none|thumb|Infinite JI Lattice 50/49 comma]] | |||
[[ | |||
That’s essentially a subgroup temperament. Not sure the name, but the names aren’t important. | That’s essentially a subgroup temperament. Not sure the name, but the names aren’t important. | ||
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I took the 7-limit diamond and removed a prime to get the 2.5.7 subgroup, then generated a lattice using the intervals those primes generate and found a small interval I wanted to squash out. That interval is called a comma. Squashing it out is called “tempering it out,” and what results is the temperament. | I took the 7-limit diamond and removed a prime to get the 2.5.7 subgroup, then generated a lattice using the intervals those primes generate and found a small interval I wanted to squash out. That interval is called a comma. Squashing it out is called “tempering it out,” and what results is the temperament. | ||
Now let’s look at 27edo. Let’s take the 27edo tempered representation of the 2.5.7 subgroup intervals. Hmm, if we superimpose the JI lattice together, we see the spot where 50/49 would be 1\27 not 0\27. 27edo does not temper out 50/49, but what does it temper? Whatever JI intetrval the red dots of the tempered lattice overlay are related to the commas that 27edo tempers out. [[ | Now let’s look at 27edo. Let’s take the 27edo tempered representation of the 2.5.7 subgroup intervals. Hmm, if we superimpose the JI lattice together, we see the spot where 50/49 would be 1\27 not 0\27. 27edo does not temper out 50/49, but what does it temper? Whatever JI intetrval the red dots of the tempered lattice overlay are related to the commas that 27edo tempers out. | ||
[[File:Infinite JI Lattice 128-125 Comma.png|none|thumb|Infinite JI Lattice 128/125 Comma]] | |||
Here we see that 27 tempers out 128/125 and you can make music in that temperament in 27edo by using 4:6:7 and 1/(7:6:4) chords. | |||
=== Temperaments Ext. === | === Temperaments Ext. === | ||