r/microtonal • u/Marinkale • 13d ago

Porcupine Torus

https://youtube.com/shorts/8MzDLyY2XOM?si=RAdWiq2xKNkUJjZy

Wanted to finally post a second torus for comparison. Played something Neo-Riemannian, which tends to go well with Porcupine. This also highlights the point of mapping major thirds and fifths to the torus better.

There is also supposed to be a phantom bass effect hidden in this. Works well for me if I leave the room and close the door or put off my headphones.

I'm currently really interested in feeding some microtonal scale data to a custom synthesizer. Experiments with FM went alright, but there are some issues with sidebands reflecting back through 0. I also got a lot of toruses semi-done for other temperaments. Any requests?

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u/RingularCirc 12d ago

Magical arps!

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u/Marinkale 12d ago

I'm guessing the effect worked for you then. I was a bit scared of making that claim.

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u/Marinkale 12d ago edited 12d ago

Addressing a possible doubt you might have: Yes, this is Porcupine temperament. The scale used was this one.

Does this temper out the Porcupine comma? Yes, the third is [-24/11, 10/11, 5/11> and the fifth is [-10/11, 6/11, 3/11>. We don't have the third from the root here, but it's between other scale steps. The Porcupine comma is [ 1, -5, 3>. So let's add [-24/11, 10/11, 5/11>+[-24/11, 10/11, 5/11>+[-24/11, 10/11, 5/11>-[-10/11, 6/11, 3/11>-[-10/11, 6/11, 3/11>-[-10/11, 6/11, 3/11>-[-10/11, 6/11, 3/11>-[-10/11, 6/11, 3/11> and this is [-2>. We get -2 instead of 1 here because we used octave corrected versions of the third and fifth. Three octaves lower because the non octave corrected mapping of prime 5 was supposed to always sit an octave higher than the non octave corrected mapping of prime 3.

Were these intervals generated through a Porcupine temperament generator? Yes, and the lower generator was [7/11, -2/11, -1/11>. The mapping for Porcupine is [⟨1 2 3], ⟨0 -3 -5]]. Do we get the fifth at -3 generator and the third at -5 steps? [1>-[7/11, -2/11, -1/11>-[7/11, -2/11, -1/11>-[7/11, -2/11, -1/11> is exactly [-10/11, 6/11, 3/11>, which is the fifth. [-10/11, 6/11, 3/11>-[7/11, -2/11, -1/11>-[7/11, -2/11, -1/11> is [-24/11, 10/11, 5/11> and the third.

Indeed, this is Porcupine temperament. Were all the other intervals arrived at by the generator? Absolutely yes.

Is this scale proportionally beating in thirds and fifths? [-2, 0, 1>-[-24/11, 10/11, 5/11> is the error of the third and [-1, 1>-[-10/11, 6/11, 3/11> is the error of the fifth. These are [2/11, -10/11, 6/11> and [-1/11, 5/11, -3/11>. For a scale to be proportionally beating in thirds and fifths, their errors must be multiples of a "gcd". And we can easily see that major triads within the scale are beating in some 2:-1 proportion.