Kite's thoughts on pergens: Difference between revisions

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For any 2.3.7 comma (a,b,0,c), the three JI ratios of interest are 3/2, 7/4 and 7/6. The sweet spot results from the exact same three ranges of ∆w. However, the tipping point formula changes, to reflect the new mapping comma 64/63. The 3-limit comma is (a,b,0,c) + c·(6,-2,0,-1) = (a+6c, b-2c). At the tipping point, ∆w = -K' / (b-2c).

The tipping point can be inferred directly from the way the comma is notated. Semaphore's 49/48 comma is a minor 2nd, and any m2 comma implies 5-edo. Since 720¢ is the sharpest possible 5th that heptatonic notation can support, no m2 comma will tip over. Porcupine's 250/243 comma is an A1, which always implies 7-edo, and a 685.7¢ tipping point. Any d2 comma implies 12-edo, with a 700¢ tipping point.

In theory, a few temperaments could tip over. Diminished (2.3.5 and (8,4,-4) = 648/625) will tip at 700¢, where ∆w = -2.0¢ and ∆y = 13.7¢, nearly a quarter comma. 5/4 becomes 400¢, and 6/5 is locked at 300¢. But in practice, there's no reason for ∆y to be that high. It should be at most only 2 or 3 times greater than ∆w. Another tipping temperament is quadruple red, 2.3.7 with (5,4,0,-4). It has the same tipping point, at 700¢, where ∆w = -2¢ and 7/4's error = 31¢, again far greater than ∆w. 7/4 is 1000¢, far sharper than it need be. For both these temperaments, the half-comma restriction creates an overly broad range for ∆w.

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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.&lt;br /&gt;

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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;

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Screenshot of the first 38 pergens:&lt;br /&gt;

For any 2.3.7 comma (a,b,0,c), the three JI ratios of interest are 3/2, 7/4 and 7/6. The sweet spot results from the exact same three ranges of ∆w. However, the tipping point formula changes, to reflect the new mapping comma 64/63. The 3-limit comma is (a,b,0,c) + c·(6,-2,0,-1) = (a+6c, b-2c). At the tipping point, ∆w = -K' / (b-2c).&lt;br /&gt;

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The tipping point can be inferred directly from the way the comma is notated. Semaphore's 49/48 comma is a minor 2nd, and any m2 comma implies 5-edo. Since 720¢ is the sharpest possible 5th that heptatonic notation can support, no m2 comma will tip over. Porcupine's 250/243 comma is an A1, which always implies 7-edo, and a 685.7¢ tipping point. Any d2 comma implies 12-edo, with a 700¢ tipping point.&lt;br /&gt;

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In theory, a few temperaments could tip over. Diminished (2.3.5 and (8,4,-4) = 648/625) will tip at 700¢, where ∆w = -2.0¢ and ∆y = 13.7¢, nearly a quarter comma. 5/4 becomes 400¢, and 6/5 is locked at 300¢. But in practice, there's no reason for ∆y to be that high. It should be at most only 2 or 3 times greater than ∆w. Another tipping temperament is quadruple red, 2.3.7 with (5,4,0,-4). It has the same tipping point, at 700¢, where ∆w = -2¢ and 7/4's error = 31¢, again far greater than ∆w. 7/4 is 1000¢, far sharper than it need be. For both these temperaments, the half-comma restriction creates an overly broad range for ∆w.&lt;br /&gt;

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