Kite's thoughts on pergens: Difference between revisions
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== Addenda (Spring 2026) == | == Addenda (Spring 2026) == | ||
=== Initial commas === | |||
As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave. | As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave. | ||
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For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v<sup>12</sup>d4, and ^<sup>12</sup>C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^<sup>12</sup>d<sup>9</sup>4. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation. | For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v<sup>12</sup>d4, and ^<sup>12</sup>C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^<sup>12</sup>d<sup>9</sup>4. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation. | ||
Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v<sup>3</sup>m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. | Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v<sup>3</sup>m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. Fortunately, the next comma on the genchain is 21G - 2P = A1/2 = 50¢ - 3.5c, which implies a double-pair notation with EU' = \\A1. This is the notation found by pergenLister. | ||
True doubles require double-pair notation and thus require finding two commas. | True doubles require double-pair notation and thus require finding two commas. | ||