Kite's thoughts on pergens: Difference between revisions

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upspan

liftspan

chain number

single-chain

multi-chain

arrow comma

==Miscellaneous Notes==

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== Addenda (late 2023) ==

~~WORK IN PROGRESS~~

=== New terminology===

All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

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===Simplifying a "squared" EI===

Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If ~~inflected by~~ an even number of ~~arrows~~, it would obviously be an invalid EI, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the ~~number of arrows~~ must be odd.

Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EI, for if v4AA1 = 0¢, then so does vvA1, and v4AA1 could be replaced with vvA1. So the upspan must be odd.

Consider an EI of v3AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].

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</math>

The two red numbers in the lower left are both even, ~~a direct result of~~ AA1 ~~being~~ a squared ratio. ~~The~~ two red numbers in the upper right must both be even to ensure their rows' dot products with the EI are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EI are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:

<math>

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</math>

~~Note that~~ while the EI has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EI to it. Changes are in red:

But while the EI has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EI to it. Changes are in red:

<math>

@@ Line 4,376: / Line 4,376: @@
 upspan<br />
 liftspan
+chain number<br />
+single-chain<br />
+multi-chain<br />
+arrow comma
 ==Miscellaneous Notes==
@@ Line 4,419: / Line 4,424: @@
 == Addenda (late 2023) ==
-WORK IN PROGRESS
 === New terminology===
 All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.
@@ Line 4,432: / Line 4,435: @@
 ===Simplifying a "squared" EI===
-Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If inflected by an even number of arrows, it would obviously be an invalid EI, for if v<sup>4</sup>AA1 = 0¢, then so does vvA1, and v<sup>4</sup>AA1 could be replaced with vvA1. So the number of arrows must be odd.
+Consider an uninflected EI of AA1. AA1 is "squared" in the sense that AA1 = A1 + A1, thus both numbers in its ratio are square numbers. If it had an even upspan (the number of ups), it would obviously be an invalid EI, for if v<sup>4</sup>AA1 = 0¢, then so does vvA1, and v<sup>4</sup>AA1 could be replaced with vvA1. So the upspan must be odd.
 Consider an EI of v<sup>3</sup>AA1. The pergen is (P8, P4/3). Here are the [[twin squares]].
@@ Line 4,457: / Line 4,460: @@
 </math>
-The two red numbers in the lower left are both even, a direct result of AA1 being a squared ratio. The two red numbers in the upper right must both be even to ensure their rows' dot products with the EI are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:
+The two red numbers in the lower left are both even, because AA1 is a squared ratio. As a result, the two red numbers in the upper right must both be even to ensure their rows' dot products with the EI are zero, which is an even number. If we halve all the red numbers and double all the green numbers, all the various row dot products will be unchanged, and the twin squares will remain valid:
 <math>
@@ Line 4,480: / Line 4,483: @@
 </math>
-Note that while the EI has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EI to it. Changes are in red:
+But while the EI has become simpler, the generator has become more complex in that it is dup not up. This is remedied by adding the EI to it. Changes are in red:
 <math>