Kite's thoughts on pergens: Difference between revisions

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[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]

===~~alt-pergenLister~~===

===PergenLister===

~~Alt-pergenLister~~ lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.

http://www.tallkite.com/~~misc_files~~/~~alt-~~pergenLister.~~zip~~ (written in Jesusonic, runs inside Reaper)

http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.

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==Various proofs (unfinished)==

Although not yet rigorously proven, the two false-double tests have been empirically verified by ~~alt-~~pergenLister.

Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.

The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

@@ Line 4,192: / Line 4,192: @@
 [[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]]
-===alt-pergenLister===
+===PergenLister===
-Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.
+PergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair.
-http://www.tallkite.com/misc_files/alt-pergenLister.zip (written in Jesusonic, runs inside Reaper)
+http://www.tallkite.com/apps/pergenLister.html (written in Jesusonic, runs inside Reaper or ReaJS)
 The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.
@@ Line 4,234: / Line 4,234: @@
 ==Various proofs (unfinished)==
-Although not yet rigorously proven, the two false-double tests have been empirically verified by alt-pergenLister.
+Although not yet rigorously proven, the two false-double tests have been empirically verified by pergenLister.
 The interval P8/2 has a "ratio" of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the '''pergen matrix''' [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.