Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-~~25 06~~:17:21 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-26 04:03:26 UTC</tt>.

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Enharmonic = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = ``//``d4. Generator = ^\M3 = v34 = ^``//``d4.

P1 — \M3 — \\A5=/m6 — P8C — E\ — Ab/ — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — FC — E^\ — Ab^^/=Avv\ — Dbv/ — F

P1 — \M3 — \\A5=/m6 — P8C — E\ — Ab/ — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11

~~This is a lot of math, but it only needs to be done once for each pergen!~~

C — E^\ — Ab^^/=Avv\ — Dbv/ — F

It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

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__**Combining pergens**__

Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).

Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension.

General rules for combining pergens:

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* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. If ~~adding~~ a ~~comma to~~ a ~~temperament doesn't change the pergen~~, it's a ~~strong extension~~, ~~otherwise it's~~ a ~~weak extension~~.

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an enharmonic of a 2nd or less. For example, sixth-4th's single pair notation has an E of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good E, sixth-4th can be notated with one pair from half-4th and another from third-4th.

__**Expanding gedras to 5-limit**__

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Enharmonic = v12m3 = /3d2 = v4/m2 = v4\\A1. Period = \M3 = v44 = //d4. Generator = ^\M3 = v34 = ^//d4.

P1 — \M3 — \\A5=/m6 — P8C — E\ — Ab/ — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — FC — E^\ — Ab^^/=Avv\ — Dbv/ — F~~ ~~

P1 — \M3 — \\A5=/m6 — P8C — E\ — Ab/ — CP1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^38=v/m9 — P11

~~This is a lot of math, but it only needs to be done once for each pergen! ~~

C — E^\ — Ab^^/=Avv\ — Dbv/ — F

It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the &quot;half-step glitch&quot;. This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

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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. It includes alternate enharmonics for many pergens.

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a>

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a>

Screenshots of the first 2 pages:

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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. Written in Jesusonic, runs inside Reaper.

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a>

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a>

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. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.

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Combining pergens