Kite's thoughts on pergens: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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||~ __temperament__ ||~ __comma__ ||~ __sweet spot__ ||||~ __no ups or downs__ ||||||~ __with ups and downs__ ||||~ __up symbol__ ||

||~  ||~  ||~ (5th = 700¢ + c) ||~ 5/4 is ||~ 4:5:6 chord ||~ 5/4 is ||~ 4:5:6 chord ||~ E ||~ ratio ||~ cents ||

||= mavila ||= 135/128 = A1 ||= c = -21¢ to -22¢ ||= m3 ||= C Eb G ||= ^M3 ||= C E^ G ||= ^A1 ||= 80/81 = d1 ||= -100¢ - 7c = 47¢-54¢ ||

||= large green ||= (-15,11,1) = A1 ||= c = -10¢ to -12¢ ||= A3 ||= C E# G ||= ^M3 ||= C E^ G ||= vA1 ||= 80/81 = A1 ||= 100¢ + 7c = 26¢-30¢ ||

||= superpyth ||= (12,-9,1) = m2 ||= c = 9¢ to 10¢ ||= A2 ||= C D# G ||= vM3 ||= C Ev G ||= vm2 ||= 81/80 = m2 ||= 100¢ - 5c = 50-55¢ ||

The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch.

Rank-1 edos sometimes require ups and downs, and sometimes don't. Rank-2 pergens sometimes require double-pair notation, and sometimes require single-pair, and occasionally (meantone) don't. Rank-3 pergens sometimes require triple-pair notation, and sometimes only require double- or single-pair notation. They can't ever be notated conventionally.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two pairs. There may be very false triples that can arise from a single comma, who knows? All true triples require triple-pair notation, but some false triples and some double-splits will require triple-pair as well, to avoid awkward enharmonics of a 3rd or more.

All the previous rank-3 examples had the 2nd generator (gen2 or G2) be a comma, represented by a up or a high. There is no enharmonic for that accidental, just as JI has no enharmonics.

Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups. G1 =

Breedsmic (P8, P5/2, ^1) half-5th with ups, G2 = ^1 = 64/63, G1 = /m3 = 49/40, E = ``\\``A1

The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch.&lt;br /&gt;

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Rank-1 edos sometimes require ups and downs, and sometimes don't. Rank-2 pergens sometimes require double-pair notation, and sometimes require single-pair, and occasionally (meantone) don't. Rank-3 pergens sometimes require triple-pair notation, and sometimes only require double- or single-pair notation. They can't ever be notated conventionally.&lt;br /&gt;

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A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two pairs. There may be very false triples that can arise from a single comma, who knows? All true triples require triple-pair notation, but some false triples and some double-splits will require triple-pair as well, to avoid awkward enharmonics of a 3rd or more.&lt;br /&gt;

All the previous rank-3 examples had the 2nd generator (gen2 or G2) be a comma, represented by a up or a high. There is no enharmonic for that accidental, just as JI has no enharmonics.&lt;br /&gt;

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Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups. G1 =&lt;br /&gt;

Breedsmic (P8, P5/2, ^1) half-5th with ups, G2 = ^1 = 64/63, G1 = /m3 = 49/40, E = &lt;!-- ws:start:WikiTextRawRule:052:``\\`` --&gt;\\&lt;!-- ws:end:WikiTextRawRule:052 --&gt;A1&lt;br /&gt;

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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.&lt;br /&gt;

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Alt-pergenLister lists out thousands of 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. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;

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