Kite's thoughts on pergens: Difference between revisions
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=<u>'''Definition'''</u>= | =<u>'''Definition'''</u>= | ||
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<span style="display: block; text-align: center;">'''<span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x</span>''' | <span style="display: block; text-align: center;">'''<span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x</span>''' | ||
A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen. | A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen. | ||
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This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize: | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize: | ||
<ul><li>'''A double-split pergen is <u>explicitly false</u> if m = |b|, and not explicitly false if m > |b|. | <ul><li>'<nowiki/>''A double-split pergen is <u>explicitly false</u> if m = |b|, and not explicitly false if m > |b|.''</li><li>''A double-split pergen is a <u>true double</u> if and only ifneither it nor its unreduced form is explicitly false'<nowiki/>'''''.'''</li><li>'''A double-split pergen is a <u>true double</u> if''' '''GCD (m, n) > |b|,''' '''and a false double if GCD (m, n) = |b|.'''</li></ul> | ||
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | ||
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<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — P11 | <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — P11 | ||
<span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span> | |||
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. | 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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<span style="display: block; text-align: center;">P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4 | <span style="display: block; text-align: center;">P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4 | ||
<span style="display: block; text-align: center;">C — C^/=Dbv/ — Db//=D\ — D^=Ebv — Eb/=E\\ — E^/=Fv\ — F | |||
==Alternate enharmonics== | ==Alternate enharmonics== | ||
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C -- Dv<span style="vertical-align: super;">3</span> -- Ev<span style="vertical-align: super;">6</span>=Db^<span style="vertical-align: super;">6</span> -- Eb^<span style="vertical-align: super;">3</span> -- F | C -- Dv<span style="vertical-align: super;">3</span> -- Ev<span style="vertical-align: super;">6</span>=Db^<span style="vertical-align: super;">6</span> -- Eb^<span style="vertical-align: super;">3</span> -- F | ||
Because G is a M2 and E is an A2, the equivalent generator G - E is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that Dv<span style="vertical-align: super;">3</span> -- Db^<span style="vertical-align: super;">6</span> is ascending. Double-pair notation may be preferable. This makes P = vM3, E = ^3d2, G = /m2, and E' = /4dd2. | Because G is a M2 and E is an A2, the equivalent generator G - E is a descending A1. Ascending intervals that look descending can be confusing, one has to take into account the eighth-tone ups to see that Dv<span style="vertical-align: super;">3</span> -- Db^<span style="vertical-align: super;">6</span> is ascending. Double-pair notation may be preferable. This makes P = vM3, E = ^3d2, G = /m2, and E' = /4dd2. | ||
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<span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8 | <span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8 | ||
<span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C | |||
<span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4 | |||
<span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span> | |||
Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. | Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. | ||
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[[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]] | [[File:pergens_2.png|alt=pergens 2.png|704x760px|pergens 2.png]] | ||
===alt-pergenLister=== | ===alt-pergenLister=== | ||
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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. | 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. | ||
http://www.tallkite.com/misc_files/alt-pergenLister.zip | |||
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. | 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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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. | 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. | ||
<u>'''Expanding gedras | <u>'''Expanding gedras'''</u> | ||
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]: | Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]: | ||
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d = -r | d = -r | ||
Gedras can also be expanded by adding an entry for ups/downs, and perhaps one for highs/lows too. For example, vM3 = [4,2,-1], and ^^\P4 = [5,3,2,-1]. | |||
<u>'''Height of a pergen'''</u> | <u>'''Height of a pergen'''</u> | ||