Kite's thoughts on pergens: Difference between revisions
→Miscellaneous Notes: added link to people in ''section'' "Credits" |
Added "Notating a pergen tuned to an EDO". Added "The pergen for an EDO paired with 7edo indicate the EDO's natural heptatonic generator." Uploaded a new alt-PergenLister, easily lets EDOs 13 & 18 be either tweaked or untweaked, improved EDO-pair output. |
||
| Line 1: | Line 1: | ||
__FORCETOC__ | __FORCETOC__ | ||
=<u>'''Definition'''</u>= | ==<u>'''Definition'''</u>== | ||
A '''pergen''' (pronounced "peer-gen") is a way of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. | A '''pergen''' (pronounced "peer-gen") is a way of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. | ||
| Line 3,011: | Line 3,011: | ||
|} | |} | ||
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2). | The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2). | ||
'''<u>Notating a pergen tuned to an EDO</u>''' | |||
If both the pergen and the EDO are notated with ups and downs, what is the relationship between the two kinds of up? The pergen's up equals some multiple of the EDO's up, i.e. some number of edosteps. For third-4th in 22edo or 29edo, the pergen's up = 1 edostep. But in 37edo or 44edo, ^1 = 2 edosteps. For half-8ve in 12edo, ^1 = 0 edosteps, and the ups and downs in the score can simply be ignored. | |||
When notating the piece, either the pergen notation or the EDO notation can be used. The former is more general, because the piece can be easily played in other supporting EDOs. But it may require the musician to mentally double every up and down symbol. | |||
Half-8ve in 22edo has P = vA4. But in 16edo, P = A4 + 2 edosteps, and ^1 = -2 edosteps. Negative edosteps means up is down, and should be avoided. The notation has tipped, and the period should be notated as ^A4. | |||
<u>'''Pergens Within An EDO'''</u> | <u>'''Pergens Within An EDO'''</u> | ||
| Line 3,717: | Line 3,725: | ||
If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. | If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. | ||
If the 1st edo is 7edo, the pergen indicates the natural heptatonic notation for the 2nd edo, e.g. 12edo = unsplit, 15edo = third-4th and 17edo = half-5th. | |||
The closer two edos are in the scale tree, the simpler the pergen they make. Examples:: | The closer two edos are in the scale tree, the simpler the pergen they make. Examples:: | ||
| Line 3,982: | Line 3,992: | ||
===alt-pergenLister=== | ===alt-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 | 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. | ||
http://www.tallkite.com/misc_files/alt-pergenLister.zip | http://www.tallkite.com/misc_files/alt-pergenLister.zip (written in Jesusonic, runs inside Reaper) | ||
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. | ||
| Line 4,142: | Line 4,152: | ||
P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G | P8 = d·b·P8 + c·b·(qr·G - P12) = b · (d·P8 + cqr·G - c·P12) = b · (d,-c) + bcqr·G | ||
== | == Glossary == | ||
to find the definition, use control-F to search for the first (bolded) occurrence of the word on this page. | |||
pergen | pergen | ||
| Line 4,193: | Line 4,202: | ||
edomapping | edomapping | ||
==Miscellaneous Notes== | |||
<u>'''Combining pergens'''</u> | <u>'''Combining pergens'''</u> | ||