Kite's thoughts on pergens: Difference between revisions

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The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[~~Ups_and_Downs_Notation~~|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

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==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[~~Ups_and_Downs_Notation~~|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

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 The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.
-All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups_and_Downs_Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.
+All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.
 One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.
@@ Line 1,261: / Line 1,261: @@
 ==Chord names and staff notation==
-Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups_and_Downs_Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
+Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
 In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.