Kite's thoughts on pergens: Difference between revisions

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<ul><li>For (P8/m, M/n), P8 = mP + xEI and M = nG + yEI', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>x is the count for EI, with EI occurring x times in one octave, and xEI is the octave's '''multi-enharmonic''', or '''multi-EI''' for short</li><li>y is the count for EI', with EI' occurring y times in one multigen, and yEI' is the multigen's multi-EI</li><li>For false doubles using single-pair notation, EI = EI', but x and y are usually different, making different multi-enharmonics</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic EI" and new counts, P8 = mP + x'EI", and M' = n'G' + y'EI"</li></ul>

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''stepspan''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:

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 <ul><li>For (P8/m, M/n), P8 = mP + xEI and M = nG + yEI', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for EI, with EI occurring x times in one octave, and xEI is the octave's '''multi-enharmonic''', or '''multi-EI''' for short</li><li>y is the count for EI', with EI' occurring y times in one multigen, and yEI' is the multigen's multi-EI</li><li>For false doubles using single-pair notation, EI = EI', but x and y are usually different, making different multi-enharmonics</li><li>The unreduced pergen is (P8/m, M'/n'), with a new enharmonic EI" and new counts, P8 = mP + x'EI", and M' = n'G' + y'EI"</li></ul>
-The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''stepspan''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.
+The '''keyspan''' of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The '''[[stepspan]]''' of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.
 Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a '''gedra''', analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa: