4M: Difference between revisions

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For MOS with equave period the 4M is the [[MOS scale|MOS]] with the second note in the [[chroma-positive]] generator chain raised by a chroma, or the second last note in the generator chain lowered by a chroma. Using Meantone[7] 4M as an example - the Meantone melodic scale - We start with the [[Meantone]][7] generator chain F-C-G-D-A-E-B. Replacing C with C# (moving it up 7 generators) leads to the chain F-.-G-D-A-E-B-.-C#, which is the notes of the D melodic scale D E F G A B C# D. Alternatively, replacing E with Eb (moving it down 7 generators) leads to the chain Eb-.-F-C-G-D-A-.-B, which we can see is the same scale on a different pitch, i.e., C D Eb F G A B C. The generator chain of the 4M of MOS with one period per equave always takes this shape, with a generator span length 2 generators longer than the number of notes in the scale. Like MOS, 4Ms of MOS with equave period are [[Chirality|achiral]], or mirror-symmetric.

The 4M of a multi-MOS is found by raising the first note or lowering the last note of one generator chain by a chroma. For example, ~~take~~ the pentachordal decatonic, 4M of [[Pajara]][10]. In [[22edo]], Pajara[10] has gen chroma-positive gen chains 0-9-7-5-3 and 11-20-18-16-14. Raising '11' by a chroma to '12' leads to the scale 0-3-5-7-9-12-14-16-18-20, or 3222322222, which is the pentachord decatonic, which has chroma-positive gen chains 0-9-7-5-3 and 20-18-16-14-12.

The 4M of a multi-MOS is found by raising the first note or lowering the last note of one generator chain by a chroma. For example, the pentachordal decatonic, 4M of [[Pajara]][10]: In [[22edo]], Pajara[10] has gen chroma-positive gen chains 0-9-7-5-3 and 11-20-18-16-14. Raising '11' by a chroma to '12' leads to the scale 0-3-5-7-9-12-14-16-18-20, or 3222322222, which is the pentachord decatonic, which has chroma-positive gen chains 0-9-7-5-3 and 20-18-16-14-12.

'''Conjecture:''' Abstract 4Ms are the abstract MODMOS with the lowest variety.

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 For MOS with equave period the 4M is the [[MOS scale|MOS]] with the second note in the [[chroma-positive]] generator chain raised by a chroma, or the second last note in the generator chain lowered by a chroma. Using Meantone[7] 4M as an example - the Meantone melodic scale - We start with the [[Meantone]][7] generator chain F-C-G-D-A-E-B. Replacing C with C# (moving it up 7 generators) leads to the chain F-.-G-D-A-E-B-.-C#, which is the notes of the D melodic scale D E F G A B C# D. Alternatively, replacing E with Eb (moving it down 7 generators) leads to the chain Eb-.-F-C-G-D-A-.-B, which we can see is the same scale on a different pitch, i.e., C D Eb F G A B C. The generator chain of the 4M of MOS with one period per equave always takes this shape, with a generator span length 2 generators longer than the number of notes in the scale. Like MOS, 4Ms of MOS with equave period are [[Chirality|achiral]], or mirror-symmetric.
-The 4M of a multi-MOS is found by raising the first note or lowering the last note of one generator chain by a chroma. For example, take the pentachordal decatonic, 4M of [[Pajara]][10]. In [[22edo]], Pajara[10] has gen chroma-positive gen chains 0-9-7-5-3 and 11-20-18-16-14. Raising '11' by a chroma to '12' leads to the scale 0-3-5-7-9-12-14-16-18-20, or 3222322222, which is the pentachord decatonic, which has chroma-positive gen chains 0-9-7-5-3 and 20-18-16-14-12.
+The 4M of a multi-MOS is found by raising the first note or lowering the last note of one generator chain by a chroma. For example, the pentachordal decatonic, 4M of [[Pajara]][10]: In [[22edo]], Pajara[10] has gen chroma-positive gen chains 0-9-7-5-3 and 11-20-18-16-14. Raising '11' by a chroma to '12' leads to the scale 0-3-5-7-9-12-14-16-18-20, or 3222322222, which is the pentachord decatonic, which has chroma-positive gen chains 0-9-7-5-3 and 20-18-16-14-12.
 '''Conjecture:''' Abstract 4Ms are the abstract MODMOS with the lowest variety.