Interval variety: Difference between revisions

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Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{{nowrap|{{(}}1/1, 10/9, 3/2, 5/3{{)}}}}, {{nowrap|{{(}}1/1, 5/4, 3/2, 5/3{{)}}}}, {{nowrap|{{(}}1/1, 5/4, 3/2, 15/8{{)}}}}, {{nowrap|{{(}}1/1, 45/32, 3/2, 125/64{{)}}}}... Only the ones where the two chains are lined up closely have good potential for harmony.

The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the ~~same~~ pattern as {{nowrap|{{(}}1/1, 9/8, 3/2{{)}}}}, and ~~in order for it~~ to make ~~an actual~~ max-variety-3 scale, the other 2 notes must ~~fall in~~ between 9/8 and 3/2, and in between 3/2 and 2/1. If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.

The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the pattern {{nowrap|{{(}}1/1, 9/8, 3/2{{)}}}}, and to make a max-variety-3 scale, the other chain of 2 notes must be separated by 4/3, falling between the large (L = 4/3) steps of the 3-note chain (between 9/8 and 3/2, and between 3/2 and 2/1). If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.

1/1 9/8 40/27 3/2 160/81 ... max variety 3

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1/1 9/8 5/4 3/2 5/3 ... max variety 3

1/1 9/8 5/4 3/2 15/8 ... max variety 3

1/1 9/8 5/4 3/2 15/8 ... fails, max variety 5

1/1 9/8 45/32 3/2 15/8 ... max variety 3

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 Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{{nowrap|{{(}}1/1, 10/9, 3/2, 5/3{{)}}}}, {{nowrap|{{(}}1/1, 5/4, 3/2, 5/3{{)}}}}, {{nowrap|{{(}}1/1, 5/4, 3/2, 15/8{{)}}}}, {{nowrap|{{(}}1/1, 45/32, 3/2, 125/64{{)}}}}... Only the ones where the two chains are lined up closely have good potential for harmony.
-The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the same pattern as {{nowrap|{{(}}1/1, 9/8, 3/2{{)}}}}, and in order for it to make an actual max-variety-3 scale, the other 2 notes must fall in between 9/8 and 3/2, and in between 3/2 and 2/1. If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.
+The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the pattern {{nowrap|{{(}}1/1, 9/8, 3/2{{)}}}}, and to make a max-variety-3 scale, the other chain of 2 notes must be separated by 4/3, falling between the large (L = 4/3) steps of the 3-note chain (between 9/8 and 3/2, and between 3/2 and 2/1). If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.
 /1 9/8 40/27 3/2 160/81 ... max variety 3
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 /1 9/8 5/4 3/2 5/3 ... max variety 3
-/1 9/8 5/4 3/2 15/8 ... max variety 3
+/1 9/8 5/4 3/2 15/8 ... fails, max variety 5
 /1 9/8 45/32 3/2 15/8 ... max variety 3