Maximum variety: Difference between revisions

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An "interval class" is the set of all intervals of a scale which span the same number of steps. For example, in the [[5L_2s|diatonic scale]], the 2-step class consists of the minor third and the major third, because those are the only intervals that are divided into 2 steps of the scale. The "variety" of an interval class is the number of different intervals in it. The maximum variety of a scale is simply the maximum variety of any interval class.

Any scale with all equal steps (such as an [[EDO|EDO]]) has maximum variety 1. All [[MOSScales|MOS~~]] and [[Distributional_Evenness|distributionally even~~]] scales have maximum variety 2 ~~(in fact this can be taken as the definition of distributional evenness)~~. An example of a scale with high max variety is the [[harmonic_series|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.

Any scale with all equal steps (such as an [[EDO|EDO]]) has maximum variety 1. All [[MOSScales|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.

==Max-variety-3 scales==

'''Max-variety-3''' scales are an attempt to generalize ~~distributional evenness (closely related to~~ the MOS property) to scales with three different step sizes rather than two (for example, those related to rank-3 [[Regular_Temperaments|regular temperaments]]). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.

'''Max-variety-3''' scales are an attempt to generalize the MOS property to scales with three different step sizes rather than two (for example, those related to rank-3 [[Regular_Temperaments|regular temperaments]]). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.

When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.

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 An "interval class" is the set of all intervals of a scale which span the same number of steps. For example, in the [[5L_2s|diatonic scale]], the 2-step class consists of the minor third and the major third, because those are the only intervals that are divided into 2 steps of the scale. The "variety" of an interval class is the number of different intervals in it. The maximum variety of a scale is simply the maximum variety of any interval class.
-Any scale with all equal steps (such as an [[EDO|EDO]]) has maximum variety 1. All [[MOSScales|MOS]] and [[Distributional_Evenness|distributionally even]] scales have maximum variety 2 (in fact this can be taken as the definition of distributional evenness). An example of a scale with high max variety is the [[harmonic_series|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
+Any scale with all equal steps (such as an [[EDO|EDO]]) has maximum variety 1. All [[MOSScales|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
 ==Max-variety-3 scales==
-'''Max-variety-3''' scales are an attempt to generalize distributional evenness (closely related to the MOS property) to scales with three different step sizes rather than two (for example, those related to rank-3 [[Regular_Temperaments|regular temperaments]]). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.
+'''Max-variety-3''' scales are an attempt to generalize the MOS property to scales with three different step sizes rather than two (for example, those related to rank-3 [[Regular_Temperaments|regular temperaments]]). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.
 When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.