Golden sequences and tuning: Difference between revisions

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For example, let's take the sequence (3,2) that generates golden meantone. We can continue the sequence into the negative numbers as ...23, -14, 9, -5, 4, -1, 3, 2, 5.... Note that this extension is not symmetrical, unlike those of the Fibonacci and Lucas sequences (which is actually a property unique to both sequences and their multiples). Instead, if we make all the terms positive and flip it around, we get a different generalized Fibonacci sequence: the sequence (3,1) (corresponding to the series of MOSes generated by the golden 1L 3s generator), which can be considered the "complementary" sequence of (3,2). In general, for a sequence (m, n), its complement is (m, m-n), corresponding to oligolarge MOSes nL ms and (m-n)L ms. The one exception is the family of scales (m, 0), corresponding to the Fibonacci sequence and its multiples, which apparently have complements of (m, m), which isn't oligolarge at all but instead belongs to the "wood" category of MOSes with the same number of large and small steps. However, it can be shown by observing the terms of the Fibonacci sequence that these two sequences are, in fact, identical.

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	For example, let's take the sequence (3,2) that generates golden meantone. We can continue the sequence into the negative numbers as ...23, -14, 9, -5, 4, -1, 3, 2, 5.... Note that this extension is not symmetrical, unlike those of the Fibonacci and Lucas sequences (which is actually a property unique to both sequences and their multiples). Instead, if we make all the terms positive and flip it around, we get a different generalized Fibonacci sequence: the sequence (3,1) (corresponding to the series of MOSes generated by the golden 1L 3s generator), which can be considered the "complementary" sequence of (3,2). In general, for a sequence (m, n), its complement is (m, m-n), corresponding to oligolarge MOSes nL ms and (m-n)L ms. The one exception is the family of scales (m, 0), corresponding to the Fibonacci sequence and its multiples, which apparently have complements of (m, m), which isn't oligolarge at all but instead belongs to the "wood" category of MOSes with the same number of large and small steps. However, it can be shown by observing the terms of the Fibonacci sequence that these two sequences are, in fact, identical.		For example, let's take the sequence (3,2) that generates golden meantone. We can continue the sequence into the negative numbers as ...23, -14, 9, -5, 4, -1, 3, 2, 5.... Note that this extension is not symmetrical, unlike those of the Fibonacci and Lucas sequences (which is actually a property unique to both sequences and their multiples). Instead, if we make all the terms positive and flip it around, we get a different generalized Fibonacci sequence: the sequence (3,1) (corresponding to the series of MOSes generated by the golden 1L 3s generator), which can be considered the "complementary" sequence of (3,2). In general, for a sequence (m, n), its complement is (m, m-n), corresponding to oligolarge MOSes nL ms and (m-n)L ms. The one exception is the family of scales (m, 0), corresponding to the Fibonacci sequence and its multiples, which apparently have complements of (m, m), which isn't oligolarge at all but instead belongs to the "wood" category of MOSes with the same number of large and small steps. However, it can be shown by observing the terms of the Fibonacci sequence that these two sequences are, in fact, identical.

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