Golden sequences and tuning

Golden sequences (the generalization of the Fibonacci sequence to have any two starting values) have a number of interesting properties relating to the tuning of MOS scales, and potentially can be used to determine a way to "naturally" tune a MOS (and thus generate a line of daughter MOSes).

Glossary

• Golden sequence: A sequence generated by the recurrence relation F(n) = F(n − 1) + F(n − 2), dependent on an initial pair (F(0), F(1)). Examples are the Fibonacci (1, 0) and Lucas (2, 1) numbers.
• Oligolarge MOS: A MOS with more small steps than large steps. For example: 2L 5s.
• Oligosmall MOS: A MOS with more large steps than small steps. For example: 5L 2s.

Theory

Say that we want to tune a moment-of-symmetry scale, such as diatonic, such that continuing the chain of generators never results in extremely close dieses. An example that does not meet this goal is Pythagorean tuning, which encounters the Pythagorean comma after 12 steps, which is much smaller than the diatonic semitone; many generators across the octave encounter similar problems much earlier.

An approach to solving this problem is to use the properties of the golden ratio (in this case, logarithmic phi). The golden ratio is as far as possible from any simple rational number. Thus, by successively stacking the golden ratio, one avoids having intervals coincide. By setting the ratio of step sizes in any MOS to the golden ratio, the generator can then be characterized by an expression in terms of logarithmic phi. By continuing to stack generators that are tuned this way, one never runs into overly small commas.

If we tune diatonic this way, the result is golden meantone, with a generator around 696.2 cents. However, we can, in any case, define an "original" scale that is the first place that particular golden tuning is encountered. For example, the golden tuning for diatonic is also the golden tuning for pentic, but not the golden tuning for trial (2L 1s), which is simply logarithmic phi. Notably, in the general case, any MOS always shares its golden generator with its last oligolarge ancestor. For example, the golden tuning of 7L 2s is the same as that of 2L 5s, but not the same as that of 3L 2s.

Interestingly, if we take the step sizes of any oligolarge MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that oligolarge MOS! (Note that this also happens to be the list of EDOs that approximate the golden tuning for this series of MOSes, by the definition of a golden sequence.)

Oligolarge	Sequence	MOSes	Golden generator	Notes
0L 1s	1, 0, 1, 1, 2, 3, 5, 8, 13..	1L 0s, 1L 1s, 2L 1s, 3L 2s, 5L 3s, 8L 5s, 13L 8s…	458.36, 741.64	Logarithmic phi MOSes
1L 2s	2, 1, 3, 4, 7, 11, 18..	3L 1s, 4L 3s, 7L 4s, 11L 7s, 18L 11s…	331.67, 868.33
1L 3s	3, 1, 4, 5, 9, 14, 23…	4L 1s, 5L 4s, 9L 5s, 14L 9s, 23L 14s…	259.85, 940.15
1L 4s	4, 1, 5, 6, 11, 17, 28…	5L 1s, 6L 5s, 11L 6s, 17L 11s, 28L 17s…	213.60, 986.40
2L 3s	3, 2, 5, 7, 12, 19, 31…	5L 2s, 7L 5s, 12L 7s, 19L 12s, 31L 19s…	503.79, 696.21	Golden meantone
2L 5s	5, 2, 7, 9, 16, 25, 41…	7L 2s, 9L 7s, 16L 9s, 25L 16s, 41L 25s…	527.15, 672.85
2L 7s	7, 2, 9, 11, 20, 31, 51…	9L 2s, 11L 9s, 20L 11s, 31L 20s, 51L 30s…	541.38, 658.62
3L 4s	4, 3, 7, 10, 17, 27…	7L 3s, 10L 7s, 17L 10s, 27L 17s…	354.82, 845.18
3L 5s	5, 3, 8, 11, 19, 30…	8L 3s, 11L 8s, 19L 11s, 30L 19s…	440.59, 759.41

This means that there is an inherent relationship between MOSes and golden sequences, and moreover that any given golden sequence can be uniquely identified by a pair of descending positive numbers.

But things get even more interesting when we extend golden sequences to negative terms.

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 golden 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.

Notes

Todo: categorize