Golden sequences and tuning

A visualization of the golden generators. Each cell is a MOS, containing its step formula on the top and its generator's step formula on the bottom. For visual clarity, they have been organized into eight branches, the colored squares on the right of which contain the tunings of the generators in cents.

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 2L 3s.

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.

Soft scales are a natural tendency for musical cultures around the world: Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.

Golden operations and MOS height

Any MOS may be constructed from 1L 1s using two "golden operations": chromaticizing, or "stepping" (taking the soft child of the MOS, thus continuing the golden sequence) and inverting, or "flipping" (inverting the number of large and small steps, switching to a new golden sequence). In effect, this makes taking the hard child "cost" 2 instead of 1. This table shows the number of such operations required to reach any MOS under 15 notes (that MOS' "height"): The total number of MOSes at each height corresponds to the Fibonacci sequence.

MOS height for MOSes under 15 notes

Height	MOS	Total notes
1	1L 1s	2
2	2L 1s	3
3	1L 2s	3
3	3L 2s	5
4	3L 1s	4
4	2L 3s	5
4	5L 3s	8
5	4L 3s	7
5	1L 3s	4
5	5L 2s	7
5	3L 5s	8
5	8L 5s	13
6	7L 4s	11
6	3L 4s	7
6	4L 1s	5
6	2L 5s	7
6	7L 5s	12
6	8L 3s	11
6	5L 8s	13
7	4L 7s	11
7	7L 3s	10
7	5L 4s	9
7	1L 4s	5
7	7L 2s	9
7	5L 7s	12
7	3L 8s	11
8	3L 7s	10
8	4L 5s	9
8	9L 5s	14
8	5L 1s	6
8	2L 7s	9
8	11L 3s	14
9	10L 3s	13
9	9L 4s	13
9	5L 9s	14
9	1L 5s	6
9	6L 5s	11
9	9L 2s	11
9	3L 11s	14
10	3L 10s	13
10	4L 9s	13
10	6L 1s	7
10	5L 6s	11
10	2L 9s	11
11	1L 6s	7
11	7L 6s	13
11	11L 2s	13
12	7L 1s	8
12	6L 7s	13
12	2L 11s	13
13	1L 7s	8
14	8L 1s	9
15	1L 8s	9
16	9L 1s	10
17	1L 9s	10
18	10L 1s	11
19	1L 10s	11
20	11L 1s	12
21	1L 11s	12
22	12L 1s	13
23	1L 12s	13
24	13L 1s	14
25	1L 13s	14

Finding the generator

For any given MOS pattern, one can easily determine what generator creates it, by the power of golden sequences. This is done by working back from the MOS to 1L 1s (which has the generator of 1L + 0s) and then working forward with the generator size as if it is a MOS itself.

Non-octave-periodic MOSes can be seen as their reduced patterns with a fractional-octave period (for example, 5L 5s can be seen as 240c-periodic 1L 1s); for this trick to work the numbers of large and small steps must be coprime.

For example, let's take the MOS 11L 6s, and interpret it as the Fibonacci sequence fragment [6, 11]. Next, we will "step" backwards, moving our two-number window back so that 6 becomes the second element and the previous entry (which is trivial to calculate as 11-6 = 5) is the first element. So, we reach [5, 6]. Then, we proceed to [1, 5], and then [4, 1]. At this point, we've reached the "beginning" of a sequence, where the two elements are descending. So, we flip: [1, 4], then proceed back to [3, 1]. Continue on until you reach [1, 1], and log the steps you took:

Step	Sequence fragment	Sequence
(Start)	6, 11	(4, 1)
Step	5, 6	(4, 1)
Step	1, 5	(4, 1)
Step	4, 1	(4, 1)
Flip	1, 4	(3, 1)
Step	3, 1	(3, 1)
Flip	1, 3	Lucas
Step	2, 1	Lucas
Flip	1, 2	Fibonacci
Step	1, 1	Fibonacci

Then, we work back through our steps, starting with [0, 1] instead of [1, 1].

Step	Sequence fragment	Sequence
(Start)	0, 1	Fibonacci
Step	1, 1	Fibonacci
Flip	1, 1	Fibonacci
Step	1, 2	Fibonacci
Flip	2, 1	Lucas
Step	1, 3	Lucas
Flip	3, 1	(3, 1)
Step	1, 4	(3, 1)
Step	4, 5	(3, 1)
Step	5, 9	(3, 1)

Note that one flip operation leaves the ordered pair unchanged as it is [1, 1].

If we take our result, [5, 9] as a number of small and large steps, we get 9L + 5s, which is the generator.

For a simpler example, let's try diatonic, 5L 2s:

Step	Sequence fragment	Sequence
(Start)	2, 5	(3, 2)
Step	3, 2	(3, 2)
Flip	2, 3	Fibonacci
Step	1, 2	Fibonacci
Step	1, 1	Fibonacci

And then to find the generator:

Step	Sequence fragment	Sequence
(Start)	0, 1	Fibonacci
Step	1, 1	Fibonacci
Step	1, 2	Fibonacci
Flip	2, 1	Lucas
Step	1, 3	Lucas

And the generator is 3 large steps and 1 small step (which is correct).

Note that the large steps are read from the second entry, which is opposite to the convention used on the wiki where the number of large steps comes first.

Now we have the generator in steps, now how do we get to a range in cents? Well, for a generator (A)L + (B)s, and a scale (C)L + (D)s, the soft boundary (equalized tuning) is (A+B)\(C+D), and the hard boundary (collapsed tuning) is A\C. For example, the range for our first scale is between 9\11 (982 cents) and (9+5)\(11+6) = 14\17 (988 cents), and the range for diatonic is, as expected, between (3+1)\(5+2) = 4\7 (686 cents) and 3\5 (720 cents). For any hardness L/s = k, the tuning for the generator is (kA+B)/(kC+D). For the golden tuning, k is equal to the golden ratio.

Naming golden generators

Golden generators can be named after temperaments which they generate, or MOS scales they are found in, i.e. "golden meantone", "golden mavila". "The golden generator" usually refers to the golden generator of oneirotonic, or logarithmic phi.

Golden generators for common temperaments

Here are some golden generators for many rank-2 octave-periodic temperaments on this page as well as some other well-known temperaments. Buzzard and trienstonian were omitted because of their very awkward tuning ranges (both extremely close to 5edo) for the purposes of generating scales.

Common temperaments

Temperament	Generator	Golden tuning	MOS
Schismic/Garibaldi	3/2	702.75 ¢	12L 17s
Kleismic/Cata	6/5	317.17 ¢	4L 11s
Sensi	9/7	440.59 ¢	3L 5s
Wurschmidt	5/4	387.82 ¢	3L 28s
MIRACLE	16/15	115.59 ¢	10L 11s
Meantone	3/2	696.21 ¢	2L 3s
Mothra/Slendric	8/7	231.74 ¢	5L 21s
Didacus	28/25	191.13 ¢	6L 13s
Superpyth	3/2	708.05 ¢	5L 12s
Mohajira	11/9	348.91 ¢	7L 17s
Semaphore	8/7	254.04 ¢	5L 9s
Negri	16/15	124.77 ¢	1L 8s
Flattone	3/2	693.06 ¢	7L 12s
Porcupine	10/9	162.56 ¢	7L 8s
Magic	5/4	380.82 ¢	3L 16s

Exotemperaments are tuned to the simplest MOS that has a golden generator in the correct range.

Exotemperaments

Temperament	Generator	Golden tuning	MOS
Father	8/5	741.64 ¢	1L 1s
Mavila	3/2	672.85 ¢	2L 5s
Bug	5/3	940.15 ¢	1L 3s
Dicot	5/4	354.83 ¢	3L 4s

Yo shares sensi's golden generator, so it is excluded here.

Big table of golden generators

This table originated on the page "Golden MOS" and has moved here. It shows the golden generators for a large number of simple MOSes. Many have the same generator, so they've been placed in the same row.

MOS	Generator (¢)
1L 1s, 2L 1s, 3L 2s, 5L 3s, 8L 5s	741.6408
1L 2s, 3L 1s, 4L 3s, 7L 4s	868.3282
1L 3s, 4L 1s, 5L 4s	940.1492
2L 3s, 5L 2s, 7L 5s	503.7855
1L 4s, 5L 1s, 6L 5s	986.4021
1L 5s, 6L 1s, 7L 6s	1018.6773
3L 4s, 7L 3s	354.8232
2L 5s, 7L 2s	527.1497
1L 6s, 7L 1s	1042.4790
3L 5s, 8L 3s	759.4078
1L 7s, 8L 1s	1060.7571
4L 5s, 9L 4s	273.8497
2L 7s, 9L 2s	541.3837
1L 8s, 9L 1s	1075.2344
3L 7s, 10L 3s	366.2564
1L 9s, 10L 1s	1086.9847
5L 6s	222.9668
4L 7s	877.7318
3L 8s	768.8815
2L 9s, 11L 2s	550.9646
1L 10s, 11L 1s	1096.7123
5L 7s	704.0956
1L 11s, 12L 1s	1104.8980
6L 7s	188.0298
5L 8s	465.0841
4L 9s	280.6103
3L 10s	373.0714
2L 11s	557.8535
1L 12s	1111.8816

Notes

Todo: categorize