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).
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 ancestor with more small steps than large steps. 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 "root" MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that 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.)
| Root MOS | 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 pentic |
| 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.
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.
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). Additionally, 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.
Golden generators for common temperaments
Here are some golden generators for many rank-2 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.
| 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 |
