Golden generator
A golden generator is an interval that, when taken as a generator against some period (usually the octave), produces MOSes that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" TAMNAMS hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators are difficult to consider in terms of any given form.
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.
As a result of this connection to the golden ratio, MOS theory has several connections to the Fibonacci sequence and other sequences generated in a similar way (which will be called golden sequences in this article).
Soft MOS child
The best way to derive a generator tuning with the desired properties is by repeatedly taking the soft child of some MOS. The tuning ranges of the successive daughter MOSes will approach a value, which if used as a generator will generate soft MOSes infinitely.
If we repeatedly take the soft child of any MOS, this is equivalent to taking the MOS' scale pattern (for instance, LLsLLsLs) and replacing each "L" with "st" (ststsststssts). We then relabel s as L and t as s, resulting in LsLsLLsLsLLsL. Note that we have taken the number of large steps and added it to the total size of the MOS, resulting in a MOS with a number of large steps equal to the previous MOS size. Here, we went from 5L 3s (8 notes total) to 8L 5s (with 13 notes total). This is a representation of the same operation that produces the numbers of the Fibonacci sequence! If we kept going here, we'd get MOSes with 21, 34, 55, and 89 notes. If we start with a MOS whose step counts aren't in the Fibonacci sequence, we get an analogous sequence with those step sizes as the starting pair of numbers. For example, diatonic generates 2, 5, 7, 12, 19, 31, 50, 81...
This also serves as an intuitive explanation for how the golden ratio is the "most irrational" number.
Root MOS
But what happens when you take a hard child? The result has the step sizes flipped from what the "golden child" would be, and therefore becomes a new golden MOS chain with its own corresponding sequence. For example, if you take the Fibonacci sequence, (1, 1, 2, 3...) and stop it at 2 and 3, then flip them, you get (3, 2, 5, 7, 12, 19, 31...). Seem familiar? In fact, a common property of all hard child MOSes is that the number of large steps is smaller than the number of small steps. This means that any golden generator chain may be simply identified by an unordered pair of numbers, and allows us to unambiguously identify the root MOS of any given chain. Additionally, 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.
| Root MOS | Sequence | MOSes | Golden generator | Related temperaments | 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 | Aurora | 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 | ||
| 2L 3s | 3, 2, 5, 7, 12, 19, 31… | 5L 2s, 7L 5s, 12L 7s, 19L 12s, 31L 19s… | 503.79, 696.21 | Meantone | Golden pentic |
| 1L 4s | 4, 1, 5, 6, 11, 17, 28… | 5L 1s, 6L 5s, 11L 6s, 17L 11s, 28L 17s… | 213.60, 986.40 | Machine | |
| 3L 4s | 4, 3, 7, 10, 17, 27… | 7L 3s, 10L 7s, 17L 10s, 27L 17s… | 354.82, 845.18 | Rastmic | |
| 2L 5s | 5, 2, 7, 9, 16, 25, 41… | 7L 2s, 9L 7s, 16L 9s, 25L 16s, 41L 25s… | 527.15, 672.85 | Trismegistus | |
| 3L 5s | 5, 3, 8, 11, 19, 30… | 8L 3s, 11L 8s, 19L 11s, 30L 19s… | 440.59, 759.41 | Sentry | |
| 2L 7s | 7, 2, 9, 11, 20, 31, 51… | 9L 2s, 11L 9s, 20L 11s, 31L 20s, 51L 30s… | 541.38, 658.62 | Joan |
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 temperaments with simple golden MOS scales, consequently leaving out many cluster temperaments.
Names in italic are not rank-2 regular temperaments.
| Temperament | Generator | Golden tuning | MOS |
|---|---|---|---|
| Aurora | 32/21 | 741.6 | 1L 1s |
| A-team | 32/21 | 734.9 | 5L 8s |
| Sensamagic | - | 759.4 | 3L 5s |
| 768.9 | 3L 8s | ||
| Meantone | 3/2 | 696.2 | 2L 3s |
| Neogothic | - | 704.1 | 5L 7s |
| Trismegistus | 28/19 | 672.9 | 2L 5s |
| Joan | 16/11 | 658.6 | 2L 7s |
| 649.0 | 2L 9s | ||
| Daemotonic | - | 331.7 | 1L 2s |
| Orgone | 77/64 | 322.3 | 4L 7s |
| Rastmic | 11/9 | 354.8 | 3L 4s |
| Acoustic phi | - | 366.3 | 3L 7s |
| Submerged | 5/4 | 373.1 | 3L 10s |
| 259.9 | 1L 3s | ||
| Semaphore | 8/7 | 254.0 | 5L 9s |
| Orwell | 7/6 | 273.8 | 4L 5s |
| 280.6 | 4L 9s | ||
| Machine | 9/8 | 213.6 | 1L 4s |
| Shoe | 8/7 | 223.0 | 5L 6s |
| 181.3 | 1L 5s | ||
| Jugular | 10/9 | 188.0 | 6L 7s |
| 157.5 | 1L 6s | ||
| Porcupine | 11/10 | 162.6 | 7L 8s |
| 139.2 | 1L 7s | ||
| Negri | 16/15 | 124.8 | 1L 8s |
| Miracle | 16/15 | 113.0 | 1L 9s |
2.3.7 is often exotempered or of very high complexity, this is because of the fact that 2.3.7 just intonation itself functions as a sort of cluster temperament.
Argent tuning
A number with similar properties to the golden ratio is the square root of 2; 1+sqrt(2) is the "silver ratio". The "silver generator" is a near-perfect fifth of about 703 cents, and is the tuning of the diatonic generator such that the ratio between the large and small steps of the pentic scale is the same as between the large and small steps of p-chromatic, and that ratio is the square root of 2. Additionally, the ratio between the large and small steps of diatonic is the same as between the large and small steps of p-enharmonic. This tuning range is closely associated with Hemifamity temperament and is approximated by 29edo, 41edo, and 70edo.
Each scale actually has two silver generators, a hard one and a soft one; as such, silver generators naturally bifurcate in a somewhat similar way to how I've forced golden ones to. Soft silver diatonic is a flattone tuning.
Example, Golden meantone
Golden meantone is the golden tuning of pentic, being a good tuning of meantone (specifically with 11/8 as the double-diminished fifth).
