View source for Golden sequences and tuning

[[File:GoldenMOS.png|thumb|555x555px|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 {{nowrap|F(''n'') {{=}} F(''n'' − 1) + F(''n'' − 2)}}, dependent on an initial pair (F(0),&nbsp;F(1)). Examples are the Fibonacci (1,&nbsp;0) and Lucas (2,&nbsp;1) numbers.
* Oligolarge MOS: A MOS with more small steps than large steps. For example: 2L&nbsp;5s.
* Oligosmall MOS: A MOS with more large steps than small steps. For example: 5L&nbsp;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&nbsp;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&nbsp;2s is the same as that of 2L&nbsp;5s, but not the same as that of 2L&nbsp;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.)

{| class="wikitable"
|-
! Oligolarge
! Sequence
! MOSes
! Golden generator
! Notes
|-
| 0L&nbsp;1s<ref group="note">Is musically identical to 1L&nbsp;0s aka [[1edo]], but is mathematically distinct precisely in regards to this topic.</ref>
| 1, 0, 1, 1, 2, 3, 5, 8, 13..
| [[1L&nbsp;0s]], [[1L&nbsp;1s]], [[2L&nbsp;1s]], [[3L&nbsp;2s]], [[5L&nbsp;3s]], [[8L&nbsp;5s]], [[13L&nbsp;8s]]…
| 458.36, 741.64
| Logarithmic phi MOSes
|-
| 1L&nbsp;2s
| 2, 1, 3, 4, 7, 11, 18..
| [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[7L&nbsp;4s]], [[11L&nbsp;7s]], [[18L&nbsp;11s]]…
| 331.67, 868.33
| 
|-
| 1L&nbsp;3s
| 3, 1, 4, 5, 9, 14, 23…
| [[4L&nbsp;1s]], [[5L&nbsp;4s]], [[9L&nbsp;5s]], [[14L&nbsp;9s]], [[23L&nbsp;14s]]…
| 259.85, 940.15
| 
|-
| 1L&nbsp;4s
| 4, 1, 5, 6, 11, 17, 28…
| [[5L&nbsp;1s]], [[6L&nbsp;5s]], [[11L&nbsp;6s]], [[17L&nbsp;11s]], [[28L&nbsp;17s]]…
| 213.60, 986.40
| 
|-
| 2L&nbsp;3s
| 3, 2, 5, 7, 12, 19, 31…
| [[5L&nbsp;2s]], [[7L&nbsp;5s]], [[12L&nbsp;7s]], [[19L&nbsp;12s]], [[31L&nbsp;19s]]…
| 503.79, 696.21
| Golden meantone
|-
| 2L&nbsp;5s
| 5, 2, 7, 9, 16, 25, 41…
| [[7L&nbsp;2s]], [[9L&nbsp;7s]], [[16L&nbsp;9s]], [[25L&nbsp;16s]], [[41L&nbsp;25s]]…
| 527.15, 672.85
| 
|-
| 2L&nbsp;7s
| 7, 2, 9, 11, 20, 31, 51…
| [[9L&nbsp;2s]], [[11L&nbsp;9s]], [[20L&nbsp;11s]], [[31L&nbsp;20s]], [[51L&nbsp;30s]]…
| 541.38, 658.62
| 
|-
| 3L&nbsp;4s
| 4, 3, 7, 10, 17, 27…
| [[7L&nbsp;3s]], [[10L&nbsp;7s]], [[17L&nbsp;10s]], [[27L&nbsp;17s]]…
| 354.82, 845.18
| 
|-
| 3L&nbsp;5s
| 5, 3, 8, 11, 19, 30…
| [[8L&nbsp;3s]], [[11L&nbsp;8s]], [[19L&nbsp;11s]], [[30L&nbsp;19s]]…
| 440.59, 759.41
| 
|}
<references group="note" />

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,&nbsp;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,&nbsp;1) (corresponding to the series of MOSes generated by the golden 1L&nbsp;3s generator), which can be considered the "complementary" sequence of (3,&nbsp;2). In general, for a sequence (''m'',&nbsp;''n''), its complement is {{nowrap|(''m'', ''m'' − ''n'')}}, corresponding to oligolarge MOSes ''n''L&nbsp;''m''s and {{nowrap|(''m'' − ''n'')L ''m''s}}. The one exception is the family of scales (''m'',&nbsp;0), corresponding to the Fibonacci sequence and its multiples, which apparently have complements of (''m'',&nbsp;''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&nbsp;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.

{| class="wikitable sortable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | MOS height for MOSes under 15 notes
|-
! Height
! MOS
! Total notes
|-
| 1
| 1L&nbsp;1s
| 2
|-
| 2
| 2L&nbsp;1s
| 3
|-
| 3
| 1L&nbsp;2s
| 3
|-
| 3
| 3L&nbsp;2s
| 5
|-
| 4
| 3L&nbsp;1s
| 4
|-
| 4
| 2L&nbsp;3s
| 5
|-
| 4
| 5L&nbsp;3s
| 8
|-
| 5
| 4L&nbsp;3s
| 7
|-
| 5
| 1L&nbsp;3s
| 4
|-
| 5
| 5L&nbsp;2s
| 7
|-
| 5
| 3L&nbsp;5s
| 8
|-
| 5
| 8L&nbsp;5s
| 13
|-
| 6
| 7L&nbsp;4s
| 11
|-
| 6
| 3L&nbsp;4s
| 7
|-
| 6
| 4L&nbsp;1s
| 5
|-
| 6
| 2L&nbsp;5s
| 7
|-
| 6
| 7L&nbsp;5s
| 12
|-
| 6
| 8L&nbsp;3s
| 11
|-
| 6
| 5L&nbsp;8s
| 13
|-
| 7
| 4L&nbsp;7s
| 11
|-
| 7
| 7L&nbsp;3s
| 10
|-
| 7
| 5L&nbsp;4s
| 9
|-
| 7
| 1L&nbsp;4s
| 5
|-
| 7
| 7L&nbsp;2s
| 9
|-
| 7
| 5L&nbsp;7s
| 12
|-
| 7
| 3L&nbsp;8s
| 11
|-
| 8
| 3L&nbsp;7s
| 10
|-
| 8
| 4L&nbsp;5s
| 9
|-
| 8
| 9L&nbsp;5s
| 14
|-
| 8
| 5L&nbsp;1s
| 6
|-
| 8
| 2L&nbsp;7s
| 9
|-
| 8
| 11L&nbsp;3s
| 14
|-
| 9
| 10L&nbsp;3s
| 13
|-
| 9
| 9L&nbsp;4s
| 13
|-
| 9
| 5L&nbsp;9s
| 14
|-
| 9
| 1L&nbsp;5s
| 6
|-
| 9
| 6L&nbsp;5s
| 11
|-
| 9
| 9L&nbsp;2s
| 11
|-
| 9
| 3L&nbsp;11s
| 14
|-
| 10
| 3L&nbsp;10s
| 13
|-
| 10
| 4L&nbsp;9s
| 13
|-
| 10
| 6L&nbsp;1s
| 7
|-
| 10
| 5L&nbsp;6s
| 11
|-
| 10
| 2L&nbsp;9s
| 11
|-
| 11
| 1L&nbsp;6s
| 7
|-
| 11
| 7L&nbsp;6s
| 13
|-
| 11
| 11L&nbsp;2s
| 13
|-
| 12
| 7L&nbsp;1s
| 8
|-
| 12
| 6L&nbsp;7s
| 13
|-
| 12
| 2L&nbsp;11s
| 13
|-
| 13
| 1L&nbsp;7s
| 8
|-
| 14
| 8L&nbsp;1s
| 9
|-
| 15
| 1L&nbsp;8s
| 9
|-
| 16
| 9L&nbsp;1s
| 10
|-
| 17
| 1L&nbsp;9s
| 10
|-
| 18
| 10L&nbsp;1s
| 11
|-
| 19
| 1L&nbsp;10s
| 11
|-
| 20
| 11L&nbsp;1s
| 12
|-
| 21
| 1L&nbsp;11s
| 12
|-
| 22
| 12L&nbsp;1s
| 13
|-
| 23
| 1L&nbsp;12s
| 13
|-
| 24
| 13L&nbsp;1s
| 14
|-
| 25
| 1L&nbsp;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:
{| class="wikitable"
|+
!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]. 
{| class="wikitable"
!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:
{| class="wikitable"
!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:
{| class="wikitable"
!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 [[Bird's eye view of temperaments by accuracy|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.
{| class="wikitable"
|+ style="font-size: 105%; white-space: nowrap;" | Common temperaments
|-
! Temperament
! Generator
! Golden tuning
! MOS
|-
| Schismic/Garibaldi
| 3/2
| 702.75{{c}}
| 12L&nbsp;17s
|-
| Kleismic/Cata
| 6/5
| 317.17{{c}}
| 4L&nbsp;11s
|-
| Sensi
| 9/7
| 440.59{{c}}
| 3L&nbsp;5s
|-
| Wurschmidt
| 5/4
| 387.82{{c}}
| 3L&nbsp;28s
|-
| MIRACLE
| 16/15
| 115.59{{c}}
| 10L&nbsp;11s
|-
| Meantone
| 3/2
| 696.21{{c}}
| 2L&nbsp;3s
|-
| Mothra/Slendric
| 8/7
| 231.74{{c}}
| 5L&nbsp;21s
|-
| Didacus
| 28/25
| 191.13{{c}}
| 6L&nbsp;13s
|-
| Superpyth
| 3/2
| 708.05{{c}}
| 5L&nbsp;12s
|-
| Mohajira
| 11/9
| 348.91{{c}}
| 7L&nbsp;17s
|-
| Semaphore
| 8/7
| 254.04{{c}}
| 5L&nbsp;9s
|-
| Negri
| 16/15
| 124.77{{c}}
| 1L&nbsp;8s
|-
| Flattone
| 3/2
| 693.06{{c}}
| 7L&nbsp;12s
|-
| Porcupine
| 10/9
| 162.56{{c}}
| 7L&nbsp;8s
|-
| Magic
| 5/4
| 380.82{{c}}<ref group="note">Also has another tuning of 377.6 cents for 3L&nbsp;13s (interpretable as [[muggles]] rather than septimal magic). This results in a fifth almost as flat as in 7edo, but is a simpler scale of 16 notes rather than 19.</ref>
| 3L&nbsp;16s
|}
<references group="note" />

Exotemperaments are tuned to the simplest MOS that has a golden generator in the correct range.
{| class="wikitable"
|+ style="font-size: 105%; white-space: nowrap;" | Exotemperaments
|-
! Temperament
! Generator
! Golden tuning
! MOS
|-
| Father
| 8/5
| 741.64{{c}}
| 1L&nbsp;1s
|-
| Mavila
| 3/2
| 672.85{{c}}
| 2L&nbsp;5s
|-
| Bug
| 5/3
| 940.15{{c}}
| 1L&nbsp;3s
|-
| Dicot
| 5/4
| 354.83{{c}}
| 3L&nbsp;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.

{| class="wikitable sortable"
|-
! MOS
! Generator (¢)
|-
| 1L&nbsp;1s, 2L&nbsp;1s, 3L&nbsp;2s, 5L&nbsp;3s, 8L&nbsp;5s
| 741.6408
|-
| 1L&nbsp;2s, 3L&nbsp;1s, 4L&nbsp;3s, 7L&nbsp;4s
| 868.3282
|-
| 1L&nbsp;3s, 4L&nbsp;1s, 5L&nbsp;4s
| 940.1492
|-
| 2L&nbsp;3s, 5L&nbsp;2s, 7L&nbsp;5s
| 503.7855
|-
| 1L&nbsp;4s, 5L&nbsp;1s, 6L&nbsp;5s
| 986.4021
|-
| 1L&nbsp;5s, 6L&nbsp;1s, 7L&nbsp;6s
| 1018.6773
|-
| 3L&nbsp;4s, 7L&nbsp;3s
| 354.8232
|-
| 2L&nbsp;5s, 7L&nbsp;2s
| 527.1497
|-
| 1L&nbsp;6s, 7L&nbsp;1s
| 1042.4790
|-
| 3L&nbsp;5s, 8L&nbsp;3s
| 759.4078
|-
| 1L&nbsp;7s, 8L&nbsp;1s
| 1060.7571
|-
| 4L&nbsp;5s, 9L&nbsp;4s
| 273.8497
|-
| 2L&nbsp;7s, 9L&nbsp;2s
| 541.3837
|-
| 1L&nbsp;8s, 9L&nbsp;1s
| 1075.2344
|-
| 3L&nbsp;7s, 10L&nbsp;3s
| 366.2564
|-
| 1L&nbsp;9s, 10L&nbsp;1s
| 1086.9847
|-
| 5L&nbsp;6s
| 222.9668
|-
| 4L&nbsp;7s
| 877.7318
|-
| 3L&nbsp;8s
| 768.8815
|-
| 2L&nbsp;9s, 11L&nbsp;2s
| 550.9646
|-
| 1L&nbsp;10s, 11L&nbsp;1s
| 1096.7123
|-
| 5L&nbsp;7s
| 704.0956
|-
| 1L&nbsp;11s, 12L&nbsp;1s
| 1104.8980
|-
| 6L&nbsp;7s
| 188.0298
|-
| 5L&nbsp;8s
| 465.0841
|-
| 4L&nbsp;9s
| 280.6103
|-
| 3L&nbsp;10s
| 373.0714
|-
| 2L&nbsp;11s
| 557.8535
|-
| 1L&nbsp;12s
| 1111.8816
|}

[[Category:Golden ratio]]
[[Category:Tuning]]