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Golden sequences and tuning: Difference between revisions

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[[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).  
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).  


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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.
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.  
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.)
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.)
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== Golden operations and MOS height ==
== Golden operations and MOS height ==
Any MOS may be constructed from 1L 1s using two "golden operations": chromaticizing (taking the soft child of the MOS, thus continuing the golden sequence) and inverting (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.
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.
 
{| class="wikitable sortable mw-collapsible mw-collapsed"
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+MOS height for MOSes under 15 notes
|+ style="font-size: 105%; white-space: nowrap;" | MOS height for MOSes under 15 notes
!Height
|-
!MOS
! Height
!Total notes
! 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
|-
|-
|1
| 8
|1L 1s
| 9L 5s
|2
| 14
|-
|-
|2
| 8
|2L 1s
| 5L 1s
|3
| 6
|-
|-
|3
| 8
|1L 2s
| 2L 7s
|3
| 9
|-
|-
|3
| 8
|3L 2s
| 11L 3s
|5
| 14
|-
|-
|4
| 9
|3L 1s
| 10L 3s
|4
| 13
|-
|-
|4
| 9
|2L 3s
| 9L 4s
|5
| 13
|-
|-
|4
| 9
|5L 3s
| 5L 9s
|8
| 14
|-
|-
|5
| 9
|4L 3s
| 1L 5s
|7
| 6
|-
|-
|5
| 9
|1L 3s
| 6L 5s
|4
| 11
|-
|-
|5
| 9
|5L 2s
| 9L 2s
|7
| 11
|-
|-
|5
| 9
|3L 5s
| 3L 11s
|8
| 14
|-
|-
|5
| 10
|8L 5s
| 3L 10s
|13
| 13
|-
|-
|6
| 10
|7L 4s
| 4L 9s
|11
| 13
|-
|-
|6
| 10
|3L 4s
| 6L 1s
|7
| 7
|-
|-
|6
| 10
|4L 1s
| 5L 6s
|5
| 11
|-
|-
|6
| 10
|2L 5s
| 2L 9s
|7
| 11
|-
|-
|6
| 11
|7L 5s
| 1L 6s
|12
| 7
|-
|-
|6
| 11
|8L 3s
| 7L 6s
|11
| 13
|-
|-
|6
| 11
|5L 8s
| 11L 2s
|13
| 13
|-
|-
|7
| 12
|4L 7s
| 7L 1s
|11
| 8
|-
|-
|7
| 12
|7L 3s
| 6L 7s
|10
| 13
|-
|-
|7
| 12
|5L 4s
| 2L 11s
|9
| 13
|-
|-
|7
| 13
|1L 4s
| 1L 7s
|5
| 8
|-
|-
|7
| 14
|7L 2s
| 8L 1s
|9
| 9
|-
|-
|7
| 15
|5L 7s
| 1L 8s
|12
| 9
|-
|-
|7
| 16
|3L 8s
| 9L 1s
|11
| 10
|-
|-
|8
| 17
|3L 7s
| 1L 9s
|10
| 10
|-
|-
|8
| 18
|4L 5s
| 10L 1s
|9
| 11
|-
|-
|8
| 19
|9L 5s
| 1L 10s
|14
| 11
|-
|-
|8
| 20
|5L 1s
| 11L 1s
|6
| 12
|-
|-
|8
| 21
|2L 7s
| 1L 11s
|9
| 12
|-
|-
|8
| 22
|11L 3s
| 12L 1s
|14
| 13
|-
|-
|9
| 23
|10L 3s
| 1L 12s
|13
| 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:
{| class="wikitable"
|+
!Step
!Sequence fragment
!Sequence
|-
|-
|9
|(Start)
|9L 4s
|6, 11
|13
|(4, 1)
|-
|-
|9
|Step
|5L 9s
|5, 6
|14
|(4, 1)
|-
|-
|9
|Step
|1L 5s
|1, 5
|6
|(4, 1)
|-
|-
|9
|Step
|6L 5s
|4, 1
|11
|(4, 1)
|-
|-
|9
|Flip
|9L 2s
|1, 4
|11
|(3, 1)
|-
|-
|9
|Step
|3L 11s
|3, 1
|14
|(3, 1)
|-
|-
|10
|Flip
|3L 10s
|1, 3
|13
|Lucas
|-
|-
|10
|Step
|4L 9s
|2, 1
|13
|Lucas
|-
|-
|10
|Flip
|6L 1s
|1, 2
|7
|Fibonacci
|-
|-
|10
|Step
|5L 6s
|1, 1
|11
|Fibonacci
|}
Then, we work back through our steps, starting with [0, 1] instead of [1, 1].
{| class="wikitable"
!Step
!Sequence fragment
!Sequence
|-
|-
|10
|(Start)
|2L 9s
|0, 1
|11
|Fibonacci
|-
|-
|11
|Step
|1L 6s
|1, 1
|7
|Fibonacci
|-
|-
|11
|Flip
|7L 6s
|1, 1
|13
|Fibonacci
|-
|-
|11
|Step
|11L 2s
|1, 2
|13
|Fibonacci
|-
|-
|12
|Flip
|7L 1s
|2, 1
|8
|Lucas
|-
|-
|12
|Step
|6L 7s
|1, 3
|13
|Lucas
|-
|-
|12
|Flip
|2L 11s
|3, 1
|13
|(3, 1)
|-
|-
|13
|Step
|1L 7s
|1, 4
|8
|(3, 1)
|-
|-
|14
|Step
|8L 1s
|4, 5
|9
|(3, 1)
|-
|-
|15
|Step
|1L 8s
|5, 9
|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
|-
|-
|16
|(Start)
|9L 1s
|2, 5
|10
|(3, 2)
|-
|-
|17
|Step
|1L 9s
|3, 2
|10
|(3, 2)
|-
|-
|18
|Flip
|10L 1s
|2, 3
|11
|Fibonacci
|-
|-
|19
|Step
|1L 10s
|1, 2
|11
|Fibonacci
|-
|-
|20
|Step
|11L 1s
|1, 1
|12
|Fibonacci
|}
And then to find the generator:
{| class="wikitable"
!Step
!Sequence fragment
!Sequence
|-
|-
|21
|(Start)
|1L 11s
|0, 1
|12
|Fibonacci
|-
|-
|22
|Step
|12L 1s
|1, 1
|13
|Fibonacci
|-
|-
|23
|Step
|1L 12s
|1, 2
|13
|Fibonacci
|-
|-
|24
|Flip
|13L 1s
|2, 1
|14
|Lucas
|-
|-
|25
|Step
|1L 13s
|1, 3
|14
|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 ==
== 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 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 17s
|-
| Kleismic/Cata
| 6/5
| 317.17{{c}}
| 4L 11s
|-
| Sensi
| 9/7
| 440.59{{c}}
| 3L 5s
|-
| Wurschmidt
| 5/4
| 387.82{{c}}
| 3L 28s
|-
| MIRACLE
| 16/15
| 115.59{{c}}
| 10L 11s
|-
| Meantone
| 3/2
| 696.21{{c}}
| 2L 3s
|-
| Mothra/Slendric
| 8/7
| 231.74{{c}}
| 5L 21s
|-
| Didacus
| 28/25
| 191.13{{c}}
| 6L 13s
|-
| Superpyth
| 3/2
| 708.05{{c}}
| 5L 12s
|-
| Mohajira
| 11/9
| 348.91{{c}}
| 7L 17s
|-
| Semaphore
| 8/7
| 254.04{{c}}
| 5L 9s
|-
| Negri
| 16/15
| 124.77{{c}}
| 1L 8s
|-
| Flattone
| 3/2
| 693.06{{c}}
| 7L 12s
|-
| Porcupine
| 10/9
| 162.56{{c}}
| 7L 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
|}
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
|}


== Notes ==
== Notes ==
Line 355: Line 730:


{{todo|inline=1|categorize}}
{{todo|inline=1|categorize}}
[[Category:Golden ratio]]
[[Category:Golden ratio]]
[[Category:Tuning]]
[[Category:Tuning]]
Retrieved from "https://en.xen.wiki/w/Golden_sequences_and_tuning"