Metallic MOS: Difference between revisions
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and the other half, | and the other half, | ||
<math>\qquad L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = | <math>\qquad L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = … = δ_s - 1 | ||
</math> | </math> | ||
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We can think of a mediant like a bizarro average of two ratios: however we may choose to weight one, it will always lie somewhere between the two ratios. That is why we can call these two ratios its bounds. This fact is easy enough to intuit: as the weight tends toward zero, we approach num2/den2, and as it tends toward infinity, we approach num1/den1. | We can think of a mediant like a bizarro average of two ratios: however we may choose to weight one, it will always lie somewhere between the two ratios. That is why we can call these two ratios its bounds. This fact is easy enough to intuit: as the weight tends toward zero, we approach num2/den2, and as it tends toward infinity, we approach num1/den1. | ||
<math>\qquad | |||
\lim_{w\to 0} \frac{a_1w + b_1}{a_2w + b_2} = \frac{a_1}{a_2} | |||
\lim_{w\to \infty} \frac{a_1w + b_1}{a_2w + b_2} = \frac{b_1}{b_2} | |||
</math> | |||
=== Checking our earlier results === | |||
To confirm that the weighted mediant formula gives the same result for the golden generator as we were using before, we can plug in 0/1 and 1/1 for our bounds: | To confirm that the weighted mediant formula gives the same result for the golden generator as we were using before, we can plug in 0/1 and 1/1 for our bounds: | ||
<math>\qquad \frac{1φ + 0}{1φ + 1} ≈ 0.618034 | |||
</math> | |||
And finally we can show how we got approximately 0.419821 as the value for the noble generator between 1/3 and 1/2: | And finally we can show how we got approximately 0.419821 as the value for the noble generator between 1/3 and 1/2: | ||
<math>\qquad \frac{1φ + 1}{2φ + 3} ≈ 0.419821 | |||
</math> | |||
=== About diagram labels === | === About diagram labels === | ||
In the generator diagrams earlier in this discussion — as we disclaimed at that time — the labels on the segments were not indicating length. We know now that what they were actually indicating was the mediant weight. For our purposes, mediant weight is more helpful information, so we’re going to continue labeling the segments this way. | In the generator diagrams earlier in this discussion — as we disclaimed at that time — the labels on the segments were not indicating length. We know now that what they were actually indicating was the mediant weight. For our purposes, mediant weight is more helpful information, so we’re going to continue labeling the segments this way. | ||
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=== Finding by child ratio === | === Finding by child ratio === | ||
Finding by the child ratio is helpful when reasoning about the levels of the tree and its intervals. It’s just good recursive design. | Finding by the child ratio is helpful when reasoning about the levels of the tree and its intervals. It’s just good recursive design. | ||
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For example, we’ll look at our noble generator example again. This one is weighted parentward, | For example, we’ll look at our noble generator example again. This one is weighted parentward, | ||
<math>\qquad \frac{1φ + 1}{2φ + 3} ≈ 0.419821 | |||
</math> | |||
because — of its two bounding ratios, 1/2 and 1/ | because — of its two bounding ratios, 1/2 and 1/3 — 1/2 is the parent. | ||
We could weight childward instead though: | We could weight childward instead though: | ||
<math>\qquad \frac{1 + 1φ}{2 + 3φ} ≈ 0.381966 | |||
</math> | |||
which, interestingly, gives us the same value as the golden generator. | which, interestingly, gives us the same value as the golden generator. | ||
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In Wilson’s work, he only used childward lean; had he included parentward lean for his noble generators, he would have found every single generator in one earlier level of the tree than he had previously. For example, he names the golden generator, leaning toward 1/2, which of the two ratios is the child: | In Wilson’s work, he only used childward lean; had he included parentward lean for his noble generators, he would have found every single generator in one earlier level of the tree than he had previously. For example, he names the golden generator, leaning toward 1/2, which of the two ratios is the child: | ||
<math>\qquad \frac{1φ + 0}{2φ + 1} ≈ 0.381966 | |||
</math> | |||
whereas we would name it, leaning toward 0/1, which of our two ratios is the parent: | whereas we would name it, leaning toward 0/1, which of our two ratios is the parent: | ||
<math>\qquad \frac{0φ + 1}{1φ + 1} ≈ 0.381966 | |||
</math> | |||
=== Levels of metallicity === | === Levels of metallicity === | ||
The Stern-Brocot tree can be recursed indefinitely, so an infinite number of metallic generators exist. And since each parent ratio branches into two child ratios, each new recursive level of the Stern-Brocot tree offers the next power of 2 more intervals. | The Stern-Brocot tree can be recursed indefinitely, so an infinite number of metallic generators exist. And since each parent ratio branches into two child ratios, each new recursive level of the Stern-Brocot tree offers the next power of 2 more intervals. | ||
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And the noble generator between 0/1 and 1/3 | And the noble generator between 0/1 and 1/3 | ||
<math>\qquad \frac{1φ + 0}{3φ + 1} ≈ 0.276393 | |||
</math> | |||
is very close to the root of the tree; it has initial L:s ratio of φ + 2, then attains L:s = φ after only one iteration. And it begins the golden interval pattern after just one iteration too. | is very close to the root of the tree; it has initial L:s ratio of φ + 2, then attains L:s = φ after only one iteration. And it begins the golden interval pattern after just one iteration too. | ||
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=== Naming generators === | === Naming generators === | ||
While not every interval can make a noble number, every noble number can be found as the weighted mediant of many intervals. For example, | While not every interval can make a noble number, every noble number can be found as the weighted mediant of many intervals. For example, | ||
<math>\qquad 0.216542 ≈ \frac{0φ + 1}{1φ + 3} ≈ \frac{1φ + 0}{4φ + 1} ≈ \frac{2φ + 1}{9φ + 5} ≈ … | |||
</math> | |||
This ambiguity presents a naming problem. Naming generators after combinations of an interval and the metallic value used in the weighted mediant formula definitely makes sense, but we need to know which of these possibilities to choose. | This ambiguity presents a naming problem. Naming generators after combinations of an interval and the metallic value used in the weighted mediant formula definitely makes sense, but we need to know which of these possibilities to choose. | ||
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=== Generator complements === | === Generator complements === | ||
Generators are equivalent to their ''complements'': | Generators are equivalent to their ''complements'': | ||
<math>\qquad g ≡ 1 - g | |||
</math> | |||
By “equivalent”, we mean that they will generate essentially the same scale sequence. And by “essentially the same” we mean that the scales will be mirror images of each other, which for an MOS scale, happens to also mean that they are simply transpositions of each other (different modes of the same scale). | By “equivalent”, we mean that they will generate essentially the same scale sequence. And by “essentially the same” we mean that the scales will be mirror images of each other, which for an MOS scale, happens to also mean that they are simply transpositions of each other (different modes of the same scale). | ||
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== Isotopic arithmetic progression == | == Isotopic arithmetic progression == | ||
Now we’ll explain why the L:s sequences for metallic means cycle through their isotopes. | Now we’ll explain why the L:s sequences for metallic means cycle through their isotopes. | ||
=== Why they decrease by 1 === | === Why they decrease by 1 === | ||
First we need to review some MOS concepts. The mechanics of scale generation are such that — when iterating from one scale to the next densest one — all large steps in the preceding scale become one large step and one small step in the new scale. | First we need to review some MOS concepts. The mechanics of scale generation are such that — when iterating from one scale to the next densest one — all large steps in the preceding scale become one large step and one small step in the new scale. | ||
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So the L:s ratio decreases by 1 because if an s-sized chunk has been sliced off L, and s’s size is 1, then 1 should be subtracted from L. | So the L:s ratio decreases by 1 because if an s-sized chunk has been sliced off L, and s’s size is 1, then 1 should be subtracted from L. | ||
<math>\qquad L’:s’ = (L - s):s = (L - 1):1 = L - 1 | |||
</math> | |||
=== Why they cycle === | === Why they cycle === | ||
That is true of scale iterations where L - s > s. For the other type of scale iteration, where L - s < s, the result is simply reciprocated: | That is true of scale iterations where L - s > s. For the other type of scale iteration, where L - s < s, the result is simply reciprocated: | ||
<math>\qquad L’:s’ = s:(L - s) = 1:(L - 1) = 1 / (L - 1) | |||
</math> | |||
This alone would not suffice to explain how the L:s sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. Phi has the property that | This alone would not suffice to explain how the L:s sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. Phi has the property that | ||
<math>\qquad φ - 1 = 1 / φ | |||
</math> | |||
So, in the case of phi: | So, in the case of phi: | ||
<math>\qquad L’:s’ = 1 / (L - 1) = 1 / (φ - 1) = 1 / (1 / φ) = φ | |||
</math> | |||
That’s why the golden L:s sequence locks into L:s = φ forever. | That’s why the golden L:s sequence locks into L:s = φ forever. | ||
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A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value’s reciprocal is equal to the silver mean. | A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value’s reciprocal is equal to the silver mean. | ||
<math>\qquad δ_s - 1 - 1 = 1 / δ_s | |||
</math> | |||
And for the bronze ratio, we must subtract thrice: | And for the bronze ratio, we must subtract thrice: | ||
<math>\qquad δ_s - 1 - 1 - 1 = 1 / δ_s | |||
</math> | |||
Basically, we are subtracting 1 from the means until only their decimal part remains. One way to describe the metallic means, then, would be the set of values for which the reciprocal of their decimal part equals themselves. | Basically, we are subtracting 1 from the means until only their decimal part remains. One way to describe the metallic means, then, would be the set of values for which the reciprocal of their decimal part equals themselves. | ||
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For example, the golden mean does have an isotope, ≈ 0.618034, however, because phi minus one is the inverse of phi, | For example, the golden mean does have an isotope, ≈ 0.618034, however, because phi minus one is the inverse of phi, | ||
<math>\qquad 1:φ = (φ - 1):1 | |||
</math> | |||
we’ve ended up splitting the period up into segments of the same length, just swapped which one is on which side. | we’ve ended up splitting the period up into segments of the same length, just swapped which one is on which side. | ||
== Continued fractions == | == Continued fractions == | ||
An understanding of continued fractions unlocks many insights about metallic means, and also many insights about MOS scales, so it should be no surprise that they are particularly rife with insights about met-MOS scales, the intersection of the two concepts. | An understanding of continued fractions unlocks many insights about metallic means, and also many insights about MOS scales, so it should be no surprise that they are particularly rife with insights about met-MOS scales, the intersection of the two concepts. | ||
=== Behavior === | === Behavior === | ||
First we’ll document some behavior of continued fractions. Then we’ll get into applications. | First we’ll document some behavior of continued fractions. Then we’ll get into applications. | ||
The metallic means are all irrational numbers. Therefore their continued fractions are infinite, with a periodic pattern at the end. An advantage of continued fractions over the decimal system is that we can easily determine whether a number is rational by whether a periodic pattern at the end is required. Decimals, on the other hand, sometimes require periodic patterns at the end even when the number is rational, such as 1/3 which is 0. | The metallic means are all irrational numbers. Therefore their continued fractions are infinite, with a periodic pattern at the end. An advantage of continued fractions over the decimal system is that we can easily determine whether a number is rational by whether a periodic pattern at the end is required. Decimals, on the other hand, sometimes require periodic patterns at the end even when the number is rational, such as 1/3 which is 0.\overline{3}, or 1/7 which is 0.\overline{142857}; as continued fractions these two values are, respectively, [0; 3] and [0; 7]. | ||
The golden mean has the continued fraction [1; | The golden mean has the continued fraction [1; \overline{1}]. The larger a term in a continued fraction, the closer the approximation of the value at that point; by this conception of irrationality, the golden mean is sometimes said to be the most irrational number possible, eluding close approximation by any ratio as much as possible at every turn. | ||
The silver mean follows it closely with continued fraction [2; | The silver mean follows it closely with continued fraction [2; \overline{2}], and the bronze mean with [3; \overline{3}]. | ||
A handy way to quickly find the reciprocal of a number is to prepend its continued fraction with a 0. So, we can find the golden mean’s isotope 0.618034 is [0; | A handy way to quickly find the reciprocal of a number is to prepend its continued fraction with a 0. So, we can find the golden mean’s isotope 0.618034 is [0; \overline{1}]. | ||
Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; | Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are [1; \overline{2}] and [0; \overline{2}]. The initial term of a continued fraction, the one to the left of the semicolon, carries the same information as the digit of a decimal just to the left of the decimal point, i.e. any number starting with [0;] is between 0 and 1. | ||
The continued fraction for any abstract generator should start with 0, then, because it must be less than the period, which is 1 (if not less than 0.5, as we’ve been preferring, because of the generator complement effect). | The continued fraction for any abstract generator should start with 0, then, because it must be less than the period, which is 1 (if not less than 0.5, as we’ve been preferring, because of the generator complement effect). | ||
Noble generators start with other numbers but then settle on all 1’s; for example, our earlier example of 0.419821 is [0; 2, 2, | Noble generators start with other numbers but then settle on all 1’s; for example, our earlier example of 0.419821 is [0; 2, 2, \overline{1}]. | ||
Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2’s, 3’s, or n if we base our noble on the n<sup>th</sup> metallic mean. For example, our earlier example 0.226541 is [0; 4, | Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2’s, 3’s, or n if we base our noble on the n<sup>th</sup> metallic mean. For example, our earlier example 0.226541 is [0; 4, \overline{2}] | ||
=== Application: L:s sequences === | === Application: L:s sequences === | ||
[ | Continued fractions can compute L:s sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we’ll look at the L:s sequence for g = [0; 2, 2, \overline{1}] ≈ 0.419821. | ||
<math>\qquad | |||
\begin{equation} | |||
\begin{cases} | |||
L:s sequence = | L:s sequence = | ||
[2; 2, \overline{1}] ≈ 2.381966 | |||
[1; 2, \overline{1}] ≈ 1.381966 | |||
[2; \overline{1}] ≈ 2.618034 | |||
[2; 2, | [1; \overline{1}] ≈ 1.618034 = φ | ||
… | |||
[1; 2, | \end{equation} | ||
\end{cases} | |||
[2; | </math> | ||
[1; | |||
We’ve stated that L:s = φ for every golden scale, while L:s for noble scales eventually do, just not at first. Noble L:s sequences lock onto phi at the point where depleting the continued fraction more no longer changes it (removing a 1 from the beginning of an infinite string of 1’s is a no-op). | We’ve stated that L:s = φ for every golden scale, while L:s for noble scales eventually do, just not at first. Noble L:s sequences lock onto phi at the point where depleting the continued fraction more no longer changes it (removing a 1 from the beginning of an infinite string of 1’s is a no-op). | ||
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[ | [ | ||
[1; | [1; \overline{1}] ≈ 1.618034 = φ | ||
… | |||
] | ] | ||
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[ | [ | ||
[2; | [2; \overline{2}] ≈ 2.414214 = δ_s | ||
[1; | [1; \overline{2}] ≈ 1.414214 = δ_s - 1 | ||
… | |||
] | ] | ||
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[0; 2, 1, 1, 1, 1] = 5/13 | [0; 2, 1, 1, 1, 1] = 5/13 | ||
… | |||
[ math ] | [ math ] | ||
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|phi, golden ratio, golden section, golden proportion, greater phi, divine proportion | |phi, golden ratio, golden section, golden proportion, greater phi, divine proportion | ||
|Φ, φ, τ, σ<sub>g</sub> | |Φ, φ, τ, σ<sub>g</sub> | ||
|1. | |1.618034… | ||
|[1; 1] | |[1; 1] | ||
|<sub>1</sub>μ<sub>1</sub> | |<sub>1</sub>μ<sub>1</sub> | ||
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|lesser phi, golden ratio conjugate, silver ratio | |lesser phi, golden ratio conjugate, silver ratio | ||
|φ | |φ | ||
|0. | |0.618034… | ||
|[0; 1] | |[0; 1] | ||
|<sub>0</sub>μ<sub>1</sub> | |<sub>0</sub>μ<sub>1</sub> | ||
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|silver ratio | |silver ratio | ||
|δ<sub>s</sub>, σ<sub>s</sub> | |δ<sub>s</sub>, σ<sub>s</sub> | ||
|2. | |2.414214… | ||
|[2; 2] | |[2; 2] | ||
|<sub>2</sub>μ<sub>2</sub> | |<sub>2</sub>μ<sub>2</sub> | ||
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silver ratio | silver ratio | ||
|√2 | |√2 | ||
|1. | |1.414214… | ||
|[1; 2] | |[1; 2] | ||
|<sub>1</sub>μ<sub>2</sub> | |<sub>1</sub>μ<sub>2</sub> | ||
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|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|0. | |0.414214… | ||
|[0; 2] | |[0; 2] | ||
|<sub>0</sub>μ<sub>2</sub> | |<sub>0</sub>μ<sub>2</sub> | ||
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|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|δ<sub>b</sub>, σ<sub>b</sub> | |δ<sub>b</sub>, σ<sub>b</sub> | ||
|3. | |3.302776… | ||
|[3; 3] | |[3; 3] | ||
|<sub>3</sub>μ<sub>3</sub> | |<sub>3</sub>μ<sub>3</sub> | ||
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|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|2. | |2.302776… | ||
|[2; 3] | |[2; 3] | ||
|<sub>2</sub>μ<sub>3</sub> | |<sub>2</sub>μ<sub>3</sub> | ||
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|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|1. | |1.302776… | ||
|[1; 3] | |[1; 3] | ||
|<sub>1</sub>μ<sub>3</sub> | |<sub>1</sub>μ<sub>3</sub> | ||
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|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|<nowiki>-</nowiki> | |<nowiki>-</nowiki> | ||
|0. | |0.302776… | ||
|[0; 3] | |[0; 3] | ||
|<sub>0</sub>μ<sub>3</sub> | |<sub>0</sub>μ<sub>3</sub> | ||
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5L+3s | 5L+3s | ||
… | |||
[ math ] | [ math ] | ||
| Line 758: | Line 776: | ||
3L+2s | 3L+2s | ||
… | |||
[ math ] | [ math ] | ||
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[ math ] | [ math ] | ||
L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = | L:s = (L+2s):(L+s) = (3L+4s):(2L+3s) = (7L+10s):(5L+7s) = (17L+24s):(12L+17s) = … = δ<sub>s</sub> - 1 | ||
[[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|right|396x396px|horogram for g = 0.292893, 7 iterations]] | [[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|right|396x396px|horogram for g = 0.292893, 7 iterations]] | ||
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29L+12s | 29L+12s | ||
… | |||
[ math ] | [ math ] | ||
| Line 820: | Line 838: | ||
12L+5s | 12L+5s | ||
… | |||
[ math ] | [ math ] | ||
| Line 840: | Line 858: | ||
17L+24s | 17L+24s | ||
… | |||
[ math ] | [ math ] | ||
| Line 856: | Line 874: | ||
12L+17s | 12L+17s | ||
… | |||
[ math ] | [ math ] | ||
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'''generator, bronze''' | '''generator, bronze''' | ||
The generator found using the weighted mediant formula on the period interval with the golden mean, equal to [0; 4, | The generator found using the weighted mediant formula on the period interval with the golden mean, equal to [0; 4, \overline{3}] ≈ 0.232408. | ||
'''generator, complement''' | '''generator, complement''' | ||
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'''generator, golden''' | '''generator, golden''' | ||
The generator found using the weighted mediant formula on the period interval with the golden mean, equal to [0; 2, | The generator found using the weighted mediant formula on the period interval with the golden mean, equal to [0; 2, \overline{1}] ≈ 0.381966. | ||
'''generator, isotopic''' | '''generator, isotopic''' | ||
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'''generator, silver''' | '''generator, silver''' | ||
The generator found using the weighted mediant formula on the period interval with the silver mean, equal to [0; 3, | The generator found using the weighted mediant formula on the period interval with the silver mean, equal to [0; 3, \overline{2}] ≈ 0.292893 | ||
'''interval pattern''' | '''interval pattern''' | ||