Douglas Blumeyer's RTT How-To: Difference between revisions

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To fully understand why this is happening, we need a crash course in [[mediants]], and the [[Stern-Brocot_ancestors_and_rank_2_temperaments|The Stern-Brocot tree]].

The mediant of two fractions a/b and c/d is a+c/b+d. It’s sometimes called the freshman’s sum because it’s an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we’re leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of 3/5 and 2/3 is 5/8, and it’s easy to see in decimal form that 0.625 is between 0.6 and 0.666.

The mediant of two fractions <math>\frac ab</math> and <math>\frac cd</math> is <math>\frac{a+c}{b+d}</math>. It’s sometimes called the freshman’s sum because it’s an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we’re leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of <math>\frac 35</math> and <math>\frac 23</math> is <math>\frac 58</math>, and it’s easy to see in decimal form that 0.625 is between 0.6 and 0.666.

The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions 0/1 and 1/1. Taking the mediant of these two gives our first node: 1/2. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So 0/1 and 1/2 become 1/3, and 1/2 and 1/1 become 2/3. In the next tier, 0/1 and 1/3 become 1/4, 1/3 and 1/2 become 2/5, 1/2 and 2/3 become 3/5, and 2/3 and 1/1 become 3/4.<ref>Each tier of the Stern-Brocot tree is the next Farey sequence.</ref> The tree continues forever.

The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions <math>\frac 01</math> and <math>\frac 11</math>. Taking the mediant of these two gives our first node: <math>\frac 12</math>. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So <math>\frac 01</math> and <math>\frac 12</math> become <math>\frac 13</math>, and <math>\frac 12</math> and <math>\frac 11</math> become <math>\frac 23</math>. In the next tier, <math>\frac 01</math> and <math>\frac 13</math> become <math>\frac 14</math>, <math>\frac 13</math> and <math>\frac 12</math> become <math>\frac 25</math>, <math>\frac 12</math> and <math>\frac 23</math> become <math>\frac 35</math>, and <math>\frac 23</math> and <math>\frac 11</math> become <math>\frac 34</math>.<ref>Each tier of the Stern-Brocot tree is the next [https://en.wikipedia.org/wiki/Farey_sequence Farey sequence].</ref> The tree continues forever.

So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there’s more to 7-ET than simply the number 7. The 7 is just a fraction’s denominator. The numerator in this case is 3. So imagine a 3/7 floating on top of the 7-ET there. And there’s more to 5-ET than simply the number 5, in that case, the fraction is the 2/5. So the mediant of 2/5 and 3/7 is 5/12. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: 5/12 is between 2/5 and 3/7 on the meantone line. You may verify yourself that the mediant of 5/12 and 3/7, 8/19, is between them in size, as well as 7/17 being between 2/5 and 5/12 in size.

So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there’s more to 7-ET than simply the number 7. The 7 is just a fraction’s denominator. The numerator in this case is 3. So imagine a <math>\frac 37</math> floating on top of the 7-ET there. And there’s more to 5-ET than simply the number 5, in that case, the fraction is the <math>\frac 25</math>. So the mediant of <math>\frac 25</math> and <math>\frac 37</math> is <math>\frac{5}{12}</math>. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: <math>\frac{5}{12}</math> is between <math>\frac 25</math> and <math>\frac 37</math> on the meantone line. You may verify yourself that the mediant of <math>\frac{5}{12}</math> and <math>\frac 37</math>, <math>\frac{8}{19}</math>, is between them in size, as well as <math>\frac{7}{17}</math> being between <math>\frac 25</math> and <math>\frac{5}{12}</math> in size.

In fact, if you followed this value along the meantone line all the way from 2/5 to 3/7, it would vary continuously from 0.4 to 0.429; the ET points are the spots where the value happens to be rational.

In fact, if you followed this value along the meantone line all the way from <math>\frac 25</math> to <math>\frac 37</math>, it would vary continuously from 0.4 to 0.429; the ET points are the spots where the value happens to be rational.

~~Okay, so it’s easy to see how all this follows from here. But where the heck did I get 2/5 and 3/7 in the first place? I seemed to pull them out of thin air. And what the heck is this value?~~

Okay, so it’s easy to see how all this follows from here. But where the heck did I get <math>\frac 25</math> and <math>\frac 37</math> in the first place? I seemed to pull them out of thin air. And what the heck is this value?

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 To fully understand why this is happening, we need a crash course in [[mediants]], and the [[Stern-Brocot_ancestors_and_rank_2_temperaments|The Stern-Brocot tree]].
-The mediant of two fractions a/b and c/d is a+c/b+d. It’s sometimes called the freshman’s sum because it’s an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we’re leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of 3/5 and 2/3 is 5/8, and it’s easy to see in decimal form that 0.625 is between 0.6 and 0.666.
+The mediant of two fractions <span><math>\frac ab</math></span> and <span><math>\frac cd</math></span> is <span><math>\frac{a+c}{b+d}</math></span>. It’s sometimes called the freshman’s sum because it’s an easy mistake to make when first learning how to add fractions. And while this operation is certainly not equivalent to adding two fractions, it does turn out to have other important mathematical properties. The one we’re leveraging here is that the mediant of two numbers is always greater than one and less than the other. For example, the mediant of <span><math>\frac 35</math></span> and <span><math>\frac 23</math></span> is <span><math>\frac 58</math></span>, and it’s easy to see in decimal form that 0.625 is between 0.6 and 0.666.
-The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions 0/1 and 1/1. Taking the mediant of these two gives our first node: 1/2. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So 0/1 and 1/2 become 1/3, and 1/2 and 1/1 become 2/3. In the next tier, 0/1 and 1/3 become 1/4, 1/3 and 1/2 become 2/5, 1/2 and 2/3 become 3/5, and 2/3 and 1/1 become 3/4.<ref>Each tier of the Stern-Brocot tree is the next Farey sequence.</ref> The tree continues forever.
+The Stern-Brocot tree is a helpful visualization of all these mediant relations. Flanking the part of the tree we care about — which comes up in the closely-related theory of [[MOS_scale|MOS scales]], where it is often referred to as the “scale tree” — are the extreme fractions <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span>. Taking the mediant of these two gives our first node: <span><math>\frac 12</math></span>. Each new node on the tree drops an infinitely descending line of copies of itself on each new tier. Then, each node branches to either side, connecting itself to a new node which is the mediant of its two adjacent values. So <span><math>\frac 01</math></span> and <span><math>\frac 12</math></span> become <span><math>\frac 13</math></span>, and <span><math>\frac 12</math></span> and <span><math>\frac 11</math></span> become <span><math>\frac 23</math></span>. In the next tier, <span><math>\frac 01</math></span> and <span><math>\frac 13</math></span> become <span><math>\frac 14</math></span>, <span><math>\frac 13</math></span> and <span><math>\frac 12</math></span> become <span><math>\frac 25</math></span>, <span><math>\frac 12</math></span> and <span><math>\frac 23</math></span> become <span><math>\frac 35</math></span>, and <span><math>\frac 23</math></span> and <span><math>\frac 11</math></span> become <span><math>\frac 34</math></span>.<ref>Each tier of the Stern-Brocot tree is the next [https://en.wikipedia.org/wiki/Farey_sequence Farey sequence].</ref> The tree continues forever.
-So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there’s more to 7-ET than simply the number 7. The 7 is just a fraction’s denominator. The numerator in this case is 3. So imagine a 3/7 floating on top of the 7-ET there. And there’s more to 5-ET than simply the number 5, in that case, the fraction is the 2/5. So the mediant of 2/5 and 3/7 is 5/12. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: 5/12 is between 2/5 and 3/7 on the meantone line. You may verify yourself that the mediant of 5/12 and 3/7, 8/19, is between them in size, as well as 7/17 being between 2/5 and 5/12 in size.
+So what does this have to do with the patterns along the temperament lines in PTS? Well, each temperament line is kind of like its own section of the scale tree. The key insight here is that in terms of meantone temperament, there’s more to 7-ET than simply the number 7. The 7 is just a fraction’s denominator. The numerator in this case is 3. So imagine a <span><math>\frac 37</math></span> floating on top of the 7-ET there. And there’s more to 5-ET than simply the number 5, in that case, the fraction is the <span><math>\frac 25</math></span>. So the mediant of <span><math>\frac 25</math></span> and <span><math>\frac 37</math></span> is <span><math>\frac{5}{12}</math></span>. And if you compare the decimal values of these numbers, we have 0.4, 0.429, and 0.417. Success: <span><math>\frac{5}{12}</math></span> is between <span><math>\frac 25</math></span> and <span><math>\frac 37</math></span> on the meantone line. You may verify yourself that the mediant of <span><math>\frac{5}{12}</math></span> and <span><math>\frac 37</math></span>, <span><math>\frac{8}{19}</math></span>, is between them in size, as well as <span><math>\frac{7}{17}</math></span> being between <span><math>\frac 25</math></span> and <span><math>\frac{5}{12}</math></span> in size.
-In fact, if you followed this value along the meantone line all the way from 2/5 to 3/7, it would vary continuously from 0.4 to 0.429; the ET points are the spots where the value happens to be rational.
+In fact, if you followed this value along the meantone line all the way from <span><math>\frac 25</math></span> to <span><math>\frac 37</math></span>, it would vary continuously from 0.4 to 0.429; the ET points are the spots where the value happens to be rational.
-Okay, so it’s easy to see how all this follows from here. But where the heck did I get 2/5 and 3/7 in the first place? I seemed to pull them out of thin air. And what the heck is this value?
+Okay, so it’s easy to see how all this follows from here. But where the heck did I get <span><math>\frac 25</math></span> and <span><math>\frac 37</math></span> in the first place? I seemed to pull them out of thin air. And what the heck is this value?
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