Patent val

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[[toc|flat]]
=Definition= 
Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding [[p-limit]] [[val]] in a canonical manner by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplying]] <1 [[log2]](3) log2(5) ... log(p)| by N and rounding to the nearest integer. In general this is not guaranteed to be the most accurate available val, but if N-edo has enough relative accuracy in the p-limit, it will be. The name //patent// comes from the fact that "patent" in one sense of the word is a synonym for "obvious"; the patent val may or may not be the best choice but it's the obvious choice.

=A 12 EDO Example= 
Multiplying 12 times <1 1.585 2.322 2.807 3.459|
yields <12 19.020 27.863 33.688 41.513|,
rounded to <12 19 28 34 42|,
which is the **11-limit patent val for [[12edo]]**.

=An expanded example for 31 EDO= 
The val contains the number of steps it takes to get to a given prime number, in prime number order:
< [2/1] [3/1] [5/1] [7/1] [etc.] |
By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is < 31 |.

What's the number of steps to 3/1?
The step size for 31 EDO is 38.70967742 cents.
3/1 is 1901.96 in cents.
1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.
This is an EDO, so we can't take 0.13383752 steps. Instead, we round. This is clearly closer to 49 steps, so that's the "obvious" or "patent" choice. The 3-limit patent val is 
< 31 49 |. Doing the same thing up through 17, and we get an 17-limit patent val of
< 31 49 72 87 107 115 127 |

To see how to extend from one limit to another, we may look at what to do for 19/1 and use that to go from the 17-limit to the 19-limit.

19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is
< 31 49 72 87 107 115 127 132 |

Note that these are the same answers you would get if you multiplied 31 times <1 1.585 2.322 2.807 3.459 | and rounded the result.

=Why this defines a regular temperament= 
A val defines a regular temperament, which is the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2^(700/1200), or 1.4983070769.

As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, <12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, <31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave.

In the patent val for 12 EDO, the number 19 is in the second spot -- the place reserved for 3/1. That implies that it takes 19 steps to get to 3/1. The 49 in the patent val for 31 EDO implies that it takes 49 steps to get to 3/1. We know those aren't precisely true, because we had to round to get these numbers in the first place. In essence, we're //pretending// 19 steps gets you to 3/1 in 12 EDO; in other words, we're deliberately introducing an error into 3/1. Likewise with 49 steps of 31 EDO.

We can calculate the error we're introducing into 3/1 as follows for 12 EDO:
12 EDO steps are 100.0 cents.
19 steps of 12 EDO = 19 steps * 100.0 cents/step = 1900.0 cents.
1900.0 cents => 2^(1900/1200), or 2.9966141538. This is the value that 12 EDO uses in place of prime 3.
That means that the 12 EDO patent val substitutes 2.9966141538 any time you would have had prime 3 in a ratio. 9/8 and 3/2 will be somewhat flat. 4/3 will be somewhat sharp.

(Note that, for now, the example intervals (3/2, 4/3, 9/8) deliberately avoid any primes other than 3 and 2, and 2 is pure, so the sharpness and flatness comes only from the impure 3/1 value.)

Likewise for 31 EDO.
31 EDO steps are 38.70967742 cents.
49 steps of 31 EDO = 49 steps * 38.70967742 cents/step = 1896.774194 cents.
1896.774194 cents => 2^(1896.774194/1200), or 2.991035765. This is what 31 EDO uses in place of prime 3. Again, 9/8 and 3/2 will be somewhat flat, and 4/3 will be somewhat sharp.

Note that 31 EDO's prime 3 is a little farther away from 3/1 than 12 EDO's 3/1 -- i.e., it has a greater error. That means 31 EDO's 3/2 will be even flatter, and its 4/3 will be even sharper, than in 12 EDO.

That doesn't make 31 EDO better or worse; it just means there's more error in the 3/1 ratio in 31 EDO than in 12 EDO. If you run these calculations for 5/1 using the patent vals for 12 EDO and 31 EDO, you'll find that 5/1 has more error in 12 EDO than in 31 EDO: 5.0396842 vs. 5.002262078, respectively. 31 EDO may therefore be preferred by people who like sweeter thirds (5/4 ratios) and are willing to have flatter fifths (3/2 ratios).

==How this relates to commas== 
These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:
81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 >.
80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 >.
Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.
Substitute in the values for 81/80 in 31 EDO and get this: (2.991035765^4) / (2^4)*(5.002262078) = 80.036193 / 80.036193 = 1/1.

The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.

Original HTML content:

<html><head><title>Patent val</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#A 12 EDO Example">A 12 EDO Example</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#xAn expanded example for 31 EDO">An expanded example for 31 EDO</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#xWhy this defines a regular temperament">Why this defines a regular temperament</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: -->
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1>
 Given N-edo, the equal division of the octave into N parts, we may for any prime p find a corresponding <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> in a canonical manner by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow">scalar multiplying</a> &lt;1 <a class="wiki_link" href="/log2">log2</a>(3) log2(5) ... log(p)| by N and rounding to the nearest integer. In general this is not guaranteed to be the most accurate available val, but if N-edo has enough relative accuracy in the p-limit, it will be. The name <em>patent</em> comes from the fact that &quot;patent&quot; in one sense of the word is a synonym for &quot;obvious&quot;; the patent val may or may not be the best choice but it's the obvious choice.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="A 12 EDO Example"></a><!-- ws:end:WikiTextHeadingRule:2 -->A 12 EDO Example</h1>
 Multiplying 12 times &lt;1 1.585 2.322 2.807 3.459|<br />
yields &lt;12 19.020 27.863 33.688 41.513|,<br />
rounded to &lt;12 19 28 34 42|,<br />
which is the <strong>11-limit patent val for <a class="wiki_link" href="/12edo">12edo</a></strong>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="xAn expanded example for 31 EDO"></a><!-- ws:end:WikiTextHeadingRule:4 -->An expanded example for 31 EDO</h1>
 The val contains the number of steps it takes to get to a given prime number, in prime number order:<br />
&lt; [2/1] [3/1] [5/1] [7/1] [etc.] |<br />
By definition, for any EDO, the number of steps to 2/1 is the EDO division: 31 for 31 EDO. The 2-limit patent val is &lt; 31 |.<br />
<br />
What's the number of steps to 3/1?<br />
The step size for 31 EDO is 38.70967742 cents.<br />
3/1 is 1901.96 in cents.<br />
1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.<br />
This is an EDO, so we can't take 0.13383752 steps. Instead, we round. This is clearly closer to 49 steps, so that's the &quot;obvious&quot; or &quot;patent&quot; choice. The 3-limit patent val is <br />
&lt; 31 49 |. Doing the same thing up through 17, and we get an 17-limit patent val of<br />
&lt; 31 49 72 87 107 115 127 |<br />
<br />
To see how to extend from one limit to another, we may look at what to do for 19/1 and use that to go from the 17-limit to the 19-limit.<br />
<br />
19/1 = 5097.51 cents, 5097.51 / 38.70967742 cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is<br />
&lt; 31 49 72 87 107 115 127 132 |<br />
<br />
Note that these are the same answers you would get if you multiplied 31 times &lt;1 1.585 2.322 2.807 3.459 | and rounded the result.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="xWhy this defines a regular temperament"></a><!-- ws:end:WikiTextHeadingRule:6 -->Why this defines a regular temperament</h1>
 A val defines a regular temperament, which is the deliberate introduction of an error into one or more primes. In 12 EDO, for instance, the perfect fifth (ratio 3/2, or exactly 1.5) is mapped to 700 cents, which is actually just barely flat: a ratio of 2^(700/1200), or 1.4983070769.<br />
<br />
As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, &lt;12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, &lt;31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave.<br />
<br />
In the patent val for 12 EDO, the number 19 is in the second spot -- the place reserved for 3/1. That implies that it takes 19 steps to get to 3/1. The 49 in the patent val for 31 EDO implies that it takes 49 steps to get to 3/1. We know those aren't precisely true, because we had to round to get these numbers in the first place. In essence, we're <em>pretending</em> 19 steps gets you to 3/1 in 12 EDO; in other words, we're deliberately introducing an error into 3/1. Likewise with 49 steps of 31 EDO.<br />
<br />
We can calculate the error we're introducing into 3/1 as follows for 12 EDO:<br />
12 EDO steps are 100.0 cents.<br />
19 steps of 12 EDO = 19 steps * 100.0 cents/step = 1900.0 cents.<br />
1900.0 cents =&gt; 2^(1900/1200), or 2.9966141538. This is the value that 12 EDO uses in place of prime 3.<br />
That means that the 12 EDO patent val substitutes 2.9966141538 any time you would have had prime 3 in a ratio. 9/8 and 3/2 will be somewhat flat. 4/3 will be somewhat sharp.<br />
<br />
(Note that, for now, the example intervals (3/2, 4/3, 9/8) deliberately avoid any primes other than 3 and 2, and 2 is pure, so the sharpness and flatness comes only from the impure 3/1 value.)<br />
<br />
Likewise for 31 EDO.<br />
31 EDO steps are 38.70967742 cents.<br />
49 steps of 31 EDO = 49 steps * 38.70967742 cents/step = 1896.774194 cents.<br />
1896.774194 cents =&gt; 2^(1896.774194/1200), or 2.991035765. This is what 31 EDO uses in place of prime 3. Again, 9/8 and 3/2 will be somewhat flat, and 4/3 will be somewhat sharp.<br />
<br />
Note that 31 EDO's prime 3 is a little farther away from 3/1 than 12 EDO's 3/1 -- i.e., it has a greater error. That means 31 EDO's 3/2 will be even flatter, and its 4/3 will be even sharper, than in 12 EDO.<br />
<br />
That doesn't make 31 EDO better or worse; it just means there's more error in the 3/1 ratio in 31 EDO than in 12 EDO. If you run these calculations for 5/1 using the patent vals for 12 EDO and 31 EDO, you'll find that 5/1 has more error in 12 EDO than in 31 EDO: 5.0396842 vs. 5.002262078, respectively. 31 EDO may therefore be preferred by people who like sweeter thirds (5/4 ratios) and are willing to have flatter fifths (3/2 ratios).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="xWhy this defines a regular temperament-How this relates to commas"></a><!-- ws:end:WikiTextHeadingRule:8 -->How this relates to commas</h2>
 These deliberate errors ensure that certain commas get tempered out. The patent vals for both 12 EDO and 31 EDO temper out 81/80. Here are the calculations:<br />
81 = 3*3*3*3. This can also be written as a power of a prime -- 3^4 -- or as a monzo -- | 0 3 &gt;.<br />
80 = 2*2*2*2*5. This can also be written as a product of powers of primes -- (2^4)*(5^1) -- or as a monzo -- | 2 0 1 &gt;.<br />
Substitute in the values for 81/80 in 12 EDO and get this: (2.9966141538^4) / (2^4)*(5.0396842) = 80.6349472 / 80.6349472 = 1/1.<br />
Substitute in the values for 81/80 in 31 EDO and get this: (2.991035765^4) / (2^4)*(5.002262078) = 80.036193 / 80.036193 = 1/1.<br />
<br />
The lesson here is that even though the errors in the primes are different for each EDO + patent val in these cases, 81/80 is still tempered out. However, that's not true for all commas; for instance, 12 EDO tempers out 128/125, the diesis, while 31 EDO does not; and 31 EDO tempers out 393216/390625, the Wuerschmidt comma, while 12 EDO does not.</body></html>