Xenharmonic Wiki

Patent val: Difference between revisions

← Older editNewer edit →
VisualWikitext
Wikispaces>jdfreivald
**Imported revision 247283603 - Original comment: **
Wikispaces>jdfreivald
**Imported revision 247284699 - Original comment: **
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-08-20 22:09:24 UTC</tt>.<br>
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2011-08-20 22:15:32 UTC</tt>.<br>
: The original revision id was <tt>247283603</tt>.<br>
: The original revision id was <tt>247284699</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 49: Line 49:
A val defines a rank 1 temperament by defining 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.
A val defines a rank 1 temperament by defining 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, &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.
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.


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.
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. We know this isn'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.


We can calculate the error we're introducing into 3/1 as follows for 12 EDO:
We can calculate the error we're introducing into 3/1 as follows for 12 EDO:
12 EDO steps are 100.0 cents.
 
12 EDO steps are 100.0 cents each.
19 steps of 12 EDO = 19 steps * 100.0 cents/step = 1900.0 cents.
19 steps of 12 EDO = 19 steps * 100.0 cents/step = 1900.0 cents.
1900.0 cents =&gt; 2^(1900/1200), or 2.9966141538. This is the value that 12 EDO uses in place of prime 3.
1900.0 cents =&gt; 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.)
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.)
 
We can do the same calculations for 31 EDO.


Likewise for 31 EDO.
The patent val for 31 EDO is &lt;31 49 72 87 107 (etc) |. The 49 implies that it takes 49 steps to get to 3/1.
31 EDO steps are 38.70967742 cents.
31 EDO steps are 38.70967742 cents each.
49 steps of 31 EDO = 49 steps * 38.70967742 cents/step = 1896.774194 cents.
49 steps of 31 EDO = 49 steps * 38.70967742 cents/step = 1896.774194 cents.
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.
1896.774194 cents =&gt; 2^(1896.774194/1200), or 2.991035765. This is what 31 EDO uses in place of prime 3.


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.
Again, as in 12 EDO, it's less than 3, so 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).
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).
Line 122: Line 124:
  A val defines a rank 1 temperament by defining 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.&lt;br /&gt;
  A val defines a rank 1 temperament by defining 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, &amp;lt;12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave. The patent val for 31 EDO, &amp;lt;31 49 72 87 107 (etc) |, implies that it takes 31 steps to get to the octave.&lt;br /&gt;
As stated above, the 2/1 in the patent val is perfect. The patent val for 12 EDO, &amp;lt;12 19 28 34 42 (etc) |, implies that it takes 12 steps to get the octave.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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 &lt;em&gt;pretending&lt;/em&gt; 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.&lt;br /&gt;
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. We know this isn'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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
We can calculate the error we're introducing into 3/1 as follows for 12 EDO:&lt;br /&gt;
We can calculate the error we're introducing into 3/1 as follows for 12 EDO:&lt;br /&gt;
12 EDO steps are 100.0 cents.&lt;br /&gt;
&lt;br /&gt;
12 EDO steps are 100.0 cents each.&lt;br /&gt;
19 steps of 12 EDO = 19 steps * 100.0 cents/step = 1900.0 cents.&lt;br /&gt;
19 steps of 12 EDO = 19 steps * 100.0 cents/step = 1900.0 cents.&lt;br /&gt;
1900.0 cents =&amp;gt; 2^(1900/1200), or 2.9966141538. This is the value that 12 EDO uses in place of prime 3.&lt;br /&gt;
1900.0 cents =&amp;gt; 2^(1900/1200), or 2.9966141538. This is the value that 12 EDO uses in place of prime 3.&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(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.)&lt;br /&gt;
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.)&lt;br /&gt;
&lt;br /&gt;
We can do the same calculations for 31 EDO.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Likewise for 31 EDO.&lt;br /&gt;
The patent val for 31 EDO is &amp;lt;31 49 72 87 107 (etc) |. The 49 implies that it takes 49 steps to get to 3/1.&lt;br /&gt;
31 EDO steps are 38.70967742 cents.&lt;br /&gt;
31 EDO steps are 38.70967742 cents each.&lt;br /&gt;
49 steps of 31 EDO = 49 steps * 38.70967742 cents/step = 1896.774194 cents.&lt;br /&gt;
49 steps of 31 EDO = 49 steps * 38.70967742 cents/step = 1896.774194 cents.&lt;br /&gt;
1896.774194 cents =&amp;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.&lt;br /&gt;
1896.774194 cents =&amp;gt; 2^(1896.774194/1200), or 2.991035765. This is what 31 EDO uses in place of prime 3.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
Again, as in 12 EDO, it's less than 3, so 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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).&lt;br /&gt;
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).&lt;br /&gt;
Retrieved from "https://en.xen.wiki/w/Patent_val"