Fudging: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-~~04 16~~:40:53 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-08 12:37:49 UTC</tt>.<br>

: The original revision id was <tt>~~515601368~~</tt>.<br>

: The original revision id was <tt>515811552</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>

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">//Fudging//, or virtual tempering, is the use of one [[just intonation]] interval to approximate another. Below are listed fudging intervals ("fudgers") which are all ratios between two 15-limit [[tonality diamond]] intervals (listed in the fifth column) which come within a comma of less than eight cents (listed in the fourth column) of a 15-limit tonality diamond interval (listed in the third column). If the comma is less than 1, the fudger is flat of the interval it approximates; if greater than 1, sharp of it.

A limit-raising fudger is a p-limit interval which approximates to a q-limit consonance, with p<q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of [[13_10|13/10]], and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony "for free", so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.

A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p<q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of [[13_10|13/10]], and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony "for free", so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.

By tempering out the fudging commas, the error in the fudging may be distributed evenly. Since 225/224, 385/384 and 540/539 occur so commonly as fudging commas, marvel tempering in particular is often an excellent means to introduce smooth 7- and 11-limit harmonies into 5- or 7- limit scales. Adding 441/440 to that list results in miracle tempering, another excellent smoothing option. However, indiscriminate tempering can lead to problems. For example, 112/75, 121/81, 338/225, 182/121 and 176/117 are all fudged fifths, with commas 676/675, 364/363, 352/351, 243/242 and 225/224. Tempering them all out leads to 34d tempering, a rather crude system by comparison.

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Fudging</title></head><body><em>Fudging</em>, or virtual tempering, is the use of one <a class="wiki_link" href="/just%20intonation">just intonation</a> interval to approximate another. Below are listed fudging intervals (&quot;fudgers&quot;) which are all ratios between two 15-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a> intervals (listed in the fifth column) which come within a comma of less than eight cents (listed in the fourth column) of a 15-limit tonality diamond interval (listed in the third column). If the comma is less than 1, the fudger is flat of the interval it approximates; if greater than 1, sharp of it.<br />

<br />

A limit-raising fudger is a p-limit interval which approximates to a q-limit consonance, with p&lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of <a class="wiki_link" href="/13_10">13/10</a>, and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony &quot;for free&quot;, so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.<br />

A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p&lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of <a class="wiki_link" href="/13_10">13/10</a>, and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony &quot;for free&quot;, so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.<br />

<br />

By tempering out the fudging commas, the error in the fudging may be distributed evenly. Since 225/224, 385/384 and 540/539 occur so commonly as fudging commas, marvel tempering in particular is often an excellent means to introduce smooth 7- and 11-limit harmonies into 5- or 7- limit scales. Adding 441/440 to that list results in miracle tempering, another excellent smoothing option. However, indiscriminate tempering can lead to problems. For example, 112/75, 121/81, 338/225, 182/121 and 176/117 are all fudged fifths, with commas 676/675, 364/363, 352/351, 243/242 and 225/224. Tempering them all out leads to 34d tempering, a rather crude system by comparison.<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-04 16:40:53 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-08 12:37:49 UTC</tt>.<br>
-: The original revision id was <tt>515601368</tt>.<br>
+: The original revision id was <tt>515811552</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>
@@ Line 8: / Line 8: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">//Fudging//, or virtual tempering, is the use of one [[just intonation]] interval to approximate another. Below are listed fudging intervals ("fudgers") which are all ratios between two 15-limit [[tonality diamond]] intervals (listed in the fifth column) which come within a comma of less than eight cents (listed in the fourth column) of a 15-limit tonality diamond interval (listed in the third column). If the comma is less than 1, the fudger is flat of the interval it approximates; if greater than 1, sharp of it.
-A limit-raising fudger is a p-limit interval which approximates to a q-limit consonance, with p&lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of [[13_10|13/10]], and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony "for free", so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.
+A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p&lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of [[13_10|13/10]], and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony "for free", so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.
 By tempering out the fudging commas, the error in the fudging may be distributed evenly. Since 225/224, 385/384 and 540/539 occur so commonly as fudging commas, marvel tempering in particular is often an excellent means to introduce smooth 7- and 11-limit harmonies into 5- or 7- limit scales. Adding 441/440 to that list results in miracle tempering, another excellent smoothing option. However, indiscriminate tempering can lead to problems. For example, 112/75, 121/81, 338/225, 182/121 and 176/117 are all fudged fifths, with commas 676/675, 364/363, 352/351, 243/242 and 225/224. Tempering them all out leads to 34d tempering, a rather crude system by comparison.
@@ Line 232: / Line 232: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Fudging&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;Fudging&lt;/em&gt;, or virtual tempering, is the use of one &lt;a class="wiki_link" href="/just%20intonation"&gt;just intonation&lt;/a&gt; interval to approximate another. Below are listed fudging intervals (&amp;quot;fudgers&amp;quot;) which are all ratios between two 15-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt; intervals (listed in the fifth column) which come within a comma of less than eight cents (listed in the fourth column) of a 15-limit tonality diamond interval (listed in the third column). If the comma is less than 1, the fudger is flat of the interval it approximates; if greater than 1, sharp of it.&lt;br /&gt;
 &lt;br /&gt;
-A limit-raising fudger is a p-limit interval which approximates to a q-limit consonance, with p&amp;lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of &lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt;, and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony &amp;quot;for free&amp;quot;, so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.&lt;br /&gt;
+A limit-raising fudger is a p prime limit interval which approximates to a q prime limit consonance, with p&amp;lt;q. An example is 100/77, which is 1001/1000 (1.7 cents) flat of &lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt;, and which arises in 11-limit scales as the interval between 11/10 and 10/7, giving 13-limit harmony &amp;quot;for free&amp;quot;, so to speak. A limit-lowering fudger is an interval such as the marvelous fourth, 75/56, which is 225/224 (7.7 cents) sharp of 4/3, and which arises very often in 7-limit JI scales as the interval between 16/15 and 10/7, 7/5 and 15/8, 8/5 and 15/7, and 28/15 and 5/2, giving a 3-limit interval approximated in the 7-limit.&lt;br /&gt;
 &lt;br /&gt;
 By tempering out the fudging commas, the error in the fudging may be distributed evenly. Since 225/224, 385/384 and 540/539 occur so commonly as fudging commas, marvel tempering in particular is often an excellent means to introduce smooth 7- and 11-limit harmonies into 5- or 7- limit scales. Adding 441/440 to that list results in miracle tempering, another excellent smoothing option. However, indiscriminate tempering can lead to problems. For example, 112/75, 121/81, 338/225, 182/121 and 176/117 are all fudged fifths, with commas 676/675, 364/363, 352/351, 243/242 and 225/224. Tempering them all out leads to 34d tempering, a rather crude system by comparison.&lt;br /&gt;