Interval matrix: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
An ''interval matrix'' is a tabular representation of all possible [[interval|interval]]s in a [[scale|scale]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-05-24 19:27:25 UTC</tt>.<br>
: The original revision id was <tt>511025642</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>
<h4>Original Wikitext content:</h4>
<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">An //interval matrix// is a tabular representation of all possible [[interval]]s in a [[scale]].


To create an interval matrix, start with a table with (at least) as many rows and columns are there are pitches in your scale.
To create an interval matrix, start with a table with (at least) as many rows and columns are there are pitches in your scale.
Let's call the pitches of the scale "a" (1/1), "b" (second pitch), "c" (third), etc.
Let's call the pitches of the scale "a" (1/1), "b" (second pitch), "c" (third), etc.


* In the first row, list the pitches of the scale as-is: a, b, c, ...
<ul><li>In the first row, list the pitches of the scale as-is: a, b, c, ...</li><li>In the second row, list the intervals: (b-b), (c-b), (d-b), ..., (a-b), (b-b).  (Reduce (a-b) by an octave, or whatever the period of the scale is.)</li><li>In the third row, list the intervals: (c-c), (d-c), ..., (a-c), (b-c), (c-c).  (Reduce (a-c) and (b-c).)</li><li>etc.</li></ul>
* In the second row, list the intervals: (b-b), (c-b), (d-b), ..., (a-b), (b-b).  (Reduce (a-b) by an octave, or whatever the period of the scale is.)
* In the third row, list the intervals: (c-c), (d-c), ..., (a-c), (b-c), (c-c).  (Reduce (a-c) and (b-c).)
* etc.


Optionally (as in the examples below), you may have a "header row" of degrees (1, 2, ...) and a "header column" of the original scale pitches.
Optionally (as in the examples below), you may have a "header row" of degrees (1, 2, ...) and a "header column" of the original scale pitches.


=Examples=
=Examples=
Line 23: Line 13:
==JI==
==JI==
Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1
Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1
Here is the interval matrix of this scale:
Here is the interval matrix of this scale:
||   || **1** || **2** || **3** || **4** || **5** || **(6)** || **comments** ||
 
|| **1/1** || 1/1 || 9/8 || 5/4 || 3/2 || 5/3 || 2/1 || original scale ||
{| class="wikitable"
|| **9/8** || 1/1 || 10/9 || 4/3 || 40/27 || 16/9 || 2/1 || 10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc. ||
|-
|| **5/4** || 1/1 || 6/5 || 4/3 || 8/5 || 9/5 || 2/1 || 6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4 ||
| |  
|| **3/2** || 1/1 || 10/9 || 4/3 || 3/2 || 5/3 || 2/1 || ||
| | '''1'''
|| **5/3** || 1/1 || 6/5 || 27/20 || 3/2 || 9/5 || 2/1 || ||
| | '''2'''
//Note that the distance between (for example) 3/2 and 5/4 is written above as 3/2 - 5/4, as is common for JI intervals, but actually calculated as 3/2 ÷ 5/4 .//
| | '''3'''
| | '''4'''
| | '''5'''
| | '''(6)'''
| | '''comments'''
|-
| | '''1/1'''
| | 1/1
| | 9/8
| | 5/4
| | 3/2
| | 5/3
| | 2/1
| | original scale
|-
| | '''9/8'''
| | 1/1
| | 10/9
| | 4/3
| | 40/27
| | 16/9
| | 2/1
| | 10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc.
|-
| | '''5/4'''
| | 1/1
| | 6/5
| | 4/3
| | 8/5
| | 9/5
| | 2/1
| | 6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4
|-
| | '''3/2'''
| | 1/1
| | 10/9
| | 4/3
| | 3/2
| | 5/3
| | 2/1
| |  
|-
| | '''5/3'''
| | 1/1
| | 6/5
| | 27/20
| | 3/2
| | 9/5
| | 2/1
| |  
|}
''Note that the distance between (for example) 3/2 and 5/4 is written above as 3/2 - 5/4, as is common for JI intervals, but actually calculated as 3/2 ÷ 5/4 .''


==Cents==
==Cents==
Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents
Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents
||   || **1** || **2** || **3** || **4** || **5** || **(6)** ||
 
|| **0.0**  || 0.0 || 226.4 || 486.8 || 713.2 || 939.5 || 1200.0 ||
{| class="wikitable"
|| **226.4** || 0.0 || 260.5 || 486.8 || 713.2 || 973.6 || 1200.0 ||
|-
|| **486.8** || 0.0 || 226.4 || 452.7 || 713.2 || 939.5 || 1200.0 ||
| |  
|| **713.2** || 0.0 || 226.4 || 486.8 || 713.2 || 973.6 || 1200.0 ||
| | '''1'''
|| **939.5** || 0.0 || 260.5 || 486.8 || 747.3 || 973.6 || 1200.0 ||
| | '''2'''
| | '''3'''
| | '''4'''
| | '''5'''
| | '''(6)'''
|-
| | '''0.0'''
| | 0.0
| | 226.4
| | 486.8
| | 713.2
| | 939.5
| | 1200.0
|-
| | '''226.4'''
| | 0.0
| | 260.5
| | 486.8
| | 713.2
| | 973.6
| | 1200.0
|-
| | '''486.8'''
| | 0.0
| | 226.4
| | 452.7
| | 713.2
| | 939.5
| | 1200.0
|-
| | '''713.2'''
| | 0.0
| | 226.4
| | 486.8
| | 713.2
| | 973.6
| | 1200.0
|-
| | '''939.5'''
| | 0.0
| | 260.5
| | 486.8
| | 747.3
| | 973.6
| | 1200.0
|}


==Scala==
==Scala==
To show the interval matrix of the current scale using [[Scala]]:
To show the interval matrix of the current scale using [[Scala|Scala]]:
* command line: "show/line intervals"
 
* graphical interface: View &gt; Interval matrix
<ul><li>command line: "show/line intervals"</li><li>graphical interface: View &gt; Interval matrix</li><li>keyboard shortcut: Shift+Alt+I</li></ul>
* keyboard shortcut: Shift+Alt+I


For example, Scala will display for this scale:
For example, Scala will display for this scale:
[[code]]
 
  0:          1/1              0.000000 unison, perfect prime
<pre>  0:          1/1              0.000000 unison, perfect prime
   1:          9/8            203.910002 major whole tone
   1:          9/8            203.910002 major whole tone
   2:          5/4            386.313714 major third
   2:          5/4            386.313714 major third
   3:          3/2            701.955001 perfect fifth
   3:          3/2            701.955001 perfect fifth
   4:          5/3            884.358713 major sixth, BP sixth
   4:          5/3            884.358713 major sixth, BP sixth
   5:          2/1            1200.000000 octave
   5:          2/1            1200.000000 octave</pre>
[[code]]


this interval matrix:
this interval matrix:
[[code]]
 
      1    2    3    4    5   
<pre>      1    2    3    4    5   
  1/1: 9/8  5/4  3/2  5/3  2/1
  1/1: 9/8  5/4  3/2  5/3  2/1
  9/8: 10/9 4/3  40/27 16/9 2/1
  9/8: 10/9 4/3  40/27 16/9 2/1
Line 65: Line 150:
  3/2: 10/9 4/3  3/2  5/3  2/1
  3/2: 10/9 4/3  3/2  5/3  2/1
  5/3: 6/5  27/20 3/2  9/5  2/1
  5/3: 6/5  27/20 3/2  9/5  2/1
  2/1
  2/1</pre>
[[code]]


(Note that Scala omits the "1/1" column, and the column numbers are offset by 1 relative to the other examples above.)
(Note that Scala omits the "1/1" column, and the column numbers are offset by 1 relative to the other examples above.)
[[media type="custom" key="26024358"]]</pre></div>
<h4>Original HTML content:</h4>
<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;interval matrix&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An &lt;em&gt;interval matrix&lt;/em&gt; is a tabular representation of all possible &lt;a class="wiki_link" href="/interval"&gt;interval&lt;/a&gt;s in a &lt;a class="wiki_link" href="/scale"&gt;scale&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
To create an interval matrix, start with a table with (at least) as many rows and columns are there are pitches in your scale.&lt;br /&gt;
Let's call the pitches of the scale &amp;quot;a&amp;quot; (1/1), &amp;quot;b&amp;quot; (second pitch), &amp;quot;c&amp;quot; (third), etc.&lt;br /&gt;
&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;In the first row, list the pitches of the scale as-is: a, b, c, ...&lt;/li&gt;&lt;li&gt;In the second row, list the intervals: (b-b), (c-b), (d-b), ..., (a-b), (b-b).  (Reduce (a-b) by an octave, or whatever the period of the scale is.)&lt;/li&gt;&lt;li&gt;In the third row, list the intervals: (c-c), (d-c), ..., (a-c), (b-c), (c-c).  (Reduce (a-c) and (b-c).)&lt;/li&gt;&lt;li&gt;etc.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
Optionally (as in the examples below), you may have a &amp;quot;header row&amp;quot; of degrees (1, 2, ...) and a &amp;quot;header column&amp;quot; of the original scale pitches.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Examples&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Examples-JI"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;JI&lt;/h2&gt;
Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1&lt;br /&gt;
Here is the interval matrix of this scale:&lt;br /&gt;
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;2&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;3&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;4&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;5&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;(6)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;comments&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;1/1&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;original scale&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;9/8&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc.&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;5/4&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;3/2&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;5/3&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
&lt;em&gt;Note that the distance between (for example) 3/2 and 5/4 is written above as 3/2 - 5/4, as is common for JI intervals, but actually calculated as 3/2 ÷ 5/4 .&lt;/em&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Examples-Cents"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Cents&lt;/h2&gt;
Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents&lt;br /&gt;
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;1&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;2&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;3&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;4&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;5&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;(6)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;0.0&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;226.4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;486.8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;713.2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;939.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200.0&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;226.4&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;260.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;486.8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;713.2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;973.6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200.0&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;486.8&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;226.4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;452.7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;713.2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;939.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200.0&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;713.2&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;226.4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;486.8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;713.2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;973.6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200.0&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;strong&gt;939.5&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;260.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;486.8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;747.3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;973.6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200.0&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Examples-Scala"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Scala&lt;/h2&gt;
To show the interval matrix of the current scale using &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt;:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;command line: &amp;quot;show/line intervals&amp;quot;&lt;/li&gt;&lt;li&gt;graphical interface: View &amp;gt; Interval matrix&lt;/li&gt;&lt;li&gt;keyboard shortcut: Shift+Alt+I&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
For example, Scala will display for this scale:&lt;br /&gt;
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&amp;lt;pre class=&amp;quot;text&amp;quot;&amp;gt;  0:          1/1              0.000000 unison, perfect prime&amp;lt;br/&amp;gt;  1:          9/8            203.910002 major whole tone&amp;lt;br/&amp;gt;  2:          5/4            386.313714 major third&amp;lt;br/&amp;gt;  3:          3/2            701.955001 perfect fifth&amp;lt;br/&amp;gt;  4:          5/3            884.358713 major sixth, BP sixth&amp;lt;br/&amp;gt;  5:          2/1            1200.000000 octave&amp;lt;/pre&amp;gt;
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* GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
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&lt;/style&gt;&lt;pre class="text"&gt;  0:          1/1              0.000000 unison, perfect prime
  1:          9/8            203.910002 major whole tone
  2:          5/4            386.313714 major third
  3:          3/2            701.955001 perfect fifth
  4:          5/3            884.358713 major sixth, BP sixth
  5:          2/1            1200.000000 octave&lt;/pre&gt;
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this interval matrix:&lt;br /&gt;
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&amp;lt;pre class=&amp;quot;text&amp;quot;&amp;gt;      1    2    3    4    5  &amp;lt;br/&amp;gt; 1/1: 9/8  5/4  3/2  5/3  2/1&amp;lt;br/&amp;gt; 9/8: 10/9 4/3  40/27 16/9 2/1&amp;lt;br/&amp;gt; 5/4: 6/5  4/3  8/5  9/5  2/1&amp;lt;br/&amp;gt; 3/2: 10/9 4/3  3/2  5/3  2/1&amp;lt;br/&amp;gt; 5/3: 6/5  27/20 3/2  9/5  2/1&amp;lt;br/&amp;gt; 2/1&amp;lt;/pre&amp;gt;
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* GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
* (http://qbnz.com/highlighter/ and http://geshi.org/)
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&lt;/style&gt;&lt;pre class="text"&gt;      1    2    3    4    5 
1/1: 9/8  5/4  3/2  5/3  2/1
9/8: 10/9 4/3  40/27 16/9 2/1
5/4: 6/5  4/3  8/5  9/5  2/1
3/2: 10/9 4/3  3/2  5/3  2/1
5/3: 6/5  27/20 3/2  9/5  2/1
2/1&lt;/pre&gt;
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(Note that Scala omits the &amp;quot;1/1&amp;quot; column, and the column numbers are offset by 1 relative to the other examples above.)&lt;br /&gt;
&lt;br /&gt;
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