Interval matrix
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author spt3125 and made on 2014-05-24 19:27:25 UTC.
- The original revision id was 511025642.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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.
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, ...
* 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.
=Examples=
==JI==
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:
|| || **1** || **2** || **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==
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 ||
|| **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==
To show the interval matrix of the current scale using [[Scala]]:
* command line: "show/line intervals"
* graphical interface: View > Interval matrix
* keyboard shortcut: Shift+Alt+I
For example, Scala will display for this scale:
[[code]]
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
[[code]]
this interval matrix:
[[code]]
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
[[code]]
(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"]]Original HTML content:
<html><head><title>interval matrix</title></head><body>An <em>interval matrix</em> is a tabular representation of all possible <a class="wiki_link" href="/interval">interval</a>s in a <a class="wiki_link" href="/scale">scale</a>.<br />
<br />
To create an interval matrix, start with a table with (at least) as many rows and columns are there are pitches in your scale.<br />
Let's call the pitches of the scale "a" (1/1), "b" (second pitch), "c" (third), etc.<br />
<br />
<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><br />
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.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:3 -->Examples</h1>
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h2> --><h2 id="toc1"><a name="Examples-JI"></a><!-- ws:end:WikiTextHeadingRule:5 -->JI</h2>
Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1<br />
Here is the interval matrix of this scale:<br />
<table class="wiki_table">
<tr>
<td><br />
</td>
<td><strong>1</strong><br />
</td>
<td><strong>2</strong><br />
</td>
<td><strong>3</strong><br />
</td>
<td><strong>4</strong><br />
</td>
<td><strong>5</strong><br />
</td>
<td><strong>(6)</strong><br />
</td>
<td><strong>comments</strong><br />
</td>
</tr>
<tr>
<td><strong>1/1</strong><br />
</td>
<td>1/1<br />
</td>
<td>9/8<br />
</td>
<td>5/4<br />
</td>
<td>3/2<br />
</td>
<td>5/3<br />
</td>
<td>2/1<br />
</td>
<td>original scale<br />
</td>
</tr>
<tr>
<td><strong>9/8</strong><br />
</td>
<td>1/1<br />
</td>
<td>10/9<br />
</td>
<td>4/3<br />
</td>
<td>40/27<br />
</td>
<td>16/9<br />
</td>
<td>2/1<br />
</td>
<td>10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc.<br />
</td>
</tr>
<tr>
<td><strong>5/4</strong><br />
</td>
<td>1/1<br />
</td>
<td>6/5<br />
</td>
<td>4/3<br />
</td>
<td>8/5<br />
</td>
<td>9/5<br />
</td>
<td>2/1<br />
</td>
<td>6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4<br />
</td>
</tr>
<tr>
<td><strong>3/2</strong><br />
</td>
<td>1/1<br />
</td>
<td>10/9<br />
</td>
<td>4/3<br />
</td>
<td>3/2<br />
</td>
<td>5/3<br />
</td>
<td>2/1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>5/3</strong><br />
</td>
<td>1/1<br />
</td>
<td>6/5<br />
</td>
<td>27/20<br />
</td>
<td>3/2<br />
</td>
<td>9/5<br />
</td>
<td>2/1<br />
</td>
<td><br />
</td>
</tr>
</table>
<em>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 .</em><br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:<h2> --><h2 id="toc2"><a name="Examples-Cents"></a><!-- ws:end:WikiTextHeadingRule:7 -->Cents</h2>
Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents<br />
<table class="wiki_table">
<tr>
<td><br />
</td>
<td><strong>1</strong><br />
</td>
<td><strong>2</strong><br />
</td>
<td><strong>3</strong><br />
</td>
<td><strong>4</strong><br />
</td>
<td><strong>5</strong><br />
</td>
<td><strong>(6)</strong><br />
</td>
</tr>
<tr>
<td><strong>0.0</strong><br />
</td>
<td>0.0<br />
</td>
<td>226.4<br />
</td>
<td>486.8<br />
</td>
<td>713.2<br />
</td>
<td>939.5<br />
</td>
<td>1200.0<br />
</td>
</tr>
<tr>
<td><strong>226.4</strong><br />
</td>
<td>0.0<br />
</td>
<td>260.5<br />
</td>
<td>486.8<br />
</td>
<td>713.2<br />
</td>
<td>973.6<br />
</td>
<td>1200.0<br />
</td>
</tr>
<tr>
<td><strong>486.8</strong><br />
</td>
<td>0.0<br />
</td>
<td>226.4<br />
</td>
<td>452.7<br />
</td>
<td>713.2<br />
</td>
<td>939.5<br />
</td>
<td>1200.0<br />
</td>
</tr>
<tr>
<td><strong>713.2</strong><br />
</td>
<td>0.0<br />
</td>
<td>226.4<br />
</td>
<td>486.8<br />
</td>
<td>713.2<br />
</td>
<td>973.6<br />
</td>
<td>1200.0<br />
</td>
</tr>
<tr>
<td><strong>939.5</strong><br />
</td>
<td>0.0<br />
</td>
<td>260.5<br />
</td>
<td>486.8<br />
</td>
<td>747.3<br />
</td>
<td>973.6<br />
</td>
<td>1200.0<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:9:<h2> --><h2 id="toc3"><a name="Examples-Scala"></a><!-- ws:end:WikiTextHeadingRule:9 -->Scala</h2>
To show the interval matrix of the current scale using <a class="wiki_link" href="/Scala">Scala</a>:<br />
<ul><li>command line: "show/line intervals"</li><li>graphical interface: View > Interval matrix</li><li>keyboard shortcut: Shift+Alt+I</li></ul><br />
For example, Scala will display for this scale:<br />
<!-- ws:start:WikiTextCodeRule:0:
<pre class="text"> 0: 1/1 0.000000 unison, perfect prime<br/> 1: 9/8 203.910002 major whole tone<br/> 2: 5/4 386.313714 major third<br/> 3: 3/2 701.955001 perfect fifth<br/> 4: 5/3 884.358713 major sixth, BP sixth<br/> 5: 2/1 1200.000000 octave</pre>
-->
<style type="text/css"><!--
/**
* GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
* (http://qbnz.com/highlighter/ and http://geshi.org/)
*/
.text {font-family:monospace;}
.text .imp {font-weight: bold; color: red;}
.text span.xtra { display:block; }
-->
</style><pre class="text"> 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</pre>
<!-- ws:end:WikiTextCodeRule:0 --><br />
this interval matrix:<br />
<!-- ws:start:WikiTextCodeRule:1:
<pre class="text"> 1 2 3 4 5 <br/> 1/1: 9/8 5/4 3/2 5/3 2/1<br/> 9/8: 10/9 4/3 40/27 16/9 2/1<br/> 5/4: 6/5 4/3 8/5 9/5 2/1<br/> 3/2: 10/9 4/3 3/2 5/3 2/1<br/> 5/3: 6/5 27/20 3/2 9/5 2/1<br/> 2/1</pre>
-->
<style type="text/css"><!--
/**
* GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
* (http://qbnz.com/highlighter/ and http://geshi.org/)
*/
.text {font-family:monospace;}
.text .imp {font-weight: bold; color: red;}
.text span.xtra { display:block; }
-->
</style><pre class="text"> 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</pre>
<!-- ws:end:WikiTextCodeRule:1 --><br />
(Note that Scala omits the "1/1" column, and the column numbers are offset by 1 relative to the other examples above.)<br />
<br />
<!-- ws:start:WikiTextMediaRule:2:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/26024358?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;26024358&quot;" title="Custom Media"/> --><!-- ws:end:WikiTextMediaRule:2 --></body></html>