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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]].
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|
| 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. |
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| |
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| * 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).)
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| * etc.
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|
| 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. |
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| =Examples= | | =Examples= |
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| ==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 .'' |
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| ==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 |
| | |} |
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| ==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 > Interval matrix
| | <ul><li>command line: "show/line intervals"</li><li>graphical interface: View > Interval matrix</li><li>keyboard shortcut: Shift+Alt+I</li></ul> |
| * keyboard shortcut: Shift+Alt+I
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| 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]]
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|
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|
| 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 |
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| 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]]
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| (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.) |
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| [[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"><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 &quot;a&quot; (1/1), &quot;b&quot; (second pitch), &quot;c&quot; (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 &quot;header row&quot; of degrees (1, 2, ...) and a &quot;header column&quot; of the original scale pitches.<br />
| |
| <br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:3 -->Examples</h1>
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><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 />
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|
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|
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| <table class="wiki_table">
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| <tr>
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| <td><br />
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| </td>
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| <td><strong>1</strong><br />
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| </td>
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| <td><strong>2</strong><br />
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| </td>
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| <td><strong>3</strong><br />
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| </td>
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| <td><strong>4</strong><br />
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| </td>
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| <td><strong>5</strong><br />
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| </td>
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| <td><strong>(6)</strong><br />
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| </td>
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| <td><strong>comments</strong><br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>1/1</strong><br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>9/8<br />
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| </td>
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| <td>5/4<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>5/3<br />
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| </td>
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| <td>2/1<br />
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| </td>
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| <td>original scale<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>9/8</strong><br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>10/9<br />
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| </td>
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| <td>4/3<br />
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| </td>
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| <td>40/27<br />
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| </td>
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| <td>16/9<br />
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| </td>
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| <td>2/1<br />
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| </td>
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| <td>10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc.<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>5/4</strong><br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>6/5<br />
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| </td>
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| <td>4/3<br />
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| </td>
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| <td>8/5<br />
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| </td>
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| <td>9/5<br />
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| </td>
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| <td>2/1<br />
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| </td>
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| <td>6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>3/2</strong><br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>10/9<br />
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| </td>
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| <td>4/3<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>5/3<br />
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| </td>
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| <td>2/1<br />
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| </td>
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| <td><br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>5/3</strong><br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>6/5<br />
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| </td>
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| <td>27/20<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>9/5<br />
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| </td>
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| <td>2/1<br />
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| </td>
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| <td><br />
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| </td>
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| </tr>
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| </table>
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|
| |
| <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 />
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| <!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc2"><a name="Examples-Cents"></a><!-- ws:end:WikiTextHeadingRule:7 -->Cents</h2>
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| Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents<br />
| |
|
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|
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| <table class="wiki_table">
| |
| <tr>
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| <td><br />
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| </td>
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| <td><strong>1</strong><br />
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| </td>
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| <td><strong>2</strong><br />
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| </td>
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| <td><strong>3</strong><br />
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| </td>
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| <td><strong>4</strong><br />
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| </td>
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| <td><strong>5</strong><br />
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| </td>
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| <td><strong>(6)</strong><br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>0.0</strong><br />
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| </td>
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| <td>0.0<br />
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| </td>
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| <td>226.4<br />
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| </td>
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| <td>486.8<br />
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| </td>
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| <td>713.2<br />
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| </td>
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| <td>939.5<br />
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| </td>
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| <td>1200.0<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>226.4</strong><br />
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| </td>
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| <td>0.0<br />
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| </td>
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| <td>260.5<br />
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| </td>
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| <td>486.8<br />
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| </td>
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| <td>713.2<br />
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| </td>
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| <td>973.6<br />
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| </td>
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| <td>1200.0<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>486.8</strong><br />
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| </td>
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| <td>0.0<br />
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| </td>
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| <td>226.4<br />
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| </td>
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| <td>452.7<br />
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| </td>
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| <td>713.2<br />
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| </td>
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| <td>939.5<br />
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| </td>
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| <td>1200.0<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>713.2</strong><br />
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| </td>
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| <td>0.0<br />
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| </td>
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| <td>226.4<br />
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| </td>
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| <td>486.8<br />
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| </td>
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| <td>713.2<br />
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| </td>
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| <td>973.6<br />
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| </td>
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| <td>1200.0<br />
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| </td>
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| </tr>
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| <tr>
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| <td><strong>939.5</strong><br />
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| </td>
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| <td>0.0<br />
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| </td>
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| <td>260.5<br />
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| </td>
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| <td>486.8<br />
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| </td>
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| <td>747.3<br />
| |
| </td>
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| <td>973.6<br />
| |
| </td>
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| <td>1200.0<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><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: &quot;show/line intervals&quot;</li><li>graphical interface: View &gt; 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:
| |
| &lt;pre class=&quot;text&quot;&gt; 0: 1/1 0.000000 unison, perfect prime&lt;br/&gt; 1: 9/8 203.910002 major whole tone&lt;br/&gt; 2: 5/4 386.313714 major third&lt;br/&gt; 3: 3/2 701.955001 perfect fifth&lt;br/&gt; 4: 5/3 884.358713 major sixth, BP sixth&lt;br/&gt; 5: 2/1 1200.000000 octave&lt;/pre&gt;
| |
| -->
| |
| <style type="text/css"><!--
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| /**
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| * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
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| * (http://qbnz.com/highlighter/ and http://geshi.org/)
| |
| */
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| .text {font-family:monospace;}
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| .text .imp {font-weight: bold; color: red;}
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| .text span.xtra { display:block; }
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|
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| -->
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| </style><pre class="text"> 0: 1/1 0.000000 unison, perfect prime
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| 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
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| 5: 2/1 1200.000000 octave</pre>
| |
|
| |
| <!-- ws:end:WikiTextCodeRule:0 --><br />
| |
| this interval matrix:<br />
| |
| <!-- ws:start:WikiTextCodeRule:1:
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| &lt;pre class=&quot;text&quot;&gt; 1 2 3 4 5 &lt;br/&gt; 1/1: 9/8 5/4 3/2 5/3 2/1&lt;br/&gt; 9/8: 10/9 4/3 40/27 16/9 2/1&lt;br/&gt; 5/4: 6/5 4/3 8/5 9/5 2/1&lt;br/&gt; 3/2: 10/9 4/3 3/2 5/3 2/1&lt;br/&gt; 5/3: 6/5 27/20 3/2 9/5 2/1&lt;br/&gt; 2/1&lt;/pre&gt;
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| -->
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| <style type="text/css"><!--
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| /**
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| * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
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| * (http://qbnz.com/highlighter/ and http://geshi.org/)
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| */
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| .text {font-family:monospace;}
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| .text .imp {font-weight: bold; color: red;}
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| -->
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| </style><pre class="text"> 1 2 3 4 5
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| 1/1: 9/8 5/4 3/2 5/3 2/1
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| 9/8: 10/9 4/3 40/27 16/9 2/1
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| 5/4: 6/5 4/3 8/5 9/5 2/1
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| 3/2: 10/9 4/3 3/2 5/3 2/1
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| 5/3: 6/5 27/20 3/2 9/5 2/1
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| 2/1</pre>
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|
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| <!-- ws:end:WikiTextCodeRule:1 --><br />
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| (Note that Scala omits the &quot;1/1&quot; column, and the column numbers are offset by 1 relative to the other examples above.)<br />
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| <br />
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| <!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/26024358?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;26024358&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><!-- ws:end:WikiTextMediaRule:2 --></body></html></pre></div>
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