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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <span style="display: block; text-align: right;">[[Monzo(Esp)|Español]] - [[モンゾ|日本語]]</span> |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2017-01-02 12:28:05 UTC</tt>.<br>
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| : The original revision id was <tt>602965630</tt>.<br>
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| : The revision comment was: <tt>tables for examples, added one space as padding as is used in most monzo expressions, for instance on the comma page</tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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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"><span style="display: block; text-align: right;">[[Monzo(Esp)|Español]] - [[モンゾ|日本語]]
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| </span>
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| This page gives a pragmatic introduction to **monzos**. For the formal mathematical definition of visit the page [[Monzos and Interval Space]].
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| =Definition= | | This page gives a pragmatic introduction to '''monzos'''. For the formal mathematical definition of visit the page [[Monzos_and_Interval_Space|Monzos and Interval Space]]. |
| A **monzo** is a way of notating a JI interval that allows us to express directly how any "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation | a b c d e f ... >, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some [[harmonic limit|prime limit]]. | | |
| | =Definition= |
| | A '''monzo''' is a way of notating a JI interval that allows us to express directly how any "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation | a b c d e f ... >, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some [[Harmonic_Limit|prime limit]]. |
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| Monzos can be thought of as counterparts to [[Vals|vals]]. | | Monzos can be thought of as counterparts to [[Vals|vals]]. |
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| For a more mathematical discussion, see also [[Monzos and Interval Space]]. | | For a more mathematical discussion, see also [[Monzos_and_Interval_Space|Monzos and Interval Space]]. |
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| =Examples= | | =Examples= |
| For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the | ... > brackets, hence yielding | -3 1 1 >. | | For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the | ... > brackets, hence yielding | -3 1 1 >. |
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| Here are some common 5-limit monzos, for your reference: | | Here are some common 5-limit monzos, for your reference: |
| ||~ Ratio ||~ Monzo || | | |
| ||= 3/2 || | -1 1 0 > || | | {| class="wikitable" |
| ||= 5/4 || | -2 0 1 > || | | |- |
| ||= 9/8 || | -3 2 0 > || | | ! | Ratio |
| ||= 81/80 || | -4 4 -1 > || | | ! | Monzo |
| | |- |
| | | style="text-align:center;" | 3/2 |
| | | | | -1 1 0 > |
| | |- |
| | | style="text-align:center;" | 5/4 |
| | | | | -2 0 1 > |
| | |- |
| | | style="text-align:center;" | 9/8 |
| | | | | -3 2 0 > |
| | |- |
| | | style="text-align:center;" | 81/80 |
| | | | | -4 4 -1 > |
| | |} |
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| Here are a few 7-limit monzos: | | Here are a few 7-limit monzos: |
| ||~ Ratio ||~ Monzo ||
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| ||= 7/4 || | -2 0 0 1 > ||
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| ||= 7/6 || | -1 -1 0 1 > ||
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| ||= 7/5 || | 0 0 -1 1 > ||
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| =Relationship with vals= | | {| class="wikitable" |
| //See also: [[Vals]], [[Keenan's explanation of vals]], [[Vals and Tuning Space]] (more mathematical)//
| | |- |
| | ! | Ratio |
| | ! | Monzo |
| | |- |
| | | style="text-align:center;" | 7/4 |
| | | | | -2 0 0 1 > |
| | |- |
| | | style="text-align:center;" | 7/6 |
| | | | | -1 -1 0 1 > |
| | |- |
| | | style="text-align:center;" | 7/5 |
| | | | | 0 0 -1 1 > |
| | |} |
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| | =Relationship with vals= |
| | ''See also: [[Vals|Vals]], [[Keenan's_explanation_of_vals|Keenan's explanation of vals]], [[Vals_and_Tuning_Space|Vals and Tuning Space]] (more mathematical)'' |
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| Monzos are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as < 12 19 28 | -4 4 -1 >. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example: | | Monzos are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as < 12 19 28 | -4 4 -1 >. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example: |
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| < 12 19 28 | -4 4 -1 > | | < 12 19 28 | -4 4 -1 > |
| (12*-4) + (19*4) + (28*1)<span class="st"> = </span>0
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| In this case, the val < 12 19 28 | is the [[patent val]] for 12-equal, and | -4 4 -1 > is 81/80, or the syntonic comma. The fact that < 12 19 28 | -4 4 -1 > tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation.
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| **In general: < a b c | d e f > = ad + be + cf**</pre></div>
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| <h4>Original HTML content:</h4>
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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>monzos</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/Monzo%28Esp%29">Español</a> - <a class="wiki_link" href="/%E3%83%A2%E3%83%B3%E3%82%BE">日本語</a><br />
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| </span><br />
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| This page gives a pragmatic introduction to <strong>monzos</strong>. For the formal mathematical definition of visit the page <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1>
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| A <strong>monzo</strong> is a way of notating a JI interval that allows us to express directly how any &quot;composite&quot; interval is represented in terms of those simpler prime intervals. They are typically written using the notation | a b c d e f ... &gt;, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br />
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| <br />
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| Monzos can be thought of as counterparts to <a class="wiki_link" href="/Vals">vals</a>.<br />
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| <br />
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| For a more mathematical discussion, see also <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1>
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| For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the | ... &gt; brackets, hence yielding | -3 1 1 &gt;.<br />
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| Here are some common 5-limit monzos, for your reference:<br />
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| <table class="wiki_table">
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| <tr>
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| <th>Ratio<br />
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| </th>
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| <th>Monzo<br />
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| </th>
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| </tr>
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| <tr>
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| <td style="text-align: center;">3/2<br />
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| </td>
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| <td>| -1 1 0 &gt;<br />
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| </td>
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| </tr>
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| <tr>
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| <td style="text-align: center;">5/4<br />
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| </td>
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| <td>| -2 0 1 &gt;<br />
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| </td>
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| </tr>
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| <tr>
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| <td style="text-align: center;">9/8<br />
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| </td>
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| <td>| -3 2 0 &gt;<br />
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| </td>
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| </tr>
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| <tr>
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| <td style="text-align: center;">81/80<br />
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| </td>
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| <td>| -4 4 -1 &gt;<br />
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| </td>
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| </tr>
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| </table>
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| <br />
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| Here are a few 7-limit monzos:<br />
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| | (12*-4) + (19*4) + (28*1)<span style=""> = </span>0 |
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| <table class="wiki_table"> | | In this case, the val < 12 19 28 | is the [[Patent_val|patent val]] for 12-equal, and | -4 4 -1 > is 81/80, or the syntonic comma. The fact that < 12 19 28 | -4 4 -1 > tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation. |
| <tr>
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| <th>Ratio<br />
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| </th>
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| <th>Monzo<br />
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| </th>
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| </tr>
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| <tr>
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| <td style="text-align: center;">7/4<br />
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| </td>
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| <td>| -2 0 0 1 &gt;<br />
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| </td>
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| </tr>
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| <tr>
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| <td style="text-align: center;">7/6<br />
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| </td>
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| <td>| -1 -1 0 1 &gt;<br />
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| </td>
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| </tr>
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| <tr>
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| <td style="text-align: center;">7/5<br />
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| </td>
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| <td>| 0 0 -1 1 &gt;<br />
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| </td>
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| </tr>
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| </table>
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| <br />
| | '''In general: < a b c | d e f > = ad + be + cf''' [[Category:definition]] |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Relationship with vals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Relationship with vals</h1>
| | [[Category:intervals]] |
| <em>See also: <a class="wiki_link" href="/Vals">Vals</a>, <a class="wiki_link" href="/Keenan%27s%20explanation%20of%20vals">Keenan's explanation of vals</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a> (more mathematical)</em><br />
| | [[Category:prime_limit]] |
| <br />
| | [[Category:theory]] |
| Monzos are important because they enable us to see how any JI interval &quot;maps&quot; onto a val. This mapping is expressed by writing the val and the monzo together, such as &lt; 12 19 28 | -4 4 -1 &gt;. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:<br />
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| <br />
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| &lt; 12 19 28 | -4 4 -1 &gt;<br />
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| (12*-4) + (19*4) + (28*1)<span class="st"> = </span>0<br />
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| <br />
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| In this case, the val &lt; 12 19 28 | is the <a class="wiki_link" href="/patent%20val">patent val</a> for 12-equal, and | -4 4 -1 &gt; is 81/80, or the syntonic comma. The fact that &lt; 12 19 28 | -4 4 -1 &gt; tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation.<br />
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| <br />
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| <strong>In general: &lt; a b c | d e f &gt; = ad + be + cf</strong></body></html></pre></div>
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