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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| en = Radical interval
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<h4>Original Wikitext content:</h4>
A '''radical interval''' is an interval whose [[ratio]] can be expressed in terms of roots of integers (e.g. sqrt(2)), as opposed to [[just interval]]s which are expressed only in terms of ratios of pure integers. Radical intervals appear as the steps in [[equal tuning]]s such as [[edo]]s, and also occur in [[eigenmonzo]] tunings of [[regular temperament]]s. In terms of primes, a radical interval can be written as a product of primes raised to rational powers (such as {{nowrap| 2<sup>1/2</sup> × 3<sup>-1/13</sup> }}). Because of this, radical intervals can be expressed as monzos, like just intervals.
<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">A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26&gt; represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/16). By taking a dot product with &lt;cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.


Vectors in Tenney interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it.  
=== Fractional monzos ===
For the sake of clarity, monzos representing radical intervals are called '''fractional monzos'''. Mathematically, fractional monzos behave the same as ordinary [[monzo]]s, except that elements have been extended to allow them to be rational numbers. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> … ''e''<sub>p</sub> }} is a fractional monzo, then it represents 2<sup>''e''<sub>2</sub></sup> 3<sup>''e''<sub>3</sub></sup> … ''p''<sup>''e''<sub>''p''</sub></sup> just as with an ordinary monzo. Hence, for instance, {{monzo| 1/13 -1/13 7/26 }} represents the interval 2<sup>1/13</sup> 3<sup>-1/13</sup> 5<sup>7/26</sup>. By taking the [[least common multiple]] of the denominators, intervals represented by a fractional monzo can always be written as an ''n''-th root of a positive rational number; for instance from our example, (312500/9)<sup>1/26</sup>, which may also be written as 1\26ed312500/9.  


===Algebraic considerations===
By multiplying each monzo entry by the [[cent]] value of the corresponding prime and adding the results together (which can be represented, if the monzo is treated as a vector, by a dot product with the [[just tuning map]] in cents 1200⋅{{val| log<sub>2</sub>(2) log<sub>2</sub>(3) … log<sub>2</sub>(''p'') }}) the value in cents of a fractional monzo may be obtained, just as with an ordinary monzo. For instance, in the above example {{nowrap| (1/13)⋅1200 - (1/13)⋅1200⋅log<sub>2</sub>(3) + (7/26)⋅1200⋅log<sub>2</sub>(5) {{=}} 696.1648 cents }}.
For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (or equivalently, Z-module) of rank r equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a [[http://en.wikipedia.org/wiki/Divisible_group|divisible group]], meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a [[http://en.wikipedia.org/wiki/Vector_space|vector space]] (of dimension r) over the rational numbers. They are also torsion-free (equivalently, [[http://en.wikipedia.org/wiki/Flat_module|flat]]) abelian groups, and are the [[http://en.wikipedia.org/wiki/Injective_hull|injective hulls]] of the corresponding monzos.</pre></div>
 
<h4>Original HTML content:</h4>
Vectors in [[monzos and interval space|interval space]], where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one ''n''-th root of a positive rational number which corresponds to it.
<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;Fractional monzos&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;fractional monzo&lt;/em&gt; is like an ordinary &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzo&lt;/a&gt; except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&amp;gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26&amp;gt; represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/16). By taking a dot product with &amp;lt;cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.&lt;br /&gt;
 
&lt;br /&gt;
== Tunings in terms of radical intervals ==
Vectors in Tenney interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. &lt;br /&gt;
Any number that can be expressed as a root corresponds to a radical interval, so radical intervals can be used to express the degrees of equal tunings. For example, [[12edo]]'s fifth can be expressed as {{monzo| 7/12 }} or 2<sup>7/12</sup>, and the [[Bohlen–Pierce]] supermajor third may be expressed as {{monzo| 0 3/13 }} or 3<sup>3/13</sup>.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Algebraic considerations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Algebraic considerations&lt;/h3&gt;
What this additionally unlocks is the ability to stack intervals from multiple edo systems. For example, one could define intervals in the 12edo.13edt subgroup by specifying their monzo, for example a subminor third of about 261 cents can be generated by {{monzo| 7/12 -3/13 }}. This also introduces the potential for dividing intervals outside of pure edo systems: one method of building scales can be to divide just intervals into portions. This is similar to temperaments like [[slendric]], and is identical to defining an [[eigenmonzo]] or rational comma-fraction tuning of these temperaments, except that while those temperaments are sometimes understood through a 2-step process of (1) equally dividing a just interval and (2) assigning the divisions to another just interval, radical intervals provide a framework for skipping the second step (if you deem it unnecessary). In fact, the structure of slendric can be described as equating {{monzo| 3 0 0 -1 }} and {{monzo| -1/3 1/3 }}.
For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow"&gt;free abelian group&lt;/a&gt; (or equivalently, Z-module) of rank r equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow"&gt;divisible group&lt;/a&gt;, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow"&gt;vector space&lt;/a&gt; (of dimension r) over the rational numbers. They are also torsion-free (equivalently, &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow"&gt;flat&lt;/a&gt;) abelian groups, and are the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow"&gt;injective hulls&lt;/a&gt; of the corresponding monzos.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== Radical subgroups ===
A radical subgroup may be notated in the same manner as a normal subgroup, except where the elements are names of equal tunings. For example, quarter-comma meantone intervals can be considered to be radical intervals in the 2.4ed5 subgroup.  
 
== Fractional monzos in projection matrices ==
{{Main| Projection matrices }}
 
[[Category:Regular temperament theory]]
[[Category:Math]]
[[Category:Tuning]]
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