5-limit

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The //5-limit// consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called [[http://en.wikipedia.org/wiki/Regular_number|regular numbers]]. Some examples of 5-limit intervals are [[5_4|5/4]], [[6_5|6/5]], [[10_9|10/9]] and [[81_80|81/80]]. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, [[4_3|4/3]], [[3_2|3/2]], [[8_5|8/5]], [[5_3|5/3]], 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.

The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a [[http://en.wikipedia.org/wiki/Hexagonal_lattice|hexagonal lattice]] or as a [[http://en.wikipedia.org/wiki/Square_lattice|square lattice]]; this can be done automatically by [[http://www.huygens-fokker.org/scala/|Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[http://en.wikipedia.org/wiki/Hexagonal_tiling|hexagonal tiling]].

[[EDO]]s which do relatively well in approximating the 5-limit are [[5edo]], [[7edo]], [[12edo]], [[19edo]], [[34edo]], [[53edo]], [[65edo]], [[118edo]] and [[171edo]].

==Syntonic Comma Pairs== 

A significant interval in 5-limit JI is [[81_80|81/80]], the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby [[3-limit]] (Pythagorean) interval. 81/80 is tempered out in [[12edo]], [[meantone]], and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely [[12edo]] musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80.

||||~ 3-limit interval ||~ interval category ||||~ 5-limit interval ||
||~ ratio ||~ cents value ||~   ||~ ratio ||~ cents value ||
|| [[256_243|256/243]] || 90.225 || minor 2nd || [[16_15|16/15]] || 111.731 ||
|| [[9_8|9/8]] || 203.910 || major 2nd || [[10_9|10/9]] || 182.404 ||
|| [[32_27|32/27]] || 294.135 || minor 3rd || [[6_5|6/5]] || 315.641 ||
|| [[81_64|81/64]] || 407.820 || major 3rd || [[5_4|5/4]] || 386.314 ||
|| [[4_3|4/3]] || 498.045 || fourth || [[27_20|27/20]] || 519.551 ||
|| [[3_2|3/2]] || 701.955 || fifth || [[40_27|40/27]] || 680.449 ||
|| [[128_81|128/81]] || 792.18 || minor 6th || [[8_5|8/5]] || 813.686 ||
|| [[27_16|27/16]] || 905.865 || major 6th || [[5_3|5/3]] || 884.359 ||
|| [[16_9|16/9]] || 996.090 || minor 7th || [[9_5|9/5]] || 1017.596 ||
|| [[243_128|243/128]] || 1109.775 || major 7th || [[15_8|15/8]] || 1088.269 ||

It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means making allowing for both 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit //includes// the 3-limit -- a work in 5-limit JI will utilize intervals from both sides of the chart above.

See [[Harmonic Limit]]

=Music= 
[[http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3|Duodene2]] by [[Chris Vaisvil]]
[[http://micro.soonlabel.com/just/Ariels-JI/ariels-12-tone-ji.mp3|Ariel's 12-tone JI]] by Chris Vaisvil
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/The%20Ballad%20of%20Jed%20Clampett.mp3|The Ballad of Jed Clampett]] by [[http://en.wikipedia.org/wiki/Paul_Henning|Paul Henning]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/Do%20Wah%20Diddy.mp3|Do Wah Diddy Diddy]] by [[http://en.wikipedia.org/wiki/Jeff_Barry|Barry]] and [[http://en.wikipedia.org/wiki/Ellie_Greenwich|Greenwich]]
[[https://soundcloud.com/williamcopper/0511_1|Symphony 4, first movement]] by [[https://soundcloud.com/williamcopper/0511_1|William Copper]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Magnificat0465.mp3|Magnificat]] by [[https://soundcloud.com/copper-magnificat/magnificat-by-william-copper-complete|William Copper]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Catch%20for%20Woodwind%20Quintet-0570.mp3|Catch for Woodwin Quintet]] by [[https://soundcloud.com/williamcopper/0570_catch|William Copper]]

Original HTML content:

<html><head><title>5-limit</title></head><body>The <em>5-limit</em> consists of all <a class="wiki_link" href="/Just%20intonation">justly tuned</a> intervals whose numerators and denominators are both products of the primes 2, 3, and 5; these are sometimes called <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_number" rel="nofollow">regular numbers</a>. Some examples of 5-limit intervals are <a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/81_80">81/80</a>. The 5 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 5. Reduced to an octave, these are the ratios 1/1, 6/5, 5/4, <a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a>, <a class="wiki_link" href="/8_5">8/5</a>, <a class="wiki_link" href="/5_3">5/3</a>, 2/1. Approximating these ratios has been basic to Western common-practice music since the Renaissance.<br />
<br />
The octave equivalence classes of 5-limit intervals can usefully be depicted on a lattice diagram, either as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_lattice" rel="nofollow">hexagonal lattice</a> or as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Square_lattice" rel="nofollow">square lattice</a>; this can be done automatically by <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow">Scala</a>. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hexagonal_tiling" rel="nofollow">hexagonal tiling</a>.<br />
<br />
<a class="wiki_link" href="/EDO">EDO</a>s which do relatively well in approximating the 5-limit are <a class="wiki_link" href="/5edo">5edo</a>, <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/53edo">53edo</a>, <a class="wiki_link" href="/65edo">65edo</a>, <a class="wiki_link" href="/118edo">118edo</a> and <a class="wiki_link" href="/171edo">171edo</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Syntonic Comma Pairs"></a><!-- ws:end:WikiTextHeadingRule:0 -->Syntonic Comma Pairs</h2>
 <br />
A significant interval in 5-limit JI is <a class="wiki_link" href="/81_80">81/80</a>, the syntonic comma or Didymus' comma, which measures about 21.5¢. Although it rarely appears as an interval in a scale, it represents the difference between many 5-limit intervals and a nearby <a class="wiki_link" href="/3-limit">3-limit</a> (Pythagorean) interval. 81/80 is tempered out in <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/meantone">meantone</a>, and many other related systems, meaning that those 5- and 3-limit distinctions are obliterated and one interval stands in for each. Living in a largely <a class="wiki_link" href="/12edo">12edo</a> musical culture from birth, we are not accustomed to distinguishing two different major thirds, two different minor seconds, etc. Below is a list of some common intervals involving 3 and 5 which are distinguished by 81/80.<br />
<br />


<table class="wiki_table">
    <tr>
        <th colspan="2">3-limit interval<br />
</th>
        <th>interval category<br />
</th>
        <th colspan="2">5-limit interval<br />
</th>
    </tr>
    <tr>
        <th>ratio<br />
</th>
        <th>cents value<br />
</th>
        <th><br />
</th>
        <th>ratio<br />
</th>
        <th>cents value<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/256_243">256/243</a><br />
</td>
        <td>90.225<br />
</td>
        <td>minor 2nd<br />
</td>
        <td><a class="wiki_link" href="/16_15">16/15</a><br />
</td>
        <td>111.731<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9_8">9/8</a><br />
</td>
        <td>203.910<br />
</td>
        <td>major 2nd<br />
</td>
        <td><a class="wiki_link" href="/10_9">10/9</a><br />
</td>
        <td>182.404<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/32_27">32/27</a><br />
</td>
        <td>294.135<br />
</td>
        <td>minor 3rd<br />
</td>
        <td><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
        <td>315.641<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/81_64">81/64</a><br />
</td>
        <td>407.820<br />
</td>
        <td>major 3rd<br />
</td>
        <td><a class="wiki_link" href="/5_4">5/4</a><br />
</td>
        <td>386.314<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4_3">4/3</a><br />
</td>
        <td>498.045<br />
</td>
        <td>fourth<br />
</td>
        <td><a class="wiki_link" href="/27_20">27/20</a><br />
</td>
        <td>519.551<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td>701.955<br />
</td>
        <td>fifth<br />
</td>
        <td><a class="wiki_link" href="/40_27">40/27</a><br />
</td>
        <td>680.449<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/128_81">128/81</a><br />
</td>
        <td>792.18<br />
</td>
        <td>minor 6th<br />
</td>
        <td><a class="wiki_link" href="/8_5">8/5</a><br />
</td>
        <td>813.686<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/27_16">27/16</a><br />
</td>
        <td>905.865<br />
</td>
        <td>major 6th<br />
</td>
        <td><a class="wiki_link" href="/5_3">5/3</a><br />
</td>
        <td>884.359<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_9">16/9</a><br />
</td>
        <td>996.090<br />
</td>
        <td>minor 7th<br />
</td>
        <td><a class="wiki_link" href="/9_5">9/5</a><br />
</td>
        <td>1017.596<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/243_128">243/128</a><br />
</td>
        <td>1109.775<br />
</td>
        <td>major 7th<br />
</td>
        <td><a class="wiki_link" href="/15_8">15/8</a><br />
</td>
        <td>1088.269<br />
</td>
    </tr>
</table>

<br />
It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means making allowing for both 3's and 5's in generating harmonic material, and so it is an interplay between both. The 5-limit <em>includes</em> the 3-limit -- a work in 5-limit JI will utilize intervals from both sides of the chart above.<br />
<br />
See <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:2 -->Music</h1>
 <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3" rel="nofollow">Duodene2</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/just/Ariels-JI/ariels-12-tone-ji.mp3" rel="nofollow">Ariel's 12-tone JI</a> by Chris Vaisvil<br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/The%20Ballad%20of%20Jed%20Clampett.mp3" rel="nofollow">The Ballad of Jed Clampett</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Henning" rel="nofollow">Paul Henning</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/5limit/Do%20Wah%20Diddy.mp3" rel="nofollow">Do Wah Diddy Diddy</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Jeff_Barry" rel="nofollow">Barry</a> and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Ellie_Greenwich" rel="nofollow">Greenwich</a><br />
<a class="wiki_link_ext" href="https://soundcloud.com/williamcopper/0511_1" rel="nofollow">Symphony 4, first movement</a> by <a class="wiki_link_ext" href="https://soundcloud.com/williamcopper/0511_1" rel="nofollow">William Copper</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Magnificat0465.mp3" rel="nofollow">Magnificat</a> by <a class="wiki_link_ext" href="https://soundcloud.com/copper-magnificat/magnificat-by-william-copper-complete" rel="nofollow">William Copper</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Copper/Catch%20for%20Woodwind%20Quintet-0570.mp3" rel="nofollow">Catch for Woodwin Quintet</a> by <a class="wiki_link_ext" href="https://soundcloud.com/williamcopper/0570_catch" rel="nofollow">William Copper</a></body></html>