Gallery of just intervals

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==Introduction== 

In [[JustIntonation|Just Intonation]], a musical interval is specified as a ratio of two frequencies. For instance, if we measure one frequency at 300 Hz (Hertz -- cycles per second) and another at 200 Hz, the interval between them would be written as 3:2. When two (or more) pitches are sounded that are in simple proportions to one another, there is a "fusing" quality to the sound which is often described as pleasing; hence the interest in tuning the pitches of musical systems according to such proportions.

There is much debate as to what "consonance" means in a musical system, but in Just Intonation, it is generally assumed that lower numbers in frequency ratios lead to greater consonance. In the actual performance of a piece of music, the number of factors involved are enormous, and it is not often helpful to reduce a musical experience to a one-dimensional description of "consonance versus dissonance." Hence the need for this gallery, to give life to conversation about what an interval means beyond the numerical description: "5:3" or "21:16" or what have you.

What follows is a Gallery of Just Intervals in ascending order from 1:1 to 2:1 and beyond (compound intervals being fair game). No such list could possibly be complete (as there are infinite possible ratios), so I (----) seed it with a few important ones while I invite wiki authors to add intervals of interest as they see fit. Any frequency ratio is welcome, including extremely complex ones, as long as the wiki author has some interest in it. I welcome contributions of all sorts to the interval lore: descriptions of common usage, technical notes, poetry, links, reservations, complaints, chords that feature it, edos that approximate it, intervals that are functionally (or emotionally) related to it, nicknames, love letters, fan art, etc. As the experience of an interval is deeply personal and depends hugely on experience (listening and composer), I particularly recommend that wiki authors sign their names.

This page will list links to dedicated pages for each interval. I offer the convention exemplified by 3:2 for the perfect fifth (rather than 2:3 or 3/2 or something else), not because that way is right, but because it is common and it seems helpful to agree for consistency sake. However, the wiki page names will need to be formatted "3_2" because both colons and slashes cannot be part of page names on wikispaces.

I am personally enamored with many intervals: both just and tempered. I don't think I am the only such interval-phile. I am hoping this section will prove fun to contribute to and fun to peruse. I look forward to your contribution!

~Andrew Heathwaite, September 14, 2010

==Gallery of Just Intervals== 

|| frequency ratio || cents value (three decimal places) ||
|| [[1_1|1:1]] || 0.000 ||
|| [[81_80|81:80]] || 21.506 ||
|| [[33_32|33:32]] || 53.273 ||
|| [[21_20|21:20]] || 84.467 ||
|| [[16_15|16:15]] || 111.713 ||
|| [[12_11|12:11]] || 150.637 ||
|| [[11_10|11:10]] || 165.004 ||
|| [[10_9|10:9]] || 182.404 ||
|| [[9_8|9:8]] || 203.910 ||
|| [[8_7|8:7]] || 231.174 ||
|| [[7_6|7:6]] || 266.871 ||
|| [[32_27|32:27]] || 294.135 ||
|| [[6_5|6:5]] || 315.641 ||
|| [[11_9|11:9]] || 347.408 ||
|| [[5_4|5:4]] || 386.314 ||
|| [[14_11|14:11]] || 417.508 ||
|| [[9_7|9:7]] || 435.084 ||
|| [[21_16|21:16]] || 470.781 ||
|| [[4_3|4:3]] || 498.045 ||
|| [[27_20|27:20]] || 519.551 ||
|| [[11_8|11:8]] || 551.318 ||
|| [[7_5|7:5]] || 582.512 ||
|| [[10_7|10:7]] || 617.488 ||
|| [[16_11|16:11]] || 648.682 ||
|| [[40_27|40:27]] || 680.449 ||
|| [[3_2|3:2]] || 701.955 ||
|| [[32_21|32:21]] || 729.219 ||
|| [[14_9|14:9]] || 764.916 ||
|| [[11_7|11:7]] || 782.492 ||
|| [[8_5|8:5]] || 813.686 ||
|| [[18_11|18:11]] || 852.592 ||
|| [[5_3|5:3]] || 884.359 ||
|| [[27_16|27:16]] || 905.865 ||
|| [[12_7|12:7]] || 933.129 ||
|| [[7_4|7:4]] || 968.826 ||
|| [[16_9|16:9]] || 996.090 ||
|| [[9_5|9:5]] || 1017.596 ||
|| [[20_11|20:11]] || 1034.996 ||
|| [[11_6|11:6]] || 1049.363 ||
|| [[15_8|15:8]] || 1088.269 ||
|| [[40_21|40:21]] || 1115.533 ||
|| [[64_33|64:33]] || 1146.727 ||
|| [[160_81|160:81]] || 1178.494 ||
|| [[2_1|2:1]] || 1200.000 ||

Original HTML content:

<html><head><title>Gallery of Just Intervals</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h2>
 <br />
In <a class="wiki_link" href="/JustIntonation">Just Intonation</a>, a musical interval is specified as a ratio of two frequencies. For instance, if we measure one frequency at 300 Hz (Hertz -- cycles per second) and another at 200 Hz, the interval between them would be written as 3:2. When two (or more) pitches are sounded that are in simple proportions to one another, there is a &quot;fusing&quot; quality to the sound which is often described as pleasing; hence the interest in tuning the pitches of musical systems according to such proportions.<br />
<br />
There is much debate as to what &quot;consonance&quot; means in a musical system, but in Just Intonation, it is generally assumed that lower numbers in frequency ratios lead to greater consonance. In the actual performance of a piece of music, the number of factors involved are enormous, and it is not often helpful to reduce a musical experience to a one-dimensional description of &quot;consonance versus dissonance.&quot; Hence the need for this gallery, to give life to conversation about what an interval means beyond the numerical description: &quot;5:3&quot; or &quot;21:16&quot; or what have you.<br />
<br />
What follows is a Gallery of Just Intervals in ascending order from 1:1 to 2:1 and beyond (compound intervals being fair game). No such list could possibly be complete (as there are infinite possible ratios), so I (----) seed it with a few important ones while I invite wiki authors to add intervals of interest as they see fit. Any frequency ratio is welcome, including extremely complex ones, as long as the wiki author has some interest in it. I welcome contributions of all sorts to the interval lore: descriptions of common usage, technical notes, poetry, links, reservations, complaints, chords that feature it, edos that approximate it, intervals that are functionally (or emotionally) related to it, nicknames, love letters, fan art, etc. As the experience of an interval is deeply personal and depends hugely on experience (listening and composer), I particularly recommend that wiki authors sign their names.<br />
<br />
This page will list links to dedicated pages for each interval. I offer the convention exemplified by 3:2 for the perfect fifth (rather than 2:3 or 3/2 or something else), not because that way is right, but because it is common and it seems helpful to agree for consistency sake. However, the wiki page names will need to be formatted &quot;3_2&quot; because both colons and slashes cannot be part of page names on wikispaces.<br />
<br />
I am personally enamored with many intervals: both just and tempered. I don't think I am the only such interval-phile. I am hoping this section will prove fun to contribute to and fun to peruse. I look forward to your contribution!<br />
<br />
~Andrew Heathwaite, September 14, 2010<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Gallery of Just Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Gallery of Just Intervals</h2>
 <br />


<table class="wiki_table">
    <tr>
        <td>frequency ratio<br />
</td>
        <td>cents value (three decimal places)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/1_1">1:1</a><br />
</td>
        <td>0.000<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/81_80">81:80</a><br />
</td>
        <td>21.506<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/33_32">33:32</a><br />
</td>
        <td>53.273<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/21_20">21:20</a><br />
</td>
        <td>84.467<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_15">16:15</a><br />
</td>
        <td>111.713<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/12_11">12:11</a><br />
</td>
        <td>150.637<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_10">11:10</a><br />
</td>
        <td>165.004<br />
</td>
    </tr>
    <tr>
        <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="/9_8">9:8</a><br />
</td>
        <td>203.910<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/8_7">8:7</a><br />
</td>
        <td>231.174<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_6">7:6</a><br />
</td>
        <td>266.871<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/32_27">32:27</a><br />
</td>
        <td>294.135<br />
</td>
    </tr>
    <tr>
        <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="/11_9">11:9</a><br />
</td>
        <td>347.408<br />
</td>
    </tr>
    <tr>
        <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="/14_11">14:11</a><br />
</td>
        <td>417.508<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9_7">9:7</a><br />
</td>
        <td>435.084<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/21_16">21:16</a><br />
</td>
        <td>470.781<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4_3">4:3</a><br />
</td>
        <td>498.045<br />
</td>
    </tr>
    <tr>
        <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="/11_8">11:8</a><br />
</td>
        <td>551.318<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_5">7:5</a><br />
</td>
        <td>582.512<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/10_7">10:7</a><br />
</td>
        <td>617.488<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_11">16:11</a><br />
</td>
        <td>648.682<br />
</td>
    </tr>
    <tr>
        <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="/3_2">3:2</a><br />
</td>
        <td>701.955<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/32_21">32:21</a><br />
</td>
        <td>729.219<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/14_9">14:9</a><br />
</td>
        <td>764.916<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_7">11:7</a><br />
</td>
        <td>782.492<br />
</td>
    </tr>
    <tr>
        <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="/18_11">18:11</a><br />
</td>
        <td>852.592<br />
</td>
    </tr>
    <tr>
        <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="/27_16">27:16</a><br />
</td>
        <td>905.865<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/12_7">12:7</a><br />
</td>
        <td>933.129<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_4">7:4</a><br />
</td>
        <td>968.826<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_9">16:9</a><br />
</td>
        <td>996.090<br />
</td>
    </tr>
    <tr>
        <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="/20_11">20:11</a><br />
</td>
        <td>1034.996<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_6">11:6</a><br />
</td>
        <td>1049.363<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/15_8">15:8</a><br />
</td>
        <td>1088.269<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/40_21">40:21</a><br />
</td>
        <td>1115.533<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/64_33">64:33</a><br />
</td>
        <td>1146.727<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/160_81">160:81</a><br />
</td>
        <td>1178.494<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/2_1">2:1</a><br />
</td>
        <td>1200.000<br />
</td>
    </tr>
</table>

</body></html>