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 composing), 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) ||~ some common names ||
|| [[1_1|1:1]] || 0.000 || unison, perfect prime ||
|| [[81_80|81:80]] || 21.506 || syntonic comma, Didymus comma ||
|| [[33_32|33:32]] || 53.273 || undecimal comma, (large) undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced) ||
|| [[21_20|21:20]] || 84.467 || minor semitone, (large) (septimal) chromatic semitone ||
|| [[16_15|16:15]] || 111.713 || (classic) diatonic semitone, classic minor second, minor diatonic semitone, 15th subharmonic (octave reduced) ||
|| [[12_11|12:11]] || 150.637 || (small) (undecimal) neutral second, 3/4-tone ||
|| [[11_10|11:10]] || 165.004 || (large) (undecimal) neutral second, 4/5-tone, Ptolemy's second ||
|| [[10_9|10:9]] || 182.404 || classic (whole) tone, classic major second, minor whole tone ||
|| [[9_8|9:8]] || 203.910 || (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced) ||
|| [[8_7|8:7]] || 231.174 || (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic ||
|| [[7_6|7:6]] || 266.871 || (septimal) subminor third, septimal minor third, augmented second ||
|| [[32_27|32:27]] || 294.135 || Pythagorean minor third, 27th subharmonic (octave reduced) ||
|| [[6_5|6:5]] || 315.641 || (classic) minor third ||
|| [[11_9|11:9]] || 347.408 || (undecimal) neutral third ||
|| [[5_4|5:4]] || 386.314 || (classic) major third, 5th harmonic (octave reduced) ||
|| [[14_11|14:11]] || 417.508 || (undecimal) supermajor third, undecimal major third, (undecimal) diminished fourth ||
|| [[9_7|9:7]] || 435.084 || (septimal) supermajor third, septimal major third, BP third, (septimal) diminished fourth ||
|| [[21_16|21:16]] || 470.781 || sub fourth, narrow fourth, augmented third, 21st harmonic or septimal 11th (octave reduced) ||
|| [[4_3|4:3]] || 498.045 || perfect fourth, 3rd subharmonic (octave reduced) ||
|| [[27_20|27:20]] || 519.551 || acute fourth ||
|| [[11_8|11:8]] || 551.318 || super fourth, undecimal semi-augmented fourth, 11th harmonic or harmonic 11th (octave reduced) ||
|| [[7_5|7:5]] || 582.512 || augmented fourth, septimal tritone, Huygen's tritone, BP fourth, subdiminished fifth ||
|| [[10_7|10:7]] || 617.488 || diminished fifth, Euler's tritone, superaugmented fourth ||
|| [[16_11|16:11]] || 648.682 || sub fifth, undecimal semi-diminished fifth, 11th subharmonic (octave reduced) ||
|| [[40_27|40:27]] || 680.449 || grave fifth ||
|| [[3_2|3:2]] || 701.955 || [[perfect fifth]], 3rd harmonic (octave reduced) ||
|| [[32_21|32:21]] || 729.219 || super fifth, wide fifth, diminished sixth, 21st subharmonic (octave reduced) ||
|| [[14_9|14:9]] || 764.916 || (septimal) subminor sixth, septimal minor sixth, augmented fifth ||
|| [[11_7|11:7]] || 782.492 || (undecimal) subminor sixth, undecimal augmented fifth ||
|| [[8_5|8:5]] || 813.686 || (classic) minor sixth, 5th subharmonic (octave reduced) ||
|| [[18_11|18:11]] || 852.592 || (undecimal) neutral sixth ||
|| [[5_3|5:3]] || 884.359 || (classic) major sixth ||
|| [[27_16|27:16]] || 905.865 || Pythagorean major sixth, 27th harmonic (octave reduced) ||
|| [[12_7|12:7]] || 933.129 || (septimal) supermajor sixth, septimal major sixth, diminished seventh ||
|| [[7_4|7:4]] || 968.826 || (septimal) subminor seventh, harmonic seventh, augmented sixth, 7th harmonic (octave reduced) ||
|| [[16_9|16:9]] || 996.090 || (Pythagorean) minor seventh, 9th subharmonic (octave reduced) ||
|| [[9_5|9:5]] || 1017.596 || (classic) minor seventh, just minor seventh, BP seventh ||
|| [[20_11|20:11]] || 1034.996 || (small) undecimal neutral seventh, large minor seventh ||
|| [[11_6|11:6]] || 1049.363 || (large) (undecimal) neutral seventh, 21/4-tone ||
|| [[15_8|15:8]] || 1088.269 || (classic) major seventh, 15th harmonic (octave reduced) ||
|| [[40_21|40:21]] || 1115.533 || acute major seventh ||
|| [[64_33|64:33]] || 1146.727 || 33rd subharmonic (octave reduced) ||
|| [[160_81|160:81]] || 1178.494 || octave minus syntonic comma ||
|| [[2_1|2:1]] || 1200.000 || octave ||

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 composing), 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 />
<br />
<hr />
<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>
        <th>frequency ratio<br />
</th>
        <th>cents value (three decimal places)<br />
</th>
        <th>some common names<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/1_1">1:1</a><br />
</td>
        <td>0.000<br />
</td>
        <td>unison, perfect prime<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/81_80">81:80</a><br />
</td>
        <td>21.506<br />
</td>
        <td>syntonic comma, Didymus comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/33_32">33:32</a><br />
</td>
        <td>53.273<br />
</td>
        <td>undecimal comma, (large) undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/21_20">21:20</a><br />
</td>
        <td>84.467<br />
</td>
        <td>minor semitone, (large) (septimal) chromatic semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_15">16:15</a><br />
</td>
        <td>111.713<br />
</td>
        <td>(classic) diatonic semitone, classic minor second, minor diatonic semitone, 15th subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/12_11">12:11</a><br />
</td>
        <td>150.637<br />
</td>
        <td>(small) (undecimal) neutral second, 3/4-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_10">11:10</a><br />
</td>
        <td>165.004<br />
</td>
        <td>(large) (undecimal) neutral second, 4/5-tone, Ptolemy's second<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/10_9">10:9</a><br />
</td>
        <td>182.404<br />
</td>
        <td>classic (whole) tone, classic major second, minor whole tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9_8">9:8</a><br />
</td>
        <td>203.910<br />
</td>
        <td>(Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/8_7">8:7</a><br />
</td>
        <td>231.174<br />
</td>
        <td>(septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_6">7:6</a><br />
</td>
        <td>266.871<br />
</td>
        <td>(septimal) subminor third, septimal minor third, augmented second<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/32_27">32:27</a><br />
</td>
        <td>294.135<br />
</td>
        <td>Pythagorean minor third, 27th subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/6_5">6:5</a><br />
</td>
        <td>315.641<br />
</td>
        <td>(classic) minor third<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_9">11:9</a><br />
</td>
        <td>347.408<br />
</td>
        <td>(undecimal) neutral third<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/5_4">5:4</a><br />
</td>
        <td>386.314<br />
</td>
        <td>(classic) major third, 5th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/14_11">14:11</a><br />
</td>
        <td>417.508<br />
</td>
        <td>(undecimal) supermajor third, undecimal major third, (undecimal) diminished fourth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9_7">9:7</a><br />
</td>
        <td>435.084<br />
</td>
        <td>(septimal) supermajor third, septimal major third, BP third, (septimal) diminished fourth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/21_16">21:16</a><br />
</td>
        <td>470.781<br />
</td>
        <td>sub fourth, narrow fourth, augmented third, 21st harmonic or septimal 11th (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4_3">4:3</a><br />
</td>
        <td>498.045<br />
</td>
        <td>perfect fourth, 3rd subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/27_20">27:20</a><br />
</td>
        <td>519.551<br />
</td>
        <td>acute fourth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_8">11:8</a><br />
</td>
        <td>551.318<br />
</td>
        <td>super fourth, undecimal semi-augmented fourth, 11th harmonic or harmonic 11th (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_5">7:5</a><br />
</td>
        <td>582.512<br />
</td>
        <td>augmented fourth, septimal tritone, Huygen's tritone, BP fourth, subdiminished fifth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/10_7">10:7</a><br />
</td>
        <td>617.488<br />
</td>
        <td>diminished fifth, Euler's tritone, superaugmented fourth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_11">16:11</a><br />
</td>
        <td>648.682<br />
</td>
        <td>sub fifth, undecimal semi-diminished fifth, 11th subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/40_27">40:27</a><br />
</td>
        <td>680.449<br />
</td>
        <td>grave fifth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/3_2">3:2</a><br />
</td>
        <td>701.955<br />
</td>
        <td><a class="wiki_link" href="/perfect%20fifth">perfect fifth</a>, 3rd harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/32_21">32:21</a><br />
</td>
        <td>729.219<br />
</td>
        <td>super fifth, wide fifth, diminished sixth, 21st subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/14_9">14:9</a><br />
</td>
        <td>764.916<br />
</td>
        <td>(septimal) subminor sixth, septimal minor sixth, augmented fifth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_7">11:7</a><br />
</td>
        <td>782.492<br />
</td>
        <td>(undecimal) subminor sixth, undecimal augmented fifth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/8_5">8:5</a><br />
</td>
        <td>813.686<br />
</td>
        <td>(classic) minor sixth, 5th subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/18_11">18:11</a><br />
</td>
        <td>852.592<br />
</td>
        <td>(undecimal) neutral sixth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/5_3">5:3</a><br />
</td>
        <td>884.359<br />
</td>
        <td>(classic) major sixth<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/27_16">27:16</a><br />
</td>
        <td>905.865<br />
</td>
        <td>Pythagorean major sixth, 27th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/12_7">12:7</a><br />
</td>
        <td>933.129<br />
</td>
        <td>(septimal) supermajor sixth, septimal major sixth, diminished seventh<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_4">7:4</a><br />
</td>
        <td>968.826<br />
</td>
        <td>(septimal) subminor seventh, harmonic seventh, augmented sixth, 7th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_9">16:9</a><br />
</td>
        <td>996.090<br />
</td>
        <td>(Pythagorean) minor seventh, 9th subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9_5">9:5</a><br />
</td>
        <td>1017.596<br />
</td>
        <td>(classic) minor seventh, just minor seventh, BP seventh<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/20_11">20:11</a><br />
</td>
        <td>1034.996<br />
</td>
        <td>(small) undecimal neutral seventh, large minor seventh<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_6">11:6</a><br />
</td>
        <td>1049.363<br />
</td>
        <td>(large) (undecimal) neutral seventh, 21/4-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/15_8">15:8</a><br />
</td>
        <td>1088.269<br />
</td>
        <td>(classic) major seventh, 15th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/40_21">40:21</a><br />
</td>
        <td>1115.533<br />
</td>
        <td>acute major seventh<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/64_33">64:33</a><br />
</td>
        <td>1146.727<br />
</td>
        <td>33rd subharmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/160_81">160:81</a><br />
</td>
        <td>1178.494<br />
</td>
        <td>octave minus syntonic comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/2_1">2:1</a><br />
</td>
        <td>1200.000<br />
</td>
        <td>octave<br />
</td>
    </tr>
</table>

</body></html>