5-limit: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-~~11 03~~:40:03 UTC</tt>.

: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-09-10 10:51:54 UTC</tt>.

: The original revision id was <tt>~~589203844~~</tt>.

: The original revision id was <tt>591567350</tt>.

: The revision comment was: <tt></tt>

: The revision comment was: <tt>expanded table</tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<h4>Original Wikitext content:</h4>

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==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.

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. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). **Bold** inter are

||||~ 3-limit interval ||~ interval category ||||~ 5-limit interval ||

||||~ 3-limit interval ||||~ interval category ||||~ |5-limit interval (81/80) ||||~ |Another 5-limit (6561/6400) ||

||~ ratio ||~ cents value ||~ ||~ ratio ||~ cents value ||

||~ ratio ||~ cents value ||~ ||~ ||~ ratio ||~ cents value ||~ ratio ||~ cents value ||

|| [[256_243|256/243]] || 90.225 || minor 2nd || [[16_15|16/15]] || 111.731 ||

|| [[1_1|1/1]] || 0.000 || unison || C || [[81_80|81/80]] || 21.506 || [[6561_6400|6561/6400]] || 43.013 ||

|| [[9_8|9/8]] || 203.910 || major 2nd || [[10_9|10/9]] || 182.404 ||

|| [[2187_2048|2187/2048]] || 113.685 || aug. unison || C# || [[135_128|135/128]] || 92.179 || [[25_24|25/24]] || 70.672 ||

|| [[32_27|32/27]] || 294.135 || minor 3rd || [[6_5|6/5]] || 315.641 ||

|| [[256_243|256/243]] || 90.225 || minor 2nd || Db || [[16_15|16/15]] || 111.731 || [[27_25|27/25]] || 133.238 ||

|| [[81_64|81/64]] || 407.820 || major 3rd || [[5_4|5/4]] || 386.314 ||

|| [[9_8|9/8]] || 203.910 || major 2nd || D || [[10_9|10/9]] || 182.404 || [[800_729|800/729]] || 160.897 ||

|| [[4_3|4/3]] || 498.045 || fourth || [[27_20|27/20]] || 519.551 ||

|| [[19683_16384|19683/16384]] || 317.595 || aug. 2nd || D# || [[1215_1024|1215/1024]] || 296.089 || [[75_64|75/64]] || 274.582 ||

|| [[3_2|3/2]] || 701.955 || fifth || [[40_27|40/27]] || 680.449 ||

|| [[32_27|32/27]] || 294.135 || minor 3rd || Eb || [[6_5|6/5]] || 315.641 || [[243_200|243/200]] || 337.148 ||

|| [[128_81|128/81]] || 792.18 || minor 6th || [[8_5|8/5]] || 813.686 ||

|| [[81_64|81/64]] || 407.820 || major 3rd || E || [[5_4|5/4]] || 386.314 || [[100_81|100/81]] || 364.807 ||

|| [[27_16|27/16]] || 905.865 || major 6th || [[5_3|5/3]] || 884.359 ||

|| [[8192_6561|8192/6561]] || 384.360 || dim. fourth || Fb || [[512_405|512/405]] || 405.866 || [[32_25|32/25]] || 427.373 ||

|| [[16_9|16/9]] || 996.090 || minor 7th || [[9_5|9/5]] || 1017.596 ||

|| [[4_3|4/3]] || 498.045 || fourth || F || [[27_20|27/20]] || 519.551 || [[2187_1600|2187/1600]] || 541.058 ||

|| [[243_128|243/128]] || 1109.775 || major 7th || [[15_8|15/8]] || 1088.269 ||

|| [[729_512|729/512]] || 611.730 || aug. fourth || F# || [[45_32|45/32]] || 590.224 || [[25_18|25/18]] || 568.717 ||

|| [[1024_729|1024/729]] || 588.270 || dim. fifth || Gb || [[64_45|64/45]] || 609.776 || [[36_25|36/25]] || 631.283 ||

|| [[3_2|3/2]] || 701.955 || fifth || G || [[40_27|40/27]] || 680.449 || [[3200_2187|3200/2187]] || 658.942 ||

|| [[6561_4096|6561/4096]] || 815.640 || aug. fifth || G# || [[405_256|405/256]] || 794.134 || [[25_16|25/16]] || 772.627 ||

|| [[128_81|128/81]] || 792.180 || minor 6th || Ab || [[8_5|8/5]] || 813.686 || [[81_50|81/50]] || 835.193 ||

|| [[27_16|27/16]] || 905.865 || major 6th || A || [[5_3|5/3]] || 884.359 || [[400_243|400/243]] || 862.852 ||

|| [[32768_19683|32768/19683]] || 882.405 || dim. 7th || Bbb || [[2048_1215|2048/1215]] || 903.911 || [[128_75|128/75]] || 925.418 ||

|| [[16_9|16/9]] || 996.090 || minor 7th || Bb || [[9_5|9/5]] || 1017.596 || [[729_400|729/400]] || 1039.103 ||

|| [[243_128|243/128]] || 1109.775 || major 7th || B || [[15_8|15/8]] || 1088.269 || [[50_27|50/27]] || 1066.762 ||

|| [[4096_2187|4096/2187]] || 1086.315 || dim. octave || Cb || [[256_135|256/135]] || 1107.821 || [[48_25|48/25]] || 1129.328 ||

|| [[2_1|2/1]] || 1200.000 || octave || C || [[160_81|160/81]] || 1178.494 || [[12800_6561|12800/6561]] || 1156.987 ||

It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means 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.

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<h2 id="toc0"><a name="x-Syntonic Comma Pairs"></a>Syntonic Comma Pairs</h2>

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.

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. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). Bold inter are

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<th colspan="2">3-limit interval

</th>

<th>interval category

<th colspan="2">interval category

</th>

<th colspan="2">5-limit interval

<th colspan="2">|5-limit interval (81/80)

</th>

<th colspan="2">|Another 5-limit (6561/6400)

</th>

</tr>

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</th>

<th>cents value

</th>

<th>

</th>

<th>

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<th>cents value

</th>

<th>ratio

</th>

<th>cents value

</th>

</tr>

<tr>

</td>

<td>0.000

</td>

<td>unison

</td>

<td>C

</td>

</td>

<td>21.506

</td>

</td>

<td>43.013

</td>

</tr>

<tr>

</td>

<td>113.685

</td>

<td>aug. unison

</td>

<td>C#

</td>

</td>

<td>92.179

</td>

</td>

<td>70.672

</td>

</tr>

<tr>

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</td>

<td>minor 2nd

</td>

<td>Db

</td>

</td>

<td>111.731

</td>

</td>

<td>133.238

</td>

</tr>

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</td>

<td>major 2nd

</td>

<td>D

</td>

</td>

<td>182.404

</td>

</td>

<td>160.897

</td>

</tr>

<tr>

</td>

<td>317.595

</td>

<td>aug. 2nd

</td>

<td>D#

</td>

</td>

<td>296.089

</td>

</td>

<td>274.582

</td>

</tr>

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</td>

<td>minor 3rd

</td>

<td>Eb

</td>

</td>

<td>315.641

</td>

</td>

<td>337.148

</td>

</tr>

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</td>

<td>major 3rd

</td>

<td>E

</td>

</td>

<td>386.314

</td>

</td>

<td>364.807

</td>

</tr>

<tr>

</td>

<td>384.360

</td>

<td>dim. fourth

</td>

<td>Fb

</td>

</td>

<td>405.866

</td>

</td>

<td>427.373

</td>

</tr>

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</td>

<td>fourth

</td>

<td>F

</td>

</td>

<td>519.551

</td>

</td>

<td>541.058

</td>

</tr>

<tr>

</td>

<td>611.730

</td>

<td>aug. fourth

</td>

<td>F#

</td>

</td>

<td>590.224

</td>

</td>

<td>568.717

</td>

</tr>

<tr>

</td>

<td>588.270

</td>

<td>dim. fifth

</td>

<td>Gb

</td>

</td>

<td>609.776

</td>

</td>

<td>631.283

</td>

</tr>

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</td>

<td>fifth

</td>

<td>G

</td>

</td>

<td>680.449

</td>

</td>

<td>658.942

</td>

</tr>

<tr>

</td>

<td>815.640

</td>

<td>aug. fifth

</td>

<td>G#

</td>

</td>

<td>794.134

</td>

</td>

<td>772.627

</td>

</tr>

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</td>

<td>792.18

<td>792.180

</td>

<td>minor 6th

</td>

<td>Ab

</td>

</td>

<td>813.686

</td>

</td>

<td>835.193

</td>

</tr>

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</td>

<td>major 6th

</td>

<td>A

</td>

</td>

<td>884.359

</td>

</td>

<td>862.852

</td>

</tr>

<tr>

</td>

<td>882.405

</td>

<td>dim. 7th

</td>

<td>Bbb

</td>

</td>

<td>903.911

</td>

</td>

<td>925.418

</td>

</tr>

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</td>

<td>minor 7th

</td>

<td>Bb

</td>

</td>

<td>1017.596

</td>

</td>

<td>1039.103

</td>

</tr>

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</td>

<td>major 7th

</td>

<td>B

</td>

</td>

<td>1088.269

</td>

</td>

<td>1066.762

</td>

</tr>

<tr>

</td>

<td>1086.315

</td>

<td>dim. octave

</td>

<td>Cb

</td>

</td>

<td>1107.821

</td>

</td>

<td>1129.328

</td>

</tr>

<tr>

</td>

<td>1200.000

</td>

<td>octave

</td>

<td>C

</td>

</td>

<td>1178.494

</td>

</td>

<td>1156.987

</td>

</tr>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-11 03:40:03 UTC</tt>.<br>
+: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-09-10 10:51:54 UTC</tt>.<br>
-: The original revision id was <tt>589203844</tt>.<br>
+: The original revision id was <tt>591567350</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>expanded table</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 14: / Line 14: @@
 ==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.
+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. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). **Bold** inter are
-||||~ 3-limit interval ||~ interval category ||||~ 5-limit interval ||
+||||~ 3-limit interval ||||~ interval category ||||~ |5-limit interval (81/80) ||||~ |Another 5-limit (6561/6400) ||
-||~ ratio ||~ cents value ||~   ||~ ratio ||~ cents value ||
+||~ ratio ||~ cents value ||~   ||~   ||~ ratio ||~ cents value ||~ ratio ||~ cents value ||
-|| [[256_243|256/243]] || 90.225 || minor 2nd || [[16_15|16/15]] || 111.731 ||
+|| [[1_1|1/1]] || 0.000 || unison || C || [[81_80|81/80]] || 21.506 || [[6561_6400|6561/6400]] || 43.013 ||
-|| [[9_8|9/8]] || 203.910 || major 2nd || [[10_9|10/9]] || 182.404 ||
+|| [[2187_2048|2187/2048]] || 113.685 || aug. unison || C# || [[135_128|135/128]] || 92.179 || [[25_24|25/24]] || 70.672 ||
-|| [[32_27|32/27]] || 294.135 || minor 3rd || [[6_5|6/5]] || 315.641 ||
+|| [[256_243|256/243]] || 90.225 || minor 2nd || Db || [[16_15|16/15]] || 111.731 || [[27_25|27/25]] || 133.238 ||
-|| [[81_64|81/64]] || 407.820 || major 3rd || [[5_4|5/4]] || 386.314 ||
+|| [[9_8|9/8]] || 203.910 || major 2nd || D || [[10_9|10/9]] || 182.404 || [[800_729|800/729]] || 160.897 ||
-|| [[4_3|4/3]] || 498.045 || fourth || [[27_20|27/20]] || 519.551 ||
+|| [[19683_16384|19683/16384]] || 317.595 || aug. 2nd || D# || [[1215_1024|1215/1024]] || 296.089 || [[75_64|75/64]] || 274.582 ||
-|| [[3_2|3/2]] || 701.955 || fifth || [[40_27|40/27]] || 680.449 ||
+|| [[32_27|32/27]] || 294.135 || minor 3rd || Eb || [[6_5|6/5]] || 315.641 || [[243_200|243/200]] || 337.148 ||
-|| [[128_81|128/81]] || 792.18 || minor 6th || [[8_5|8/5]] || 813.686 ||
+|| [[81_64|81/64]] || 407.820 || major 3rd || E || [[5_4|5/4]] || 386.314 || [[100_81|100/81]] || 364.807 ||
-|| [[27_16|27/16]] || 905.865 || major 6th || [[5_3|5/3]] || 884.359 ||
+|| [[8192_6561|8192/6561]] || 384.360 || dim. fourth || Fb || [[512_405|512/405]] || 405.866 || [[32_25|32/25]] || 427.373 ||
-|| [[16_9|16/9]] || 996.090 || minor 7th || [[9_5|9/5]] || 1017.596 ||
+|| [[4_3|4/3]] || 498.045 || fourth || F || [[27_20|27/20]] || 519.551 || [[2187_1600|2187/1600]] || 541.058 ||
-|| [[243_128|243/128]] || 1109.775 || major 7th || [[15_8|15/8]] || 1088.269 ||
+|| [[729_512|729/512]] || 611.730 || aug. fourth || F# || [[45_32|45/32]] || 590.224 || [[25_18|25/18]] || 568.717 ||
+|| [[1024_729|1024/729]] || 588.270 || dim. fifth || Gb || [[64_45|64/45]] || 609.776 || [[36_25|36/25]] || 631.283 ||
+|| [[3_2|3/2]] || 701.955 || fifth || G || [[40_27|40/27]] || 680.449 || [[3200_2187|3200/2187]] || 658.942 ||
+|| [[6561_4096|6561/4096]] || 815.640 || aug. fifth || G# || [[405_256|405/256]] || 794.134 || [[25_16|25/16]] || 772.627 ||
+|| [[128_81|128/81]] || 792.180 || minor 6th || Ab || [[8_5|8/5]] || 813.686 || [[81_50|81/50]] || 835.193 ||
+|| [[27_16|27/16]] || 905.865 || major 6th || A || [[5_3|5/3]] || 884.359 || [[400_243|400/243]] || 862.852 ||
+|| [[32768_19683|32768/19683]] || 882.405 || dim. 7th || Bbb || [[2048_1215|2048/1215]] || 903.911 || [[128_75|128/75]] || 925.418 ||
+|| [[16_9|16/9]] || 996.090 || minor 7th || Bb || [[9_5|9/5]] || 1017.596 || [[729_400|729/400]] || 1039.103 ||
+|| [[243_128|243/128]] || 1109.775 || major 7th || B || [[15_8|15/8]] || 1088.269 || [[50_27|50/27]] || 1066.762 ||
+|| [[4096_2187|4096/2187]] || 1086.315 || dim. octave || Cb || [[256_135|256/135]] || 1107.821 || [[48_25|48/25]] || 1129.328 ||
+|| [[2_1|2/1]] || 1200.000 || octave || C || [[160_81|160/81]] || 1178.494 || [[12800_6561|12800/6561]] || 1156.987 ||
 It is important to note that 5-limit music does not mean favoring intervals of 5 over intervals of 3. It means 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.
@@ Line 50: / Line 60: @@
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Syntonic Comma Pairs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Syntonic Comma Pairs&lt;/h2&gt;
   &lt;br /&gt;
-A significant interval in 5-limit JI is &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; (Pythagorean) interval. 81/80 is tempered out in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; 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.&lt;br /&gt;
+A significant interval in 5-limit JI is &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/3-limit"&gt;3-limit&lt;/a&gt; (Pythagorean) interval. 81/80 is tempered out in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; 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. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). &lt;strong&gt;Bold&lt;/strong&gt; inter are&lt;br /&gt;
 &lt;br /&gt;
@@ Line 58: / Line 68: @@
          &lt;th colspan="2"&gt;3-limit interval&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;interval category&lt;br /&gt;
+         &lt;th colspan="2"&gt;interval category&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th colspan="2"&gt;5-limit interval&lt;br /&gt;
+         &lt;th colspan="2"&gt;|5-limit interval (81/80)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th colspan="2"&gt;|Another 5-limit (6561/6400)&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
@@ Line 67: / Line 79: @@
 &lt;/th&gt;
          &lt;th&gt;cents value&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;&lt;br /&gt;
@@ Line 74: / Line 88: @@
          &lt;th&gt;cents value&lt;br /&gt;
 &lt;/th&gt;
+        &lt;th&gt;ratio&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;cents value&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1_1"&gt;1/1&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0.000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;unison&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;21.506&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6561_6400"&gt;6561/6400&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;43.013&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2187_2048"&gt;2187/2048&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;113.685&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;aug. unison&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/135_128"&gt;135/128&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;92.179&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/25_24"&gt;25/24&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;70.672&lt;br /&gt;
+&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
@@ Line 81: / Line 135: @@
 &lt;/td&gt;
          &lt;td&gt;minor 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;111.731&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/27_25"&gt;27/25&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;133.238&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 93: / Line 153: @@
 &lt;/td&gt;
          &lt;td&gt;major 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;182.404&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/800_729"&gt;800/729&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;160.897&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/19683_16384"&gt;19683/16384&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;317.595&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;aug. 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;D#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1215_1024"&gt;1215/1024&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;296.089&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/75_64"&gt;75/64&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;274.582&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 105: / Line 189: @@
 &lt;/td&gt;
          &lt;td&gt;minor 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Eb&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;315.641&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/243_200"&gt;243/200&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;337.148&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 117: / Line 207: @@
 &lt;/td&gt;
          &lt;td&gt;major 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;E&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;386.314&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/100_81"&gt;100/81&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;364.807&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/8192_6561"&gt;8192/6561&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;384.360&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim. fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Fb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/512_405"&gt;512/405&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;405.866&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/32_25"&gt;32/25&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;427.373&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 129: / Line 243: @@
 &lt;/td&gt;
          &lt;td&gt;fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;F&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/27_20"&gt;27/20&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;519.551&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2187_1600"&gt;2187/1600&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;541.058&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/729_512"&gt;729/512&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;611.730&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;aug. fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;F#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/45_32"&gt;45/32&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;590.224&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/25_18"&gt;25/18&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;568.717&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/1024_729"&gt;1024/729&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;588.270&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim. fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Gb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/64_45"&gt;64/45&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;609.776&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/36_25"&gt;36/25&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;631.283&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 141: / Line 297: @@
 &lt;/td&gt;
          &lt;td&gt;fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;G&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/40_27"&gt;40/27&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;680.449&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/3200_2187"&gt;3200/2187&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;658.942&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/6561_4096"&gt;6561/4096&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;815.640&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;aug. fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;G#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/405_256"&gt;405/256&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;794.134&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/25_16"&gt;25/16&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;772.627&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 150: / Line 330: @@
          &lt;td&gt;&lt;a class="wiki_link" href="/128_81"&gt;128/81&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;792.18&lt;br /&gt;
+         &lt;td&gt;792.180&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;minor 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Ab&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;813.686&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/81_50"&gt;81/50&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;835.193&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 165: / Line 351: @@
 &lt;/td&gt;
          &lt;td&gt;major 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;884.359&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/400_243"&gt;400/243&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;862.852&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/32768_19683"&gt;32768/19683&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;882.405&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim. 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Bbb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2048_1215"&gt;2048/1215&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;903.911&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/128_75"&gt;128/75&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;925.418&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 177: / Line 387: @@
 &lt;/td&gt;
          &lt;td&gt;minor 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Bb&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1017.596&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/729_400"&gt;729/400&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1039.103&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 189: / Line 405: @@
 &lt;/td&gt;
          &lt;td&gt;major 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;B&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;a class="wiki_link" href="/15_8"&gt;15/8&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1088.269&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/50_27"&gt;50/27&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1066.762&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/4096_2187"&gt;4096/2187&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1086.315&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim. octave&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Cb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/256_135"&gt;256/135&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1107.821&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/48_25"&gt;48/25&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1129.328&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/2_1"&gt;2/1&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200.000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;octave&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/160_81"&gt;160/81&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1178.494&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;a class="wiki_link" href="/12800_6561"&gt;12800/6561&lt;/a&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1156.987&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;