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>2016-09-10 10:51:54 UTC</tt>.

: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-09-10 10:57:19 UTC</tt>.

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

: The original revision id was <tt>591567562</tt>.

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

: The revision comment was: <tt></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. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). **Bold** ~~inter~~ are

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** fractions are simplest for this interval category.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

|| [[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 ||

|| [[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 ||

|| **[[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 ||

|| [[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 ||

|| [[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 ||

|| [[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 ||

|| [[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 ||

|| [[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 ||

|| [[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 ||

|| [[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 ||

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

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 fractions are simplest for this interval category.

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</tr>

<tr>

</td>

<td>0.000

<td>0.000

</td>

<td>unison

Line 124:

<td>92.179

</td>

</td>

<td>70.672

<td>70.672

</td>

</tr>

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<td>Db

</td>

</td>

<td>111.731

<td>111.731

</td>

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</tr>

<tr>

</td>

<td>203.910

<td>203.910

</td>

<td>major 2nd

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<td>296.089

</td>

</td>

<td>274.582

<td>274.582

</td>

</tr>

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<td>Eb

</td>

</td>

<td>315.641

<td>315.641

</td>

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<td>E

</td>

</td>

<td>386.314

<td>386.314

</td>

Line 232:

<td>405.866

</td>

</td>

<td>427.373

<td>427.373

</td>

</tr>

<tr>

</td>

<td>498.045

<td>498.045

</td>

<td>fourth

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<td>590.224

</td>

</td>

<td>568.717

<td>568.717

</td>

</tr>

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<td>609.776

</td>

</td>

<td>631.283

<td>631.283

</td>

</tr>

<tr>

</td>

<td>701.955

<td>701.955

</td>

<td>fifth

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<td>794.134

</td>

</td>

<td>772.627

<td>772.627

</td>

</tr>

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<td>Ab

</td>

</td>

<td>813.686

<td>813.686

</td>

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<td>A

</td>

</td>

<td>884.359

<td>884.359

</td>

Line 376:

<td>903.911

</td>

</td>

<td>925.418

<td>925.418

</td>

</tr>

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<td>Bb

</td>

</td>

<td>1017.596

<td>1017.596

</td>

Line 408:

<td>B

</td>

</td>

<td>1088.269

<td>1088.269

</td>

Line 430:

<td>1107.821

</td>

</td>

<td>1129.328

<td>1129.328

</td>

</tr>

<tr>

</td>

<td>1200.000

<td>1200.000

</td>

<td>octave

@@ 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>2016-09-10 10:51:54 UTC</tt>.<br>
+: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-09-10 10:57:19 UTC</tt>.<br>
-: The original revision id was <tt>591567350</tt>.<br>
+: The original revision id was <tt>591567562</tt>.<br>
-: The revision comment was: <tt>expanded table</tt><br>
+: The revision comment was: <tt></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. The next column modifies intervals by another 81/80, for a total of 6561/6400 (43 cents). **Bold** inter are
+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** fractions are simplest for this interval category.
 ||||~ 3-limit interval ||||~ interval category ||||~ |5-limit interval (81/80) ||||~ |Another 5-limit (6561/6400) ||
 ||~ ratio ||~ cents value ||~   ||~   ||~ ratio ||~ cents value ||~ ratio ||~ cents value ||
-|| [[1_1|1/1]] || 0.000 || unison || C || [[81_80|81/80]] || 21.506 || [[6561_6400|6561/6400]] || 43.013 ||
+|| **[[1_1|1/1]]** || **0.000** || unison || C || [[81_80|81/80]] || 21.506 || [[6561_6400|6561/6400]] || 43.013 ||
-|| [[2187_2048|2187/2048]] || 113.685 || aug. unison || C# || [[135_128|135/128]] || 92.179 || [[25_24|25/24]] || 70.672 ||
+|| [[2187_2048|2187/2048]] || 113.685 || aug. unison || C# || [[135_128|135/128]] || 92.179 || **[[25_24|25/24]]** || **70.672** ||
-|| [[256_243|256/243]] || 90.225 || minor 2nd || Db || [[16_15|16/15]] || 111.731 || [[27_25|27/25]] || 133.238 ||
+|| [[256_243|256/243]] || 90.225 || minor 2nd || Db || **[[16_15|16/15]]** || **111.731** || [[27_25|27/25]] || 133.238 ||
-|| [[9_8|9/8]] || 203.910 || major 2nd || D || [[10_9|10/9]] || 182.404 || [[800_729|800/729]] || 160.897 ||
+|| **[[9_8|9/8]]** || **203.910** || major 2nd || D || [[10_9|10/9]] || 182.404 || [[800_729|800/729]] || 160.897 ||
-|| [[19683_16384|19683/16384]] || 317.595 || aug. 2nd || D# || [[1215_1024|1215/1024]] || 296.089 || [[75_64|75/64]] || 274.582 ||
+|| [[19683_16384|19683/16384]] || 317.595 || aug. 2nd || D# || [[1215_1024|1215/1024]] || 296.089 || **[[75_64|75/64]]** || **274.582** ||
-|| [[32_27|32/27]] || 294.135 || minor 3rd || Eb || [[6_5|6/5]] || 315.641 || [[243_200|243/200]] || 337.148 ||
+|| [[32_27|32/27]] || 294.135 || minor 3rd || Eb || **[[6_5|6/5]]** || **315.641** || [[243_200|243/200]] || 337.148 ||
-|| [[81_64|81/64]] || 407.820 || major 3rd || E || [[5_4|5/4]] || 386.314 || [[100_81|100/81]] || 364.807 ||
+|| [[81_64|81/64]] || 407.820 || major 3rd || E || **[[5_4|5/4]]** || **386.314** || [[100_81|100/81]] || 364.807 ||
-|| [[8192_6561|8192/6561]] || 384.360 || dim. fourth || Fb || [[512_405|512/405]] || 405.866 || [[32_25|32/25]] || 427.373 ||
+|| [[8192_6561|8192/6561]] || 384.360 || dim. fourth || Fb || [[512_405|512/405]] || 405.866 || **[[32_25|32/25]]** || **427.373** ||
-|| [[4_3|4/3]] || 498.045 || fourth || F || [[27_20|27/20]] || 519.551 || [[2187_1600|2187/1600]] || 541.058 ||
+|| **[[4_3|4/3]]** || **498.045** || fourth || F || [[27_20|27/20]] || 519.551 || [[2187_1600|2187/1600]] || 541.058 ||
-|| [[729_512|729/512]] || 611.730 || aug. fourth || F# || [[45_32|45/32]] || 590.224 || [[25_18|25/18]] || 568.717 ||
+|| [[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 ||
+|| [[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 ||
+|| **[[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 ||
+|| [[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 ||
+|| [[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 ||
+|| [[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 ||
+|| [[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 ||
+|| [[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 ||
+|| [[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 ||
+|| [[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 ||
+|| **[[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 60: / 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. 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;
+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; fractions are simplest for this interval category.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 94: / Line 94: @@
      &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;strong&gt;&lt;a class="wiki_link" href="/1_1"&gt;1/1&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;0.000&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;0.000&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;unison&lt;br /&gt;
@@ Line 124: / Line 124: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/25_24"&gt;25/24&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;70.672&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;70.672&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 138: / Line 138: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;111.731&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;111.731&lt;/strong&gt;&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;
@@ Line 148: / Line 148: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;203.910&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;203.910&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;major 2nd&lt;br /&gt;
@@ Line 178: / Line 178: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/75_64"&gt;75/64&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;274.582&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;274.582&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 192: / Line 192: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;315.641&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;315.641&lt;/strong&gt;&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;
@@ Line 210: / Line 210: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;386.314&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;386.314&lt;/strong&gt;&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;
@@ Line 232: / Line 232: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/32_25"&gt;32/25&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;427.373&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;427.373&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;498.045&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;498.045&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;fourth&lt;br /&gt;
@@ Line 268: / Line 268: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/25_18"&gt;25/18&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;568.717&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;568.717&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 286: / Line 286: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/36_25"&gt;36/25&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;631.283&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;631.283&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;701.955&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;701.955&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;fifth&lt;br /&gt;
@@ Line 322: / Line 322: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/25_16"&gt;25/16&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;772.627&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;772.627&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 336: / Line 336: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;813.686&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;813.686&lt;/strong&gt;&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;
@@ Line 354: / Line 354: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;884.359&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;884.359&lt;/strong&gt;&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;
@@ Line 376: / Line 376: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/128_75"&gt;128/75&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;925.418&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;925.418&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 390: / Line 390: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1017.596&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1017.596&lt;/strong&gt;&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;
@@ Line 408: / Line 408: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/15_8"&gt;15/8&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1088.269&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1088.269&lt;/strong&gt;&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;
@@ Line 430: / Line 430: @@
          &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;strong&gt;&lt;a class="wiki_link" href="/48_25"&gt;48/25&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1129.328&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1129.328&lt;/strong&gt;&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;strong&gt;&lt;a class="wiki_link" href="/2_1"&gt;2/1&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1200.000&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;1200.000&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;octave&lt;br /&gt;