13edo: Difference between revisions
Wikispaces>igliashon **Imported revision 243300347 - Original comment: ** |
Wikispaces>igliashon **Imported revision 243312155 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-28 | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-28 18:45:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>243312155</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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=13 tone equal temperament / 13edo= | =13 tone equal temperament / 13edo= | ||
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). | 13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). | ||
|| Degree || Cents ||= Approximate Ratios* || 6L1s Names || | || Degree || Cents ||= Approximate Ratios* || 6L1s Names || 5L3s Names || | ||
|| 0 || 0 ||= 1/1 || C || | || 0 || 0 ||= 1/1 || C || C || | ||
|| 1 || 92.3077 ||= 22/21, 55/52, 117/110, 26/25 || C#/Db || | || 1 || 92.3077 ||= 22/21, 55/52, 117/110, 26/25 || C#/Db || C#/Db || | ||
|| 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D || | || 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D || D || | ||
|| 3 || 276.9231 ||= 13/11, 7/6 || D#/Eb || | || 3 || 276.9231 ||= 13/11, 7/6 || D#/Eb || D#/Eb || | ||
|| 4 || 369.2308 ||= 5/4, 16/13, 11/9, 26/21 || E || | || 4 || 369.2308 ||= 5/4, 16/13, 11/9, 26/21 || E || E || | ||
|| 5 || 461.5385 ||= 13/10, 21/16 || E#/Fb || | || 5 || 461.5385 ||= 13/10, 21/16 || E#/Fb || F || | ||
|| 6 || 553.84 ||= 11/8, 18/13 || F || | || 6 || 553.84 ||= 11/8, 18/13 || F || F#/Gb || | ||
|| 7 || 646.15 ||= 16/11, 13/9 || F#/Gb || | || 7 || 646.15 ||= 16/11, 13/9 || F#/Gb || G || | ||
|| 8 || 738.46 ||= 20/13, 32/21 || G || | || 8 || 738.46 ||= 20/13, 32/21 || G || G#/Hb || | ||
|| 9 || 830.77 ||= 8/5, 13/8, 18/11, 21/13 || G#/Ab || | || 9 || 830.77 ||= 8/5, 13/8, 18/11, 21/13 || G#/Ab || H || | ||
|| 10 || 923.08 ||= 22/13, 12/7 || A || | || 10 || 923.08 ||= 22/13, 12/7 || A || A || | ||
|| 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb || | || 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb || A#/Bb || | ||
|| 12 || 1107.69 ||= 21/11, 25/13, 104/55 || B/Cb || | || 12 || 1107.69 ||= 21/11, 25/13, 104/55 || B/Cb || B || | ||
|| 13 || 1200 ||= 2/1 || C/B# || | || 13 || 1200 ||= 2/1 || C/B# || C || | ||
*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | ||
=Harmony in 13edo= | =Harmony in 13edo= | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. | ||
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=Igliashon's 13-EDO diatonic approaches= | =Igliashon's 13-EDO diatonic approaches= | ||
From a temperament perspective, we can probably make the best use of 13-EDO as a 2.5.9.11.13 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, | From a temperament perspective, we can probably make the "best" use of 13-EDO as a 2.5.9.11.13.21 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping <1 -1| (for 5 and 13), corresponding to the 3rd horogram above. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping <2 3| (for 11 and 13). This corresponds to the 2nd horogram above. | ||
2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of <2 1| (for 5 and 9) and <2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating | 2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of <2 1| (for 5 and 9) and <2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating a full 2.5.9.11.13 pentad, which it achieves in the space of 5 generators via the octave-equivalent mapping <2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it as a tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for 13-EDO notation, leading to a notation very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart (in the 6L1s column). It can be thought of as a "circle of major 2nds" rather than a circle of 5ths. | ||
For the 2.9.21 subgroup, we can use the 5\13 generator, the closest thing 13-EDO has to a "perfect 4th", giving an octave-equivalent mapping of <3 1| and MOS scales corresponding to the 4th horogram above. The 8-note MOS scale of 5L3s, independently discovered by Easley Blackwood Jr, Paul Rapoport, and Erv Wilson (among others), is excellent for melody, being somewhat similar to the 12-TET diatonic scale but with an extra semitone added. It is also a conceivable basis for 13-EDO notation, using a modified "circle of fifths" (8\13, the octave inversion of 5\13) including an H: B#-G#-D#-A#-F#-C#-H-E-B-G-D-A-F-C-Hb-Eb-Bb-Gb-Db-Ab, which when arranged in order of ascending pitch within the octave gives the 5L3s names in the above interval chart. This notation has the advantage of preserving some familiar features: diatonic semitones still occur between B and C and E and F, and the dyads E-B, G-D, D-A, and F-C (and associated accidentals) sound approximately like "fifths". Also, the 5L3s scale on C somewhat approximates a 12-TET C major scale (if H is omitted). | |||
=Commas= | =Commas= | ||
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</td> | </td> | ||
<td>6L1s Names<br /> | <td>6L1s Names<br /> | ||
</td> | |||
<td>5L3s Names<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 97: | Line 99: | ||
</td> | </td> | ||
<td style="text-align: center;">1/1<br /> | <td style="text-align: center;">1/1<br /> | ||
</td> | |||
<td>C<br /> | |||
</td> | </td> | ||
<td>C<br /> | <td>C<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">22/21, 55/52, 117/110, 26/25<br /> | <td style="text-align: center;">22/21, 55/52, 117/110, 26/25<br /> | ||
</td> | |||
<td>C#/Db<br /> | |||
</td> | </td> | ||
<td>C#/Db<br /> | <td>C#/Db<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">10/9, 9/8, 11/10<br /> | <td style="text-align: center;">10/9, 9/8, 11/10<br /> | ||
</td> | |||
<td>D<br /> | |||
</td> | </td> | ||
<td>D<br /> | <td>D<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">13/11, 7/6<br /> | <td style="text-align: center;">13/11, 7/6<br /> | ||
</td> | |||
<td>D#/Eb<br /> | |||
</td> | </td> | ||
<td>D#/Eb<br /> | <td>D#/Eb<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">5/4, 16/13, 11/9, 26/21<br /> | <td style="text-align: center;">5/4, 16/13, 11/9, 26/21<br /> | ||
</td> | |||
<td>E<br /> | |||
</td> | </td> | ||
<td>E<br /> | <td>E<br /> | ||
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</td> | </td> | ||
<td>E#/Fb<br /> | <td>E#/Fb<br /> | ||
</td> | |||
<td>F<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>F<br /> | <td>F<br /> | ||
</td> | |||
<td>F#/Gb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>F#/Gb<br /> | <td>F#/Gb<br /> | ||
</td> | |||
<td>G<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>G<br /> | <td>G<br /> | ||
</td> | |||
<td>G#/Hb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>G#/Ab<br /> | <td>G#/Ab<br /> | ||
</td> | |||
<td>H<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td style="text-align: center;">22/13, 12/7<br /> | <td style="text-align: center;">22/13, 12/7<br /> | ||
</td> | |||
<td>A<br /> | |||
</td> | </td> | ||
<td>A<br /> | <td>A<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">9/5, 16/9, 20/11<br /> | <td style="text-align: center;">9/5, 16/9, 20/11<br /> | ||
</td> | |||
<td>A#/Bb<br /> | |||
</td> | </td> | ||
<td>A#/Bb<br /> | <td>A#/Bb<br /> | ||
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</td> | </td> | ||
<td>B/Cb<br /> | <td>B/Cb<br /> | ||
</td> | |||
<td>B<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td>C/B#<br /> | <td>C/B#<br /> | ||
</td> | |||
<td>C<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.<br /> | *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h1> | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br /> | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br /> | ||
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Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br /> | Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:345:&lt;img src=&quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><!-- ws:end:WikiTextLocalImageRule:345 --><br /> | ||
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~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br /> | ~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br /> | ||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Igliashon's 13-EDO diatonic approaches"></a><!-- ws:end:WikiTextHeadingRule:8 -->Igliashon's 13-EDO diatonic approaches</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Igliashon's 13-EDO diatonic approaches"></a><!-- ws:end:WikiTextHeadingRule:8 -->Igliashon's 13-EDO diatonic approaches</h1> | ||
<br /> | <br /> | ||
From a temperament perspective, we can probably make the best use of 13-EDO as a 2.5.9.11.13 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, | From a temperament perspective, we can probably make the &quot;best&quot; use of 13-EDO as a 2.5.9.11.13.21 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping &lt;1 -1| (for 5 and 13), corresponding to the 3rd horogram above. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping &lt;2 3| (for 11 and 13). This corresponds to the 2nd horogram above.<br /> | ||
<br /> | <br /> | ||
2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of &lt;2 1| (for 5 and 9) and &lt;2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating | 2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of &lt;2 1| (for 5 and 9) and &lt;2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating a full 2.5.9.11.13 pentad, which it achieves in the space of 5 generators via the octave-equivalent mapping &lt;2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it as a tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for 13-EDO notation, leading to a notation very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart (in the 6L1s column). It can be thought of as a &quot;circle of major 2nds&quot; rather than a circle of 5ths.<br /> | ||
<br /> | <br /> | ||
For the 2.9.21 subgroup, we can use the 5\13 generator, the closest thing 13-EDO has to a &quot;perfect 4th&quot;, giving an octave-equivalent mapping of &lt;3 1| and MOS scales corresponding to the 4th horogram above. The 8-note MOS scale of 5L3s, independently discovered by Easley Blackwood Jr, Paul Rapoport, and Erv Wilson (among others), is excellent for melody, being somewhat similar to the 12-TET diatonic scale but with an extra semitone added. It is also a conceivable basis for 13-EDO notation, using a modified &quot;circle of fifths&quot; (8\13, the octave inversion of 5\13) including an H: B#-G#-D#-A#-F#-C#-H-E-B-G-D-A-F-C-Hb-Eb-Bb-Gb-Db-Ab, which when arranged in order of ascending pitch within the octave gives the 5L3s names in the above interval chart. This notation has the advantage of preserving some familiar features: diatonic semitones still occur between B and C and E and F, and the dyads E-B, G-D, D-A, and F-C (and associated accidentals) sound approximately like &quot;fifths&quot;. Also, the 5L3s scale on C somewhat approximates a 12-TET C major scale (if H is omitted).<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->Commas</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->Commas</h1> | ||