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Septimal comma
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<h2 id="specific-commas">Specific commas</h2> <p>The 64/63 septimal comma, also known as <i><a href="/facts/Archytas/xx8DLsgx">Archytas</a>' Comma</i>,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> is the interval equal to the difference between a major and septimal whole tone (with 9/8 and 8/7 ratios, respectively). Alternatively, it can be viewed as the difference between the 16/9 <a href="/facts/Pythagorean_tuning/VjGlelBG">Pythagorean</a> <a href="/facts/Minor_seventh/Aa7YIt0e">minor seventh</a> (the composition of two 4/3 perfect fourths) and the 7/4 <a href="/facts/Harmonic_seventh/2BLT7C25">harmonic seventh</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Its size is 27.264 <a href="/facts/Cent_(music)/lsm8TJgX">cents</a>, slightly larger than the <a href="/facts/Pythagorean_comma/xsj2MvI5">Pythagorean comma</a>. </p><p>The composition of the septimal comma and the <a href="/facts/Syntonic_comma/UnE0nzpq">syntonic comma</a> is 36/35, known as the <i><a href="/facts/Septimal_diesis/uv0dyNh6">septimal diesis</a></i>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Its size is 48.8 cents, making it practically a <a href="/facts/Quarter_tone/9QwftMsC">quarter tone</a>. The septimal diesis appears as the difference between many septimal intervals and their <a href="/facts/Limit_(music)/WYGN7KSr">5-limit</a> counterparts: the minor seventh (9/5) and the seventh harmonic (7/4),<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> the 8/7 septimal <a href="/facts/Whole_tone/x94d53Fo">whole tone</a> and the 10/9 minor whole tone, the 7/6 <a href="/facts/Septimal_minor_third/eNuNrSXe">septimal minor third</a> and the 6/5 <a href="/facts/Minor_third/4y8zBxwY">minor third</a>, the 9/7 <a href="/facts/Septimal_major_third/ygaqB6TF">septimal major third</a> and the 5/4 <a href="/facts/Major_third/7CPRURg4">major third</a>, and many more. </p> <p>Other septimal commas include 49/48 (occasionally called the <i>slendro diesis</i><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>), which commonly appears as the difference between a ratio with 7 in the denominator and another with 7 in the numerator, like 8/7 and 7/6; and 50/49, called the <i>tritonic diesis</i>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> because it is the difference between the two septimal <a href="/facts/Tritone/fH2JqArb">tritones</a>, 7/5 and 10/7, or <i>Erlich's decatonic comma</i>, because it plays an important role in the ten-tone scales of <a href="/facts/Paul_Erlich/HhYKRWCz">Paul Erlich</a> (the intervals are tempered so that 50/49 vanishes). </p><p>The <a href="/facts/Septimal_kleisma/USQ5tx2k">septimal kleisma</a> and the <a href="/facts/Septimal_semicomma/CjyMh1c6">septimal semicomma</a> are smaller septimal commas. </p> <h2 id="summary">Summary</h2> <table><tbody><tr><th>Ratio</th><th>Size in cents</th><th>Ben Johnston'snotation</th><th>Names</th></tr><tr><td>64/63</td><td>27.26</td><td>C-</td><td>Septimal comma, Archytas' comma</td></tr><tr><td>50/49</td><td>34.98</td><td>B♯-</td><td><a href="/facts/Septimal_sixth-tone/CPNrN2Dk">Septimal sixth-tone</a>, tritonic diesis, Erlich's decatonic comma</td></tr><tr><td>49/48</td><td>35.7</td><td>D♭</td><td><a href="/facts/Septimal_diesis/uv0dyNh6">Slendro diesis</a></td></tr><tr><td>36/35</td><td>48.77</td><td>C</td><td><a href="/facts/Septimal_quarter_tone/BAgJM9Kp">Septimal quarter tone</a></td></tr></tbody></table> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Manuel Op de Coul. "List of intervals". Huygens-Fokker Foundation. Retrieved 2006-07-29. <a href="http://www.huygens-fokker.org/docs/intervals.html" target="_blank">http://www.huygens-fokker.org/docs/intervals.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Perrett, Wilfrid (April 1932). "The Heritage of Greece in Music". Proceedings of the Musical Association. 58: 85–103. doi:10.1093/jrma/58.1.85. JSTOR 09588442. <a href="/wiki/Proceedings_of_the_Musical_Association" target="_blank">/wiki/Proceedings_of_the_Musical_Association</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>John Fonville. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 113, Perspectives of New Music, vol. 29, no. 2 (Summer 1991), pp. 106–137. <a href="/wiki/John_Fonville" target="_blank">/wiki/John_Fonville</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Manuel Op de Coul. "List of intervals". Huygens-Fokker Foundation. Retrieved 2006-07-29. <a href="http://www.huygens-fokker.org/docs/intervals.html" target="_blank">http://www.huygens-fokker.org/docs/intervals.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Benson, Dave (2006). Music: A Mathematical Offering, p. 171. ISBN 0-521-85387-7. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Manuel Op de Coul. "List of intervals". Huygens-Fokker Foundation. Retrieved 2006-07-29. <a href="http://www.huygens-fokker.org/docs/intervals.html" target="_blank">http://www.huygens-fokker.org/docs/intervals.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>John Fonville. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 113, Perspectives of New Music, vol. 29, no. 2 (Summer 1991), pp. 106–137. <a href="/wiki/John_Fonville" target="_blank">/wiki/John_Fonville</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Manuel Op de Coul. "List of intervals". Huygens-Fokker Foundation. Retrieved 2006-07-29. <a href="http://www.huygens-fokker.org/docs/intervals.html" target="_blank">http://www.huygens-fokker.org/docs/intervals.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Manuel Op de Coul. "List of intervals". Huygens-Fokker Foundation. Retrieved 2006-07-29. <a href="http://www.huygens-fokker.org/docs/intervals.html" target="_blank">http://www.huygens-fokker.org/docs/intervals.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>