7-limit tuning

<h2 id="lattice-and-tonality-diamond">Lattice and tonality diamond</h2>
<p>The <a href="/facts/Tonality_diamond/SFx7AXc2">7-limit tonality diamond</a>:
</p>
<table><tbody><tr><td></td><td></td><td></td><td><a href="/facts/Harmonic_seventh/2BLT7C25">7/4</a></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td><a href="/facts/Perfect_fifth/PTIaHPuI">3/2</a></td><td></td><td><a href="/facts/Septimal_tritone/Kj0TCmbJ">7/5</a></td><td></td><td></td></tr><tr><td></td><td><a href="/facts/Major_third/7CPRURg4">5/4</a></td><td></td><td><a href="/facts/Just_minor_third/4y8zBxwY">6/5</a></td><td></td><td><a href="/facts/Septimal_minor_third/eNuNrSXe">7/6</a></td><td></td></tr><tr><td><a href="/facts/Unison/cGTkFxgu">1/1</a></td><td></td><td>1/1</td><td></td><td>1/1</td><td></td><td>1/1</td></tr><tr><td></td><td><a href="/facts/Just_minor_sixth/lyMBj5By">8/5</a></td><td></td><td><a href="/facts/Major_sixth/noVTP950">5/3</a></td><td></td><td><a href="/facts/Septimal_major_sixth/2VuYKeQo">12/7</a></td><td></td></tr><tr><td></td><td></td><td><a href="/facts/Perfect_fourth/IA1GPHBw">4/3</a></td><td></td><td><a href="/facts/Septimal_tritone/Kj0TCmbJ">10/7</a></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td><a href="/facts/Septimal_major_second/Ih7AfbPc">8/7</a></td><td></td><td></td><td></td></tr></tbody></table>
<p>This diamond contains four <a href="/facts/Limit_(music)/WYGN7KSr">identities</a> (1, 3, 5, 7 [P8, P5, M3, H7]). Similarly, the 2,3,5,7 <a href="/facts/Pitch_lattice/DeSI0bAC">pitch lattice</a> contains four identities and thus 3-4 axes, but a potentially infinite number of pitches. LaMonte Young created a lattice containing only identities 3 and 7, thus requiring only two axes, for <i>The Well-Tuned Piano</i>.
</p>
<h3>Approximation using equal temperament</h3>
<p>It is possible to approximate 7-limit music using equal temperament, for example <a href="/facts/31_equal_temperament/XfXmqNrR">31-ET</a>.
</p>
<table><tbody><tr><th>Fraction</th><th>Cents</th><th>Degree (31-ET)</th><th>Name (31-ET)</th></tr><tr><td>1/1</td><td>0</td><td>0.0</td><td>C</td></tr><tr><td>8/7</td><td>231</td><td>6.0</td><td>D or E</td></tr><tr><td>7/6</td><td>267</td><td>6.9</td><td>D♯</td></tr><tr><td>6/5</td><td>316</td><td>8.2</td><td>E♭</td></tr><tr><td>5/4</td><td>386</td><td>10.0</td><td>E</td></tr><tr><td>4/3</td><td>498</td><td>12.9</td><td>F</td></tr><tr><td>7/5</td><td>583</td><td>15.0</td><td>F♯</td></tr><tr><td>10/7</td><td>617</td><td>16.0</td><td>G♭</td></tr><tr><td>3/2</td><td>702</td><td>18.1</td><td>G</td></tr><tr><td>8/5</td><td>814</td><td>21.0</td><td>A♭</td></tr><tr><td>5/3</td><td>884</td><td>22.8</td><td>A</td></tr><tr><td>12/7</td><td>933</td><td>24.1</td><td>A or B</td></tr><tr><td>7/4</td><td>969</td><td>25.0</td><td>A♯</td></tr><tr><td>2/1</td><td>1200</td><td>31.0</td><td>C</td></tr></tbody></table>
<h2 id="ptolemys-harmonikon">Ptolemy's <i>Harmonikon</i></h2>
<p><a href="/facts/Ptolemy/htudGQTQ">Claudius Ptolemy</a> of Alexandria described several 7-limit tuning systems for the <a href="/facts/Diatonic_scale/ejXSu379">diatonic</a> and <a href="/facts/Chromatic/gyD81SHE">chromatic</a> genera. He describes several "soft" (μαλακός) diatonic tunings which all use 7-limit intervals.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> One, called by Ptolemy the "<a href="https://sevish.com/scaleworkshop/?n=Tonic%20Diatonic&l=sFr_wFr_4F3_3F2_eF9_gF9_2F1&version=2.1.0">tonic diatonic</a>," is ascribed to the <a href="/facts/Pythagoreanism/25BjMN2z">Pythagorean</a> philosopher and statesman <a href="/facts/Archytas/xx8DLsgx">Archytas of Tarentum</a>. It used the following <a href="/facts/Tetrachord/MQdCkYry">tetrachord:</a> 28:27, 8:7, 9:8. Ptolemy also shares the "<a href="https://sevish.com/scaleworkshop/?n=Aristoxenus%20soft%20diatonic&l=kFj_8F7_4F3_3F2_uFj_cF7_2F1__&version=2.1.0">soft diatonic</a>" according to <a href="/facts/Peripatetic_school/KeaVEvQG">peripatetic philosopher</a> <a href="/facts/Aristoxenus/BBcQrubX">Aristoxenus</a> of Tarentum: 20:19, 38:35, 7:6.  Ptolemy offers his own "<a href="https://sevish.com/scaleworkshop/?n=soft%20diatonic&l=lFk_7F6_4F3_3F2_1rF14_7F4_2F1&version=2.1.0">soft diatonic</a>" as the best alternative to Archytas and Aristoxenus, with a tetrachord of: 21:20, 10:9, 8:7. 
</p><p>Ptolemy also describes a "tense chromatic" tuning that utilizes the following tetrachord: 22:21, 12:11, 7:6.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/%25C4%2590%25C3%25A0n_b%25E1%25BA%25A7u/bSy3kEus">Đàn bầu</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://anaphoria.com/centaur.html">Centaur a 7 limit tuning</a> shows Centaur tuning plus other related 7 tone tunings by others</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Benson, Dave (2007). Music: A Mathematical Offering, p. 212. ISBN 9780521853873. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Partch, Harry (2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. ISBN 9780786751006. <a href="/wiki/Harry_Partch" target="_blank">/wiki/Harry_Partch</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Shirlaw, Matthew (1900). Theory of Harmony, p. 32. ISBN 978-1-4510-1534-8. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Partch, Harry (2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. ISBN 9780786751006. <a href="/wiki/Harry_Partch" target="_blank">/wiki/Harry_Partch</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Hindemith, Paul (1942). Craft of Musical Composition, vol. 1, p. 38. ISBN 0901938300. <a href="/wiki/Paul_Hindemith" target="_blank">/wiki/Paul_Hindemith</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Partch, Harry (2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, pp. 90–91. ISBN 9780786751006. <a href="/wiki/Harry_Partch" target="_blank">/wiki/Harry_Partch</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Barker, Andrew (1989). Greek Musical Writings: II Harmonic and Acoustic Theory. Cambridge: Cambridge University Press. ISBN 0521616972. <a href="0521616972" target="_blank">0521616972</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

7-limit tuning open-in-new

7-limit tuning