https://xenreference.com/wiki/index.php?action=history&feed=atom&title=Generator_gronutGenerator gronut - Revision history2026-10-21T09:34:18ZRevision history for this page on the wikiMediaWiki 1.44.2https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8295&oldid=prevInthar at 01:45, 20 October 20262026-10-20T01:45:24Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:45, 20 October 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">See [[{{FULLPAGENAME}}/Mathematics]] for a mathematical treatment.</ins></div></td></tr>
</table>Intharhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8277&oldid=prevGround at 23:30, 19 October 20262026-10-19T23:30:11Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:30, 19 October 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 <del style="font-weight: bold; text-decoration: none;">turns down </del>in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside.]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 <ins style="font-weight: bold; text-decoration: none;">of a turn clockwise </ins>in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8276&oldid=prevGround at 23:28, 19 October 20262026-10-19T23:28:49Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. <!--It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also supposedly. --> <del style="font-weight: bold; text-decoration: none;">The self-intersection may also be resolved by adding a fourth spatial dimension.</del>]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. <!--It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also supposedly.-->]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8275&oldid=prevGround at 23:28, 19 October 20262026-10-19T23:28:17Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:28, 19 October 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. <!--It may be adjusted into a type of torus knot, supposedly. --> The self-intersection may also be resolved by adding a fourth spatial dimension.]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. <!--It may be adjusted into a type of torus knot, <ins style="font-weight: bold; text-decoration: none;">supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also </ins>supposedly. --> The self-intersection may also be resolved by adding a fourth spatial dimension.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude. Center-crossing faces twist upside-down so the outside of the shape remains outside.]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude. Center-crossing faces <ins style="font-weight: bold; text-decoration: none;">approach a width of zero near the center, then </ins>twist upside-down so the outside of the shape remains outside.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8274&oldid=prevGround at 23:26, 19 October 20262026-10-19T23:26:15Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:26, 19 October 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude.]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude<ins style="font-weight: bold; text-decoration: none;">. Center-crossing faces twist upside-down so the outside of the shape remains outside</ins>.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8273&oldid=prevGround at 23:21, 19 October 20262026-10-19T23:21:40Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:21, 19 October 2026</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">Line 6:</td>
<td colspan="2" class="diff-lineno">Line 6:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude.]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8271&oldid=prevInthar at 16:40, 19 October 20262026-10-19T16:40:41Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:40, 19 October 2026</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l15">Line 15:</td>
<td colspan="2" class="diff-lineno">Line 15:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>My version of this concept uses a torus traced with a curve resembling a lemniscate <del style="font-weight: bold; text-decoration: none;">or torus knot </del>in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>My version of this concept uses a torus <ins style="font-weight: bold; text-decoration: none;">(actually a surface of revolution about the intersection point of a lemniscate) </ins>traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td></tr>
</table>Intharhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8270&oldid=prevGround at 03:27, 19 October 20262026-10-19T03:27:34Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 03:27, 19 October 2026</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td>
<td colspan="2" class="diff-lineno">Line 11:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The <del style="font-weight: bold; text-decoration: none;">next </del>and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <ins style="font-weight: bold; text-decoration: none;">second </ins>and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Tori have been used for applications like this elsewhere, such as the one </del>documented in the [[15edo#The harmonic table|15edo article]], involving a circle of <del style="font-weight: bold; text-decoration: none;">one generator </del>along the longitudes and a circle of <del style="font-weight: bold; text-decoration: none;">another </del>along the meridians.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The third is to map a harmonic table onto a torus. One of these is </ins>documented in the [[15edo#The harmonic table|15edo article]], involving a circle of <ins style="font-weight: bold; text-decoration: none;">fifths </ins>along the longitudes and a circle of <ins style="font-weight: bold; text-decoration: none;">major thirds </ins>along the meridians<ins style="font-weight: bold; text-decoration: none;">. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>My version of this concept uses a torus traced with a lemniscate<del style="font-weight: bold; text-decoration: none;">-like curve</del>, connecting notes of adjacent pitches across the center of the torus. I call this shape a <del style="font-weight: bold; text-decoration: none;">'''lemniscated torus''', or </del>'''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle. When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>My version of this concept uses a torus traced with <ins style="font-weight: bold; text-decoration: none;">a curve resembling </ins>a lemniscate <ins style="font-weight: bold; text-decoration: none;">or torus knot in 3D</ins>, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle<ins style="font-weight: bold; text-decoration: none;">, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8269&oldid=prevGround at 02:25, 19 October 20262026-10-19T02:25:43Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:25, 19 October 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. Aberrismic theory is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. <ins style="font-weight: bold; text-decoration: none;">[[Aberrisma|</ins>Aberrismic theory<ins style="font-weight: bold; text-decoration: none;">]] </ins>is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The next and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a straddled generator. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The next and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a <ins style="font-weight: bold; text-decoration: none;">[[Straddle primes|</ins>straddled generator<ins style="font-weight: bold; text-decoration: none;">]]</ins>. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Tori have been used for applications like this elsewhere, such as the one documented in the [[15edo#The harmonic table|15edo article]], involving a circle of one generator along the longitudes and a circle of another along the meridians.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Tori have been used for applications like this elsewhere, such as the one documented in the [[15edo#The harmonic table|15edo article]], involving a circle of one generator along the longitudes and a circle of another along the meridians.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>My version of this concept uses a torus traced with a lemniscate-like curve, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''lemniscated torus''', or '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle. When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>My version of this concept uses a torus traced with a lemniscate-like curve, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''lemniscated torus''', or '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle. When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div></td></tr>
</table>Groundhttps://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8266&oldid=prevInthar at 02:14, 19 October 20262026-10-19T02:14:18Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:14, 19 October 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension.]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. <ins style="font-weight: bold; text-decoration: none;"><!--</ins>It may be adjusted into a type of torus knot, supposedly. <ins style="font-weight: bold; text-decoration: none;">--> </ins>The self-intersection may also be resolved by adding a fourth spatial dimension.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]</div></td></tr>
</table>Inthar