https://xenreference.com/wiki/index.php?action=history&feed=atom&title=Regular_temperament 2026-09-16T13:11:12Z Revision history for this page on the wiki MediaWiki 1.44.2 https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=7695&oldid=prev 2026-06-28T18:39:00Z

<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 18:39, 28 June 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that deforms JI intervals in a way consistent with stacking. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, <del style="font-weight: bold; text-decoration: none;">if you stack </del>the <del style="font-weight: bold; text-decoration: none;">tempered version </del>of <del style="font-weight: bold; text-decoration: none;">[[9/8]], it must always produce </del>the tempered versions of <del style="font-weight: bold; text-decoration: none;">81</del>/<del style="font-weight: bold; text-decoration: none;">64, 729</del>/<del style="font-weight: bold; text-decoration: none;">512, 6561/4096, etc. The </del>tempered version of 6/5 <del style="font-weight: bold; text-decoration: none;">always differs from the tempered version of </del>4/3 <del style="font-weight: bold; text-decoration: none;">by the tempered version of 10/9, and so on</del>. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, <del style="font-weight: bold; text-decoration: none;">like </del>in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that deforms JI intervals in a way consistent with stacking. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, the <ins style="font-weight: bold; text-decoration: none;">sum </ins>of the tempered versions of <ins style="font-weight: bold; text-decoration: none;">10</ins>/<ins style="font-weight: bold; text-decoration: none;">9 and 6</ins>/<ins style="font-weight: bold; text-decoration: none;">5 must be the </ins>tempered version of <ins style="font-weight: bold; text-decoration: none;">(10/9)*(</ins>6/5<ins style="font-weight: bold; text-decoration: none;">) = </ins>4/3. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, <ins style="font-weight: bold; text-decoration: none;">just as </ins>in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </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>In order to simplify JI, [[Comma|commas]] are tempered out: small (or not-so-small) differences between intervals that the regular temperament represents with the unison. For example, in [[Meantone]], [[5/4]] (the classical major third) and [[Diatonic major third|81/64]] (the <del style="font-weight: bold; text-decoration: none;">pythagorean </del>major third) are equated. Their difference in just intonation is the ratio [[81/80]], so we say that Meantone '''tempers out''' 81/80. As a consequence, due to the rules of a regular temperament, 6561/6400, 531441/512000, etc (<del style="font-weight: bold; text-decoration: none;">ratios obtained by stacking </del>81/80) are also tempered out, but knowing that Meantone tempers out 81/80 is enough to determine this information, so it is usually left unstated.</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>In order to simplify JI, [[Comma|commas]] are tempered out: small (or not-so-small) differences between intervals that the regular temperament represents with the unison. For example, in [[Meantone]], [[5/4]] (the classical major third) and [[Diatonic major third|81/64]] (the <ins style="font-weight: bold; text-decoration: none;">Pythagorean </ins>major third) are equated. Their difference in just intonation is the ratio [[81/80]], so we say that Meantone '''tempers out''' 81/80. As a consequence, due to the rules of a regular temperament, 6561/6400, 531441/512000, etc (<ins style="font-weight: bold; text-decoration: none;">which are powers of </ins>81/80) are also tempered out, but knowing that Meantone tempers out 81/80 is enough to determine this information, so it is usually left unstated.</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>It turns out that tempering out a single comma reduces the <del style="font-weight: bold; text-decoration: none;">dimensionality </del>of an [[interval space]] by 1 (as long as that comma or any multiple of it can't be reached by stacking the other intervals you temper out)<del style="font-weight: bold; text-decoration: none;">; that is</del>, if a JI subgroup can be reached by stacking any multiples of a minimum of three distinct intervals (such as 2, 3, and 5 in the [[5-limit]]), tempering out 1 comma leads to a system built out of two distinct intervals. This means that a [[rank-2 temperament]] (one with two generators, or a period and a generator) must always temper out ''p''-2 commas in a subgroup with ''p'' primes.</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>It turns out that tempering out a single comma reduces the <ins style="font-weight: bold; text-decoration: none;">number of dimensions (also called the ''rank'') </ins>of an [[interval space]] by 1 (as long as that comma or any multiple of it can't be reached by stacking the other intervals you temper out)<ins style="font-weight: bold; text-decoration: none;">. For example</ins>, if a JI subgroup can be reached by stacking any multiples of a minimum of three distinct intervals (such as 2, 3, and 5 in the [[5-limit]]), tempering out 1 comma leads to a system built out of two distinct intervals. This means that a [[rank-2 temperament]] (one with two generators, or a period and a generator) must always temper out ''p''-2 commas in a subgroup with ''p'' primes.</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>{{adv|Mathematically, regular temperaments are commonly treated as ''abstract'' tunings where the intervals have no concrete cent values until such are assigned to the generators. This is because abstract regular temperaments, such as "Magic", are meant to abstract features that concrete regular temperaments, such as 7-limit <del style="font-weight: bold; text-decoration: none;">19edo </del>and 7-limit <del style="font-weight: bold; text-decoration: none;">22edo</del>, have in common; see the "Temperament joining" section.}}</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>{{adv|Mathematically, regular temperaments are commonly treated as ''abstract'' tunings where the intervals have no concrete cent values until such are assigned to the generators. This is because abstract regular temperaments, such as "Magic", are meant to abstract features that concrete regular temperaments, such as 7-limit <ins style="font-weight: bold; text-decoration: none;">pure-octave 19et </ins>and 7-limit <ins style="font-weight: bold; text-decoration: none;">pure-octave 22et</ins>, have in common; see the "Temperament joining" section.}}</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>== Temperament joining ==</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>== Temperament joining ==</div></td></tr> </table>

Hkm https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=5564&oldid=prev 2026-04-03T22:45:47Z

<p><span class="autocomment">See also</span></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 22:45, 3 April 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</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>* [[List of regular temperaments]]</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>* [[List of regular temperaments]]</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 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;">{{Navbox regtemp}}</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;"><div>{{Cat|Temperaments{{!}}*}}</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>{{Cat|Temperaments{{!}}*}}</div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=4306&oldid=prev 2026-02-26T00:55:47Z

<p><span class="autocomment">See also</span></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 00:55, 26 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l19">Line 19:</td> <td colspan="2" class="diff-lineno">Line 19:</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>* [[List of regular temperaments]]</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>* [[List of regular temperaments]]</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>{{Cat|<del style="font-weight: bold; text-decoration: none;">regular temperaments</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>{{Cat|<ins style="font-weight: bold; text-decoration: none;">Temperaments</ins>{{!}}*}}</div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=4305&oldid=prev 2026-02-26T00:55:38Z

<p><span class="autocomment">See also</span></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 00:55, 26 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l18">Line 18:</td> <td colspan="2" class="diff-lineno">Line 18:</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>== See also ==</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>== See also ==</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>* [[List of regular temperaments]]</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>* [[List of regular temperaments]]</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;">{{Cat|regular temperaments{{!}}*}}</ins></div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=4199&oldid=prev 2026-02-24T09:40:54Z

<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 09:40, 24 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13">Line 13:</td> <td colspan="2" class="diff-lineno">Line 13:</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>One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament&#039;s generator. This means that, while 12 &amp; 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 &amp; 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.</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>One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament&#039;s generator. This means that, while 12 &amp; 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 &amp; 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.</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;">Note that this presents a currently rather idiosyncratic way of conceptualizing higher-rank temperaments as, analogously to equal temperaments, arbitrary in [[subgroup]].</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>== See also ==</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>== See also ==</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>* [[List of regular temperaments]]</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>* [[List of regular temperaments]]</div></td></tr> </table>

Vector https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=3654&oldid=prev 2026-02-11T01:31:18Z

<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 01:31, 11 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that deforms <del style="font-weight: bold; text-decoration: none;">a given </del>JI <del style="font-weight: bold; text-decoration: none;">interval </del>in a way consistent with stacking. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that deforms JI <ins style="font-weight: bold; text-decoration: none;">intervals </ins>in a way consistent with stacking. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=3653&oldid=prev 2026-02-11T01:30:58Z

<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 01:30, 11 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that <del style="font-weight: bold; text-decoration: none;">is [[</del>consistent<del style="font-weight: bold; text-decoration: none;">]]</del>. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that <ins style="font-weight: bold; text-decoration: none;">deforms a given JI interval in a way </ins>consistent <ins style="font-weight: bold; text-decoration: none;">with stacking</ins>. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=3051&oldid=prev 2026-01-27T08:13:44Z

<p><span class="autocomment">Temperament joining</span></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 08:13, 27 January 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10">Line 10:</td> <td colspan="2" class="diff-lineno">Line 10:</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>== Temperament joining ==</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>== Temperament joining ==</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>Any two [[equal temperament]]s have one specific rank-2 temperament that they both support, which can be seen as "what those equal temperaments have in common". The temperament shared by ''x''-ET and ''y''-ET is denoted by ''x'' &amp; ''y''. For example, [[Porcupine]] is 15 &amp; 22. This means 15-ET and 22-ET (and their sum, 37-ET) have the characteristics of Porcupine in common (note 15 &amp; 37 or 22 &amp; 37 are other valid denotations of Porcupine). And they do: [[6/5]] is tuned sharp, [[4/3]] and [[10/9]] are tuned flat, (4/3)^2 is [[7/4]], ([[11/10]])^3 is 4/3, the soft tuning of the [[onyx]] scale (1L 6s MOS pattern) is supported, etc. Moreover, it means that you can take it the other way: Porcupine can be ''defined'' not by what commas it tempers out, but as "what 15-ET and 22-ET have in common". Similarly, a rank-3 temperament may be defined by three equal temperaments, and so on.</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>Any two [[equal temperament]]s have one specific rank-2 temperament that they both support, which can be seen as "what those equal temperaments have in common". The temperament shared by ''x''-ET and ''y''-ET is denoted by ''x'' &amp; ''y''. For example, [[Porcupine]] is 15 &amp; 22. This means <ins style="font-weight: bold; text-decoration: none;">[[15edo|</ins>15-ET<ins style="font-weight: bold; text-decoration: none;">]] </ins>and <ins style="font-weight: bold; text-decoration: none;">[[22edo|</ins>22-ET<ins style="font-weight: bold; text-decoration: none;">]] </ins>(and their sum, <ins style="font-weight: bold; text-decoration: none;">[[37edo|</ins>37-ET<ins style="font-weight: bold; text-decoration: none;">]]</ins>) have the characteristics of Porcupine in common (note 15 &amp; 37 or 22 &amp; 37 are other valid denotations of Porcupine). And they do: [[6/5]] is tuned sharp, [[4/3]] and [[10/9]] are tuned flat, (4/3)^2 is [[7/4]], ([[11/10]])^3 is 4/3, the soft tuning of the [[onyx]] scale (1L 6s MOS pattern) is supported, etc. Moreover, it means that you can take it the other way: Porcupine can be ''defined'' not by what commas it tempers out, but as "what 15-ET and 22-ET have in common". Similarly, a rank-3 temperament may be defined by three equal temperaments, and so on.</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>One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament&#039;s generator. This means that, while 12 &amp; 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 &amp; 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.</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>One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament&#039;s generator. This means that, while 12 &amp; 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 &amp; 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.</div></td></tr> </table>

Lériendil https://xenreference.com/wiki/index.php?title=Regular_temperament&diff=3050&oldid=prev 2026-01-27T08:13:01Z

<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 08:13, 27 January 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8">Line 8:</td> <td colspan="2" class="diff-lineno">Line 8:</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>{{adv|Mathematically, regular temperaments are commonly treated as &#039;&#039;abstract&#039;&#039; tunings where the intervals have no concrete cent values until such are assigned to the generators. This is because abstract regular temperaments, such as &quot;Magic&quot;, are meant to abstract features that concrete regular temperaments, such as 7-limit 19edo and 7-limit 22edo, have in common; see the &quot;Temperament joining&quot; section.}}</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>{{adv|Mathematically, regular temperaments are commonly treated as &#039;&#039;abstract&#039;&#039; tunings where the intervals have no concrete cent values until such are assigned to the generators. This is because abstract regular temperaments, such as &quot;Magic&quot;, are meant to abstract features that concrete regular temperaments, such as 7-limit 19edo and 7-limit 22edo, have in common; see the &quot;Temperament joining&quot; section.}}</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 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>== Temperament joining ==</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>== Temperament joining ==</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>Any two [[equal temperament]]s have one specific rank-2 temperament that they both support, which can be seen as "what those equal temperaments have in common". The temperament shared by ''x''-ET and ''y''-ET is denoted by ''x'' &amp; ''y''. For example, Porcupine is 15 &amp; 22. This means 15-ET and 22-ET (and their sum, 37-ET) have the characteristics of Porcupine in common (note 15 &amp; 37 or 22 &amp; 37 are other valid denotations of Porcupine). And they do: [[6/5]] is tuned sharp, [[4/3]] and [[10/9]] are tuned flat, (4/3)^2 is [[7/4]], ([[11/10]])^3 is 4/3, the soft tuning of the [[onyx]] scale (1L 6s MOS pattern) is supported, etc. Moreover, it means that you can take it the other way: Porcupine can be ''defined'' not by what commas it tempers out, but as "what 15-ET and 22-ET have in common". Similarly, a rank-3 temperament may be defined by three equal temperaments, and so on.</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>Any two [[equal temperament]]s have one specific rank-2 temperament that they both support, which can be seen as "what those equal temperaments have in common". The temperament shared by ''x''-ET and ''y''-ET is denoted by ''x'' &amp; ''y''. For example, <ins style="font-weight: bold; text-decoration: none;">[[</ins>Porcupine<ins style="font-weight: bold; text-decoration: none;">]] </ins>is 15 &amp; 22. This means 15-ET and 22-ET (and their sum, 37-ET) have the characteristics of Porcupine in common (note 15 &amp; 37 or 22 &amp; 37 are other valid denotations of Porcupine). And they do: [[6/5]] is tuned sharp, [[4/3]] and [[10/9]] are tuned flat, (4/3)^2 is [[7/4]], ([[11/10]])^3 is 4/3, the soft tuning of the [[onyx]] scale (1L 6s MOS pattern) is supported, etc. Moreover, it means that you can take it the other way: Porcupine can be ''defined'' not by what commas it tempers out, but as "what 15-ET and 22-ET have in common". Similarly, a rank-3 temperament may be defined by three equal temperaments, and so on.</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>One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament&#039;s generator. This means that, while 12 &amp; 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 &amp; 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.</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>One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament&#039;s generator. This means that, while 12 &amp; 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 &amp; 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.</div></td></tr> </table>

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<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 00:16, 9 January 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that is [[consistent]]. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that is [[consistent]]. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc<ins style="font-weight: bold; text-decoration: none;">. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on</ins>. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''.  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </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>[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]).  </div></td></tr> </table>

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