https://xenreference.com/wiki/index.php?action=history&feed=atom&title=Comma 2026-07-30T18:21:53Z Revision history for this page on the wiki MediaWiki 1.44.2 https://xenreference.com/wiki/index.php?title=Comma&diff=4206&oldid=prev 2026-02-24T20:38:22Z

<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 20:38, 24 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td> <td colspan="2" class="diff-lineno">Line 3:</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be &#039;&#039;observed&#039;&#039; (less commonly &#039;&#039;emancipated&#039;&#039;) if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be &#039;&#039;observed&#039;&#039; (less commonly &#039;&#039;emancipated&#039;&#039;) if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which observes this comma, it does not.]]</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which observes this comma, it does not.]]</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis''), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a ''comma pump''; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis''), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a ''<ins style="font-weight: bold; text-decoration: none;">[[</ins>comma pump<ins style="font-weight: bold; text-decoration: none;">]]</ins>''; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Finding where useful commas are in a structure focusing on particular intervals — fundamentally implying mathematical &quot;coincidences&quot; —  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</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>Finding where useful commas are in a structure focusing on particular intervals — fundamentally implying mathematical &quot;coincidences&quot; —  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Comma&diff=783&oldid=prev 2025-12-15T16:57:14Z

<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 16:57, 15 December 2025</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"></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>A &#039;&#039;&#039;comma&#039;&#039;&#039; is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), most of their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>A &#039;&#039;&#039;comma&#039;&#039;&#039; is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), most of their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is ''tempered out''), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be ''observed'' (<del style="font-weight: bold; text-decoration: none;">or </del>''emancipated'') if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is ''tempered out''), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be ''observed'' (<ins style="font-weight: bold; text-decoration: none;">less commonly </ins>''emancipated'') if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which <del style="font-weight: bold; text-decoration: none;">emancipates </del>this comma, it does not.]]</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which <ins style="font-weight: bold; text-decoration: none;">observes </ins>this comma, it does not.]]</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Finding where useful commas are in a structure focusing on particular intervals — fundamentally implying mathematical &quot;coincidences&quot; —  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</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>Finding where useful commas are in a structure focusing on particular intervals — fundamentally implying mathematical &quot;coincidences&quot; —  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Comma&diff=782&oldid=prev 2025-12-15T16:56:15Z

<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 16:56, 15 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td> <td colspan="2" class="diff-lineno">Line 5:</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Finding where useful commas are in a structure focusing on particular intervals <del style="font-weight: bold; text-decoration: none;">- </del>fundamentally implying mathematical "coincidences" <del style="font-weight: bold; text-decoration: none;">- </del> is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</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>Finding where useful commas are in a structure focusing on particular intervals <ins style="font-weight: bold; text-decoration: none;">— </ins>fundamentally implying mathematical "coincidences" <ins style="font-weight: bold; text-decoration: none;">— </ins> is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Comma&diff=778&oldid=prev 2025-12-15T16:53:17Z

<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 16:53, 15 December 2025</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"></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>A &#039;&#039;&#039;comma&#039;&#039;&#039; is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), most of their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>A &#039;&#039;&#039;comma&#039;&#039;&#039; is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), most of their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is ''tempered out''), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be ''emancipated'' if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is ''tempered out''), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be <ins style="font-weight: bold; text-decoration: none;">''observed'' (or </ins>''emancipated''<ins style="font-weight: bold; text-decoration: none;">) </ins>if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which emancipates this comma, it does not.]]</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which emancipates this comma, it does not.]]</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Finding where useful commas are in a structure focusing on particular intervals - fundamentally implying mathematical &quot;coincidences&quot; -  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</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>Finding where useful commas are in a structure focusing on particular intervals - fundamentally implying mathematical &quot;coincidences&quot; -  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</div></td></tr> </table>

Inthar https://xenreference.com/wiki/index.php?title=Comma&diff=763&oldid=prev 2025-12-15T13:24:25Z

<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 13:24, 15 December 2025</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 '''comma''' is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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 '''comma''' is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), <ins style="font-weight: bold; text-decoration: none;">most of </ins>their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be &#039;&#039;emancipated&#039;&#039; if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system. A comma is said to be &#039;&#039;emancipated&#039;&#039; if it is not tempered out, i.e. represents a salient distinction within a tuning system.</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which emancipates this comma, it does not.]]</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in [[12edo]], which tempers out 81/80, this progression returns to the starting note; in [[15edo]], which emancipates this comma, it does not.]]</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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;">Finding where useful commas are in a structure focusing on particular intervals - fundamentally implying mathematical "coincidences" -  is highly nontrivial, but it can be said for example that a large number of commas can be expressed in terms of differences between nearby [[superparticular]] intervals. See [[Mathematics of commas]] for a deeper exploration of this subject.</ins></div></td></tr> </table>

Lériendil https://xenreference.com/wiki/index.php?title=Comma&diff=762&oldid=prev 2025-12-15T13:18:25Z

<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 13:18, 15 December 2025</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"></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>A &#039;&#039;&#039;comma&#039;&#039;&#039; is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>A &#039;&#039;&#039;comma&#039;&#039;&#039; is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[Interval space#Monzos|monzos]]), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is ''tempered out''), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is ''tempered out''), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the <ins style="font-weight: bold; text-decoration: none;">tuning system. A comma is said to be ''emancipated'' if it is not tempered out, i.e. represents a salient distinction within a </ins>tuning system.</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in 12edo, this progression returns to the starting note; in 15edo, it does not.]]</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in <ins style="font-weight: bold; text-decoration: none;">[[</ins>12edo<ins style="font-weight: bold; text-decoration: none;">]], which tempers out 81/80</ins>, this progression returns to the starting note; in <ins style="font-weight: bold; text-decoration: none;">[[</ins>15edo<ins style="font-weight: bold; text-decoration: none;">]], which emancipates this comma</ins>, it does not.]]</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</div></td></tr> </table>

Lériendil https://xenreference.com/wiki/index.php?title=Comma&diff=757&oldid=prev 2025-12-15T07:35:12Z

<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 07:35, 15 December 2025</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 '''comma''' is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[<del style="font-weight: bold; text-decoration: none;">monzo</del>]]<del style="font-weight: bold; text-decoration: none;">s</del>), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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 '''comma''' is a small difference between two [[stack]]s of similar or dissimilar simpler [[Just intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[<ins style="font-weight: bold; text-decoration: none;">Interval space#Monzos|monzos</ins>]]), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in 12edo, this progression returns to the starting note; in 15edo, it does not.]]</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:Qeyqry.png|thumb|459x459px|An example of a comma pump: in 12edo, this progression returns to the starting note; in 15edo, it does not.]]</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[Regular temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the &#039;&#039;comma basis&#039;&#039;), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a &#039;&#039;comma pump&#039;&#039;; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</div></td></tr> </table>

Vector https://xenreference.com/wiki/index.php?title=Comma&diff=756&oldid=prev 2025-12-15T07:33:00Z

<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 07:33, 15 December 2025</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 '''comma''' is a small difference between two [[stack]]s of similar or dissimilar simpler [[<del style="font-weight: bold; text-decoration: none;">just </del>intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[monzo]]s), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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 '''comma''' is a small difference between two [[stack]]s of similar or dissimilar simpler [[<ins style="font-weight: bold; text-decoration: none;">Just </ins>intonation|just intervals]], or a difference between intervals belonging to the same category but reached differently. For instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80, the [[syntonic comma]]. While commas are written as fractions (or [[monzo]]s), their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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> </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;">[[File:Qeyqry.png|thumb|459x459px|An example of a comma pump: in 12edo, this progression returns to the starting note; in 15edo, it does not.]]</ins></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>Every [[<del style="font-weight: bold; text-decoration: none;">regular </del>temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis''), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a ''comma pump''; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[<ins style="font-weight: bold; text-decoration: none;">Regular </ins>temperament|regular]] way of setting equivalences between intervals, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis''), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a ''comma pump''; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</div></td></tr> </table>

Vector https://xenreference.com/wiki/index.php?title=Comma&diff=750&oldid=prev 2025-12-15T07:12:52Z

<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 07:12, 15 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td> <td colspan="2" class="diff-lineno">Line 3:</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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>Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by 81/80, but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do [[5/3]] and [[27/16]], and so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma is &#039;&#039;tempered out&#039;&#039;), structural distinctions are lost in very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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>Every [[regular temperament]], including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis''), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a ''comma pump''; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</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>Every [[regular temperament<ins style="font-weight: bold; text-decoration: none;">|regular</ins>]] <ins style="font-weight: bold; text-decoration: none;">way of setting equivalences between intervals</ins>, including [[equal temperament]]s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis''), which provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence is called a ''comma pump''; knowing if a progression pumps a comma helps determine which tuning systems it can be played in without issue and which systems require modification to it.</div></td></tr> </table>

Lériendil https://xenreference.com/wiki/index.php?title=Comma&diff=749&oldid=prev 2025-12-15T07:06:29Z

<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 07:06, 15 December 2025</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 '''comma''' is <del style="font-weight: bold; text-decoration: none;">typically </del>a small <del style="font-weight: bold; text-decoration: none;">interval that, in a </del>[[<del style="font-weight: bold; text-decoration: none;">temperament</del>]], <del style="font-weight: bold; text-decoration: none;">defines the (logarithmic) </del>difference between <del style="font-weight: bold; text-decoration: none;">two other </del>intervals <del style="font-weight: bold; text-decoration: none;">which are assigned </del>to the same <del style="font-weight: bold; text-decoration: none;">approximate interval</del>. For <del style="font-weight: bold; text-decoration: none;">example</del>, the [[<del style="font-weight: bold; text-decoration: none;">Meantone|</del>syntonic comma]], 81/80, <del style="font-weight: bold; text-decoration: none;">is the difference between the </del>[[<del style="font-weight: bold; text-decoration: none;">diatonic major third</del>]] and [[<del style="font-weight: bold; text-decoration: none;">classical major third</del>]], and is tempered out in [[<del style="font-weight: bold; text-decoration: none;">meantone</del>]], <del style="font-weight: bold; text-decoration: none;">which equates them. A comma can be contrasted with a </del>[[<del style="font-weight: bold; text-decoration: none;">diesis</del>]], which is <del style="font-weight: bold; text-decoration: none;">usually understood to </del>''<del style="font-weight: bold; text-decoration: none;">not</del>'' be <del style="font-weight: bold; text-decoration: none;">tempered out</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>A '''comma''' is a small <ins style="font-weight: bold; text-decoration: none;">difference between two [[stack]]s of similar or dissimilar simpler </ins>[[<ins style="font-weight: bold; text-decoration: none;">just intonation|just intervals</ins>]], <ins style="font-weight: bold; text-decoration: none;">or a </ins>difference between intervals <ins style="font-weight: bold; text-decoration: none;">belonging </ins>to the same <ins style="font-weight: bold; text-decoration: none;">category but reached differently</ins>. For <ins style="font-weight: bold; text-decoration: none;">instance, a stack of three [[6/5]] minor thirds differs from a stack of two [[4/3]] perfect fourths by the interval 250/243, the [[porcupine]] comma; or two distinct wholetones of [[9/8]] and [[10/9]] differ by the interval 81/80</ins>, the [[syntonic comma]]<ins style="font-weight: bold; text-decoration: none;">. While commas are written as fractions (or [[monzo]]s)</ins>, <ins style="font-weight: bold; text-decoration: none;">their useful properties and characteristics come from their role as a separator, rather than the literal fraction they evaluate to.</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><ins style="font-weight: bold; text-decoration: none;">Each comma serves as the separation between many different pairs of intervals. 9/8 and 10/9 differ by </ins>81/80, <ins style="font-weight: bold; text-decoration: none;">but so do [[5/1]] and a stack of four [[3/2]] perfect fifths, and so do </ins>[[<ins style="font-weight: bold; text-decoration: none;">5/3</ins>]] and [[<ins style="font-weight: bold; text-decoration: none;">27/16</ins>]], and <ins style="font-weight: bold; text-decoration: none;">so on. These therefore become an important characteristic of the way just intervals are approximated in a given [[tuning system]]: if a comma vanishes to nothing (we say that the comma </ins>is <ins style="font-weight: bold; text-decoration: none;">''</ins>tempered out<ins style="font-weight: bold; text-decoration: none;">''), structural distinctions are lost </ins>in <ins style="font-weight: bold; text-decoration: none;">very particular and consistent places; while if a comma is exaggerated beyond its justly-tuned size, these often subtle differences are made much more prominent by the tuning system.</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><ins style="font-weight: bold; text-decoration: none;">Every </ins>[[<ins style="font-weight: bold; text-decoration: none;">regular temperament</ins>]], <ins style="font-weight: bold; text-decoration: none;">including </ins>[[<ins style="font-weight: bold; text-decoration: none;">equal temperament</ins>]]<ins style="font-weight: bold; text-decoration: none;">s, can also be defined in terms of tempering out a specific set of commas (the ''comma basis'')</ins>, which <ins style="font-weight: bold; text-decoration: none;">provides the most compact formalism for a temperament at the costs of obfuscation of the generating structure and non-uniqueness of the basis. In musical terms, every time two sequences of movements by intervals, played as represented in the temperament, would differ by one of these commas if they were played in just intonation, their result is identical. A series of musical progressions relying on this equivalence </ins>is <ins style="font-weight: bold; text-decoration: none;">called a </ins>''<ins style="font-weight: bold; text-decoration: none;">comma pump</ins>''<ins style="font-weight: bold; text-decoration: none;">; knowing if a progression pumps a comma helps determine which tuning systems it can </ins>be <ins style="font-weight: bold; text-decoration: none;">played in without issue and which systems require modification to it</ins>.</div></td></tr> </table>

Lériendil