S-expression: Difference between revisions

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An '''S-expression''' is any product, or ratio of products, of the square superparticulars S''k'', where S''k'' is defined as the fraction of the form (''k''2~~)/(~~''k''2 - 1). Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.

An '''S-expression''' is any product, or ratio of products, of the square superparticulars S''k'', where S''k'' is defined as the fraction of the form {{sfrac|''k''2|''k''2 − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.

== Quick rules of S-expressions ==

As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply in terms of [[tempering out|tempering]]. The linked sections provide deeper information into each comma family.

* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form (''k''2~~)/(~~''k''2 - 1). Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).

* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''2|''k''2 − 1}}. Tempering out S''k'' equates {{nowrap|''k'' + 1|''k''}} and {{sfrac|''k''|''k'' − 1}}.

* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', superparticular fractions of the form (''k''(''k''+1)~~/2)/(~~(''k''-1)(''k''+2)~~/2)~~. Tempering out S''k'' * S(''k''+1) equates (''k''+2~~)/(~~''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1~~)/(~~''k''-1).

* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)|(''k'' − 1)(''k'' + 2)}}. Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} and {{nowrap|''k''|''k'' − 1}}, or {{nowrap|''k'' + 2|''k''}} and {{nowrap|''k'' + 1|''k'' − 1}}.

* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')2 * S(''k''+1)''' and '''(S''k'')2 * S(''k''-1)'''. Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))2, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')2.

* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')2 * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')2 * S(''k'' − 1)'''}}. Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}2, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}2.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: '''S''k''/S(''k''+1)'''. Tempering this out equates (''k''+2~~)/(~~''k''-1) with ((''k''+1)/''k'')3.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out equates {{nowrap|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: '''S''k''/S(''k''+2)'''. Tempering this out equates (''k''+3~~)/(~~''k''-1) with ((''k''+2)/''k'')2.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out equates {{sfrac|''k'' + 3|''k'' − 1}} with {{pars|{{sfrac|''k'' + 2|''k''}}}}2.

== Sk (square-particulars) ==

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<math>\Large \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k}</math>

which is square-(super)particular ''k'' for a given integer ''k ~~> 1~~''. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means S2 = [[4/3]] is the first musically meaningful square-particular, as S1 = 1/0.

which is square-(super)particular ''k'' for a given integer {{nowrap|''k'' > 1}}. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means {{nowrap|S2 {{=}} [[4/3]]}} is the first musically meaningful square-particular, as {{nowrap|S1 {{=}} 1/0}}.

Also note that we use the notation S''k''''p'' to mean (S''k'')''p'' rather than S(''k''''p'') for convenience in the practical analysis of regular temperaments using [[S-expression]]s.

=== Significance/motivation ===

Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''k'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.

Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.

=== Table of square-particulars ===

Below is a table of all [[31-limit]] square-particulars:

{| class="wikitable center-all

|-

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=== Alternatives to tempering square-particulars ===

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because (''k'' + 2~~)/(~~''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:

<math>\large {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3}</math>

This means you can read the ''k'' and ''k'' + 1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).

This means you can read the ''k'' and {{nowrap|''k'' + 1}} from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).

<nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.

Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|S''k''|S(''k'' + 2)}} leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.

Furthermore, defining another sequence of commas with [[semiparticular|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.

== Sk*S(k + 1) (triangle-particulars) ==

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(Note: after 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to 'break' as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.)

== S(k - 1)*Sk*S(k + 1) (1/3-square-particulars) ==

== S(k − 1)*Sk*S(k + 1) (1/3-square-particulars) ==

This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the ''k''th harmonic.

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|}

== Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars) ==

== Sk*S(k + 1)*...*S(k + n − 1) (1/n-square-particulars) ==

=== Motivation ===

1/n-square-particulars continue the pattern (1/2-square-particulars, 1/3-square-particulars, ...) to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1).

Line 2,042:

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(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)

== Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas) ==

== Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas) ==

=== Significance ===

1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.

@@ Line 1: / Line 1: @@
-An '''S-expression''' is any product, or ratio of products, of the square superparticulars S''k'', where S''k'' is defined as the fraction of the form (''k''<sup>2</sup>)/(''k''<sup>2</sup> - 1). Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.
+An '''S-expression''' is any product, or ratio of products, of the square superparticulars S''k'', where S''k'' is defined as the fraction of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.
 == Quick rules of S-expressions ==
 As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply in terms of [[tempering out|tempering]].  The linked sections provide deeper information into each comma family.
-* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form (''k''<sup>2</sup>)/(''k''<sup>2</sup> - 1). <br /> Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).
+* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}.<br />Tempering out S''k'' equates {{nowrap|''k'' + 1|''k''}} and {{sfrac|''k''|''k'' − 1}}.
-* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', superparticular fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). <br /> Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).
+* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)|(''k'' − 1)(''k'' + 2)}}.<br />Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} and {{nowrap|''k''|''k'' − 1}}, or {{nowrap|''k'' + 2|''k''}} and {{nowrap|''k'' + 1|''k'' − 1}}.
-* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')<sup>2</sup> * S(''k''+1)''' and '''(S''k'')<sup>2</sup> * S(''k''-1)'''. <br /> Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))<sup>2</sup>, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')<sup>2</sup>.
+* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' − 1)'''}}.<br />Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup>.
-* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: '''S''k''/S(''k''+1)'''. Tempering this out equates (''k''+2)/(''k''-1) with ((''k''+1)/''k'')<sup>3</sup>.
+* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out equates {{nowrap|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
-* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: '''S''k''/S(''k''+2)'''. Tempering this out equates (''k''+3)/(''k''-1) with ((''k''+2)/''k'')<sup>2</sup>.
+* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out equates {{sfrac|''k'' + 3|''k'' − 1}} with {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
 == Sk (square-particulars) ==
@@ Line 15: / Line 15: @@
 <math>\Large \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k}</math>
-which is square-(super)particular ''k'' for a given integer ''k > 1''. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means S2 = [[4/3]] is the first musically meaningful square-particular, as S1 = 1/0.
+which is square-(super)particular ''k'' for a given integer {{nowrap|''k'' &gt; 1}}. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means {{nowrap|S2 {{=}} [[4/3]]}} is the first musically meaningful square-particular, as {{nowrap|S1 {{=}} 1/0}}.
 Also note that we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of regular temperaments using [[S-expression]]s.
 === Significance/motivation ===
-Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''k'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.
+Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.
 === Table of square-particulars ===
 Below is a table of all [[31-limit]] square-particulars:
 {| class="wikitable center-all
 |-
@@ Line 483: / Line 484: @@
 === Alternatives to tempering square-particulars ===
-It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:
+It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:
 <math>\large {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3}</math>
-This means you can read the ''k'' and ''k'' + 1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).
+This means you can read the ''k'' and {{nowrap|''k'' + 1}} from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).
-<nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.
+Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|S''k''|S(''k'' + 2)}} leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.
-Furthermore, defining another sequence of commas with [[semiparticular|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.
 == Sk*S(k + 1) (triangle-particulars) ==
@@ Line 783: / Line 782: @@
 (Note: after 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to 'break' as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.)
-== S(k - 1)*Sk*S(k + 1) (1/3-square-particulars) ==
+== S(k − 1)*Sk*S(k + 1) (1/3-square-particulars) ==
 This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the ''k''th harmonic.
@@ Line 1,176: / Line 1,175: @@
 |}
-== Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars) ==
+== Sk*S(k + 1)*...*S(k + n − 1) (1/n-square-particulars) ==
 === Motivation ===
 /n-square-particulars continue the pattern (1/2-square-particulars, 1/3-square-particulars, ...) to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1).
@@ Line 2,042: / Line 2,041: @@
 (Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)
-== Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) ==
+== Sk<sup>2</sup> * S(k + 1) and S(k − 1) * Sk<sup>2</sup> (lopsided commas) ==
 === Significance ===
 . Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.