S-expression: Difference between revisions

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== {{nowrap|S(''k'' ~~− 1)~~*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ==

== {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} (1/''n''-square-particulars) ==

~~This section concerns commas~~ of the form {{nowrap|S(''k'' ~~− 1~~) * S''k'' * S(''k'' + 1~~) {{=~~}} ~~{{sfrac| ~~{{sfrac|''k'' ~~− 1~~|''k'' − 2}}~~ ~~|~~ ~~{{~~sfrac~~|''k'' ~~+ 2~~|''k'' + 1}}~~ ~~}}}} which ~~therefore do not (directly) involve~~ the ''k''~~th harmonic~~.

=== Motivation ===

1/''n''-square-particulars are a generalization of square- and 1/2-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 {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.

In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).

These commas are important in a few ways:

1. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.

=== Significance ===

2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.

3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities.

4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.

5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …

6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.

=== Significance/implications for consistency ===

1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:

If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:

Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:

<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math>

Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' > ''x'' > ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:

<math>\displaystyle\begin{align}

\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\

\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)

\end{align}</math>

Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.

=== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ===

This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac| {{sfrac|''k'' − 1|''k'' − 2}} | {{sfrac|''k'' + 2|''k'' + 1}} }}}} which therefore do not (directly) involve the ''k''th harmonic. These are a special case of 1/''n''-square-particulars.

==== Significance ====

1. Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)

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4. Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s.

=== Proof of simplification of 1/3-square-particulars ===

==== Proof of simplification of 1/3-square-particulars ====

We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:

Line 815:

Line 853:

In other words, what this shows is all {{frac|1|3}}-square-particulars of the form {{frac|S(''k'' − 1) * S''k'' * S(''k'' + 1)}} are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd).

=== Tables ===

{| class="wikitable center-all

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|}

~~== {{nowrap|S''k''*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 {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.

In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).

~~These commas are important in a few ways:~~

1. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.

2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.

3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities.

4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.

5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …

~~6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.~~

~~=== Significance/implications for consistency ===~~

1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:

If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:

~~Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:~~

~~<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math>~~

Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' > ''x'' > ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:

~~<math>\displaystyle\begin{align}~~

~~\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\~~

~~\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)~~

~~\end{align}</math>~~

Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.

{| class="wikitable center-all\

@@ Line 780: / Line 780: @@
 |}
-== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ==
+== {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} (1/''n''-square-particulars) ==
-This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic.
+=== Motivation ===
+/''n''-square-particulars are a generalization of square- and 1/2-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 {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.
+In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).
+These commas are important in a few ways:
+. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.
-=== Significance ===
+. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.
+. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities.
+. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.
+. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …
+. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.
+=== Significance/implications for consistency ===
+/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:
+If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:
+Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:
+<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math>
+Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' &gt; ''x'' &gt; ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:
+<math>\displaystyle\begin{align}
+\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\
+\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)
+\end{align}</math>
+Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.
+=== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ===
+This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic. These are a special case of 1/''n''-square-particulars.
+==== Significance ====
 . Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)
@@ Line 792: / Line 830: @@
 . Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s.
-=== Proof of simplification of 1/3-square-particulars ===
+==== Proof of simplification of 1/3-square-particulars ====
 We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:
@@ Line 815: / Line 853: @@
 In other words, what this shows is all {{frac|1|3}}-square-particulars of the form {{frac|S(''k'' − 1) * S''k'' * S(''k'' + 1)}} are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd).
+=== Tables ===
 {| class="wikitable center-all
@@ Line 1,180: / Line 1,220: @@
 |}
-== {{nowrap|S''k''*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 {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.
-In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).
-These commas are important in a few ways:
-. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.
-. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.
-. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities.
-. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.
-. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …
-. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.
-=== Significance/implications for consistency ===
-/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:
-If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:
-Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:
-<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math>
-Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' &gt; ''x'' &gt; ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:
-<math>\displaystyle\begin{align}
-\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\
-\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)
-\end{align}</math>
-Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.
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