S-expression: Difference between revisions
m →Examples: order fix |
Sintel bot (talk | contribs) Fix <math> environments for MathJax v4 (align->aligned, etc.) |
||
| (16 intermediate revisions by 3 users not shown) | |||
| Line 814: | Line 814: | ||
Because of tempering out {{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: | Because of tempering out {{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{ | <math>\displaystyle\begin{aligned} | ||
\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\ | \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) | \operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right) | ||
\end{ | \end{aligned}</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. | 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. | ||
| Line 836: | Line 836: | ||
<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\ | S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\ | ||
&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\ | &= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\ | ||
&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\ | &= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\ | ||
&= \frac{k^2 - 1}{k^2 - 4} | &= \frac{k^2 - 1}{k^2 - 4} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
| Line 1,619: | Line 1,619: | ||
== S''k''/S(''k'' + 1) (ultraparticulars) == | == S''k''/S(''k'' + 1) (ultraparticulars) == | ||
Often it is desirable to make consecutive superparticular intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page. | Often it is desirable to make consecutive superparticular intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page. | ||
| Line 1,628: | Line 1,627: | ||
From this, it can be deduced that {{nowrap| (6/5)<sup>3</sup> ~ [[7/4]] }}, because one of the 6/5's can be lowered by S6 to 7/6 and another of the 6/5's can be raised by S5 to 5/4. Then because we have tempered S5 and S6 together, we have lowered and raised by the same amount, so the result of {{nowrap| (7/6)⋅(6/5)⋅(5/4) {{=}} 7/4 }} must be the same as the result of {{nowrap| (6/5)⋅(6/5)⋅(6/5)}} in this temperament. | From this, it can be deduced that {{nowrap| (6/5)<sup>3</sup> ~ [[7/4]] }}, because one of the 6/5's can be lowered by S6 to 7/6 and another of the 6/5's can be raised by S5 to 5/4. Then because we have tempered S5 and S6 together, we have lowered and raised by the same amount, so the result of {{nowrap| (7/6)⋅(6/5)⋅(5/4) {{=}} 7/4 }} must be the same as the result of {{nowrap| (6/5)⋅(6/5)⋅(6/5)}} in this temperament. | ||
The reader is encouraged to familiarize themself with the structure of this argument, as it generalizes to arbitrary S''k'' (→ [[S-expression/ | The reader is encouraged to familiarize themself with the structure of this argument, as it generalizes to arbitrary S''k'' (→ [[S-expression/Advanced results #Mathematical derivations]]); the algebraic proof is tedious, but the intuition is the same: | ||
$$ \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1} $$ | $$ \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1} $$ | ||
| Line 1,638: | Line 1,637: | ||
# Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their product. A rather-interesting arithmetic of square-particular and related commas exists. This arithmetic can be described compactly with S-expressions, which is to say, expressions composed of square superparticulars multiplied and divided together, using the S''k'' notation to achieve that compactness. | # Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their product. A rather-interesting arithmetic of square-particular and related commas exists. This arithmetic can be described compactly with S-expressions, which is to say, expressions composed of square superparticulars multiplied and divided together, using the S''k'' notation to achieve that compactness. | ||
# Tempering out the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k''⋅S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup>⋅S(''k'' + 1)}} or {{nowrap|S''k''⋅S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and {{nowrap|S(''k'' + 1)}}. This only gets better if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another motivation of the use of S-expressions. For example, {{nowrap|{S16, S17} → {S16⋅S17, S16/S17, S16<sup>2</sup>⋅S17, S16⋅S17<sup>2</sup>} }}, and any of the two commas in the latter set imply all the other commas too. | # Tempering out the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k''⋅S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup>⋅S(''k'' + 1)}} or {{nowrap|S''k''⋅S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and {{nowrap|S(''k'' + 1)}}. This only gets better if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another motivation of the use of S-expressions. For example, {{nowrap|{S16, S17} → {S16⋅S17, S16/S17, S16<sup>2</sup>⋅S17, S16⋅S17<sup>2</sup>} }}, and any of the two commas in the latter set imply all the other commas too. | ||
=== C''k'' and Cp''k'' (cube-particulars) === | |||
This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}} (for ''cube-particular complement''), and is closely related to ultraparticulars via the cube-relation identity: S''k''/S(''k'' + 1) = C''k''⋅Cp(''k'' + 1), which suggests structurally induced extensions for temperaments that temper out ultraparticulars. Applying the term ''S-expression'' to these is debatable, but they are noted here anyway because of that relation. | |||
Both C''k'' and Cp''k'' are notable because ''k''<sup>3</sup> + 1 and ''k''<sup>3</sup> − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''<sup>2</sup> + 1 can be prime. Note that as ''k'' increases, the maximal prime limit of a cube-particular grows more quickly than that of a square-particular; cube-particulars essentially rely on the factorizability of the term (''k''<sup>2</sup> + ''k'' + 1) for C''k'' or (''k''<sup>2</sup> − ''k'' + 1) for Cp''k'' to get to a reasonable prime limit. | |||
Besides being related to ultraparticulars via the cube-relation identity, cube-particulars are very sensitive to the tuning of the harmonic in question, so they can be used to keep track of how the harmonic is tuned across systems. | |||
Cube-particulars have the following properties: | |||
# Like square-particulars, all cube-particulars involve prime 2, but unlike square-particulars, some cube-particulars are no-3. These are Cp(3''k'' + 1) and C(3''k'' + 2) for any integer ''k''. | |||
# C''k''/Cp''k'' = C(''k''<sup>2</sup>). Note S''k''/Sp''k'' = S(k<sup>2</sup>) holds too if Sp''k'' notation is used for (''k''<sup>2</sup> + 1)/k<sup>2</sup>. | |||
# C(''k''<sup>2</sup>) = S(''k''<sup>3</sup>). In other words, C''k'' is a square-particular if ''k'' is a perfect square, and conversely S''k'' is a cube-particular if ''k'' is a perfect cube. | |||
=== Table of ultraparticulars === | === Table of ultraparticulars === | ||
| Line 1,887: | Line 1,898: | ||
$$ \frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3} $$ | $$ \frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3} $$ | ||
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of the bulk of the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next. | Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of the bulk of the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed in the next section. | ||
== | === Table of cube-particulars === | ||
<div><div style="display: inline-grid; margin-right: 25px;"> | |||
{| class="wikitable center-all left-3 left-6" | |||
|+ style="font-size: 105%;" | 31-limit cube-particulars – C''k'' | |||
|- | |||
! S-expression | |||
! Ratio | |||
! Subgroup | |||
|- | |||
| – | |||
| – | |||
| – | |||
|- | |||
| C2 | |||
| [[8/7]] | |||
| 2.7 | |||
|- | |||
| C3 | |||
= | | [[27/26]] | ||
| 2.3.13 | |||
|- | |||
| C4 | |||
| [[64/63]] | |||
| 2.3.7 | |||
|- | |||
| C5 | |||
| [[125/124]] | |||
| 2.5.31 | |||
|- | |||
| C7 | |||
| [[343/342]] | |||
| 2.3.7.19 | |||
|- | |- | ||
| C9 | |||
| [[729/728]] | |||
| 2.3.7.13 | |||
|- | |- | ||
| | | C11 | ||
| | | [[1331/1330]] | ||
| 2.5.7.11.19 | |||
| 2.5 | |||
|- | |- | ||
| | | C16 | ||
| [[4096/4095]] | |||
| [[ | | 2.3.5.7.13 | ||
| 3.5 | |||
|- | |- | ||
| | | C18 | ||
| [[5832/5831]] | |||
| [[ | | 2.3.7.17 | ||
| 2.3.7 | |||
|- | |- | ||
| | | C22 | ||
| | | [[10648/10647]] | ||
| 2.3.7.11.13 | |||
| 2. | |||
|- | |- | ||
| | | C25 | ||
| | | [[15625/15624]] | ||
| 2.3.5.7.31 | |||
| 2.3.5 | |||
|- | |- | ||
| | | C30 | ||
| [[27000/26999]] | |||
| [[ | | 2.3.5.7.19.29 | ||
| 3.5.7 | |||
|- | |- | ||
| | | – | ||
| | | – | ||
| | | – | ||
|- | |- | ||
| | | – | ||
| | | – | ||
| | | – | ||
| | |} | ||
</div> | |||
<div style="display: inline-grid;"> | |||
{| class="wikitable center-all left-3 left-6" | |||
|+ style="font-size: 105%;" | 31-limit cube-particulars – Cp''k'' | |||
|- | |- | ||
! S-expression | |||
! Ratio | |||
! Subgroup | |||
|- | |- | ||
| | | Cp2 | ||
| [[9/8]] | |||
| [[ | | 2.3 | ||
| | |||
|- | |- | ||
| | | Cp3 | ||
| [[28/27]] | |||
| [[ | | 2.3.7 | ||
| 2.3 | |||
|- | |- | ||
| | | Cp4 | ||
| [[65/64]] | |||
| [[ | | 2.5.13 | ||
| 2 | |||
|- | |- | ||
| | | Cp5 | ||
| [[126/125]] | |||
| [[ | | 2.3.5.7 | ||
| 2. | |||
|- | |- | ||
| | | Cp6 | ||
| | | [[217/216]] | ||
| 2.3.7.31 | |||
| 3 | |||
|- | |- | ||
| | | Cp8 | ||
| [[513/512]] | |||
| [[ | | 2.3.19 | ||
| 2.3 | |||
|- | |- | ||
| | | Cp10 | ||
| | | [[1001/1000]] | ||
| 2.5.7.11.13 | |||
| 2.5. | |||
|- | |- | ||
| | | Cp12 | ||
| | | [[1729/1728]] | ||
| 2.3.7.13.19 | |||
| 2.3. | |||
|- | |- | ||
| | | Cp17 | ||
| | | [[4914/4913]] | ||
| 2.3.7.13.17 | |||
| 3.7. | |||
|- | |- | ||
| | | Cp19 | ||
| | | [[6860/6859]] | ||
| 2.5.7.19 | |||
| 2.5. | |||
|- | |- | ||
| | | Cp23 | ||
| [[12168/12167]] | |||
| [[ | | 2.3.13.23 | ||
| 2.3. | |||
|- | |- | ||
| | | Cp26 | ||
| | | [[17577/17576]] | ||
| 2.3.7.13.31 | |||
| 2.3. | |||
|- | |- | ||
| | | Cp31 | ||
| [[29792/29791]] | |||
| [[ | | 2.7.19.31 | ||
| | |||
|- | |- | ||
| | | Cp68 | ||
| | | <small>[[314433/314432]]</small> | ||
| 2.3.7.17.23.31 | |||
| 2.3. | |||
|- | |- | ||
| | | Cp69 | ||
| <small>[[328510/328509]]</small> | |||
| 2.3.5.7.13.19.23 | |||
|} | |||
</div></div> | |||
Note that C''k'' and Cp(''k'' + 1) tend to share the same highest prime, and as we set the prime limit to 31, each C''k'' almost perfectly matches a Cp(''k'' + 1) – except for Cp68 and Cp69 at the bottom. | |||
| 2. | == S''k''/S(''k'' + 2) (semiparticulars) == | ||
=== Motivational examples === | |||
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so: | |||
* [[4/3]] is approximately half of [[9/5]] | |||
* [[9/7]] is approximately half of [[5/3]] (= 10/6) | |||
* [[5/4]] is approximately half of [[11/7]] | |||
* [[11/9]] is approximately half of [[3/2]] (= 12/8) | |||
* [[6/5]] is approximately half of [[13/9]] | |||
* [[13/11]] is approximately half of [[7/5]] (= 14/10) | |||
* [[7/6]] is approximately half of [[15/11]] | |||
* [[15/13]] is approximately half of [[4/3]] (= 16/12) | |||
* [[8/7]] is approximately half of [[17/13]] | |||
* [[17/15]] is approximately half of [[9/7]] (= 18/14) | |||
* [[9/8]] is approximately half of [[19/15]] | |||
* [[19/17]] is approximately half of [[5/4]] (= 20/16) | |||
These properties show a pattern: take some arbitrary [[#Glossary|quodd-particular]] (''k'' + 4)/''k''; observe that we can split it into (''k'' + 4)/(''k'' + 2) * (''k'' + 2)/''k''. Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''. It turns out that making this mediant — (''k'' + 3)/(''k'' + 1) — equal to half of (''k'' + 4)/''k'' is equivalent to tempering S(''k'' + 1)/S(''k'' + 3). | |||
| | |||
=== Significance === | |||
| | # For differences between square-particulars of the form S''k''/S(''k'' + 2), the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]], so these are efficient commas. (This terminology also suggests [[#Glossary|throdd-particular]] and [[#Glossary|quodd-particular]] as generalizations.) | ||
| | # Tempering out any two consecutive [[ultraparticular]]s implies tempering out a semiparticular, so from two adjacent "thirding" equivalences you get a "halving" equivalence for free. | ||
# Tempering out any two nearly-consecutive square-particulars (S''k'' and S(''k'' + 2)) implies tempering out a semiparticular; this is generally much more ideal than tempering out two consecutive S''k'' because it is of a lot lower damage (see [[lopsided comma]]s for relatively large commas implied by this higher-damage strategy). | |||
# On top of the halving equivalence, there is a number of subtler structural implications, discussed below, that may be desirable to the temperament designer. | |||
=== Meaning === | |||
In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it is equivalent to using an offset ''k''. Also keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn. | |||
Tempering out S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (= s) and ''k''/(''k'' - 2) (= L) are equidistant from (''k'' + 1)/(''k'' - 1) (= M) because, to make them equidistant, we need to temper out: | |||
<nowiki>$$ \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M²} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)²} $$</nowiki> | |||
… and notice that the latter expression is the one we have shown is equal to S(''k'' - 1)/S(''k'' + 1) up to an offset ''k'' (→ [[S-expression/Advanced results #Mathematical derivations]]). In other words, that tempering out S(''k'' - 1)/S(''k'' + 1) results in (''k'' + 1)/(''k'' - 1) being half of (''k'' + 2)/(''k'' - 2) is an implication that it makes (''k'' + 2)/''k'', (''k'' + 1)/(''k'' - 1), and ''k''/(''k'' - 2) equidistant. | |||
Also note that in the above, (''k'' + 1)/(''k'' - 1) is the mediant of the adjacent two intervals, meaning that division of an interval into two via tempering out a semiparticular is in some sense 'optimal' relative to the complexity. This also means that if ''k'' is a multiple of 2, this corresponds to a natural way to split the square superparticular S(''k''/2) into two parts. For example, if ''k'' = 10, then we have (10 + 2)/10, (10 + 1)/(10 - 1), 10/(10 - 2) as equidistant, which simplified is 6/5, 11/9, 5/4, with 11/9 being the mediant of 6/5 and 5/4, and therefore the corresponding superparticular S5 = (5/4)/(6/5) is split into two parts which are tempered together: (5/4)/(11/9) = 45/44 and (11/9)/(6/5) = 55/54. The semiparticular is therefore S(10 - 1)/S(10 + 1) = S9/S11 = 243/242 = (45/44)/(55/54) = ((10 + 2)/(10 - 2))/((10 + 1)/(10 - 1))<sup>2</sup>. | |||
This form of comma has been named ''semiparticular'' because most of the time it is superparticular but less often it is odd-particular, and because, when tempered out, they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular. Specifically: | |||
* To find out what a superparticular (''a'' + 1)/''a'' is approximately half of, temper out the semiparticular S(2''a'')/S(2''a''+2) and you can observe that (2''a'' + 3)/(2''a'' - 1) is the interval it is approximately half of. | |||
* To find out what an odd-particular (2''a'' + 1)/(2''a'' - 1) is approximately half of, temper out the semiparticular S(2''a'' - 1)/S(2''a'' + 1) and you can observe that (2''a'' + 2)/(2''a'' - 2) = (''a'' + 1)/(''a'' - 1), a superparticular or odd-particular, is the interval it is approximately half of. | |||
* To find out what splits a superparticular (''a'' + 1)/''a'' in half, temper out the semiparticular S(4''a'' + 1)/S(4''a'' + 3) and you can observe that (4''a'' + 3)/(4''a'' + 1), an odd-particular, is the interval that is approximately half of it. | |||
* To find out what splits an odd-particular (2''a'' + 1)/(2''a''-1) in half, temper out the semiparticular S(4''a'' - 2)/S(4''a'' + 2) and you can observe that (4''a'' - 1)/(4''a'' + 1), an odd-particular, is the interval that is approximately half of it. | |||
Also, the interval in the denominator of an expression of a semiparticular of the form (''a''/''b'')/(''c''/''d'')<sup>2</sup> is significant in that it has a special relationship: specifically, consider tempering out (''a''/''b'')/(''c''/''d'')<sup>2</sup> so the interval ''c''/''d'' is equal to the interval (''a''/''b'')/(''c''/''d''). This is significant because it allows the intuitive replacement of two consecutive superparticulars, whose product is a superparticular or odd-particular, with the two superparticulars directly adjacent to them. | |||
For example, from 9/8 = (18/17)⋅(17/16), we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering out S16/S18 = (19/15)/(9/8)<sup>2</sup> because we can multiply 9/8 by the vanishing comma (19/15)/(9/8)<sup>2</sup> to get (19/15)/(9/8) = (19/18)(16/15), or as 13/11 = (13/12)⋅(12/11) we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering out S11/S13 = (7/5)/(13/11)<sup>2</sup> because we can multiply 13/11 by the vanishing comma (7/5)/(13/11)<sup>2</sup> to get (7/5)/(13/11) = (14/13)⋅(11/10). Both intervals should be replaced simultaneously as this results in lower error, and note that to replace them individually we must pick the higher-error route of tempering out S16 and S18 or S11 and S13 individually, for which tempering out the semiparticular is then an implied consequence. The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas. | |||
=== Table of semiparticulars === | |||
Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing superparticular intervals up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper out the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table: | |||
{| class="wikitable center-all left-4" | |||
|- | |- | ||
! S-expression | |||
! Square relation | |||
! Ratio | |||
! Subgroup | |||
|- | |- | ||
| | | S2/S4 = ([[4/3]])/([[16/15]]) | ||
| ([[ | | ([[5/1]])/([[2/1]])<sup>2</sup> | ||
| [[ | | [[5/4]] | ||
| 2.5 | | 2.5 | ||
|- | |- | ||
| | | S3/S5 = ([[9/8]])/([[25/24]]) | ||
| ([[ | | ([[3/1]])/([[5/3]])<sup>2</sup> | ||
| [[ | | [[27/25]] | ||
| | | 3.5 | ||
|- | |- | ||
| | | S4/S6 = ([[16/15]])/([[36/35]]) | ||
| ([[ | | ([[7/3]])/([[3/2]])<sup>2</sup> | ||
| [[ | | [[28/27]] | ||
| 2.3 | | 2.3.7 | ||
|- | |- | ||
| | | S5/S7 = ([[25/24]])/([[49/48]]) | ||
| ([[ | | ([[2/1]])/([[7/5]])<sup>2</sup> | ||
| [[ | | [[50/49]] | ||
| 2 | | 2.5.7 | ||
|- | |- | ||
| | | S6/S8 = ([[36/35]])/([[64/63]]) | ||
| ([[ | | ([[9/5]])/([[4/3]])<sup>2</sup> | ||
| [[ | | [[81/80]] | ||
| 2.5 | | 2.3.5 | ||
|- | |- | ||
| | | S7/S9 = ([[49/48]])/([[81/80]]) | ||
| ([[ | | ([[5/3]])/([[9/7]])<sup>2</sup> | ||
| [[ | | [[245/243]] | ||
| | | 3.5.7 | ||
|- | |- | ||
| | | S8/S10 = ([[64/63]])/([[100/99]]) | ||
| ([[ | | ([[11/7]])/([[5/4]])<sup>2</sup> | ||
| [[ | | [[176/175]] | ||
| 2 | | 2.5.7.11 | ||
|- | |- | ||
| | | S9/S11 = ([[81/80]])/([[121/120]]) | ||
| ([[ | | ([[3/2]])/([[11/9]])<sup>2</sup> | ||
| [[ | | [[243/242]] | ||
| 2.3. | | 2.3.11 | ||
|- | |- | ||
| | | S10/S12 = ([[100/99]])/([[144/143]]) | ||
| ([[ | | ([[13/9]])/([[6/5]])<sup>2</sup> | ||
| [[ | | [[325/324]] | ||
| 2.3.5.11.13. | | 2.3.5.13 | ||
|- | |||
| S11/S13 = ([[121/120]])/([[169/168]]) | |||
| ([[7/5]])/([[13/11]])<sup>2</sup> | |||
| [[847/845]] | |||
| 5.7.11.13 | |||
|- | |||
| S12/S14 = ([[144/143]])/([[196/195]]) | |||
| ([[15/11]])/([[7/6]])<sup>2</sup> | |||
| [[540/539]] | |||
| 2.3.5.7.11 | |||
|- | |- | ||
| | | S13/S15 = ([[169/168]])/([[225/224]]) | ||
| ([[ | | ([[4/3]])/([[15/13]])<sup>2</sup> | ||
| [[ | | [[676/675]] | ||
| 2.3.5. | | 2.3.5.13 | ||
|- | |- | ||
| | | S14/S16 = ([[196/195]])/([[256/255]]) | ||
| ([[ | | ([[17/13]])/([[8/7]])<sup>2</sup> | ||
| [[ | | [[833/832]] | ||
| 2 | | 2.7.13.17 | ||
|- | |||
| S15/S17 = ([[225/224]])/([[289/288]]) | |||
| ([[9/7]])/([[17/15]])<sup>2</sup> | |||
| [[2025/2023]] | |||
| 3.5.7.17 | |||
|- | |||
| S16/S18 = ([[256/255]])/([[324/323]]) | |||
| ([[19/15]])/([[9/8]])<sup>2</sup> | |||
| [[1216/1215]] | |||
| 2.3.5.19 | |||
|- | |||
| S17/S19 = ([[289/288]])/([[361/360]]) | |||
| ([[5/4]])/([[19/17]])<sup>2</sup> | |||
| [[1445/1444]] | |||
| 2.5.17.19 | |||
< | |||
</ | |||
|- | |- | ||
| S18/S20 = ([[324/323]])/([[400/399]]) | |||
| ([[21/17]])/([[10/9]])<sup>2</sup> | |||
| [[1701/1700]] | |||
| 2.3.5.7.17 | |||
|- | |- | ||
| | | S19/S21 = ([[361/360]])/([[441/440]]) | ||
| ([[11/9]])/([[21/19]])<sup>2</sup> | |||
| [[ | | [[3971/3969]] | ||
| | | 3.7.11.19 | ||
|- | |- | ||
| | | S20/S22 = ([[400/399]])/([[484/483]]) | ||
| ([[23/19]])/([[11/10]])<sup>2</sup> | |||
| [[ | | [[2300/2299]] | ||
| 2.5. | | 2.5.11.19.23 | ||
|- | |- | ||
| | | S21/S23 = ([[441/440]])/([[529/528]]) | ||
| ([[6/5]])/([[23/21]])<sup>2</sup> | |||
| [[ | | [[2646/2645]] | ||
| 2.5.7 | | 2.3.5.7.23 | ||
|- | |- | ||
| | | S22/S24 = ([[484/483]])/([[576/575]]) | ||
| ([[25/21]])/([[12/11]])<sup>2</sup> | |||
| [[ | | [[3025/3024]] | ||
| 2.3.5.7 | | 2.3.5.7.11 | ||
|- | |- | ||
| | | S23/S25 = ([[529/528]])/([[625/624]]) | ||
| ([[13/11]])/([[25/23]])<sup>2</sup> | |||
| [[ | | [[6877/6875]] | ||
| | | 5.11.13.23 | ||
|- | |- | ||
| | | S24/S26 = ([[576/575]])/([[676/675]]) | ||
| ([[27/23]])/([[13/12]])<sup>2</sup> | |||
| [[ | | [[3888/3887]] | ||
| 2.3. | | 2.3.13.23 | ||
|- | |- | ||
| | | S25/S27 = ([[625/624]])/([[729/728]]) | ||
| ([[7/6]])/([[27/25]])<sup>2</sup> | |||
| [[ | | [[4375/4374]] | ||
| 2.3.5.7 | | 2.3.5.7 | ||
|- | |- | ||
| | | S26/S28 = ([[676/675]])/([[784/783]]) | ||
| ([[29/25]])/([[14/13]])<sup>2</sup> | |||
| [[ | | [[4901/4900]] | ||
| 2 | | 2.5.7.13.29 | ||
|- | |- | ||
| | | S27/S29 = ([[729/728]])/([[841/840]]) | ||
| ([[15/13]])/([[29/27]])<sup>2</sup> | |||
| [[ | | [[10935/10933]] | ||
| | | 3.5.13.29 | ||
|- | |- | ||
| | | S28/S30 = ([[784/783]])/([[900/899]]) | ||
| ([[31/27]])/([[15/14]])<sup>2</sup> | |||
| [[ | | [[6076/6075]] | ||
| 2.7. | | 2.3.5.7.31 | ||
|- | |- | ||
| | | S29/S31 = ([[841/840]])/([[961/960]]) | ||
| ([[8/7]])/([[31/29]])<sup>2</sup> | |||
| [[ | | [[6728/6727]] | ||
| 2 | | 2.7.29.31 | ||
|- | |- | ||
| | | S30/S32 = ([[900/899]])/([[1024/1023]]) | ||
| ([[33/29]])/([[16/15]])<sup>2</sup> | |||
| [[ | | [[7425/7424]] | ||
| 2.5. | | 2.3.5.11.29 | ||
|- | |- | ||
| | | S31/S33 = ([[961/960]])/([[1089/1088]]) | ||
| ([[17/15]])/([[33/31]])<sup>2</sup> | |||
| [[ | | [[16337/16335]] | ||
| | | 3.5.11.17.31 | ||
|- | |- | ||
| | | S32/S34 = ([[1024/1023]])/([[1156/1155]]) | ||
| ([[35/31]])/([[17/16]])<sup>2</sup> | |||
| [[ | | [[8960/8959]] | ||
| 2 | | 2.5.7.17.31 | ||
|- | |- | ||
| | | S33/S35 = ([[1089/1088]])/([[1225/1224]]) | ||
| ([[9/8]])/([[35/33]])<sup>2</sup> | |||
| [[ | | [[9801/9800]] | ||
| 2.3. | | 2.3.5.7.11 | ||
|- | |- | ||
| | | S36/S38 = ([[1296/1295]])/([[1444/1443]]) | ||
| ([[39/35]])/([[19/18]])<sup>2</sup> | |||
| [[ | | [[12636/12635]] | ||
| 2.3.5.7. | | 2.3.5.7.13.19 | ||
|- | |- | ||
| | | S37/S39 = ([[1369/1368]])/([[1521/1520]]) | ||
| ([[10/9]])/([[39/37]])<sup>2</sup> | |||
| [[ | | [[13690/13689]] | ||
| 2.3.5. | | 2.3.5.13.37 | ||
|- | |- | ||
| | | S41/S43 = ([[1681/1680]])/([[1849/1848]]) | ||
| ([[11/10]])/([[43/41]])<sup>2</sup> | |||
| [[ | | [[18491/18490]] | ||
| 2 | | 2.5.11.41.43 | ||
|- | |- | ||
| | | S45/S47 = ([[2025/2024]])/([[2209/2208]]) | ||
| ([[12/11]])/([[47/45]])<sup>2</sup> | |||
| [[ | | [[24300/24299]] | ||
| 2.5 | | 2.3.5.11.47 | ||
|- | |- | ||
| | | S46/S48 = ([[2116/2115]])/([[2304/2303]]) | ||
| ([[49/45]])/([[24/23]])<sup>2</sup> | |||
| [[ | | [[25921/25920]] | ||
| 2.5.7 | | 2.3.5.7.23 | ||
|- | |- | ||
| | | S49/S51 = ([[2401/2400]])/([[2601/2600]]) | ||
| ([[13/12]])/([[51/49]])<sup>2</sup> | |||
| [[ | | [[31213/31212]] | ||
| 2. | | 2.3.7.13.17 | ||
|- | |- | ||
| | | S52/S54 = ([[2704/2703]])/([[2916/2915]]) | ||
| ([[55/51]])/([[27/26]])<sup>2</sup> | |||
| [[ | | [[37180/37179]] | ||
| 2.3.5.11.13. | | 2.3.5.11.13.17 | ||
|- | |- | ||
| | | S66/S68 = ([[4356/4355]])/([[4624/4623]]) | ||
| ([[69/65]])/([[34/33]])<sup>2</sup> | |||
| [[ | | [[75141/75140]] | ||
| 2.3.5. | | 2.3.5.11.17.23 | ||
|- | |- | ||
| | | S78/S80 = ([[6084/6083]])/([[6400/6399]]) | ||
| ([[81/77]])/([[40/39]])<sup>2</sup> | |||
| [[ | | [[123201/123200]] | ||
| 2.3.5.7.13. | | 2.3.5.7.11.13 | ||
|- | |} | ||
| | |||
{{Note| While a lot of these have pages, not all of them do, although that does not mean they should not. 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. }} | |||
| [[ | |||
| | == {{nowrap|S''k''/S(''k'' + 3)}} (three-particulars) == | ||
|- | |||
| | === Significance === | ||
# It it surprising that this form corresponds to yet another infinite family to do with spacing of obviously-related intervals. It thusly also helps further motivate expressing comma lists in terms of chains of square-particulars that are equated, either via {{nowrap| S''x'' {{=}} S''y'' {{=}} S''z'' {{=}} ... }} or via {{nowrap| { S''x''/''y''/''z''/... }, with ''x'' < ''y'' < ''z'' }}. | |||
# Relatedly to the previous point, they are used in the construction of ''pentaparticulars'', which take the form S''k''/S(''k'' + 3) * ( S(''k'' + 1)/S(''k'' + 2) )<sup>2</sup>, which are equal to how (''k'' + 2)/(''k'' + 1) is approximately a fifth of (''k'' + 4)/(''k'' - 1), and are intuitively obvious from the idea of making superparticulars equidistant via ultraparticulars. | |||
# More trivially, they are implied in various simple S-expression-based comma lists, like [[58edo]]'s [[17-limit]] as describable by {S6/S7, S8/9/10/11/13, S12, S14, S16, S17}, where S8/9/10/11/13 is a shorthand for [[64/63|S8]] = [[81/80|S9]] = [[100/99|S10]] = [[121/120|S11]] = [[169/168|S13]], wherein we have {{nowrap| three-particulars }} {{nowrap| { S8/S11 {{=}} ( (10/7)/(11/8) )/( (11/8)/(4/3) ), S10/S13 {{=}} ( (4/3)/(13/10) )/( (13/10)/(14/11) ), S14/S17 {{=}} ( (16/13)/(17/14) )/( (17/14)/(6/5) ) }. }} | |||
# Three-particulars currently have no ''simple'' known splitting property due to the three intervals made equidistant not obviously composing to some other interval of significance, so seem to instead be about making exact spacings that feel approximately equidistant. For example, in 7:8:9:10:11:12, the closest is to notice the smallest (7:10) and largest (9:12) interval multiply to an interval which is 11/10 more than 7:12, and hence in general splits an interval (''k'' + 4)/(''k'' - 1) * (''k'' + 2)/(''k'' + 1) into two parts of (''k'' + 3)/''k''. Because the multiplication by a superparticular interval is a rather peculiar requirement, we leave it here as a note. | |||
=== Derivation === | |||
For a general introduction to method we use here, see [[#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], as we will not re-explain the method here. | |||
We start with: | |||
<pre> | |||
Sk = [k-1, k, k+1]^[-1, 2, -1] | |||
</pre> | |||
From which we can examine | |||
<pre> | |||
| | Sk / S(k+3) = [k-1, k, k+1]^[-1, 2, -1] / [k+2, k+3, k+4]^[-1, 2, -1] | ||
= [k-1, k, k+1]^[-1, 2, -1] * [k+2, k+3, k+4]^[1, -2, 1] | |||
| [ | = [k-1, k, k+1, k+2, k+3, k+4]^[-1, 2, -1, 1, -2, 1] | ||
| 2. | </pre> | ||
It is initially not obvious that this corresponds to anything interesting. However, as discovered by [[Lériendil]] (though not by this method), tempering out the comma makes (''k'' + 2)/(''k'' - 1), (''k'' + 3)/''k'', (''k'' + 4)/(''k'' + 1) equidistant, which is not arcane to understand either; we have three intervals of the form (''a'' + 3)/''a'' for adjacent values of ''a'' made equidistant, and the placement of ''a'' is obvious w.r.t the S-expression, being "the only place it could be" (in the center). Thus, we will show that the S-factorization derived from making these equidistant is equivalent: | |||
<pre> | |||
| | let F(k + 1) = ( (k + 2)/(k-1) )/( (k + 3)/k ) = [k-1, k, k+1, k+2, k+3]^[-1, 1, 0, 1, -1] | ||
| | let F(k + 2) = ( (k + 3)/k )/( (k + 4)/(k + 1) ) = [k, k+1, k+2, k+3, k+4]^[-1, 1, 0, 1, -1] | ||
then F(k + 1)/F(k + 2) = [k-1, k, k+1, k+2, k+3, k+4]^([-1, 1, 0, 1, -1, 0] - [0, -1, 1, 0, 1, -1]) | |||
= [k-1, k, k+1, k+2, k+3, k+4]^( [-1, 1, 0, 1, -1, 0] | |||
+ [0, 1, -1, 0, -1, 1] ) | |||
= [k-1, k, k+1, k+2, k+3, k+4]^[ -1, 2, -1, 1, -2, 1 ] | |||
</pre> | |||
...as expected. In fact, if you look at the expressions in the table, it becomes obvious why we have the central element squared (via a 2 and -2 in the S-factorization), as it follows from the general algebraic form of {{nowrap| ''n''<sup>2</sup> / (''n''<sup>2</sup> - 1) {{=}} ( ''n''/(''n'' - 1) )/( (''n'' + 1)/''n'' ) }}. | |||
'''(An important corollary that should be noted:''' The same derivation above can be observed as not depending on the number of 0's in the S-factorization of the expressions of {{nowrap| <code>F(k + 1)</code> and <code>F(k + 2)</code> }}, and that therefore we have S''a''/S''b'' being an equidistance relation of intervals, where the generalized versions of the terms {{nowrap| <code>F(k + 1)</code> and <code>F(k + 2)</code> }} correspond to the two differences of intervals of the form (''n'' + ''k'')/''n'', where ''k'' = 3 corresponds to one zero in the monzo and zero in {{nowrap| <code>[ -1, 2, -1, 1, -2, 1 ]</code> }}, and where the terms {{nowrap| <code>F(k + 1)</code> and <code>F(k + 2)</code> }} therefore correspond to the two differences used to make three intervals of the form (''n'' + ''k'')/''n'' equidistant. "Which three intervals" is thus answered as those of 3 consecutive/adjacent values of ''n''. '''Therefore:''' What makes three-particulars special in this more general S-comma family is that the largest and smallest interval of the three almost compose to something simple, up to a superparticular difference, s.t the corresponding splitting relation can often simplify for cases of interest, as discussed in [[S-expression#Significance_7|#Significance]]'''.)''' | |||
=== Table of three-particulars === | |||
Beyond the obvious range, 43-limit three-particulars become very sparse, hence S286/S289 is the first to break this pattern. | |||
However, as it concerns intervals so small they can be considered commas, we do not extend the table further, as it's equal to {{sfrac| ([[96/95]])/([[289/286]]) | ([[289/286]])/([[290/287]]) }}. | |||
(The 43-limit is picked due to 47/32 being close to 3/2, and 43/32 being close to 4/3, as well as to avoid spam.) | |||
{| class="wikitable center-all | |||
|- | |- | ||
! S-expression | |||
! Square relation | |||
! Ratio | |||
! Subgroup | |||
|- | |- | ||
| | | S2/S5 | ||
| ( ([[ | | ( ([[4/1]])/([[5/2]]) )/( ([[5/2]])/([[2/1]]) ) | ||
| [[ | | [[32/25]] | ||
| 2 | | 2.5 | ||
|- | |- | ||
| | | S3/S6 | ||
| ( ([[ | | ( ([[5/2]])/([[2/1]]) )/( ([[2/1]])/([[7/4]]) ) | ||
| [[ | | [[35/32]] | ||
| 2 | | 2.5.7 | ||
|- | |- | ||
| | | S4/S7 | ||
| ( ([[ | | ( ([[2/1]])/([[7/4]]) )/( ([[7/4]])/([[8/5]]) ) | ||
| [[ | | [[256/245]] | ||
| 2 | | 2.5.7 | ||
|- | |- | ||
| | | S5/S8 | ||
| ( ([[ | | ( ([[7/4]])/([[8/5]]) )/( ([[8/5]])/([[3/2]]) ) | ||
| [[ | | [[525/512]] | ||
| 2.3.7 | | 2.3.5.7 | ||
|- | |- | ||
| | | S6/S9 | ||
| ( ([[ | | ( ([[8/5]])/([[3/2]]) )/( ([[3/2]])/([[10/7]]) ) | ||
| [[ | | [[64/63]] | ||
| 2.3. | | 2.3.7 | ||
|- | |- | ||
| | | S7/S10 | ||
| ( ([[ | | ( ([[3/2]])/([[10/7]]) )/( ([[10/7]])/([[11/8]]) ) | ||
| [[ | | [[1617/1600]] | ||
| 2.5.7. | | 2.3.5.7.11 | ||
|- | |- | ||
| | | S8/S11 | ||
| ( ([[ | | ( ([[10/7]])/([[11/8]]) )/( ([[11/8]])/([[4/3]]) ) | ||
| [[ | | [[2560/2541]] | ||
| 2.5.7. | | 2.3.5.7.11 | ||
|- | |- | ||
| | | S9/S12 | ||
| ( ([[ | | ( ([[11/8]])/([[4/3]]) )/( ([[4/3]])/([[13/10]]) ) | ||
| [[ | | [[1287/1280]] | ||
| 2.5 | | 2.3.5.11.13 | ||
|- | |- | ||
| | | S10/S13 | ||
| ( ([[ | | ( ([[4/3]])/([[13/10]]) )/( ([[13/10]])/([[14/11]]) ) | ||
| [[ | | [[5600/5577]] | ||
| 2.3.7.11. | | 2.3.5.7.11.13 | ||
|- | |- | ||
| | | S11/S14 | ||
| ( ([[ | | ( ([[13/10]])/([[14/11]]) )/( ([[14/11]])/([[5/4]]) ) | ||
| [[ | | [[1573/1568]] | ||
| 2.3.5.7. | | 2.7.11.13 | ||
|- | |||
| S12/S15 | |||
| ( ([[14/11]])/([[5/4]]) )/( ([[5/4]])/([[16/13]]) ) | |||
| [[3584/3575]] | |||
| 2.5.7.11.13 | |||
|- | |||
| S13/S16 | |||
| ( ([[5/4]])/([[16/13]]) )/( ([[16/13]])/([[17/14]]) ) | |||
| [[14365/14336]] | |||
| 2.5.7.13.17 | |||
|- | |||
| S14/S17 | |||
| ( ([[16/13]])/([[17/14]]) )/( ([[17/14]])/([[6/5]]) ) | |||
| [[18816/18785]] | |||
| 2.3.5.7.13.17 | |||
|- | |- | ||
| | | S15/S18 | ||
| ( ([[ | | ( ([[17/14]])/([[6/5]]) )/( ([[6/5]])/([[19/16]]) ) | ||
| [[ | | [[8075/8064]] | ||
| 2.3.7 | | 2.3.5.7.17.19 | ||
|- | |- | ||
| S16/S19 | |||
| ( ([[6/5]])/([[19/16]]) )/( ([[19/16]])/([[20/17]]) ) | |||
| [[6144/6137]] | |||
| 2.3.17.19 | |||
|- | |- | ||
| ([[ | | S17/S20 | ||
| ( ([[19/16]])/([[20/17]]) )/( ([[20/17]])/([[7/6]]) ) | |||
| [[ | | [[38437/38400]] | ||
| 2.3 | | 2.3.5.7.17.19 | ||
|- | |- | ||
| ([[ | | S18/S21 | ||
| ( ([[20/17]])/([[7/6]]) )/( ([[7/6]])/([[22/19]]) ) | |||
| [[ | | [[15840/15827]] | ||
| 2.5.7 | | 2.3.5.7.11.17.19 | ||
|- | |- | ||
| ([[ | | S19/S22 | ||
| ( ([[7/6]])/([[22/19]]) )/( ([[22/19]])/([[23/20]]) ) | |||
| [[ | | [[58121/58080]] | ||
| 2.3.5 | | 2.3.5.7.11.19.23 | ||
|- | |- | ||
| ([[ | | S20/S23 | ||
| ( ([[22/19]])/([[23/20]]) )/( ([[23/20]])/([[8/7]]) ) | |||
| [[ | | [[70400/70357]] | ||
| 2. | | 2.5.7.11.19.23 | ||
|- | |- | ||
| ([[ | | S21/S24 | ||
| ( ([[23/20]])/([[8/7]]) )/( ([[8/7]])/([[25/22]]) ) | |||
| [[ | | [[5635/5632]] | ||
| 2.7 | | 2.5.7.11.23 | ||
|- | |- | ||
| ([[ | | S22/S25 | ||
| ( ([[8/7]])/([[25/22]]) )/( ([[25/22]])/([[26/23]]) ) | |||
| [[ | | [[100672/100625]] | ||
| 2. | | 2.5.7.11.13.23 | ||
|- | |- | ||
| ([[ | | S23/S26 | ||
| ( ([[25/22]])/([[26/23]]) )/( ([[26/23]])/([[9/8]]) ) | |||
| [[ | | [[119025/118976]] | ||
| 2.3.5. | | 2.3.5.11.13.23 | ||
|- | |- | ||
| ([[ | | S24/S27 | ||
| ( ([[26/23]])/([[9/8]]) )/( ([[9/8]])/([[28/25]]) ) | |||
| [[ | | [[46592/46575]] | ||
| 2.5. | | 2.3.5.7.13.23 | ||
|- | |- | ||
| ([[ | | S25/S28 | ||
| ( ([[9/8]])/([[28/25]]) )/( ([[28/25]])/([[29/26]]) ) | |||
| [[ | | [[163125/163072]] | ||
| 2.3.7. | | 2.3.5.7.13.29 | ||
|- | |- | ||
| ([[ | | S26/S29 | ||
| ( ([[28/25]])/([[29/26]]) )/( ([[29/26]])/([[10/9]]) ) | |||
| [[ | | [[37856/37845]] | ||
| 2.3.13 | | 2.3.5.7.13.29 | ||
|- | |- | ||
| ([[ | | S27/S30 | ||
| ( ([[29/26]])/([[10/9]]) )/( ([[10/9]])/([[31/28]]) ) | |||
| [[ | | [[72819/72800]] | ||
| 2.7. | | 2.3.5.7.13.29.31 | ||
|- | |- | ||
| ([[ | | S28/S31 | ||
| ( ([[10/9]])/([[31/28]]) )/( ([[31/28]])/([[32/29]]) ) | |||
| [[ | | [[250880/250821]] | ||
| 2.3.5.7. | | 2.3.5.7.29.31 | ||
|- | |- | ||
| ([[ | | S29/S32 | ||
| ( ([[31/28]])/([[32/29]]) )/( ([[32/29]])/([[11/10]]) ) | |||
| [[ | | [[286781/286720]] | ||
| 2. | | 2.5.7.11.29.31 | ||
|- | |- | ||
| ([[ | | S30/S33 | ||
| ( ([[32/29]])/([[11/10]]) )/( ([[11/10]])/([[34/31]]) ) | |||
| [[ | | [[108800/108779]] | ||
| 2. | | 2.5.11.17.29.31 | ||
|- | |- | ||
| ([[ | | S31/S34 | ||
| ( ([[11/10]])/([[34/31]]) )/( ([[34/31]])/([[35/32]]) ) | |||
| [[ | | [[73997/73984]] | ||
| 2. | | 2.7.11.17.31 | ||
|- | |- | ||
| ([[ | | S32/S35 | ||
| ( ([[34/31]])/([[35/32]]) )/( ([[35/32]])/([[12/11]]) ) | |||
| [[ | | [[417792/417725]] | ||
| 2.3.7. | | 2.3.5.7.11.17.31 | ||
|- | |- | ||
| ([[ | | S33/S36 | ||
| ( ([[35/32]])/([[12/11]]) )/( ([[12/11]])/([[37/34]]) ) | |||
| [[ | | [[156695/156672]] | ||
| 2.5.11.17. | | 2.3.5.7.11.17.37 | ||
|- | |- | ||
| ([[ | | S34/S37 | ||
| ( ([[12/11]])/([[37/34]]) )/( ([[37/34]])/([[38/35]]) ) | |||
| [[ | | [[527136/527065]] | ||
| 2.3.5.7. | | 2.3.5.7.11.17.19.37 | ||
|- | |- | ||
| ([[ | | S35/S38 | ||
| ( ([[37/34]])/([[38/35]]) )/( ([[38/35]])/([[13/12]]) ) | |||
| [[ | | [[589225/589152]] | ||
| 2.3.7. | | 2.3.5.7.13.17.19.37 | ||
|- | |- | ||
| ([[ | | S36/S39 | ||
| ( ([[38/35]])/([[13/12]]) )/( ([[13/12]])/([[40/37]]) ) | |||
| [[ | | [[43776/43771]] | ||
| 2. | | 2.3.7.13.19.37 | ||
|- | |- | ||
| ([[ | | S37/S40 | ||
| ( ([[13/12]])/([[40/37]]) )/( ([[40/37]])/([[41/38]]) ) | |||
| [[ | | [[729677/729600]] | ||
| 2.3. | | 2.3.5.13.19.37.41 | ||
|- | |- | ||
| ([[ | | S38/S41 | ||
| ( ([[40/37]])/([[41/38]]) )/( ([[41/38]])/([[14/13]]) ) | |||
| [[ | | [[808640/808561]] | ||
| 2. | | 2.5.7.13.19.37.41 | ||
|- | |- | ||
| ([[ | | S39/S42 | ||
| ( ([[41/38]])/([[14/13]]) )/( ([[14/13]])/([[43/40]]) ) | |||
| [[ | | [[297947/297920]] | ||
| 2.5.7.13. | | 2.5.7.13.19.41.43 | ||
|- | |- | ||
| ([[ | | S40/S43 | ||
| ( ([[14/13]])/([[43/40]]) )/( ([[43/40]])/([[44/41]]) ) | |||
| [[ | | [[985600/985517]] | ||
| 2. | | 2.5.7.11.13.41.43 | ||
|- | |- | ||
| ([[ | | S41/S44 | ||
| ( ([[43/40]])/([[44/41]]) )/( ([[44/41]])/([[15/14]]) ) | |||
| [[ | | [[216849/216832]] | ||
| 2.3. | | 2.3.7.11.41.43 | ||
|- | |- | ||
| ([[ | | S42/S45 | ||
| ( ([[44/41]])/([[15/14]]) )/( ([[15/14]])/([[46/43]]) ) | |||
| [[ | | [[396704/396675]] | ||
| 2.7. | | 2.3.5.7.11.23.41.43 | ||
|- | |- | ||
| | | S286/S289 | ||
| ([[ | | ( ([[96/95]])/([[289/286]]) )/( ([[289/286]])/([[290/287]]) ) | ||
| [[455440128/455440013]] | |||
| 2.3.7.11.13.17.19.29.41 | |||
| [[ | |||
| 2.3 | |||
|} | |} | ||
== S''k'' | == {{nowrap|(S''k''/S(''k'' + 1))<sup>2</sup> * S(''k'' - 1)/S(''k'' + 2)}} (pentaparticulars) == | ||
=== Significance === | === Significance === | ||
# | # Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2 : ''k''-1 : ''k'' : ''k''+1 : ''k''+2 : ''k''+3.}} | ||
# | # Interestingly, while three-particulars are about equidistance but (currently) have no (clean) known splitting property (due to the three intervals made equidistant not obviously composing to some other significant interval), pentaparticulars are in a sense symmetric to this and related algebraically, by instead being specifically about splitting. | ||
# | # They are obviously implied by any system that equates S''k'' with S(''k'' + 1) and S(''k'' - 1) with S(''k'' + 2) simultaneously, so are not as uncommon as might be guessed, but tend to be accurate equivalences, and 5 is a rather specific number of parts to divide an interval into, even if obviously the most natural for an interval whose numerator is 5 more than the denominator (up to simplification). | ||
=== Derivation | === Derivation === | ||
As before, for a general introduction to the method we use here, see [[#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], as we will not re-explain the method here. However, we will use a previous result from three-particulars as to immediately say that: | |||
<pre> | |||
Sk / S(k+3) = [k-1, k, k+1, k+2, k+3, k+4]^[-1, 2, -1, 1, -2, 1] | |||
</pre> | |||
and also relevant to our problem are | |||
<pre> | |||
S(k+1) / S(k+2) = [k, k+1, k+2]^[-1, 2, -1] / [k+1, k+2, k+3]^[-1, 2, -1] | |||
= [k, k+1, k+2, k+3]^( [-1, 2, -1, 0] - [0, -1, 2, -1] ) | |||
= [k, k+1, k+2, k+3]^[-1, 3, -3, 1] | |||
pentaparticular((k+2)/(k+1)) = [k-1, k, k+1, k+2, k+3, k+4]^[-1, 0, -5, 5, 0, 1] | |||
</pre> | |||
Then, dividing the pentaparticular by S''k''/S(''k'' + 3) yields: | |||
<pre> | |||
[k-1, k, k+1, k+2, k+3, k+4]^( [-1, 0, 5, -5, 0, 1] | |||
- [-1, 2, -1, 1, -2, 1] ) | |||
= [k-1, k, k+1, k+2, k+3, k+4]^[0, -2, 6, -6, 2, 0] | |||
= ( [k, k+1, k+2, k+3]^[-1, 3, -3, 1] )^2 = ( S(k+1) / S(k+2) )^2 | |||
</pre> | |||
Thus proving the general form, which is for any given superparticular, the corresponding pentaparticular is equal to the square of the ultraparticular times the correspondingly-centered three-particular. | |||
=== Table of pentaparticulars === | |||
Up to what is currently the largest pentaparticular on the wiki. | |||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|- | |- | ||
! S-expression | ! S-expression | ||
! | ! Relation | ||
! Ratio | ! Ratio | ||
! Subgroup | |||
|- | |- | ||
| | | ([[135/128|S3/S4]])<sup>2</sup> * [[32/25|S2/S5]] | ||
| ([[6/1]])/([[4/3]])<sup>5</sup> | |||
| | | [[729/512]] | ||
| 2.3 | |||
|- | |- | ||
| | | ([[128/125|S4/S5]])<sup>2</sup> * [[35/32|S3/S6]] | ||
| ([[7/2]])/([[5/4]])<sup>5</sup> | |||
| | | [[3584/3125]] | ||
| 2.5.7 | |||
|- | |- | ||
| | | ([[875/864|S5/S6]])<sup>2</sup> * [[256/245|S4/S7]] | ||
| ([[8/3]])/([[6/5]])<sup>5</sup> | |||
| | | [[3125/2916]] | ||
| 2.3.5 | |||
|- | |- | ||
| | | ([[1728/1715|S6/S7]])<sup>2</sup> * [[525/512|S5/S8]] | ||
| ([[9/4]])/([[7/6]])<sup>5</sup> | |||
| | | [[17496/16807]] | ||
| 2.3.7 | |||
|- | |- | ||
| | | ([[1029/1024|S7/S8]])<sup>2</sup> * [[64/63|S6/S9]] | ||
| ([[2/1]])/([[8/7]])<sup>5</sup> | |||
| | | [[16807/16384]] | ||
| 2.7 | |||
|- | |- | ||
| | | ([[5120/5103|S8/S9]])<sup>2</sup> * [[1617/1600|S7/S10]] | ||
| ([[11/6]])/([[9/8]])<sup>5</sup> | |||
| | | [[180224/177147]] | ||
| 2.3.11 | |||
|- | |- | ||
| | | ([[8019/8000|S9/S10]])<sup>2</sup> * [[2560/2541|S8/S11]] | ||
| ([[12/7]])/([[10/9]])<sup>5</sup> | |||
| | | [[177147/175000]] | ||
| 2.3.5.7 | |||
|- | |- | ||
| | | ([[4000/3993|S10/S11]])<sup>2</sup> * [[1287/1280|S9/S12]] | ||
| ([[13/8]])/([[11/10]])<sup>5</sup> | |||
| | | [[162500/161051]] | ||
| 2.5.11.13 | |||
|- | |- | ||
| | | ([[17303/17280|S11/S12]])<sup>2</sup> * [[5600/5577|S10/S13]] | ||
| ([[14/9]])/([[12/11]])<sup>5</sup> | |||
| | | [[1127357/1119744]] | ||
| 2.3.7.11 | |||
|- | |- | ||
| | | ([[24192/24167|S12/S13]])<sup>2</sup> * [[1573/1568|S11/S14]] | ||
| ([[3/2]])/([[13/12]])<sup>5</sup> | |||
| | | [[373248/371293]] | ||
| 2.3.13 | |||
|- | |- | ||
| | | ([[10985/10976|S13/S14]])<sup>2</sup> * [[3584/3575|S12/S15]] | ||
| ([[16/11]])/([[14/13]])<sup>5</sup> | |||
| | | [[371293/369754]] | ||
| 2.7.11.13 | |||
|- | |- | ||
| | | ([[43904/43875|S14/S15]])<sup>2</sup> * [[14365/14336|S13/S16]] | ||
| ([[17/12]])/([[15/14]])<sup>5</sup> | |||
| | | [[2285752/2278125]] | ||
| 2.3.5.7.17 | |||
|- | |- | ||
| | | ([[57375/57344|S15/S16]])<sup>2</sup> * [[18816/18785|S14/S17]] | ||
| ([[18/13]])/([[16/15]])<sup>5</sup> | |||
| | | [[6834375/6815744]] | ||
| 2.3.5.13 | |||
|- | |- | ||
| | | ([[24576/24565|S16/S17]])<sup>2</sup> * [[8075/8064|S15/S18]] | ||
| ([[19/14]])/([[17/16]])<sup>5</sup> | |||
| | | [[9961472/9938999]] | ||
| 2.7.17.19 | |||
|- | |- | ||
| | | ([[93347/93312|S17/S18]])<sup>2</sup> * [[6144/6137|S16/S19]] | ||
| ([[4/3]])/([[18/17]])<sup>5</sup> | |||
| | | [[1419857/1417176]] | ||
| 2.3.17 | |||
|- | |- | ||
| | | ([[116640/116603|S18/S19]])<sup>2</sup> * [[38437/38400|S17/S20]] | ||
| ([[21/16]])/([[19/18]])<sup>5</sup> | |||
| | | [[2480058/2476099]] | ||
| 2.3.7.19 | |||
|- | |- | ||
| | | ([[48013/48000|S19/S20]])<sup>2</sup> * [[15840/15827|S18/S21]] | ||
| ([[22/17]])/([[20/19]])<sup>5</sup> | |||
| | | [[27237089/27200000]] | ||
| 2.5.11.17.19 | |||
|- | |- | ||
| | | ([[176000/175959|S20/S21]])<sup>2</sup> * [[58121/58080|S19/S22]] | ||
| ([[23/18]])/([[21/20]])<sup>5</sup> | |||
| | | [[36800000/36756909]] | ||
| 2.3.5.7.23 | |||
|- | |- | ||
| | | ([[213003/212960|S21/S22]])<sup>2</sup> * [[70400/70357|S20/S23]] | ||
| ([[24/19]])/([[22/21]])<sup>5</sup> | |||
| | | [[12252303/12239876]] | ||
| 2.3.7.11.19 | |||
|- | |- | ||
| | | ([[85184/85169|S22/S23]])<sup>2</sup> * [[5635/5632|S21/S24]] | ||
| ([[5/4]])/([[23/22]])<sup>5</sup> | |||
| | | [[6442040/6436343]] | ||
| 2.5.11.23 | |||
|- | |- | ||
| | | ([[304175/304128|S23/S24]])<sup>2</sup> * [[100672/100625|S22/S25]] | ||
| ([[26/21]])/([[24/23]])<sup>5</sup> | |||
| | | [[83672459/83607552]] | ||
| 2.3.7.13.23 | |||
|- | |- | ||
| | | ([[359424/359375|S24/S25]])<sup>2</sup> * [[119025/118976|S23/S26]] | ||
| ([[27/22]])/([[25/24]])<sup>5</sup> | |||
| | | [[107495424/107421875]] | ||
| 2.3.5.11 | |||
|- | |- | ||
| | | ([[140625/140608|S25/S26]])<sup>2</sup> * [[46592/46575|S24/S27]] | ||
| ([[28/23]])/([[26/25]])<sup>5</sup> | |||
| | | [[68359375/68317912]] | ||
| 2.5.7.13.23 | |||
|- | |- | ||
| | | ([[492128/492075|S26/S27]])<sup>2</sup> * [[163125/163072|S25/S28]] | ||
| ([[29/24]])/([[27/26]])<sup>5</sup> | |||
| | | [[43069988/43046721]] | ||
| 2.3.13.29 | |||
|- | |- | ||
| | | ([[570807/570752|S27/S28]])<sup>2</sup> * [[37856/37845|S26/S29]] | ||
| ([[6/5]])/([[28/27]])<sup>5</sup> | |||
| | | [[43046721/43025920]] | ||
| 2.3.5.7 | |||
|- | |- | ||
| | | ([[219520/219501|S28/S29]])<sup>2</sup> * [[72819/72800|S27/S30]] | ||
| ([[31/26]])/([[29/28]])<sup>5</sup> | |||
| | | [[266760704/266644937]] | ||
| 2.7.13.29.31 | |||
|- | |- | ||
| | | ([[756059/756000|S29/S30]])<sup>2</sup> * [[250880/250821|S28/S31]] | ||
| ([[32/27]])/([[30/29]])<sup>5</sup> | |||
| | | [[20511149/20503125]] | ||
| 3.5.29 | |||
|- | |- | ||
| | | ([[864000/863939|S30/S31]])<sup>2</sup> * [[286781/286720|S29/S32]] | ||
| ([[33/28]])/([[31/30]])<sup>5</sup> | |||
| | | [[200475000/200404057]] | ||
| 2.3.5.7.11.31 | |||
|- | |- | ||
| | | ([[327701/327680|S31/S32]])<sup>2</sup> * [[108800/108779|S30/S33]] | ||
| ([[34/29]])/([[32/31]])<sup>5</sup> | |||
| | | [[486695567/486539264]] | ||
| 2.17.29.31 | |||
|- | |- | ||
| | | ([[1114112/1114047|S32/S33]])<sup>2</sup> * [[73997/73984|S31/S34]] | ||
| | | ([[7/6]])/([[33/32]])<sup>5</sup> | ||
| [[ | | [[117440512/117406179]] | ||
|- | | 2.3.7.11 | ||
|} | |||
== S''k''<sup>2</sup>⋅S(''k'' + 1) and S(''k'' − 1)⋅S''k''<sup>2</sup> (lopsided commas) == | |||
=== Significance === | |||
# Tempering out any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering out 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. | |||
# If a comma (such as the diaschisma, [[2048/2025]]), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering out the square-particulars composing it (such as [[256/255|S16]] and [[289/288|S17]] in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise. | |||
# It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form. | |||
=== Derivation of equivalence relation === | |||
Using the clarity of [[#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name: | |||
S''k''<sup>2</sup>⋅S(''k'' + 1) = [''k'' - 1, ''k'', ''k'' + 1, ''k'' + 2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies: | |||
S''k''<sup>2</sup>⋅S(''k'' + 1) = (''k''/(''k'' - 1))<sup>2</sup> / ((''k'' + 2)/''k'') through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1]. | |||
S(''k'' - 1)⋅S''k''<sup>2</sup> = [''k'' - 2, ''k'' - 1, ''k'', ''k'' + 1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies: | |||
S(''k'' - 1)⋅S''k''<sup>2</sup> = (''k''/(''k'' - 2)) / ((''k'' + 1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2]. | |||
=== Tables === | |||
Below are two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large, so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]]. | |||
==== Top-heavy lopsided commas ==== | |||
{| class="wikitable center-all | |||
|- | |- | ||
! S-expression | |||
! Square relation | |||
! Ratio | |||
|- | |- | ||
| | | S2<sup>2</sup>⋅S3 = ([[3/2]])⋅([[4/3]]) | ||
| ([[ | | ([[2/1]])<sup>2</sup>/([[2/1]]) | ||
| [[ | | [[2/1]] | ||
|- | |- | ||
| | | S3<sup>2</sup>⋅S4 = ([[6/5]])⋅([[9/8]]) | ||
| ([[ | | ([[3/2]])<sup>2</sup>/([[5/3]]) | ||
| [[ | | [[27/20]] | ||
|- | |- | ||
| | | S4<sup>2</sup>⋅S5 = ([[10/9]])⋅([[16/15]]) | ||
| ([[ | | ([[4/3]])<sup>2</sup>/([[3/2]]) | ||
| [[ | | [[32/27]] | ||
|- | |- | ||
| | | S5<sup>2</sup>⋅S6 = ([[15/14]])⋅([[25/24]]) | ||
| ([[ | | ([[5/4]])<sup>2</sup>/([[7/5]]) | ||
| [[ | | [[125/112]] | ||
|- | |- | ||
| | | S6<sup>2</sup>⋅S7 = ([[21/20]])⋅([[36/35]]) | ||
| ([[ | | ([[6/5]])<sup>2</sup>/([[4/3]]) | ||
| [[ | | [[27/25]] | ||
|- | |- | ||
| | | S7<sup>2</sup>⋅S8 = ([[28/27]])⋅([[49/48]]) | ||
| ([[ | | ([[7/6]])<sup>2</sup>/([[9/7]]) | ||
| [[ | | [[343/324]] | ||
|- | |- | ||
| | | S8<sup>2</sup>⋅S9 = ([[36/35]])([[64/63]]) | ||
| ([[ | | ([[8/7]])<sup>2</sup>/([[5/4]]) | ||
| [[ | | [[256/245]] | ||
|- | |- | ||
| | | S9<sup>2</sup>⋅S10 = ([[45/44]])([[81/80]]) | ||
| ([[ | | ([[9/8]])<sup>2</sup>/([[11/9]]) | ||
| [[ | | [[729/704]] | ||
|- | |- | ||
| | | S10<sup>2</sup>⋅S11 = ([[55/54]])⋅([[100/99]]) | ||
| ([[ | | ([[10/9]])<sup>2</sup>/([[6/5]]) | ||
| [[ | | [[250/243]] | ||
|- | |- | ||
| | | S11<sup>2</sup>⋅S12 = ([[66/65]])⋅([[121/120]]) | ||
| ([[ | | ([[11/10]])<sup>2</sup>/([[13/11]]) | ||
| [[ | | [[1331/1300]] | ||
|- | |- | ||
| | | S12<sup>2</sup>⋅S13 = ([[78/77]]⋅([[144/143]]) | ||
| ([[ | | ([[12/11]])<sup>2</sup>/([[7/6]]) | ||
| [[ | | [[864/847]] | ||
|- | |- | ||
| | | S13<sup>2</sup>⋅S14 = ([[91/90]])⋅([[169/168]]) | ||
| ([[ | | ([[13/12]])<sup>2</sup>/([[15/13]]) | ||
| [[ | | [[2197/2160]] | ||
|- | |- | ||
| | | S14<sup>2</sup>⋅S15 = ([[105/104]])⋅([[196/195]]) | ||
| ([[ | | ([[14/13]])<sup>2</sup>/([[8/7]]) | ||
| [[ | | [[343/338]] | ||
|- | |- | ||
| | | S15<sup>2</sup>⋅S16 = ([[120/119]])⋅([[225/224]]) | ||
| ([[ | | ([[15/14]])<sup>2</sup>/([[17/15]]) | ||
| [[ | | [[3375/3332]] | ||
|- | |- | ||
| | | S16<sup>2</sup>⋅S17 = ([[136/135]])⋅([[256/255]]) | ||
| ([[ | | ([[16/15]])<sup>2</sup>/([[9/8]]) | ||
| [[ | | [[2048/2025]] | ||
|- | |- | ||
| | | S17<sup>2</sup>⋅S18 = ([[153/152]])⋅([[289/288]]) | ||
| ([[ | | ([[17/16]])<sup>2</sup>/([[19/17]]) | ||
| [[ | | [[4913/4864]] | ||
|- | |- | ||
| | | S18<sup>2</sup>⋅S19 = ([[171/170]])⋅([[324/323]]) | ||
| ([[ | | ([[18/17]])<sup>2</sup>/([[10/9]]) | ||
| [[ | | [[1458/1445]] | ||
|- | |- | ||
| | | S19<sup>2</sup>⋅S20 = ([[190/189]])⋅([[361/360]]) | ||
| ([[ | | ([[19/18]])<sup>2</sup>/([[21/19]]) | ||
| [[ | | [[6859/6804]] | ||
|- | |- | ||
| | | S20<sup>2</sup>⋅S21 = ([[210/209]])⋅([[400/399]]) | ||
| ([[ | | ([[20/19]])<sup>2</sup>/([[11/10]]) | ||
| [[ | | [[4000/3971]] | ||
|- | |- | ||
| | | S21<sup>2</sup>⋅S22 = ([[231/230]])⋅([[441/440]]) | ||
| ([[ | | ([[21/20]])<sup>2</sup>/([[23/21]]) | ||
| [[ | | [[9261/9200]] | ||
|- | |- | ||
| | | S22<sup>2</sup>⋅S23 = ([[253/252]])⋅([[484/483]]) | ||
| ([[ | | ([[22/21]])<sup>2</sup>/([[12/11]]) | ||
| [[ | | [[1331/1323]] | ||
|- | |- | ||
| | | S23<sup>2</sup>⋅S24 = ([[276/275]])⋅([[529/528]]) | ||
| ([[ | | ([[23/22]])<sup>2</sup>/([[25/23]]) | ||
| [[ | | [[12167/12100]] | ||
|- | |- | ||
| | | S24<sup>2</sup>⋅S25 = ([[300/299]])⋅([[576/575]]) | ||
| ([[ | | ([[24/23]])<sup>2</sup>/([[13/12]]) | ||
| [[ | | [[6912/6877]] | ||
|- | |- | ||
| | | S25<sup>2</sup>⋅S26 = ([[325/324]])⋅([[625/624]]) | ||
| ([[ | | ([[25/24]])<sup>2</sup>/([[27/25]]) | ||
| [[ | | [[15625/15552]] | ||
|- | |- | ||
| | | S26<sup>2</sup>⋅S27 = ([[351/350]])⋅([[676/675]]) | ||
| ([[ | | ([[26/25]])<sup>2</sup>/([[14/13]]) | ||
| [[ | | [[4394/4375]] | ||
|- | |- | ||
| S27<sup>2</sup>⋅S28 = ([[378/377]])⋅([[729/728]]) | |||
| ([[27/26]])<sup>2</sup>/([[29/27]]) | |||
| [[19683/19604]] | |||
|- | |- | ||
| | | S28<sup>2</sup>⋅S29 = ([[406/405]])⋅([[784/783]]) | ||
| ([[ | | ([[28/27]])<sup>2</sup>/([[15/14]]) | ||
| [[ | | [[10976/10935]] | ||
|- | |- | ||
| | | S29<sup>2</sup>⋅S30 = ([[435/434]])⋅([[841/840]]) | ||
| ([[ | | ([[29/28]])<sup>2</sup>/([[31/29]]) | ||
| [[ | | [[24389/24304]] | ||
|- | |- | ||
| | | S30<sup>2</sup>⋅S31 = ([[465/464]])⋅([[900/899]]) | ||
| ([[ | | ([[30/29]])<sup>2</sup>/([[16/15]]) | ||
| [[ | | [[3375/3364]] | ||
|- | |- | ||
| | | S31<sup>2</sup>⋅S32 = ([[496/495]])⋅([[961/960]]) | ||
| ([[ | | ([[31/30]])<sup>2</sup>/([[33/31]]) | ||
| [[ | | [[29791/29700]] | ||
|- | |- | ||
| | | S32<sup>2</sup>⋅S33 = ([[528/527]])⋅([[1024/1023]]) | ||
| ([[ | | ([[32/31]])<sup>2</sup>/([[17/16]]) | ||
| [[ | | [[16384/16337]] | ||
|- | |- | ||
| | | S33<sup>2</sup>⋅S34 = ([[561/560]])⋅([[1089/1088]]) | ||
| ([[ | | ([[33/32]])<sup>2</sup>/([[35/33]]) | ||
| [[ | | [[35937/35840]] | ||
|- | |- | ||
| | | S34<sup>2</sup>⋅S35 = ([[595/594]])⋅([[1156/1155]]) | ||
| ([[ | | ([[34/33]])<sup>2</sup>/([[18/17]]) | ||
| [[ | | [[9826/9801]] | ||
|- | |- | ||
| | | S35<sup>2</sup>⋅S36 = ([[630/629]])⋅([[1225/1224]]) | ||
| ([[ | | ([[35/34]])<sup>2</sup>/([[37/35]]) | ||
| [[ | | [[42875/42772]] | ||
|- | |- | ||
| | | S36<sup>2</sup>⋅S37 = ([[666/665]])⋅([[1296/1295]]) | ||
| ([[ | | ([[36/35]])<sup>2</sup>/([[19/18]]) | ||
| [[ | | [[23328/23275]] | ||
|- | |- | ||
| | | S37<sup>2</sup>⋅S38 = ([[703/702]])⋅([[1369/1368]]) | ||
| ([[ | | ([[37/36]])<sup>2</sup>/([[39/37]]) | ||
| [[ | | [[50653/50544]] | ||
|- | |- | ||
| | | S38<sup>2</sup>⋅S39 = ([[741/740]])⋅([[1444/1443]]) | ||
| ([[ | | ([[38/37]])<sup>2</sup>/([[20/19]]) | ||
| [[ | | [[6859/6845]] | ||
|- | |- | ||
| | | S39<sup>2</sup>⋅S40 = ([[780/779]])⋅([[1521/1520]]) | ||
| ([[ | | ([[39/38]])<sup>2</sup>/([[41/39]]) | ||
| [[ | | [[59319/59204]] | ||
|- | |- | ||
| | | S40<sup>2</sup>⋅S41 = ([[820/819]])⋅([[1600/1599]]) | ||
| ([[ | | ([[40/39]])<sup>2</sup>/([[21/20]]) | ||
| [[ | | [[32000/31941]] | ||
|- | |- | ||
| | | S41<sup>2</sup>⋅S42 = ([[861/860]])⋅([[1681/1680]]) | ||
| ([[ | | ([[41/40]])<sup>2</sup>/([[43/41]]) | ||
| [[ | | [[68921/68800]] | ||
|- | |- | ||
| | | S42<sup>2</sup>⋅S43 = ([[903/902]])⋅([[1764/1763]]) | ||
| ([[ | | ([[42/41]])<sup>2</sup>/([[22/21]]) | ||
| [[ | | [[18522/18491]] | ||
|- | |- | ||
| | | S43<sup>2</sup>⋅S44 = ([[946/945]])⋅([[1849/1848]]) | ||
| ([[ | | ([[43/42]])<sup>2</sup>/([[45/43]]) | ||
| [[ | | [[79507/79380]] | ||
|- | |- | ||
| | | S44<sup>2</sup>⋅S45 = ([[990/989]])⋅([[1936/1935]]) | ||
| ([[ | | ([[44/43]])<sup>2</sup>/([[23/22]]) | ||
| [[ | | [[42592/42527]] | ||
|- | |- | ||
| | | S46<sup>2</sup>⋅S47 = ([[1081/1080]])⋅([[2116/2115]]) | ||
| ([[ | | ([[46/45]])<sup>2</sup>/([[24/23]]) | ||
| [[ | | [[12167/12150]] | ||
|- | |- | ||
| | | S49<sup>2</sup>⋅S50 = ([[1225/1224]])⋅([[2401/2400]]) | ||
| ([[ | | ([[49/48]])<sup>2</sup>/([[51/49]]) | ||
| [[ | | [[117649/117504]] | ||
|- | |- | ||
| | | S50<sup>2</sup>⋅S51 = ([[1275/1274]])⋅([[2500/2499]]) | ||
| ([[ | | ([[50/49]])<sup>2</sup>/([[26/25]]) | ||
| [[ | | [[31250/31213]] | ||
|- | |- | ||
| | | S52<sup>2</sup>⋅S53 = ([[1378/1377]])⋅([[2704/2703]]) | ||
| ([[ | | ([[52/51]])<sup>2</sup>/([[27/26]]) | ||
| [[ | | [[70304/70227]] | ||
|- | |- | ||
| | | S55<sup>2</sup>⋅S56 = ([[1540/1539]])⋅([[3025/3024]]) | ||
| ([[ | | ([[55/54]])<sup>2</sup>/([[57/55]]) | ||
| [[ | | [[166375/166212]] | ||
|- | |- | ||
| | | S56<sup>2</sup>⋅S57 = ([[1596/1595]])⋅([[3136/3135]]) | ||
| ([[ | | ([[56/55]])<sup>2</sup>/([[29/28]]) | ||
| [[ | | [[87808/87725]] | ||
|- | |- | ||
| | | S58<sup>2</sup>⋅S59 = ([[1711/1710]])⋅([[3364/3363]]) | ||
| ([[ | | ([[58/57]])<sup>2</sup>/([[30/29]]) | ||
| [[ | | [[48778/48735]] | ||
|- | |- | ||
| | | S63<sup>2</sup>⋅S64 = ([[2016/2015]])⋅([[3969/3968]]) | ||
| ([[ | | ([[63/62]])<sup>2</sup>/([[65/63]]) | ||
| [[ | | [[250047/249860]] | ||
|- | |- | ||
| | | S64<sup>2</sup>⋅S65 = ([[2080/2079]])⋅([[4096/4095]]) | ||
| ([[ | | ([[64/63]])<sup>2</sup>/([[33/32]]) | ||
| [[ | | [[131072/130977]] | ||
|- | |- | ||
| | | S66<sup>2</sup>⋅S67 = ([[2211/2210]])⋅([[4356/4355]]) | ||
| ([[ | | ([[66/65]])<sup>2</sup>/([[34/33]]) | ||
| [[ | | [[71874/71825]] | ||
|- | |- | ||
| | | S70<sup>2</sup>⋅S71 = ([[2485/2484]])⋅([[4900/4899]]) | ||
| ([[ | | ([[70/69]])<sup>2</sup>/([[36/35]]) | ||
| [[ | | [[42875/42849]] | ||
|- | |- | ||
| | | S75<sup>2</sup>⋅S76 = ([[2850/2849]])⋅([[5625/5624]]) | ||
| ([[ | | ([[75/74]])<sup>2</sup>/([[77/75]]) | ||
| [[ | | [[421875/421652]] | ||
|- | |- | ||
| | | S76<sup>2</sup>⋅S77 = ([[2926/2925]])⋅([[5776/5775]]) | ||
| ([[ | | ([[76/75]])<sup>2</sup>/([[39/38]]) | ||
| [[ | | [[219488/219375]] | ||
|- | |- | ||
| | | S78<sup>2</sup>⋅S79 = ([[3081/3080]])⋅([[6084/6083]]) | ||
| ([[ | | ([[78/77]])<sup>2</sup>/([[40/39]]) | ||
| [[ | | [[59319/59290]] | ||
|} | |||
==== Bottom-heavy lopsided commas ==== | |||
{| class="wikitable center-all | |||
|- | |- | ||
! S-expression | |||
! Square relation | |||
! Ratio | |||
|- | |- | ||
| | | S3<sup>2</sup>⋅S2 = ([[3/2]])⋅([[9/8]]) | ||
| ([[ | | ([[3/1]])/([[4/3]])<sup>2</sup> | ||
| [[ | | [[27/16]] | ||
|- | |- | ||
| | | S4<sup>2</sup>⋅S3 = ([[6/5]])⋅([[16/15]]) | ||
| ([[ | | ([[2/1]])/([[5/4]])<sup>2</sup> | ||
| [[ | | [[32/25]] | ||
|- | |- | ||
| | | S5<sup>2</sup>⋅S4 = ([[10/9]])⋅([[25/24]]) | ||
| ([[ | | ([[5/3]])/([[6/5]])<sup>2</sup> | ||
| [[ | | [[125/108]] | ||
|- | |- | ||
| | | S6<sup>2</sup>⋅S5 = ([[15/14]])⋅([[36/35]]) | ||
| ([[ | | ([[3/2]])/([[7/6]])<sup>2</sup> | ||
| [[ | | [[54/49]] | ||
|- | |- | ||
| | | S7<sup>2</sup>⋅S6 = ([[21/20]])⋅([[49/48]]) | ||
| ([[ | | ([[7/5]])/([[8/7]])<sup>2</sup> | ||
| [[ | | [[343/320]] | ||
|- | |- | ||
| | | S8<sup>2</sup>⋅S7 = ([[28/27]])⋅([[64/63]]) | ||
| ([[ | | ([[4/3]])/([[9/8]])<sup>2</sup> | ||
| [[ | | [[256/243]] | ||
|- | |- | ||
| | | S9<sup>2</sup>⋅S8 = ([[36/35]])⋅([[81/80]]) | ||
| ([[ | | ([[9/7]])/([[10/9]])<sup>2</sup> | ||
| [[ | | [[729/700]] | ||
|- | |- | ||
| | | S10<sup>2</sup>⋅S9 = ([[45/44]])⋅([[100/99]]) | ||
| ([[ | | ([[5/4]])/([[11/10]])<sup>2</sup> | ||
| [[ | | [[125/121]] | ||
|- | |- | ||
| | | S11<sup>2</sup>⋅S10 = ([[55/54]])⋅([[121/120]]) | ||
| ([[ | | ([[11/9]])/([[12/11]])<sup>2</sup> | ||
| [[ | | [[1331/1296]] | ||
|- | |- | ||
| | | S12<sup>2</sup>⋅S11 = ([[66/65]])⋅([[144/143]]) | ||
| ([[ | | ([[6/5]])/([[13/12]])<sup>2</sup> | ||
| [[ | | [[864/845]] | ||
|- | |- | ||
| | | S13<sup>2</sup>⋅S12 = ([[78/77]])⋅([[169/168]]) | ||
| ([[ | | ([[13/11]])/([[14/13]])<sup>2</sup> | ||
| [[ | | [[2197/2156]] | ||
|- | |- | ||
| | | S14<sup>2</sup>⋅S13 = ([[91/90]])⋅([[196/195]]) | ||
| ([[ | | ([[7/6]])/([[15/14]])<sup>2</sup> | ||
| [[ | | [[686/675]] | ||
|- | |- | ||
| | | S15<sup>2</sup>⋅S14 = ([[105/104]])⋅([[225/224]]) | ||
| ([[ | | ([[15/13]])/([[16/15]])<sup>2</sup> | ||
| [[ | | [[3375/3328]] | ||
|- | |- | ||
| | | S16<sup>2</sup>⋅S15 = ([[120/119]])⋅([[256/255]]) | ||
| ([[ | | ([[8/7]])/([[17/16]])<sup>2</sup> | ||
| [[ | | [[2048/2023]] | ||
|- | |- | ||
| | | S17<sup>2</sup>⋅S16 = ([[136/135]])⋅([[289/288]]) | ||
| ([[ | | ([[17/15]])/([[18/17]])<sup>2</sup> | ||
| [[ | | [[4913/4860]] | ||
|- | |- | ||
| | | S18<sup>2</sup>⋅S17 = ([[153/152]])⋅([[324/323]]) | ||
| ([[ | | ([[9/8]])/([[19/18]])<sup>2</sup> | ||
| [[ | | [[729/722]] | ||
|- | |- | ||
| | | S19<sup>2</sup>⋅S18 = ([[171/170]])⋅([[361/360]]) | ||
| ([[ | | ([[19/17]])/([[20/19]])<sup>2</sup> | ||
| [[ | | [[6859/6800]] | ||
|- | |- | ||
| | | S20<sup>2</sup>⋅S19 = ([[190/189]])⋅([[400/399]]) | ||
| ([[ | | ([[10/9]])/([[21/20]])<sup>2</sup> | ||
| [[ | | [[4000/3969]] | ||
|- | |- | ||
| | | S21<sup>2</sup>⋅S20 = ([[210/209]])⋅([[441/440]]) | ||
| ([[ | | ([[21/19]])/([[22/21]])<sup>2</sup> | ||
| [[ | | [[9261/9196]] | ||
|- | |- | ||
| | | S22<sup>2</sup>⋅S21 = ([[231/230]])⋅([[484/483]]) | ||
| ([[ | | ([[11/10]])/([[23/22]])<sup>2</sup> | ||
| [[ | | [[2662/2645]] | ||
|- | |- | ||
| | | S23<sup>2</sup>⋅S22 = ([[253/252]])⋅([[529/528]]) | ||
| ([[ | | ([[23/21]])/([[24/23]])<sup>2</sup> | ||
| [[ | | [[12167/12096]] | ||
|- | |- | ||
| | | S24<sup>2</sup>⋅S23 = ([[276/275]])⋅([[576/575]]) | ||
| ([[ | | ([[12/11]])/([[25/24]])<sup>2</sup> | ||
| [[ | | [[6912/6875]] | ||
|- | |- | ||
| | | S25<sup>2</sup>⋅S24 = ([[300/299]])⋅([[625/624]]) | ||
| ([[ | | ([[25/23]])/([[26/25]])<sup>2</sup> | ||
| [[ | | [[15625/15548]] | ||
|- | |- | ||
| | | S26<sup>2</sup>⋅S25 = ([[325/324]])⋅([[676/675]]) | ||
| ([[ | | ([[13/12]])/([[27/26]])<sup>2</sup> | ||
| [[ | | [[2197/2187]] | ||
|- | |- | ||
| | | S27<sup>2</sup>⋅S26 = ([[351/350]])⋅([[729/728]]) | ||
| ([[ | | ([[27/25]])/([[28/27]])<sup>2</sup> | ||
| [[ | | [[19683/19600]] | ||
|- | |- | ||
| S28<sup>2</sup>⋅S27 = ([[378/377]])⋅([[784/783]]) | |||
| ([[14/13]])/([[29/28]])<sup>2</sup> | |||
| [[10976/10933]] | |||
|- | |- | ||
| | | S29<sup>2</sup>⋅S28 = ([[406/405]])⋅([[841/840]]) | ||
| | | ([[29/27]])/([[30/29]])<sup>2</sup> | ||
| | | [[24389/24300]] | ||
|- | |- | ||
| | | S30<sup>2</sup>⋅S29 = ([[435/434]])⋅([[900/899]]) | ||
| [[ | | ([[15/14]])/([[31/30]])<sup>2</sup> | ||
| | | [[6750/6727]] | ||
|- | |- | ||
| | | S31<sup>2</sup>⋅S30 = ([[465/464]])⋅([[961/960]]) | ||
| [[ | | ([[31/29]])/([[32/31]])<sup>2</sup> | ||
| | | [[29791/29696]] | ||
|- | |- | ||
| | | S32<sup>2</sup>⋅S31 = ([[496/495]])⋅([[1024/1023]]) | ||
| [[ | | ([[16/15]])/([[33/32]])<sup>2</sup> | ||
| | | [[16384/16335]] | ||
|- | |- | ||
| | | S33<sup>2</sup>⋅S32 = ([[528/527]])⋅([[1089/1088]]) | ||
| [[ | | ([[33/31]])/([[34/33]])<sup>2</sup> | ||
| | | [[35937/35836]] | ||
|- | |- | ||
| | | S34<sup>2</sup>⋅S33 = ([[561/560]])⋅([[1156/1155]]) | ||
| [[ | | ([[17/16]])/([[35/34]])<sup>2</sup> | ||
| | | [[4913/4900]] | ||
|- | |- | ||
| | | S35<sup>2</sup>⋅S34 = ([[595/594]])⋅([[1225/1224]]) | ||
| [[ | | ([[35/33]])/([[36/35]])<sup>2</sup> | ||
| | | [[42875/42768]] | ||
|- | |- | ||
| | | S36<sup>2</sup>⋅S35 = ([[630/629]])⋅([[1296/1295]]) | ||
| [[ | | ([[18/17]])/([[37/36]])<sup>2</sup> | ||
| | | [[23328/23273]] | ||
|- | |- | ||
| | | S37<sup>2</sup>⋅S36 = ([[666/665]])⋅([[1369/1368]]) | ||
| [[ | | ([[37/35]])/([[38/37]])<sup>2</sup> | ||
| | | [[50653/50540]] | ||
|- | |- | ||
| | | S38<sup>2</sup>⋅S37 = ([[703/702]])⋅([[1444/1443]]) | ||
| [[ | | ([[19/18]])/([[39/38]])<sup>2</sup> | ||
| | | [[13718/13689]] | ||
|- | |- | ||
| | | S39<sup>2</sup>⋅S38 = ([[741/740]])⋅([[1521/1520]]) | ||
| [[ | | ([[39/37]])/([[40/39]])<sup>2</sup> | ||
| | | [[59319/59200]] | ||
|- | |- | ||
| | | S40<sup>2</sup>⋅S39 = ([[780/779]])⋅([[1600/1599]]) | ||
| [[ | | ([[20/19]])/([[41/40]])<sup>2</sup> | ||
| | | [[32000/31939]] | ||
|- | |- | ||
| | | S41<sup>2</sup>⋅S40 = ([[820/819]])⋅([[1681/1680]]) | ||
| [[ | | ([[41/39]])/([[42/41]])<sup>2</sup> | ||
| | | [[68921/68796]] | ||
|- | |- | ||
| | | S42<sup>2</sup>⋅S41 = ([[861/860]])⋅([[1764/1763]]) | ||
| | | ([[21/20]])/([[43/42]])<sup>2</sup> | ||
| | | [[9261/9245]] | ||
|- | |- | ||
| | | S43<sup>2</sup>⋅S42 = ([[903/902]])⋅([[1849/1848]]) | ||
| | | ([[43/41]])/([[44/43]])<sup>2</sup> | ||
| [[79507/79376]] | |||
< | |||
< | |||
|- | |- | ||
| S44<sup>2</sup>⋅S43 = ([[946/945]])⋅([[1936/1935]]) | |||
| ([[22/21]])/([[45/44]])<sup>2</sup> | |||
| [[42592/42525]] | |||
|- | |- | ||
| | | S45<sup>2</sup>⋅S44 = ([[990/989]])⋅([[2025/2024]]) | ||
| [[ | | ([[45/43]])/([[46/45]])<sup>2</sup> | ||
| | | [[91125/90988]] | ||
|- | |- | ||
| | | S48<sup>2</sup>⋅S47 = ([[1128/1127]])⋅([[2304/2303]]) | ||
| [[ | | ([[24/23]])/([[49/48]])<sup>2</sup> | ||
| | | [[55296/55223]] | ||
|- | |- | ||
| | | S50<sup>2</sup>⋅S49 = ([[1225/1224]])⋅([[2500/2499]]) | ||
| [[ | | ([[25/24]])/([[51/50]])<sup>2</sup> | ||
| | | [[15625/15606]] | ||
|- | |- | ||
| | | S51<sup>2</sup>⋅S50 = ([[1275/1274]])⋅([[2601/2600]]) | ||
| [[ | | ([[51/49]])/([[52/51]])<sup>2</sup> | ||
| | | [[132651/132496]] | ||
|- | |- | ||
| | | S54<sup>2</sup>⋅S53 = ([[1431/1430]])⋅([[2916/2915]]) | ||
| [[ | | ([[27/26]])/([[55/54]])<sup>2</sup> | ||
| | | [[39366/39325]] | ||
|- | |- | ||
| | | S56<sup>2</sup>⋅S55 = ([[1540/1539]])⋅([[3136/3135]]) | ||
| [[ | | ([[28/27]])/([[57/56]])<sup>2</sup> | ||
| | | [[87808/87723]] | ||
|- | |- | ||
| | | S57<sup>2</sup>⋅S56 = ([[1596/1595]])⋅([[3249/3248]]) | ||
| [[ | | ([[57/55]])/([[58/57]])<sup>2</sup> | ||
| | | [[185193/185020]] | ||
|- | |- | ||
| | | S62<sup>2</sup>⋅S61 = ([[1891/1890]])⋅([[3844/3843]]) | ||
| [[ | | ([[31/30]])/([[63/62]])<sup>2</sup> | ||
| | | [[59582/59535]] | ||
|- | |- | ||
| | | S64<sup>2</sup>⋅S63 = ([[2016/2015]])⋅([[4096/4095]]) | ||
| [[ | | ([[32/31]])/([[65/64]])<sup>2</sup> | ||
| | | [[131072/130975]] | ||
|- | |- | ||
| | | S65<sup>2</sup>⋅S64 = ([[2080/2079]])⋅([[4225/4224]]) | ||
| [[ | | ([[65/63]])/([[66/65]])<sup>2</sup> | ||
| | | [[274625/274428]] | ||
|- | |- | ||
| | | S68<sup>2</sup>⋅S67 = ([[2278/2277]])⋅([[4624/4623]]) | ||
| [[ | | ([[34/33]])/([[69/68]])<sup>2</sup> | ||
| | | [[157216/157113]] | ||
|- | |- | ||
| | | S74<sup>2</sup>⋅S73 = ([[2701/2700]])⋅([[5476/5475]]) | ||
| [[ | | ([[37/36]])/([[75/74]])<sup>2</sup> | ||
| | | [[50653/50625]] | ||
|- | |- | ||
| | | S76<sup>2</sup>⋅S75 = ([[2850/2849]])⋅([[5776/5775]]) | ||
| [[ | | ([[38/37]])/([[77/76]])<sup>2</sup> | ||
| | | [[219488/219373]] | ||
|- | |- | ||
| | | S77<sup>2</sup>⋅S76 = ([[2926/2925]])⋅([[5929/5928]]) | ||
| | | ([[77/75]])/([[78/77]])<sup>2</sup> | ||
| | | [[456533/456300]] | ||
|- | |- | ||
| | | S80<sup>2</sup>⋅S79 = ([[3160/3159]])⋅([[6400/6399]]) | ||
| | | ([[40/39]])/([[81/80]])<sup>2</sup> | ||
| | | [[256000/255879]] | ||
|} | |} | ||
== Using S-factorizations to understand the significance of S-expressions == | == Using S-factorizations to understand the significance of S-expressions == | ||
| Line 3,383: | Line 3,380: | ||
| | | | ||
|(S26⋅S27)²/S351 | |(S26⋅S27)²/S351 | ||
|- | |||
|[[15625/15552]] | |||
|S25²⋅S26 | |||
| | |||
|S15⋅(S25/S27) | |||
| | |||
|- | |- | ||
|[[225/224]] | |[[225/224]] | ||
| Line 3,405: | Line 3,408: | ||
|S12/(S9/S11) | |S12/(S9/S11) | ||
|S11⋅S12/S9 | |S11⋅S12/S9 | ||
| | |S40⋅S41⋅S42⋅S43⋅S44 | ||
|(S8/S9)/(S64⋅S65) | |(S8/S9)/(S64⋅S65) | ||
|- | |- | ||
| [[364/363]]* | | [[364/363]]* | ||
| S14/(S11/S13) | | S14/(S11/S13) | ||
|S13⋅S14/S11 | |(S13⋅S14)/S11 | ||
|S24⋅S25⋅S26/S22 | |S24⋅S25⋅S26/S22 | ||
|(S9/S11)/S27 | |(S9/S11)/S27 | ||
| Line 3,419: | Line 3,422: | ||
| | | | ||
|(S8/S9)/(S64²/S65) | |(S8/S9)/(S64²/S65) | ||
|- | |||
|[[Septischisma|<small>33554432/33480783</small>]]* | |||
|(S8/S9)²/S15 | |||
|(S8/S9)/(S27/S64) | |||
|(S16/S18)³/(S19/S20) | |||
|S49⋅S55⋅(S64²⋅S65)² | |||
|- | |- | ||
| [[540/539]] | | [[540/539]] | ||
| S12/S14 | | S12/S14 | ||
| | | | ||
|(S9⋅S10)/S7 | |(S9⋅S10)/S7 | ||
|(S6/S7)/(S8/S10) | |(S6/S7)/(S8/S10) | ||
|- | |- | ||
| [[676/675]] | |[[4000/3993]] | ||
| S13/S15 | |S10/S11 | ||
|S26 | | | ||
| | |(S12/S14)/S99 | ||
|S49⋅(S64⋅S65)²⋅S99 | |S25⋅(S64⋅S65)/S55 | ||
|- | |- | ||
|[[1001/1000]] | | [[676/675]] | ||
| | | S13/S15 | ||
| | |S26 | ||
|√{{Overline|S26⋅S49⋅S99}} | | | ||
| | |S49⋅(S64⋅S65)²⋅S99 | ||
|- | |- | ||
|[[1216/1215]] | |[[32805/32768]]* | ||
|S16/S18 | | | ||
| | |S27/S64 | ||
| | |S15/(S8/S9) | ||
| | |S19/(S16/S18)² | ||
|- | |- | ||
| [[1225/1224]] | |[[1001/1000]] | ||
| S35 | |Cp10 | ||
|S49⋅S50 | | | ||
| | |√{{Overline|S26⋅S49⋅S99}} | ||
| | |S49⋅(S64⋅S65)⋅S99 | ||
|- | |||
|[[4459/4455]] | |||
| | |||
| | |||
|S49⋅(S64⋅S65) | |||
|(S64⋅S65)/S26⋅S99 | |||
|- | |||
|[[1216/1215]] | |||
|S16/S18 | |||
| | |||
| | |||
|(S64⋅S65)⋅(S76⋅S77) | |||
|- | |||
|[[10985/10976]] | |||
|S13/S14 | |||
| | |||
|(S64⋅S65²)⋅S99 | |||
| | |||
|- | |||
| [[1225/1224]] | |||
| S35 | |||
|S49⋅S50 | |||
| | |||
| | |||
|- | |||
|[[41503/41472]]* | |||
| | |||
| | |||
|S49⋅S55 | |||
|S65⋅(S76⋅S77²) | |||
|- | |||
|<small>[[131072/130977]]</small> | |||
|S64²⋅S65 | |||
| | |||
|S32/S63 | |||
| | |||
|- | |||
|[[43904/43875]] | |||
|S14/S15 | |||
| | |||
|S49⋅S64 | |||
|S56⋅(S76⋅S77) | |||
|- | |- | ||
|[[2080/2079]] | |[[2080/2079]] | ||
| Line 3,473: | Line 3,524: | ||
| | | | ||
| | | | ||
|- | |- | ||
|[[4225/4224]] | |[[4225/4224]] | ||
| Line 3,491: | Line 3,536: | ||
|S55/S99 | |S55/S99 | ||
|S65/S351 | |S65/S351 | ||
|- | |- | ||
|[[6656/6655]] | |[[6656/6655]] | ||
| Line 3,508: | Line 3,547: | ||
|S33/S35 | |S33/S35 | ||
| | | | ||
| | | | ||
|- | |- | ||
| Line 3,520: | Line 3,553: | ||
| | | | ||
|(S76⋅S77)/S64 | |(S76⋅S77)/S64 | ||
|(S18⋅S19/S16)/(S12/S14) | |((S18⋅S19)/S16)/(S12/S14) | ||
|- | |- | ||
|[[10648/10647]]* | |[[10648/10647]]* | ||
| Line 3,526: | Line 3,559: | ||
| | | | ||
|S99/S351 | |S99/S351 | ||
| | |S55/S65 | ||
|- | |- | ||
| [[25921/25920]] | | [[25921/25920]] | ||
| Line 3,557: | Line 3,566: | ||
|S46/S48 | |S46/S48 | ||
| | | | ||
|- | |- | ||
| <small>[[123201/123200]]</small> | | <small>[[123201/123200]]</small> | ||