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== Quick rules of S-expressions ==
== Quick rules of S-expressions ==
[[File:S(expression).svg|thumb|Here is a summarized infographic of what many of these S-expression types do to a harmonic series segment when tempered out.]]
As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.
As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.


Line 1,919: Line 1,920:
$$ \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} $$
$$ \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} $$


… 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.  
… 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>.
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>.
Line 2,156: Line 2,157:
# 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'' }}.
# 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.
# 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 ===
=== Derivation ===
Line 2,166: Line 2,169:
From which we can examine
From which we can examine
<pre>
<pre>
Sk * S(k+3) = [k-1, k, k+1]^[-1, 2, -1] / [k+2, k+3, k+4]^[-1, 2, -1]
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]^[-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]
  = [k-1, k, k+1, k+2, k+3, k+4]^[-1, 2, -1, 1, -2, 1]
Line 2,181: Line 2,184:
  = [k-1, k, k+1, k+2, k+3, k+4]^[ -1, 2, -1, 1, -2, 1 ]
  = [k-1, k, k+1, k+2, k+3, k+4]^[ -1, 2, -1, 1, -2, 1 ]
</pre>
</pre>
...as expected.
...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 ===
=== Table of three-particulars ===
W.I.P.
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
{| class="wikitable center-all
|-
|-
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! Square relation
! Square relation
! Ratio
! Ratio
! Subgroup
|-
| S2/S5
| ( ([[4/1]])/([[5/2]]) )/( ([[5/2]])/([[2/1]]) )
| [[32/25]]
| 2.5
|-
| S3/S6
| ( ([[5/2]])/([[2/1]]) )/( ([[2/1]])/([[7/4]]) )
| [[35/32]]
| 2.5.7
|-
| S4/S7
| ( ([[2/1]])/([[7/4]]) )/( ([[7/4]])/([[8/5]]) )
| [[256/245]]
| 2.5.7
|-
| S5/S8
| ( ([[7/4]])/([[8/5]]) )/( ([[8/5]])/([[3/2]]) )
| [[525/512]]
| 2.3.5.7
|-
| S6/S9
| ( ([[8/5]])/([[3/2]]) )/( ([[3/2]])/([[10/7]]) )
| [[64/63]]
| 2.3.7
|-
| S7/S10
| ( ([[3/2]])/([[10/7]]) )/( ([[10/7]])/([[11/8]]) )
| [[1617/1600]]
| 2.3.5.7.11
|-
| S8/S11
| ( ([[10/7]])/([[11/8]]) )/( ([[11/8]])/([[4/3]]) )
| [[2560/2541]]
| 2.3.5.7.11
|-
| S9/S12
| ( ([[11/8]])/([[4/3]]) )/( ([[4/3]])/([[13/10]]) )
| [[1287/1280]]
| 2.3.5.11.13
|-
| S10/S13
| ( ([[4/3]])/([[13/10]]) )/( ([[13/10]])/([[14/11]]) )
| [[5600/5577]]
| 2.3.5.7.11.13
|-
| S11/S14
| ( ([[13/10]])/([[14/11]]) )/( ([[14/11]])/([[5/4]]) )
| [[1573/1568]]
| 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]]
| 2.3.5.7.17.19
|-
| 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]]
| 2.3.5.7.17.19
|-
| S18/S21
| ( ([[20/17]])/([[7/6]]) )/( ([[7/6]])/([[22/19]]) )
| [[15840/15827]]
| 2.3.5.7.11.17.19
|-
| S19/S22
| ( ([[7/6]])/([[22/19]]) )/( ([[22/19]])/([[23/20]]) )
| [[58121/58080]]
| 2.3.5.7.11.19.23
|-
| S20/S23
| ( ([[22/19]])/([[23/20]]) )/( ([[23/20]])/([[8/7]]) )
| [[70400/70357]]
| 2.5.7.11.19.23
|-
| S21/S24
| ( ([[23/20]])/([[8/7]]) )/( ([[8/7]])/([[25/22]]) )
| [[5635/5632]]
| 2.5.7.11.23
|-
| S22/S25
| ( ([[8/7]])/([[25/22]]) )/( ([[25/22]])/([[26/23]]) )
| [[100672/100625]]
| 2.5.7.11.13.23
|-
| S23/S26
| ( ([[25/22]])/([[26/23]]) )/( ([[26/23]])/([[9/8]]) )
| [[119025/118976]]
| 2.3.5.11.13.23
|-
| S24/S27
| ( ([[26/23]])/([[9/8]]) )/( ([[9/8]])/([[28/25]]) )
| [[46592/46575]]
| 2.3.5.7.13.23
|-
| S25/S28
| ( ([[9/8]])/([[28/25]]) )/( ([[28/25]])/([[29/26]]) )
| [[163125/163072]]
| 2.3.5.7.13.29
|-
| S26/S29
| ( ([[28/25]])/([[29/26]]) )/( ([[29/26]])/([[10/9]]) )
| [[37856/37845]]
| 2.3.5.7.13.29
|-
| S27/S30
| ( ([[29/26]])/([[10/9]]) )/( ([[10/9]])/([[31/28]]) )
| [[72819/72800]]
| 2.3.5.7.13.29.31
|-
| S28/S31
| ( ([[10/9]])/([[31/28]]) )/( ([[31/28]])/([[32/29]]) )
| [[250880/250821]]
| 2.3.5.7.29.31
|-
| S29/S32
| ( ([[31/28]])/([[32/29]]) )/( ([[32/29]])/([[11/10]]) )
| [[286781/286720]]
| 2.5.7.11.29.31
|-
| S30/S33
| ( ([[32/29]])/([[11/10]]) )/( ([[11/10]])/([[34/31]]) )
| [[108800/108779]]
| 2.5.11.17.29.31
|-
| S31/S34
| ( ([[11/10]])/([[34/31]]) )/( ([[34/31]])/([[35/32]]) )
| [[73997/73984]]
| 2.7.11.17.31
|-
| S32/S35
| ( ([[34/31]])/([[35/32]]) )/( ([[35/32]])/([[12/11]]) )
| [[417792/417725]]
| 2.3.5.7.11.17.31
|-
| S33/S36
| ( ([[35/32]])/([[12/11]]) )/( ([[12/11]])/([[37/34]]) )
| [[156695/156672]]
| 2.3.5.7.11.17.37
|-
| S34/S37
| ( ([[12/11]])/([[37/34]]) )/( ([[37/34]])/([[38/35]]) )
| [[527136/527065]]
| 2.3.5.7.11.17.19.37
|-
| S35/S38
| ( ([[37/34]])/([[38/35]]) )/( ([[38/35]])/([[13/12]]) )
| [[589225/589152]]
| 2.3.5.7.13.17.19.37
|-
| S36/S39
| ( ([[38/35]])/([[13/12]]) )/( ([[13/12]])/([[40/37]]) )
| [[43776/43771]]
| 2.3.7.13.19.37
|-
| S37/S40
| ( ([[13/12]])/([[40/37]]) )/( ([[40/37]])/([[41/38]]) )
| [[729677/729600]]
| 2.3.5.13.19.37.41
|-
| S38/S41
| ( ([[40/37]])/([[41/38]]) )/( ([[41/38]])/([[14/13]]) )
| [[808640/808561]]
| 2.5.7.13.19.37.41
|-
| S39/S42
| ( ([[41/38]])/([[14/13]]) )/( ([[14/13]])/([[43/40]]) )
| [[297947/297920]]
| 2.5.7.13.19.41.43
|-
| S40/S43
| ( ([[14/13]])/([[43/40]]) )/( ([[43/40]])/([[44/41]]) )
| [[985600/985517]]
| 2.5.7.11.13.41.43
|-
| S41/S44
| ( ([[43/40]])/([[44/41]]) )/( ([[44/41]])/([[15/14]]) )
| [[216849/216832]]
| 2.3.7.11.41.43
|-
| S42/S45
| ( ([[44/41]])/([[15/14]]) )/( ([[15/14]])/([[46/43]]) )
| [[396704/396675]]
| 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
|}
== {{nowrap|(S''k''/S(''k'' + 1))<sup>2</sup> * S(''k'' - 1)/S(''k'' + 2)}} (pentaparticulars) ==
=== 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 ===
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 I 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
|-
! S-expression
! Relation
! 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
|-
|-
| S2<sup>2</sup>⋅S3 = ([[3/2]])⋅([[4/3]])
| ([[1114112/1114047|S32/S33]])<sup>2</sup> * [[73997/73984|S31/S34]]
| ([[2/1]])<sup>2</sup>/([[2/1]])
| ([[7/6]])/([[33/32]])<sup>5</sup>
| [[2/1]]
| [[117440512/117406179]]
| 2.3.7.11
|}
|}


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