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| [[12376/12375]]
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| 17
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|}
=== Table of 1/5-square-particulars ===
Below is a table of 1/5-square-particulars in the 23-limit:
{| class="wikitable center-all
|-
! S-expression
! Interval relation
! Ratio
! Prime limit
|-
| S2*S3*S4*S5*S6
| ([[2/1]])/([[7/6]])
| [[12/7]]
| 7
|-
| S3*S4*S5*S6*S7
| ([[3/2]])/([[8/7]])
| [[21/16]]
| 7
|-
| S4*S5*S6*S7*S8
| ([[4/3]])/([[9/8]])
| [[32/27]]
| 3
|-
| S5*S6*S7*S8*S9
| ([[5/4]])/([[10/9]])
| [[9/8]]
| 3
|-
| S6*S7*S8*S9*S10
| ([[6/5]])/([[11/10]])
| [[12/11]]
| 11
|-
| S7*S8*S9*S10*S11
| ([[7/6]])/([[12/11]])
| [[77/72]]
| 11
|-
| S8*S9*S10*S11*S12
| ([[8/7]])/([[13/12]])
| [[96/91]]
| 13
|-
| S9*S10*S11*S12*S13
| ([[9/8]])/([[14/13]])
| [[117/112]]
| 13
|-
| S10*S11*S12*S13*S14
| ([[10/9]])/([[15/14]])
| [[28/27]]
| 7
|-
| S11*S12*S13*S14*S15
| ([[11/10]])/([[16/15]])
| [[33/32]]
| 11
|-
| S12*S13*S14*S15*S16
| ([[12/11]])/([[17/16]])
| [[192/187]]
| 17
|-
| S13*S14*S15*S16*S17
| ([[13/12]])/([[18/17]])
| [[221/216]]
| 17
|-
| S14*S15*S16*S17*S18
| ([[14/13]])/([[19/18]])
| [[252/247]]
| 19
|-
| S15*S16*S17*S18*S19
| ([[15/14]])/([[20/19]])
| [[57/56]]
| 19
|-
| S16*S17*S18*S19*S20
| ([[16/15]])/([[21/20]])
| [[64/63]]
| 7
|-
| S17*S18*S19*S20*S21
| ([[17/16]])/([[22/21]])
| [[357/352]]
| 17
|-
| S18*S19*S20*S21*S22
| ([[18/17]])/([[23/22]])
| [[396/391]]
| 23
|-
| S19*S20*S21*S22*S23
| ([[19/18]])/([[24/23]])
| [[437/432]]
| 23
|-
| S20*S21*S22*S23*S24
| ([[20/19]])/([[25/24]])
| [[96/95]]
| 19
|-
| S21*S22*S23*S24*S25
| ([[21/20]])/([[26/25]])
| [[105/104]]
| 13
|-
| S22*S23*S24*S25*S26
| ([[22/21]])/([[27/26]])
| [[572/567]]
| 13
|-
| S23*S24*S25*S26*S27
| ([[23/22]])/([[28/27]])
| [[621/616]]
| 23
|-
| S28*S29*S30*S31*S32
| ([[28/27]])/([[33/32]])
| [[896/891]]
| 11
|-
| S34*S35*S36*S37*S38
| ([[34/33]])/([[39/38]])
| [[1292/1287]]
| 19
|-
| S35*S36*S37*S38*S39
| ([[35/34]])/([[40/39]])
| [[273/272]]
| 17
|-
| S40*S41*S42*S43*S44
| ([[40/39]])/([[45/44]])
| [[352/351]]
| 13
|-
| S45*S46*S47*S48*S49
| ([[45/44]])/([[50/49]])
| [[441/440]]
| 11
|-
| S46*S47*S48*S49*S50
| ([[46/45]])/([[51/50]])
| [[460/459]]
| 23
|-
| S50*S51*S52*S53*S54
| ([[50/49]])/([[55/54]])
| [[540/539]]
| 11
|-
| S51*S52*S53*S54*S55
| ([[51/50]])/([[56/55]])
| [[561/560]]
| 17
|-
| S52*S53*S54*S55*S56
| ([[52/51]])/([[57/56]])
| [[2912/2907]]
| 19
|-
| S64*S65*S66*S67*S68
| ([[64/63]])/([[69/68]])
| [[4352/4347]]
| 23
|-
| S65*S66*S67*S68*S69
| ([[65/64]])/([[70/69]])
| [[897/896]]
| 23
|-
| S76*S77*S78*S79*S80
| ([[76/75]])/([[81/80]])
| [[1216/1215]]
| 19
|-
| S91*S92*S93*S94*S95
| ([[91/90]])/([[96/95]])
| [[1729/1728]]
| 19
|-
| <font style="font-size:0.79em">S100*S101*S102*S103*S104</font>
| <font style="font-size:0.83em">([[100/99]])/([[105/104]])</font>
| [[2080/2079]]
| 13
|-
| <font style="font-size:0.79em">S115*S116*S117*S118*S119</font>
| <font style="font-size:0.79em">([[115/114]])/([[120/119]])</font>
| [[2737/2736]]
| 23
|-
| <font style="font-size:0.79em">S121*S122*S123*S124*S125</font>
| <font style="font-size:0.79em">([[121/120]])/([[126/125]])</font>
| [[3025/3024]]
| 11
|-
| <font style="font-size:0.79em">S171*S172*S173*S174*S175</font>
| <font style="font-size:0.79em">([[171/170]])/([[176/175]])</font>
| [[5985/5984]]
| 19
|}
|}


Line 1,906: Line 2,112:
...where we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of [[regular temperament]]s using [[S-expression]]s.
...where we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of [[regular temperament]]s using [[S-expression]]s.


For tuning theory only integer ''k'' > 1 is of relevance. Technically, rational ''k'' other than 1 correspond to rational commas too; the most relevant case for tuning theory is that half-integer ''k'' work as an alternative notation for [[odd-particulars]], though for intuitively understanding the notation, the method described in [[#Abstraction]] may be recommendable as having (in a mathematical sense) exact analogues for every infinite family of commas defined in terms of an analogue of an S-expression, for which the most musically fruitful example is O''k'' = (''k'' / (''k'' - 2))/((''k'' + 2) / ''k'') for odd ''k'' as relevant to [[no-twos subgroup temperaments]].
For tuning theory only integer ''k'' > 1 is of relevance. Technically, rational ''k'' other than 1 correspond to rational commas too; the most relevant case for tuning theory is that half-integer ''k'' work as an alternative notation for [[odd-particular]]s, though for intuitively understanding the notation, the method described in [[#Abstraction]] may be recommendable as having (in a mathematical sense) exact analogues for every infinite family of commas defined in terms of an analogue of an S-expression, for which the most musically fruitful example is O''k'' = (''k'' / (''k'' - 2))/((''k'' + 2) / ''k'') for odd ''k'' as relevant to [[no-twos subgroup temperaments]].


== Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) ==
== Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) ==
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