S-expression: Difference between revisions

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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.

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

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

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

* [[#Sk2⋅S(k + 1) and S(k − 1)⋅Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')2⋅S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')2⋅S(''k'' − 1)'''}}. Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}2 and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}3, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}2 and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

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

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

* [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''3|''k''3 − 1}} and {{sfrac|''k''3 + 1|''k''3}}, respectively.

== S''k'' (square-particulars) ==

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For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of inconsistency of mappings when tempered out for the above reason.

{| class="wikitable center-all

{| class="wikitable center-all left-4"

|+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as 311edo can be described as the unique 23-limit temperament that tempers out all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers out all the square-particulars composing those triangle-particulars with the exception of S169 and S170, and maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3, meaning the mapping of that superparticular is inconsistent.</ref>

|-

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! Interval relation

! Ratio

! ~~Prime limit~~

! Subgroup

|-

| S2⋅S3

| ([[3/1]])/([[2/1]])

| [[3/2]]

| 3

| 2.3

|-

| S3⋅S4

| ([[3/2]])/([[5/4]])

| [[6/5]]

| 5

| 2.3.5

|-

| S4⋅S5

| ([[4/3]])/([[6/5]])

| [[10/9]]

| 5

| 2.3.5

|-

| S5⋅S6

| ([[5/4]])/([[7/6]])

| [[15/14]]

| 7

| 2.3.5.7

|-

| S6⋅S7

| ([[6/5]])/([[8/7]])

| [[21/20]]

| 7

| 2.3.5.7

|-

| S7⋅S8

| ([[7/6]])([[9/8]])

| [[28/27]]

| 7

| 2.3.7

|-

| S8⋅S9

| ([[8/7]])/([[10/9]])

| [[36/35]]

| 7

| 2.3.5.7

|-

| S9⋅S10

| ([[9/8]])/([[11/10]])

| [[45/44]]

| 11

| 2.3.5.11

|-

| S10⋅S11

| ([[10/9]])/([[12/11]])

| [[55/54]]

| 11

| 2.3.5.11

|-

| S11⋅S12

| ([[11/10]])/([[13/12]])

| [[66/65]]

| 13

| 2.3.5.11.13

|-

| S12⋅S13

| ([[12/11]])/([[14/13]])

| [[78/77]]

| 13

| 2.3.7.11.13

|-

| S13⋅S14

| ([[13/12]])/([[15/14]])

| [[91/90]]

| 13

| 2.3.5.7.13

|-

| S14⋅S15

| ([[14/13]])/([[16/15]])

| [[105/104]]

| 13

| 2.3.5.7.13

|-

| S15⋅S16

| ([[15/14]])/([[17/16]])

| [[120/119]]

| 17

| 2.3.5.7.17

|-

| S16⋅S17

| ([[16/15]])/([[18/17]])

| [[136/135]]

| 17

| 2.3.5.17

|-

| S17⋅S18

| ([[17/16]])/([[19/18]])

| [[153/152]]

| 19

| 2.3.17.19

|-

| S18⋅S19

| ([[18/17]])/([[20/19]])

| [[171/170]]

| 19

| 2.3.5.17.19

|-

| S19⋅S20

| ([[19/18]])/([[21/20]])

| [[190/189]]

| 19

| 2.3.5.7.19

|-

| S20⋅S21

| ([[20/19]])/([[22/21]])

| [[210/209]]

| 19

| 2.3.5.7.11.19

|-

| S21⋅S22

| ([[21/20]])/([[23/22]])

| [[231/230]]

| 23

| 2.3.5.7.11.23

|-

| S22⋅S23

| ([[22/21]])/([[24/23]])

| [[253/252]]

| 23

| 2.3.5.7.11.23

|-

| S23⋅S24

| ([[23/22]])/([[25/24]])

| [[276/275]]

| 23

| 2.3.5.11.23

|-

| S24⋅S25

| ([[24/23]])/([[26/25]])

| [[300/299]]

| 23

| 2.3.5.13.23

|-

| S25⋅S26

| ([[25/24]])/([[27/26]])

| [[325/324]]

| 13

| 2.3.5.13

|-

| S26⋅S27

| ([[26/25]])/([[28/27]])

| [[351/350]]

| 13

| 2.3.5.7.13

|-

| S27⋅S28

| ([[27/26]])/([[29/28]])

| [[378/377]]

| 29

| 2.3.5.7.13.29

|-

| S28⋅S29

| ([[28/27]])/([[30/29]])

| [[406/405]]

| 29

| 2.3.5.7.29

|-

| S29⋅S30

| ([[29/28]])/([[31/30]])

| [[435/434]]

| 31

| 2.3.5.7.29.31

|-

| S30⋅S31

| ([[30/29]])/([[32/31]])

| [[465/464]]

| 31

| 2.3.5.29.31

|-

| S31⋅S32

| ([[31/30]])/([[33/32]])

| [[496/495]]

| 31

| 2.3.5.11.31

|-

| S32⋅S33

| ([[32/31]])/([[34/33]])

| [[528/527]]

| 31

| 2.3.11.17.31

|-

| S33⋅S34

| ([[33/32]])/([[35/34]])

| [[561/560]]

| 17

| 2.3.5.7.11.17

|-

| S34⋅S35

| ([[34/33]])/([[36/35]])

| [[595/594]]

| 17

| 2.3.5.7.11.17

|-

| S49⋅S50

| ([[49/48]])/([[51/50]])

| [[1225/1224]]

| 17

| 2.3.5.7.17

|-

| S50⋅S51

| ([[50/49]])/([[52/51]])

| [[1275/1274]]

| 17

| 2.3.5.7.13.17

|-

| S55⋅S56

| ([[55/54]])/([[57/56]])

| [[1540/1539]]

| 19

| 2.3.5.7.11.19

|-

| S56⋅S57

| ([[56/55]])/([[58/57]])

| [[1596/1595]]

| 2.3.5.7.11.19.29

|-

| S63⋅S64

| ([[63/62]])/([[65/64]])

| [[2016/2015]]

| 31

| 2.3.5.7.13.31

|-

| S64⋅S65

| ([[64/63]])/([[66/65]])

| [[2080/2079]]

| 13

| 2.3.5.7.11.13

|-

| S76⋅S77

| ([[76/75]])/([[78/77]])

| [[2926/2925]]

| 19

| 2.3.5.7.11.13.19

|-

| S91⋅S92

| ([[91/90]])/([[93/92]])

| [[4186/4185]]

| 31

| 2.3.5.7.13.23.31

|-

| S115⋅S116

| S115⋅S116

| ([[115/114]])/([[117/116]])

| ([[115/114]])/([[117/116]])

| [[6670/6669]]

| 29

| 2.3.5.13.19.23.29

|-

| S153⋅S154

| S153⋅S154

| ([[153/152]])/([[155/154]])

| ([[153/152]])/([[155/154]])

| [[11781/11780]]

| [[11781/11780]]

| 31

| 2.3.5.7.11.17.19.31

|-

| S154⋅S155

| S154⋅S155

| ([[154/153]])/([[156/155]])

| ([[154/153]])/([[156/155]])

| [[11935/11934]]

| [[11935/11934]]

| 31

| 2.3.5.7.11.13.17.31

|-

| S169⋅S170

| S169⋅S170

| ([[169/168]])/([[171/170]])

| ([[169/168]])/([[171/170]])

| [[14365/14364]]

| [[14365/14364]]

| 19

| 2.3.5.7.13.17.19

|-

| S208⋅S209

| S208⋅S209

| ([[208/207]])/([[210/209]])

| ([[208/207]])/([[210/209]])

| [[21736/21735]]

| [[21736/21735]]

| 19

| 2.3.5.7.11.13.19

|-

| S323⋅S324

| S323⋅S324

| ([[323/322]])/([[325/324]])

| ([[323/322]])/([[325/324]])

| [[52326/52325]]

| [[52326/52325]]

| 23

| 2.3.5.7.13.17.19.23

|-

| S341⋅S342

| S341⋅S342

| ([[341/340]])/([[343/342]])

| ([[341/340]])/([[343/342]])

| [[58311/58310]]

| [[58311/58310]]

| 31

| 2.3.5.7.11.17.19.31

|-

| S494⋅S495

| S494⋅S495

| ([[494/493]])/([[496/495]])

| ([[494/493]])/([[496/495]])

| [[122265/122264]]

| [[122265/122264]]

| 31

| 2.3.5.7.11.13.19.29.31

|-

| S1519⋅S1520

| S1519⋅S1520

| ([[1519/1518]])/([[1521/1520]])

| ([[1519/1518]])/([[1521/1520]])

| [[1154440/1154439]]

| [[1154440/1154439]]

| 31

| 2.3.5.7.11.13.19.23.31

|}

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Line 1,801:

| [[570807/570752]]

| 0.167

| 29

| 2.3.7.13.29

|-

| S28/S29 = ([[784/783]])/([[841/840]])

Line 1,801:

Line 1,807:

| [[219520/219501]]

| 0.150

| 29

| 2.3.5.7.29

|-

| S31/S32 = ([[961/960]])/([[1024/1023]])

Line 1,807:

Line 1,813:

| [[327701/327680]]

| 0.111

| 31

| 2.5.11.31

|-

| S33/S34 = ([[1089/1088]])/([[1156/1155]])

Line 1,813:

Line 1,819:

| [[1257795/1257728]]

| 0.092

| 17

| 2.3.5.7.11.17

|-

| S34/S35 = ([[1156/1155]])/([[1225/1224]])

Line 1,819:

Line 1,825:

| [[471648/471625]]

| 0.084

| 17

| 2.3.5.7.11.17

|-

| S37/S38 = ([[1369/1368]])/([[1444/1443]])

Line 1,825:

Line 1,831:

| [[658489/658464]]

| 0.066

| 37

| 2.3.13.19.37

|-

| S40/S41 = ([[1600/1599]])/([[1681/1680]])

Line 1,831:

Line 1,837:

| [[896000/895973]]

| 0.052

| 41

| 2.5.7.13.41

|-

| S43/S44 = ([[1849/1848]])/([[1936/1935]])

Line 1,837:

Line 1,843:

| [[1192605/1192576]]

| 0.042

| 43

| 2.3.5.7.11.43

|-

| S46/S47 = ([[2116/2115]])/([[2209/2208]])

Line 1,843:

Line 1,849:

| [[1557376/1557345]]

| 0.034

| 47

| 2.3.5.23.47

|-

| S49/S50 = ([[2401/2400]])/([[2500/2499]])

Line 1,930:

Line 1,936:

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"

{| class="wikitable center-all left-4"

|-

! S-expression

! Square relation

! Ratio

! Subgroup

|-

| S2/S4 = ([[4/3]])/([[16/15]])

| ([[5/1]])/([[2/1]])2

| [[5/4]]

| 2.5

|-

| S3/S5 = ([[9/8]])/([[25/24]])

| ([[3/1]])/([[5/3]])2

| [[27/25]]

| 3.5

|-

| S4/S6 = ([[16/15]])/([[36/35]])

| ([[7/3]])/([[3/2]])2

| [[28/27]]

| 2.3.7

|-

| S5/S7 = ([[25/24]])/([[49/48]])

| ([[2/1]])/([[7/5]])2

| [[50/49]]

| 2.5.7

|-

| S6/S8 = ([[36/35]])/([[64/63]])

| ([[9/5]])/([[4/3]])2

| [[81/80]]

| 2.3.5

|-

| S7/S9 = ([[49/48]])/([[81/80]])

| ([[5/3]])/([[9/7]])2

| [[245/243]]

| 3.5.7

|-

| S8/S10 = ([[64/63]])/([[100/99]])

| ([[11/7]])/([[5/4]])2

| [[176/175]]

| 2.5.7.11

|-

| S9/S11 = ([[81/80]])/([[121/120]])

| ([[3/2]])/([[11/9]])2

| [[243/242]]

| 2.3.11

|-

| S10/S12 = ([[100/99]])/([[144/143]])

| ([[13/9]])/([[6/5]])2

| [[325/324]]

| 2.3.5.13

|-

| S11/S13 = ([[121/120]])/([[169/168]])

| ([[7/5]])/([[13/11]])2

| [[847/845]]

| 5.7.11.13

|-

| S12/S14 = ([[144/143]])/([[196/195]])

| ([[15/11]])/([[7/6]])2

| [[540/539]]

| 2.3.5.7.11

|-

| S13/S15 = ([[169/168]])/([[225/224]])

| ([[4/3]])/([[15/13]])2

| [[676/675]]

| 2.3.5.13

|-

| S14/S16 = ([[196/195]])/([[256/255]])

| ([[17/13]])/([[8/7]])2

| [[833/832]]

| 2.7.13.17

|-

| S15/S17 = ([[225/224]])/([[289/288]])

| ([[9/7]])/([[17/15]])2

| [[2025/2023]]

| 3.5.7.17

|-

| S16/S18 = ([[256/255]])/([[324/323]])

| ([[19/15]])/([[9/8]])2

| [[1216/1215]]

| 2.3.5.19

|-

| S17/S19 = ([[289/288]])/([[361/360]])

| ([[5/4]])/([[19/17]])2

| [[1445/1444]]

| 2.5.17.19

|-

| S18/S20 = ([[324/323]])/([[400/399]])

| ([[21/17]])/([[10/9]])2

| [[1701/1700]]

| 2.3.5.7.17

|-

| S19/S21 = ([[361/360]])/([[441/440]])

| ([[11/9]])/([[21/19]])2

| [[3971/3969]]

| 3.7.11.19

|-

| S20/S22 = ([[400/399]])/([[484/483]])

| ([[23/19]])/([[11/10]])2

| [[2300/2299]]

| 2.5.11.19.23

|-

| S21/S23 = ([[441/440]])/([[529/528]])

| ([[6/5]])/([[23/21]])2

| [[2646/2645]]

| 2.3.5.7.23

|-

| S22/S24 = ([[484/483]])/([[576/575]])

| ([[25/21]])/([[12/11]])2

| [[3025/3024]]

| 2.3.5.7.11

|-

| S23/S25 = ([[529/528]])/([[625/624]])

| ([[13/11]])/([[25/23]])2

| [[6877/6875]]

| 5.11.13.23

|-

| S24/S26 = ([[576/575]])/([[676/675]])

| ([[27/23]])/([[13/12]])2

| [[3888/3887]]

| 2.3.13.23

|-

| S25/S27 = ([[625/624]])/([[729/728]])

| ([[7/6]])/([[27/25]])2

| [[4375/4374]]

| 2.3.5.7

|-

| S26/S28 = ([[676/675]])/([[784/783]])

| ([[29/25]])/([[14/13]])2

| [[4901/4900]]

| 2.5.7.13.29

|-

| S27/S29 = ([[729/728]])/([[841/840]])

| ([[15/13]])/([[29/27]])2

| [[10935/10933]]

| 3.5.13.29

|-

| S28/S30 = ([[784/783]])/([[900/899]])

| ([[31/27]])/([[15/14]])2

| [[6076/6075]]

| 2.3.5.7.31

|-

| S29/S31 = ([[841/840]])/([[961/960]])

| ([[8/7]])/([[31/29]])2

| [[6728/6727]]

| 2.7.29.31

|-

| S30/S32 = ([[900/899]])/([[1024/1023]])

| ([[33/29]])/([[16/15]])2

| [[7425/7424]]

| 2.3.5.11.29

|-

| S31/S33 = ([[961/960]])/([[1089/1088]])

| ([[17/15]])/([[33/31]])2

| [[16337/16335]]

| 3.5.11.17.31

|-

| S32/S34 = ([[1024/1023]])/([[1156/1155]])

| ([[35/31]])/([[17/16]])2

| [[8960/8959]]

| 2.5.7.17.31

|-

| S33/S35 = ([[1089/1088]])/([[1225/1224]])

| ([[9/8]])/([[35/33]])2

| [[9801/9800]]

| 2.3.5.7.11

|-

| S36/S38 = ([[1296/1295]])/([[1444/1443]])

| ([[39/35]])/([[19/18]])2

| [[12636/12635]]

| 2.3.5.7.13.19

|-

| S37/S39 = ([[1369/1368]])/([[1521/1520]])

| ([[10/9]])/([[39/37]])2

| [[13690/13689]]

| 2.3.5.13.37

|-

| S41/S43 = ([[1681/1680]])/([[1849/1848]])

| ([[11/10]])/([[43/41]])2

| [[18491/18490]]

| 2.5.11.41.43

|-

| S45/S47 = ([[2025/2024]])/([[2209/2208]])

| ([[12/11]])/([[47/45]])2

| [[24300/24299]]

| 2.3.5.11.47

|-

| S46/S48 = ([[2116/2115]])/([[2304/2303]])

| ([[49/45]])/([[24/23]])2

| [[25921/25920]]

| 2.3.5.7.23

|-

| S49/S51 = ([[2401/2400]])/([[2601/2600]])

| ([[13/12]])/([[51/49]])2

| [[31213/31212]]

| 2.3.7.13.17

|-

| S52/S54 = ([[2704/2703]])/([[2916/2915]])

| ([[55/51]])/([[27/26]])2

| [[37180/37179]]

| 2.3.5.11.13.17

|-

| S66/S68 = ([[4356/4355]])/([[4624/4623]])

| ([[69/65]])/([[34/33]])2

| [[75141/75140]]

| 2.3.5.11.17.23

|-

| S78/S80 = ([[6084/6083]])/([[6400/6399]])

| ([[81/77]])/([[40/39]])2

| [[123201/123200]]

| 2.3.5.7.11.13

|}

Line 2,595:

Line 2,643:

|}

== ~~Using S~~-~~factorizations to understand the significance of S-expressions~~ ==

== C''k'' and Cp''k'' (cube-particulars) ==

This ~~section deals with the forms~~ of ~~the infinite comma families as expressed in terms~~ of ~~nearby harmonics in~~ the ~~harmonic series~~ and ~~as related to square~~-~~superparticulars; note that this uses a mathematical notation of [a, b, c,~~ .~~..]^[x~~, ~~y, z~~, .~~..]~~ to ~~denote a^x * b^y * c^z * ..~~.

This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''3|''k''3 − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''3 + 1|''k''3}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''3 + 1 and ''k''3 − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''2 + 1 can be prime. The term ''S-expression'' applies to these despite not using the letter ''S'', in avoidance of introducing additional terms.

~~If instead~~ of ~~working through things algebraically we look at~~ square-particulars ~~as describing a relationship between adjacent harmonics, we can use this~~ to ~~understand why certain simplifications and equivalences exist in~~ a ~~way that is equivalent to the sometimes harder-to-understand usual algebraic form:~~

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''2 + ''k'' + 1) for C''k'' or (''k''2 − ''k'' + 1) for Cp''k'' to get to a reasonable prime limit.

~~If we describe~~ S''k'' ~~as [~~''k''-1, ''k'', ''k''+1~~]^[-1, 2~~, -~~1] then if we write something like S''k''/S(''k'' + 2) (semiparticulars)~~ in ~~this form we get:~~

=== Significance ===

# Cube-particulars are 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.

# 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.

[''k''-1, ''k'', ''k''~~+1,~~ ''k''~~+2,~~ ''k''~~+3]^([-1, 2, -1, 0, 0] - [0, 0, -1, 2, -1]~~ = ~~[-1, 2, 0, -2, 1]) from which we can clearly see that we have two~~ (''k''+2)/''k'''~~s making up a~~ (''k''+3)/(''k''-1). ~~An exercise to the reader is to go through the~~ other ~~forms discussed on this page to derive similar expressions. (For example~~, ~~through cancellation it~~'~~s easy to prove that 1/n-~~square-~~particulars (the product of n consecutive~~ square-~~(super)particulars) are equal to the ratio of the two superparticular intervals on the ends~~.)

=== 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''2). Note S''k''/Sp''k'' = S(k2) holds too if Sp''k'' notation is used for (''k''2 + 1)/k2.

# C(''k''2) = S(''k''3). 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.

~~<pre>~~

=== Table of cube-particulars ===

Sk = [k-~~1, k, k+1]^[-1, 2, -1]~~

<~~/pre~~>

{| class="wikitable center-all left-3 left-6"

<~~pre~~>

|+ style="font-size: 105%;" | 31-limit cube-particulars – C''k''

~~Sk * S(k+1)~~ = [k-~~1, k, k+1, k+2]^[~~-~~1, 1, 1,~~ -1]

|-

= [k-~~1, k, k+1(, k+2)]^[~~-~~1, 2,~~ -~~1(, 0)] * [(~~k-~~1,) k, k+1, k+2]^[(0,)~~ -~~1, 2,~~ -1]

! S-expression

~~</pre>~~

! Ratio

~~<pre>~~

! Subgroup

~~S(k~~-~~1) * Sk * S(k+1) =~~ [k-2~~, k~~-~~1, k, k+1, k+2~~]~~^[-1, 1, 0, 1, -1~~]

|-

~~= ( (k~~-1)/~~(k-~~2~~) )( k/(k~~-~~1) ) * ( k~~/(k-~~1) )~~/~~( (k+1)/k ) * ( (k+1)/k )/( (k+~~2~~)/(k+1) )~~

| –

~~= ( (k~~-1)/(k-~~2) )~~/~~( (k+~~2~~)/(k+1) ) = ( (k~~-~~1)(k+1) )~~/~~( (k~~-~~2)(k+2) )~~

| –

k-2 k-~~1 k k+1 k+~~2

|-

-1 2 -~~1 0 0~~

| C2

~~0 -1~~ 2 -~~1 0~~

| [[8/7]]

~~0 0~~ -~~1 2 -1~~

| 2.7

~~========================~~

|-

~~-1 1 0 1 -1~~

| C3

</~~pre~~>

| [[27/26]]

<~~pre~~>

| 2.3.13

~~Sk / S(k+1)~~ = [k-~~1, k, k+1, k+2]^[~~-1, 3, -~~3, 1]~~

|-

= [k-~~1, k, k+1]^[~~-~~1, 2,~~ -1] * [~~k, k+1, k+2~~]~~^[1, -2, 1~~]

| C4

~~= (k+~~2~~)/(k~~-~~1) * ( k~~/~~(k+1) )^~~3 ~~= (k+2)/(k~~-~~1) / ((k+1)/k)^3~~

| [[64/63]]

</~~pre>~~

| 2.3.7

~~<pre>~~

|-

~~S(k~~-1) / ~~S(k+1) = [k-~~2~~, k~~-~~1, k, k+1, k+2~~]~~^[-1,~~ 2~~, 0,~~ -~~2, 1~~]

| C5

~~= [k-~~2~~, k~~-~~1, k]^~~[-1, 2, -1] * [~~k, k+1, k+2~~]~~^[ 1, -2, 1~~]

| [[125/124]]

~~= [k-~~2~~, k~~-~~1, k]^~~[~~-1, 2, -1]~~ / ~~[k, k+1, k+2~~]~~^[-1, 2, -1~~]

| 2.5.31

~~= (k+~~2~~)/(k~~-~~2) * ((k-1)~~/~~(k+1))^~~2 ~~= (k+2)/(k~~-2) / ~~((k+1)/(k-1))^2~~

|-

| C7

~~k-2 k-1 k k+1 k+~~2

| [[343/342]]

-~~1 2 -1 0 0~~

| 2.3.7.19

~~0 0 1 -2 1~~

|-

~~========================~~

| C9

-1 2 0 -~~2 1~~

| [[729/728]]

</~~pre>~~

| 2.3.7.13

|-

~~This technique will be called "'''S~~-~~factorizations'''", as it is uses a certain format for expressing factorization (analogous to~~ [[~~monzo~~]]~~s) that is uniquely suited for interpreting the relationships described by '''S~~-~~expressions'''.~~

| C11

| [[1331/1330]]

~~Note that the redundancy in these factorizations (in the sense that there are generators that are not linearly independent of the others) is a property that reflects the reality of~~ [[~~#Equivalent S-expressions|equivalent S-expressions~~]].

| 2.5.7.11.19

|-

~~The generalisation of this method using commutative group theory is discussed in~~ [[~~S-expression~~/~~Advanced_results#Abstraction~~]]~~, though the ideas are very simple for anyone with simple mathematical training willing to learn the very basics needed~~.

| C16

| [[4096/4095]]

| 2.3.5.7.13

|-

| C18

| [[5832/5831]]

| 2.3.7.17

|-

| C22

| [[10648/10647]]

| 2.3.7.11.13

|-

| C25

| [[15625/15624]]

| 2.3.5.7.31

|-

| C30

| [[27000/26999]]

| 2.3.5.7.19.29

|-

| –

|-

| –

|}

</div>

{| 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

|-

| Cp4

| [[65/64]]

| 2.5.13

|-

| Cp5

| [[126/125]]

| 2.3.5.7

|-

| Cp6

| [[217/216]]

| 2.3.7.31

|-

| Cp8

| [[513/512]]

| 2.3.19

|-

| Cp10

| [[1001/1000]]

| 2.5.7.11.13

|-

| Cp12

| [[1729/1728]]

| 2.3.7.13.19

|-

| Cp17

| [[4914/4913]]

| 2.3.7.13.17

|-

| Cp19

| [[6860/6859]]

| 2.5.7.19

|-

| Cp23

| [[12168/12167]]

| 2.3.13.23

|-

| Cp26

| [[17577/17576]]

| 2.3.7.13.31

|-

| Cp31

| [[29792/29791]]

| 2.7.19.31

|-

| Cp68

| [[314433/314432]]

| 2.3.7.17.23.31

|-

| Cp69

| [[328510/328509]]

| 2.3.5.7.13.19.23

|}

</div></div>

~~=== Using S-factorizations~~ to ~~show a useful equivalence/redundancy of S-expressions ===~~

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.

~~Absent of restrictions on~~ the ~~form that an S-expression may take~~, ~~there is no unique S-expression for any given rational number. This is in fact a huge advantage~~, ~~because it allows one to understand the landscape of commas in~~ a ~~way that sees interconnectedness of subgroups~~ and ~~corresponding tempering opportunities~~. ~~But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:~~

==~~== The general~~ S-~~expression equivalence ==~~==

== Using S-factorizations to understand the significance of S-expressions ==

~~Consider:~~

This section deals with the forms of the infinite comma families as expressed in terms of nearby harmonics in the harmonic series and as related to square-superparticulars; note that this uses a mathematical notation of [a, b, c, ...]^[x, y, z, ...] to denote a^x * b^y * c^z * ...

~~<pre>~~

~~Sk =~~ [~~k-1~~, k, ~~k+1~~]^[-1, 2, -1] ~~versus what it is claimed~~ to ~~be equivalent to:~~

~~S(2k-1)~~ * ~~S(2k)~~ * ~~S(2k)~~ * ~~S(2k+1)~~

~~= [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(~~

~~[-1, 2, -1]~~

~~+ [-2, 4, -2]~~

~~+ [-1, 2, -1]~~

~~= [-1, 0, 2, 0, -1] )~~

~~</pre>~~

From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). ~~Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':~~

~~<math>~~

If instead of working through things algebraically we look at square-particulars as describing a relationship between adjacent harmonics, we can use this to understand why certain simplifications and equivalences exist in a way that is equivalent to the sometimes harder-to-understand usual algebraic form:

~~\large {\rm S}k = \large {\rm S}(2k~~-~~1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)~~

~~</math>~~

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

If we describe S''k'' as [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] then if we write something like S''k''/S(''k'' + 2) (semiparticulars) in this form we get:

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]].

[''k''-1, ''k'', ''k''+1, ''k''+2, ''k''+3]^([-1, 2, -1, 0, 0] - [0, 0, -1, 2, -1] = [-1, 2, 0, -2, 1]) from which we can clearly see that we have two (''k''+2)/''k'''s making up a (''k''+3)/(''k''-1). An exercise to the reader is to go through the other forms discussed on this page to derive similar expressions. (For example, through cancellation it's easy to prove that 1/n-square-particulars (the product of n consecutive square-(super)particulars) are equal to the ratio of the two superparticular intervals on the ends.)

== Equivalent S-expressions ==

<pre>

~~=== Significance and meaning =~~==

Sk = [k-1, k, k+1]^[-1, 2, -1]

All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed ~~on this page~~, or otherwise makes it equal to a product or ratio between two such commas, this often means ~~exceptional and~~ nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.

</pre>

<pre>

=== A useful general rule ===

Sk * S(k+1) = [k-1, k, k+1, k+2]^[-1, 1, 1, -1]

While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:

= [k-1, k, k+1(, k+2)]^[-1, 2, -1(, 0)] * [(k-1,) k, k+1, k+2]^[(0,) -1, 2, -1]

</pre>

$$

<pre>

~~\large~~ {\rm S}k = ~~\large~~ {\rm S}(2k-1) \cdot ~~\large~~ {\rm S}(2k)^2 \cdot ~~\large~~ {\rm S}(2k+1)

S(k-1) * Sk * S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 1, 0, 1, -1]

$$

= ( (k-1)/(k-2) )( k/(k-1) ) * ( k/(k-1) )/( (k+1)/k ) * ( (k+1)/k )/( (k+2)/(k+1) )

= ( (k-1)/(k-2) )/( (k+2)/(k+1) ) = ( (k-1)(k+1) )/( (k-2)(k+2) )

This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions ~~that should be discarded from [[#Examples]]~~. See [[S-expression/Advanced results]] for mathematical details.

k-2 k-1 k k+1 k+2

=== Examples ===

-1 2 -1 0 0

Here is an incomplete list of examples.

0 -1 2 -1 0

0 0 -1 2 -1

{| class="wikitable center-1"

========================

|-

-1 1 0 1 -1

! Comma

</pre>

! S-expressions

<pre>

|-

Sk / S(k+1) = [k-1, k, k+1, k+2]^[-1, 3, -3, 1]

| [[28/27]]

= [k-1, k, k+1]^[-1, 2, -1] * [k, k+1, k+2]^[1, -2, 1]

| S7⋅S8, S4/S6

= (k+2)/(k-1) * ( k/(k+1) )^3 = (k+2)/(k-1) / ((k+1)/k)^3

|-

</pre>

| [[36/35]]

<pre>

| S6, S8⋅S9

S(k-1) / S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 2, 0, -2, 1]

|-

= [k-2, k-1, k]^[-1, 2, -1] * [k, k+1, k+2]^[ 1, -2, 1]

| [[64/63]]

= [k-2, k-1, k]^[-1, 2, -1] / [k, k+1, k+2]^[-1, 2, -1]

| S8, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3

= (k+2)/(k-2) * ((k-1)/(k+1))^2 = (k+2)/(k-2) / ((k+1)/(k-1))^2

|-

| [[81/80]]

k-2 k-1 k k+1 k+2

| S9, S6/S8

-1 2 -1 0 0

|-

0 0 1 -2 1

| [[176/175]]

========================

| S8/S10, S22⋅S23⋅S24

-1 2 0 -2 1

</pre>

This technique will be called "'''S-factorizations'''", as it is uses a certain format for expressing factorization (analogous to [[monzo]]s) that is uniquely suited for interpreting the relationships described by '''S-expressions'''.

Note that the redundancy in these factorizations (in the sense that there are generators that are not linearly independent of the others) is a property that reflects the reality of [[#Equivalent S-expressions|equivalent S-expressions]].

The generalisation of this method using commutative group theory is discussed in [[S-expression/Advanced_results#Abstraction]], though the ideas are very simple for anyone with simple mathematical training willing to learn the very basics needed.

=== Using S-factorizations to show a useful equivalence/redundancy of S-expressions ===

Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:

==== The general S-expression equivalence ====

Consider:

<pre>

Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:

S(2k-1) * S(2k) * S(2k) * S(2k+1)

= [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(

[-1, 2, -1]

+ [-2, 4, -2]

+ [-1, 2, -1]

= [-1, 0, 2, 0, -1] )

</pre>

From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':

<math>

\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)

</math>

...where we use the notation S''k''''p'' to mean (S''k'')''p'' rather than S(''k''''p'') for convenience in the practical analysis of [[regular 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-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]].

== Equivalent S-expressions ==

All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed above, or otherwise makes it equal to a product or ratio between two such commas, this often means nontrivial, deep tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.

=== A useful general rule ===

While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:

$$

{\rm S}k = {\rm S}(2k - 1) \cdot {\rm S}(2k)^2 \cdot {\rm S}(2k + 1)

$$

This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions. See [[S-expression/Advanced results]] for mathematical details.

=== Examples ===

Here is an incomplete list of examples.

{| class="wikitable center-1"

|-

! Comma

! S-expressions

|-

| [[28/27]]

| S7⋅S8, S4/S6

|-

| [[36/35]]

| S6, S8⋅S9

|-

| [[64/63]]

| S8, S6/S9, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3

|-

| [[81/80]]

| S9, S6/S8

|-

| [[176/175]]

| S8/S10, S22⋅S23⋅S24

|-

| [[243/242]]

| S9/S11, S15/(~~[[3025/3024|~~S22/S24 ~~= S55]]~~)

| S9/S11, S15/S55, S15/(S22/S24)

|-

| [[325/324]]

Line 2,721:

Line 2,924:

| [[1225/1224]]

| S35, S49⋅S50

|-

| [[2601/2600]]

| S51, S17/(S25⋅S26)

|-

| [[3025/3024]]

| S55, S22/S24, (S25/S27)⋅S99

|-

~~| [[2601/2600]]~~

~~| S51, S17/(S25⋅S26)~~

|-

| [[9801/9800]]

Line 2,734:

Line 2,937:

| S161, S46/S48

|-

| [[123201/123200]]

| [[123201/123200]]

| S351, S78/S80

|}

{{Note| Examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are not included. }}

{{Note| Where a comma written in the form ''a''/''b'' is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient. }}

{{Tip| Feel free to expand with any equivalences you find that you think are valuable. }}

Line 2,780:

Line 2,982:

: For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.

: These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk⋅S(k + 1)⋅…⋅S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].

; Ultraparticular

: An interval/comma which is the ratio of two consecutive square-particulars.

: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.

; Semiparticular

: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:

: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.

; Cube-particular

: A superparticular interval/comma whose numerator or denominator is a cube number. A shorthand (nick)name for cube superparticular.

: These are of the form {{nowrap|{{sfrac|''k''3|''k''3 − 1}} {{=}} C''k''}} or {{nowrap|{{sfrac|''k''3 + 1|''k''3}} {{=}} Cp''k''}}.

; Odd-particular

Line 2,797:

Line 3,011:

: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.

: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.

~~; Ultraparticular~~

~~: An interval/comma which is the ratio of two consecutive square-particulars.~~

~~: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.~~

~~; Semiparticular~~

~~: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:~~

~~: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.~~

; S-expression

: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.

: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only arbitrary square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars. (By extension, the expression could also include cube-particulars, C''k'' and Cp''k''.)

; S-factorization

Line 2,814:

Line 3,020:

; S-comma

: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).

: Any comma within one of the infinite families of commas discussed here, excluding 1/''n''-square-particulars for ''n'' > 5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than ''n'' > 3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).

; Indirect S-comma

@@ Line 4: / Line 4: @@
 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.
-* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.
+* [[#Sk (square-particulars)|Square-particulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.
 * [[#Sk⋅S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k''⋅S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k''⋅S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.
 * [[#Sk2⋅S(k + 1) and S(k − 1)⋅Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup>⋅S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup>⋅S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup> and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>3</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with  {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup> and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
 * [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
 * [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
+* [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} and {{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}}, respectively.
 == S''k'' (square-particulars) ==
@@ Line 527: / Line 528: @@
 For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of inconsistency of mappings when tempered out for the above reason.
-{| class="wikitable center-all
+{| class="wikitable center-all left-4"
 |+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as 311edo can be described as the unique 23-limit temperament that tempers out all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers out all the square-particulars composing those triangle-particulars with the exception of S169 and S170, and maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3, meaning the mapping of that superparticular is inconsistent.</ref>
 |-
@@ Line 533: / Line 534: @@
 ! Interval relation
 ! Ratio
-! Prime limit
+! Subgroup
 |-
 | S2⋅S3
 | ([[3/1]])/([[2/1]])
 | [[3/2]]
-| 3
+| 2.3
 |-
 | S3⋅S4
 | ([[3/2]])/([[5/4]])
 | [[6/5]]
-| 5
+| 2.3.5
 |-
 | S4⋅S5
 | ([[4/3]])/([[6/5]])
 | [[10/9]]
-| 5
+| 2.3.5
 |-
 | S5⋅S6
 | ([[5/4]])/([[7/6]])
 | [[15/14]]
-| 7
+| 2.3.5.7
 |-
 | S6⋅S7
 | ([[6/5]])/([[8/7]])
 | [[21/20]]
-| 7
+| 2.3.5.7
 |-
 | S7⋅S8
 | ([[7/6]])([[9/8]])
 | [[28/27]]
-| 7
+| 2.3.7
 |-
 | S8⋅S9
 | ([[8/7]])/([[10/9]])
 | [[36/35]]
-| 7
+| 2.3.5.7
 |-
 | S9⋅S10
 | ([[9/8]])/([[11/10]])
 | [[45/44]]
-| 11
+| 2.3.5.11
 |-
 | S10⋅S11
 | ([[10/9]])/([[12/11]])
 | [[55/54]]
-| 11
+| 2.3.5.11
 |-
 | S11⋅S12
 | ([[11/10]])/([[13/12]])
 | [[66/65]]
-| 13
+| 2.3.5.11.13
 |-
 | S12⋅S13
 | ([[12/11]])/([[14/13]])
 | [[78/77]]
-| 13
+| 2.3.7.11.13
 |-
 | S13⋅S14
 | ([[13/12]])/([[15/14]])
 | [[91/90]]
-| 13
+| 2.3.5.7.13
 |-
 | S14⋅S15
 | ([[14/13]])/([[16/15]])
 | [[105/104]]
-| 13
+| 2.3.5.7.13
 |-
 | S15⋅S16
 | ([[15/14]])/([[17/16]])
 | [[120/119]]
-| 17
+| 2.3.5.7.17
 |-
 | S16⋅S17
 | ([[16/15]])/([[18/17]])
 | [[136/135]]
-| 17
+| 2.3.5.17
 |-
 | S17⋅S18
 | ([[17/16]])/([[19/18]])
 | [[153/152]]
-| 19
+| 2.3.17.19
 |-
 | S18⋅S19
 | ([[18/17]])/([[20/19]])
 | [[171/170]]
-| 19
+| 2.3.5.17.19
 |-
 | S19⋅S20
 | ([[19/18]])/([[21/20]])
 | [[190/189]]
-| 19
+| 2.3.5.7.19
 |-
 | S20⋅S21
 | ([[20/19]])/([[22/21]])
 | [[210/209]]
-| 19
+| 2.3.5.7.11.19
 |-
 | S21⋅S22
 | ([[21/20]])/([[23/22]])
 | [[231/230]]
-| 23
+| 2.3.5.7.11.23
 |-
 | S22⋅S23
 | ([[22/21]])/([[24/23]])
 | [[253/252]]
-| 23
+| 2.3.5.7.11.23
 |-
 | S23⋅S24
 | ([[23/22]])/([[25/24]])
 | [[276/275]]
-| 23
+| 2.3.5.11.23
 |-
 | S24⋅S25
 | ([[24/23]])/([[26/25]])
 | [[300/299]]
-| 23
+| 2.3.5.13.23
 |-
 | S25⋅S26
 | ([[25/24]])/([[27/26]])
 | [[325/324]]
-| 13
+| 2.3.5.13
 |-
 | S26⋅S27
 | ([[26/25]])/([[28/27]])
 | [[351/350]]
-| 13
+| 2.3.5.7.13
 |-
 | S27⋅S28
 | ([[27/26]])/([[29/28]])
 | [[378/377]]
-| 29
+| 2.3.5.7.13.29
 |-
 | S28⋅S29
 | ([[28/27]])/([[30/29]])
 | [[406/405]]
-| 29
+| 2.3.5.7.29
 |-
 | S29⋅S30
 | ([[29/28]])/([[31/30]])
 | [[435/434]]
-| 31
+| 2.3.5.7.29.31
 |-
 | S30⋅S31
 | ([[30/29]])/([[32/31]])
 | [[465/464]]
-| 31
+| 2.3.5.29.31
 |-
 | S31⋅S32
 | ([[31/30]])/([[33/32]])
 | [[496/495]]
-| 31
+| 2.3.5.11.31
 |-
 | S32⋅S33
 | ([[32/31]])/([[34/33]])
 | [[528/527]]
-| 31
+| 2.3.11.17.31
 |-
 | S33⋅S34
 | ([[33/32]])/([[35/34]])
 | [[561/560]]
-| 17
+| 2.3.5.7.11.17
 |-
 | S34⋅S35
 | ([[34/33]])/([[36/35]])
 | [[595/594]]
-| 17
+| 2.3.5.7.11.17
 |-
 | S49⋅S50
 | ([[49/48]])/([[51/50]])
 | [[1225/1224]]
-| 17
+| 2.3.5.7.17
 |-
 | S50⋅S51
 | ([[50/49]])/([[52/51]])
 | [[1275/1274]]
-| 17
+| 2.3.5.7.13.17
 |-
 | S55⋅S56
 | ([[55/54]])/([[57/56]])
 | [[1540/1539]]
-| 19
+| 2.3.5.7.11.19
+|-
+| S56⋅S57
+| ([[56/55]])/([[58/57]])
+| [[1596/1595]]
+| 2.3.5.7.11.19.29
 |-
 | S63⋅S64
 | ([[63/62]])/([[65/64]])
 | [[2016/2015]]
-| 31
+| 2.3.5.7.13.31
 |-
 | S64⋅S65
 | ([[64/63]])/([[66/65]])
 | [[2080/2079]]
-| 13
+| 2.3.5.7.11.13
 |-
 | S76⋅S77
 | ([[76/75]])/([[78/77]])
 | [[2926/2925]]
-| 19
+| 2.3.5.7.11.13.19
 |-
 | S91⋅S92
 | ([[91/90]])/([[93/92]])
 | [[4186/4185]]
-| 31
+| 2.3.5.7.13.23.31
 |-
-| S115⋅S116
+| <small>S115⋅S116</small>
-| ([[115/114]])/([[117/116]])
+| <small>([[115/114]])/([[117/116]])</small>
 | [[6670/6669]]
-| 29
+| 2.3.5.13.19.23.29
 |-
-| S153⋅S154
+| <small>S153⋅S154</small>
-| ([[153/152]])/([[155/154]])
+| <small>([[153/152]])/([[155/154]])</small>
-| [[11781/11780]]
+| <small>[[11781/11780]]</small>
-| 31
+| 2.3.5.7.11.17.19.31
 |-
-| S154⋅S155
+| <small>S154⋅S155</small>
-| ([[154/153]])/([[156/155]])
+| <small>([[154/153]])/([[156/155]])</small>
-| [[11935/11934]]
+| <small>[[11935/11934]]</small>
-| 31
+| 2.3.5.7.11.13.17.31
 |-
-| S169⋅S170
+| <small>S169⋅S170</small>
-| ([[169/168]])/([[171/170]])
+| <small>([[169/168]])/([[171/170]])</small>
-| [[14365/14364]]
+| <small>[[14365/14364]]</small>
-| 19
+| 2.3.5.7.13.17.19
 |-
-| S208⋅S209
+| <small>S208⋅S209</small>
-| ([[208/207]])/([[210/209]])
+| <small>([[208/207]])/([[210/209]])</small>
-| [[21736/21735]]
+| <small>[[21736/21735]]</small>
-| 19
+| 2.3.5.7.11.13.19
 |-
-| S323⋅S324
+| <small>S323⋅S324</small>
-| ([[323/322]])/([[325/324]])
+| <small>([[323/322]])/([[325/324]])</small>
-| [[52326/52325]]
+| <small>[[52326/52325]]</small>
-| 23
+| 2.3.5.7.13.17.19.23
 |-
-| S341⋅S342
+| <small>S341⋅S342</small>
-| ([[341/340]])/([[343/342]])
+| <small>([[341/340]])/([[343/342]])</small>
-| [[58311/58310]]
+| <small>[[58311/58310]]</small>
-| 31
+| 2.3.5.7.11.17.19.31
 |-
-| S494⋅S495
+| <small>S494⋅S495</small>
-| ([[494/493]])/([[496/495]])
+| <small>([[494/493]])/([[496/495]])</small>
-| [[122265/122264]]
+| <small>[[122265/122264]]</small>
-| 31
+| 2.3.5.7.11.13.19.29.31
 |-
-| S1519⋅S1520
+| <small>S1519⋅S1520</small>
-| ([[1519/1518]])/([[1521/1520]])
+| <small>([[1519/1518]])/([[1521/1520]])</small>
-| [[1154440/1154439]]
+| <small>[[1154440/1154439]]</small>
-| 31
+| 2.3.5.7.11.13.19.23.31
 |}
@@ Line 1,795: / Line 1,801: @@
 | [[570807/570752]]
 | 0.167
-| 29
+| 2.3.7.13.29
 |-
 | S28/S29 = ([[784/783]])/([[841/840]])
@@ Line 1,801: / Line 1,807: @@
 | [[219520/219501]]
 | 0.150
-| 29
+| 2.3.5.7.29
 |-
 | <small>S31/S32 = ([[961/960]])/([[1024/1023]])</small>
@@ Line 1,807: / Line 1,813: @@
 | [[327701/327680]]
 | 0.111
-| 31
+| 2.5.11.31
 |-
 | <small>S33/S34 = ([[1089/1088]])/([[1156/1155]])</small>
@@ Line 1,813: / Line 1,819: @@
 | <small>[[1257795/1257728]]</small>
 | 0.092
-| 17
+| 2.3.5.7.11.17
 |-
 | <small>S34/S35 = ([[1156/1155]])/([[1225/1224]])</small>
@@ Line 1,819: / Line 1,825: @@
 | [[471648/471625]]
 | 0.084
-| 17
+| 2.3.5.7.11.17
 |-
 | <small>S37/S38 = ([[1369/1368]])/([[1444/1443]])</small>
@@ Line 1,825: / Line 1,831: @@
 | [[658489/658464]]
 | 0.066
-| 37
+| 2.3.13.19.37
 |-
 | <small>S40/S41 = ([[1600/1599]])/([[1681/1680]])</small>
@@ Line 1,831: / Line 1,837: @@
 | [[896000/895973]]
 | 0.052
-| 41
+| 2.5.7.13.41
 |-
 | <small>S43/S44 = ([[1849/1848]])/([[1936/1935]])</small>
@@ Line 1,837: / Line 1,843: @@
 | <small>[[1192605/1192576]]</small>
 | 0.042
-| 43
+| 2.3.5.7.11.43
 |-
 | <small>S46/S47 = ([[2116/2115]])/([[2209/2208]])</small>
@@ Line 1,843: / Line 1,849: @@
 | <small>[[1557376/1557345]]</small>
 | 0.034
-| 47
+| 2.3.5.23.47
 |-
 | <small>S49/S50 = ([[2401/2400]])/([[2500/2499]])</small>
@@ Line 1,930: / Line 1,936: @@
 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"
+{| 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
 |-
 | 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]]
+| 2.3.7
 |-
 | S5/S7 = ([[25/24]])/([[49/48]])
 | ([[2/1]])/([[7/5]])<sup>2</sup>
 | [[50/49]]
+| 2.5.7
 |-
 | S6/S8 = ([[36/35]])/([[64/63]])
 | ([[9/5]])/([[4/3]])<sup>2</sup>
 | [[81/80]]
+| 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]]
+| 2.5.7.11
 |-
 | S9/S11 = ([[81/80]])/([[121/120]])
 | ([[3/2]])/([[11/9]])<sup>2</sup>
 | [[243/242]]
+| 2.3.11
 |-
 | S10/S12 = ([[100/99]])/([[144/143]])
 | ([[13/9]])/([[6/5]])<sup>2</sup>
 | [[325/324]]
+| 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]]
+| 2.3.5.13
 |-
 | S14/S16 = ([[196/195]])/([[256/255]])
 | ([[17/13]])/([[8/7]])<sup>2</sup>
 | [[833/832]]
+| 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]]
+| 2.5.11.19.23
 |-
 | S21/S23 = ([[441/440]])/([[529/528]])
 | ([[6/5]])/([[23/21]])<sup>2</sup>
 | [[2646/2645]]
+| 2.3.5.7.23
 |-
 | S22/S24 = ([[484/483]])/([[576/575]])
 | ([[25/21]])/([[12/11]])<sup>2</sup>
 | [[3025/3024]]
+| 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]]
+| 2.3.13.23
 |-
 | S25/S27 = ([[625/624]])/([[729/728]])
 | ([[7/6]])/([[27/25]])<sup>2</sup>
 | [[4375/4374]]
+| 2.3.5.7
 |-
 | S26/S28 = ([[676/675]])/([[784/783]])
 | ([[29/25]])/([[14/13]])<sup>2</sup>
 | [[4901/4900]]
+| 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]]
+| 2.3.5.7.31
 |-
 | S29/S31 = ([[841/840]])/([[961/960]])
 | ([[8/7]])/([[31/29]])<sup>2</sup>
 | [[6728/6727]]
+| 2.7.29.31
 |-
 | S30/S32 = ([[900/899]])/([[1024/1023]])
 | ([[33/29]])/([[16/15]])<sup>2</sup>
 | [[7425/7424]]
+| 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]]
+| 2.5.7.17.31
 |-
 | S33/S35 = ([[1089/1088]])/([[1225/1224]])
 | ([[9/8]])/([[35/33]])<sup>2</sup>
 | [[9801/9800]]
+| 2.3.5.7.11
 |-
 | S36/S38 = ([[1296/1295]])/([[1444/1443]])
 | ([[39/35]])/([[19/18]])<sup>2</sup>
 | [[12636/12635]]
+| 2.3.5.7.13.19
 |-
 | S37/S39 = ([[1369/1368]])/([[1521/1520]])
 | ([[10/9]])/([[39/37]])<sup>2</sup>
 | [[13690/13689]]
+| 2.3.5.13.37
 |-
 | S41/S43 = ([[1681/1680]])/([[1849/1848]])
 | ([[11/10]])/([[43/41]])<sup>2</sup>
 | [[18491/18490]]
+| 2.5.11.41.43
 |-
 | S45/S47 = ([[2025/2024]])/([[2209/2208]])
 | ([[12/11]])/([[47/45]])<sup>2</sup>
 | [[24300/24299]]
+| 2.3.5.11.47
 |-
 | S46/S48 = ([[2116/2115]])/([[2304/2303]])
 | ([[49/45]])/([[24/23]])<sup>2</sup>
 | [[25921/25920]]
+| 2.3.5.7.23
 |-
 | S49/S51 = ([[2401/2400]])/([[2601/2600]])
 | ([[13/12]])/([[51/49]])<sup>2</sup>
 | [[31213/31212]]
+| 2.3.7.13.17
 |-
 | S52/S54 = ([[2704/2703]])/([[2916/2915]])
 | ([[55/51]])/([[27/26]])<sup>2</sup>
 | [[37180/37179]]
+| 2.3.5.11.13.17
 |-
 | S66/S68 = ([[4356/4355]])/([[4624/4623]])
 | ([[69/65]])/([[34/33]])<sup>2</sup>
 | [[75141/75140]]
+| 2.3.5.11.17.23
 |-
 | S78/S80 = ([[6084/6083]])/([[6400/6399]])
 | ([[81/77]])/([[40/39]])<sup>2</sup>
 | [[123201/123200]]
+| 2.3.5.7.11.13
 |}
@@ Line 2,595: / Line 2,643: @@
 |}
-== Using S-factorizations to understand the significance of S-expressions ==
+== C''k'' and Cp''k'' (cube-particulars) ==
-This section deals with the forms of the infinite comma families as expressed in terms of nearby harmonics in the harmonic series and as related to square-superparticulars; note that this uses a mathematical notation of [a, b, c, ...]^[x, y, z, ...] to denote a^x * b^y * c^z * ...
+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''). 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. The term ''S-expression'' applies to these despite not using the letter ''S'', in avoidance of introducing additional terms.
-If instead of working through things algebraically we look at square-particulars as describing a relationship between adjacent harmonics, we can use this to understand why certain simplifications and equivalences exist in a way that is equivalent to the sometimes harder-to-understand usual algebraic form:
+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.
-If we describe S''k'' as [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] then if we write something like S''k''/S(''k'' + 2) (semiparticulars) in this form we get:
+=== Significance ===
+# Cube-particulars are 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.
+# 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.
-[''k''-1, ''k'', ''k''+1, ''k''+2, ''k''+3]^([-1, 2, -1, 0, 0] - [0, 0, -1, 2, -1] = [-1, 2, 0, -2, 1]) from which we can clearly see that we have two (''k''+2)/''k'''s making up a (''k''+3)/(''k''-1). An exercise to the reader is to go through the other forms discussed on this page to derive similar expressions. (For example, through cancellation it's easy to prove that 1/n-square-particulars (the product of n consecutive square-(super)particulars) are equal to the ratio of the two superparticular intervals on the ends.)
+=== 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.
-<pre>
+=== Table of cube-particulars ===
-Sk = [k-1, k, k+1]^[-1, 2, -1]
+<div><div style="display: inline-grid; margin-right: 25px;">
-</pre>
+{| class="wikitable center-all left-3 left-6"
-<pre>
+|+ style="font-size: 105%;" | 31-limit cube-particulars – C''k''
-Sk * S(k+1) = [k-1, k, k+1, k+2]^[-1, 1, 1, -1]
+|-
- = [k-1, k, k+1(, k+2)]^[-1, 2, -1(, 0)] * [(k-1,) k, k+1, k+2]^[(0,) -1, 2, -1]
+! S-expression
-</pre>
+! Ratio
-<pre>
+! Subgroup
-S(k-1) * Sk * S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 1, 0, 1, -1]
+|-
- = ( (k-1)/(k-2) )( k/(k-1) ) * ( k/(k-1) )/( (k+1)/k ) * ( (k+1)/k )/( (k+2)/(k+1) )
+| –
- = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) = ( (k-1)(k+1) )/( (k-2)(k+2) )
+| –
+| –
-k-2  k-1   k   k+1   k+2
+|-
--1    2   -1    0    0
+| C2
-   -1    2   -1    0
+| [[8/7]]
-    0   -1    2   -1
+| 2.7
-========================
+|-
--1    1    0    1   -1
+| C3
-</pre>
+| [[27/26]]
-<pre>
+| 2.3.13
-Sk / S(k+1) = [k-1, k, k+1, k+2]^[-1, 3, -3, 1]
+|-
- = [k-1, k, k+1]^[-1, 2, -1] * [k, k+1, k+2]^[1, -2, 1]
+| C4
- = (k+2)/(k-1) * ( k/(k+1) )^3 = (k+2)/(k-1) / ((k+1)/k)^3
+| [[64/63]]
-</pre>
+| 2.3.7
-<pre>
+|-
-S(k-1) / S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 2, 0, -2, 1]
+| C5
- = [k-2, k-1, k]^[-1, 2, -1] * [k, k+1, k+2]^[ 1, -2,  1]
+| [[125/124]]
- = [k-2, k-1, k]^[-1, 2, -1] / [k, k+1, k+2]^[-1,  2, -1]
+| 2.5.31
- = (k+2)/(k-2) * ((k-1)/(k+1))^2 = (k+2)/(k-2) / ((k+1)/(k-1))^2
+|-
+| C7
-k-2  k-1   k   k+1   k+2
+| [[343/342]]
--1    2   -1    0    0
+| 2.3.7.19
-    0    1   -2    1
+|-
-========================
+| C9
--1    2    0   -2    1
+| [[729/728]]
-</pre>
+| 2.3.7.13
+|-
-This technique will be called "'''S-factorizations'''", as it is uses a certain format for expressing factorization (analogous to [[monzo]]s) that is uniquely suited for interpreting the relationships described by '''S-expressions'''.
+| C11
+| [[1331/1330]]
-Note that the redundancy in these factorizations (in the sense that there are generators that are not linearly independent of the others) is a property that reflects the reality of [[#Equivalent S-expressions|equivalent S-expressions]].
+| 2.5.7.11.19
+|-
-The generalisation of this method using commutative group theory is discussed in [[S-expression/Advanced_results#Abstraction]], though the ideas are very simple for anyone with simple mathematical training willing to learn the very basics needed.
+| C16
+| [[4096/4095]]
+| 2.3.5.7.13
+|-
+| C18
+| [[5832/5831]]
+| 2.3.7.17
+|-
+| C22
+| [[10648/10647]]
+| 2.3.7.11.13
+|-
+| C25
+| [[15625/15624]]
+| 2.3.5.7.31
+|-
+| C30
+| [[27000/26999]]
+| 2.3.5.7.19.29
+|-
+| –
+| –
+| –
+|-
+| –
+| –
+| –
+|}
+</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
+|-
+| Cp4
+| [[65/64]]
+| 2.5.13
+|-
+| Cp5
+| [[126/125]]
+| 2.3.5.7
+|-
+| Cp6
+| [[217/216]]
+| 2.3.7.31
+|-
+| Cp8
+| [[513/512]]
+| 2.3.19
+|-
+| Cp10
+| [[1001/1000]]
+| 2.5.7.11.13
+|-
+| Cp12
+| [[1729/1728]]
+| 2.3.7.13.19
+|-
+| Cp17
+| [[4914/4913]]
+| 2.3.7.13.17
+|-
+| Cp19
+| [[6860/6859]]
+| 2.5.7.19
+|-
+| Cp23
+| [[12168/12167]]
+| 2.3.13.23
+|-
+| Cp26
+| [[17577/17576]]
+| 2.3.7.13.31
+|-
+| Cp31
+| [[29792/29791]]
+| 2.7.19.31
+|-
+| Cp68
+| <small>[[314433/314432]]</small>
+| 2.3.7.17.23.31
+|-
+| Cp69
+| <small>[[328510/328509]]</small>
+| 2.3.5.7.13.19.23
+|}
+</div></div>
-=== Using S-factorizations to show a useful equivalence/redundancy of S-expressions ===
+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.
-Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:
-==== The general S-expression equivalence ====
+== Using S-factorizations to understand the significance of S-expressions ==
-Consider:
+This section deals with the forms of the infinite comma families as expressed in terms of nearby harmonics in the harmonic series and as related to square-superparticulars; note that this uses a mathematical notation of [a, b, c, ...]^[x, y, z, ...] to denote a^x * b^y * c^z * ...
-<pre>
-Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:
-S(2k-1) * S(2k) * S(2k) * S(2k+1)
- = [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(
-   [-1,    2,   -1]
-       + [-2,    4,   -2]
-             + [-1,    2,   -1]
- = [-1,    0,    2,    0,   -1] )
-</pre>
-From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':
-<math>
+If instead of working through things algebraically we look at square-particulars as describing a relationship between adjacent harmonics, we can use this to understand why certain simplifications and equivalences exist in a way that is equivalent to the sometimes harder-to-understand usual algebraic form:
-\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
-</math>
-...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.
+If we describe S''k'' as [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] then if we write something like S''k''/S(''k'' + 2) (semiparticulars) in this form we get:
-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]].
+[''k''-1, ''k'', ''k''+1, ''k''+2, ''k''+3]^([-1, 2, -1, 0, 0] - [0, 0, -1, 2, -1] = [-1, 2, 0, -2, 1]) from which we can clearly see that we have two (''k''+2)/''k'''s making up a (''k''+3)/(''k''-1). An exercise to the reader is to go through the other forms discussed on this page to derive similar expressions. (For example, through cancellation it's easy to prove that 1/n-square-particulars (the product of n consecutive square-(super)particulars) are equal to the ratio of the two superparticular intervals on the ends.)
-== Equivalent S-expressions ==
+<pre>
-=== Significance and meaning ===
+Sk = [k-1, k, k+1]^[-1, 2, -1]
-All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.
+</pre>
+<pre>
-=== A useful general rule ===
+Sk * S(k+1) = [k-1, k, k+1, k+2]^[-1, 1, 1, -1]
-While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:
+ = [k-1, k, k+1(, k+2)]^[-1, 2, -1(, 0)] * [(k-1,) k, k+1, k+2]^[(0,) -1, 2, -1]
+</pre>
-$$
+<pre>
-\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
+S(k-1) * Sk * S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 1, 0, 1, -1]
-$$
+ = ( (k-1)/(k-2) )( k/(k-1) ) * ( k/(k-1) )/( (k+1)/k ) * ( (k+1)/k )/( (k+2)/(k+1) )
+ = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) = ( (k-1)(k+1) )/( (k-2)(k+2) )
-This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.
+k-2  k-1   k   k+1   k+2
-=== Examples ===
+-1    2   -1    0    0
-Here is an incomplete list of examples.
+   -1    2   -1    0
+    0   -1    2   -1
-{| class="wikitable center-1"
+========================
-|-
+-1    1    0    1   -1
-! Comma
+</pre>
-! S-expressions
+<pre>
-|-
+Sk / S(k+1) = [k-1, k, k+1, k+2]^[-1, 3, -3, 1]
-| [[28/27]]
+ = [k-1, k, k+1]^[-1, 2, -1] * [k, k+1, k+2]^[1, -2, 1]
-| S7⋅S8, S4/S6
+ = (k+2)/(k-1) * ( k/(k+1) )^3 = (k+2)/(k-1) / ((k+1)/k)^3
-|-
+</pre>
-| [[36/35]]
+<pre>
-| S6, S8⋅S9
+S(k-1) / S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 2, 0, -2, 1]
-|-
+ = [k-2, k-1, k]^[-1, 2, -1] * [k, k+1, k+2]^[ 1, -2,  1]
-| [[64/63]]
+ = [k-2, k-1, k]^[-1, 2, -1] / [k, k+1, k+2]^[-1,  2, -1]
-| S8, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
+ = (k+2)/(k-2) * ((k-1)/(k+1))^2 = (k+2)/(k-2) / ((k+1)/(k-1))^2
-|-
-| [[81/80]]
+k-2  k-1   k   k+1   k+2
-| S9, S6/S8
+-1    2   -1    0    0
-|-
+    0    1   -2    1
-| [[176/175]]
+========================
-| S8/S10, S22⋅S23⋅S24
+-1    2    0   -2    1
+</pre>
+This technique will be called "'''S-factorizations'''", as it is uses a certain format for expressing factorization (analogous to [[monzo]]s) that is uniquely suited for interpreting the relationships described by '''S-expressions'''.
+Note that the redundancy in these factorizations (in the sense that there are generators that are not linearly independent of the others) is a property that reflects the reality of [[#Equivalent S-expressions|equivalent S-expressions]].
+The generalisation of this method using commutative group theory is discussed in [[S-expression/Advanced_results#Abstraction]], though the ideas are very simple for anyone with simple mathematical training willing to learn the very basics needed.
+=== Using S-factorizations to show a useful equivalence/redundancy of S-expressions ===
+Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:
+==== The general S-expression equivalence ====
+Consider:
+<pre>
+Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:
+S(2k-1) * S(2k) * S(2k) * S(2k+1)
+ = [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(
+   [-1,    2,   -1]
+       + [-2,    4,   -2]
+             + [-1,    2,   -1]
+ = [-1,    0,    2,    0,   -1] )
+</pre>
+From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':
+<math>
+\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
+</math>
+...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-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]].
+== Equivalent S-expressions ==
+All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed above, or otherwise makes it equal to a product or ratio between two such commas, this often means nontrivial, deep tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.
+=== A useful general rule ===
+While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:
+$$
+{\rm S}k = {\rm S}(2k - 1) \cdot {\rm S}(2k)^2 \cdot {\rm S}(2k + 1)
+$$
+This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions. See [[S-expression/Advanced results]] for mathematical details.
+=== Examples ===
+Here is an incomplete list of examples.
+{| class="wikitable center-1"
+|-
+! Comma
+! S-expressions
+|-
+| [[28/27]]
+| S7⋅S8, S4/S6
+|-
+| [[36/35]]
+| S6, S8⋅S9
+|-
+| [[64/63]]
+| S8, S6/S9, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
+|-
+| [[81/80]]
+| S9, S6/S8
+|-
+| [[176/175]]
+| S8/S10, S22⋅S23⋅S24
 |-
 | [[243/242]]
-| S9/S11, S15/([[3025/3024|S22/S24 = S55]])
+| S9/S11, S15/S55, S15/(S22/S24)
 |-
 | [[325/324]]
@@ Line 2,721: / Line 2,924: @@
 | [[1225/1224]]
 | S35, S49⋅S50
+|-
+| [[2601/2600]]
+| S51, S17/(S25⋅S26)
 |-
 | [[3025/3024]]
 | S55, S22/S24, (S25/S27)⋅S99
-|-
-| [[2601/2600]]
-| S51, S17/(S25⋅S26)
 |-
 | [[9801/9800]]
@@ Line 2,734: / Line 2,937: @@
 | S161, S46/S48
 |-
-| [[123201/123200]]
+| <small>[[123201/123200]]</small>
 | S351, S78/S80
 |}
 {{Note| Examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are not included. }}
-{{Note| Where a comma written in the form ''a''/''b'' is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient. }}
 {{Tip| Feel free to expand with any equivalences you find that you think are valuable. }}
@@ Line 2,780: / Line 2,982: @@
 : For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.
 : These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk⋅S(k + 1)⋅…⋅S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].
+; Ultraparticular
+: An interval/comma which is the ratio of two consecutive square-particulars.
+: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.
+; Semiparticular
+: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:
+: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.
+; Cube-particular
+: A superparticular interval/comma whose numerator or denominator is a cube number. A shorthand (nick)name for cube superparticular.
+: These are of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} or {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}}.
 ; Odd-particular
@@ Line 2,797: / Line 3,011: @@
 : If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.
 : If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.
-; Ultraparticular
-: An interval/comma which is the ratio of two consecutive square-particulars.
-: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.
-; Semiparticular
-: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:
-: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.
 ; S-expression
-: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.
+: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only arbitrary square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars. (By extension, the expression could also include cube-particulars, C''k'' and Cp''k''.)
 ; S-factorization
@@ Line 2,814: / Line 3,020: @@
 ; S-comma
-: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).
+: Any comma within one of the infinite families of commas discussed here, excluding 1/''n''-square-particulars for ''n'' > 5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than ''n'' > 3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).
 ; Indirect S-comma