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An '''S-expression''' is any product, or ratio of products, of the '''square superparticulars''' '''S''k''''', which are defined as the fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.
An '''S-expression''' is any product, or ratio of products, of the '''square superparticulars''' '''S''k''''', which are defined as the fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[comma]]s that appear frequently in [[JI]] and [[regular temperament|temperaments]].


== Quick rules of S-expressions ==
== Quick rules of S-expressions ==
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.


* [[#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}}.
* [[#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>.
* [[#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 + 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>.
* [[#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) ==
== S''k'' (square-particulars) ==
Line 27: Line 28:


=== Table of square-particulars ===
=== Table of square-particulars ===
{| class="wikitable center-all
{| class="wikitable center-all left-4"
|+ style="font-size: 105%;" | 31-limit square-particulars
|+ style="font-size: 105%;" | 31-limit square-particulars
|-
|-
Line 33: Line 34:
! Interval relation
! Interval relation
! Ratio
! Ratio
! Prime limit
! Subgroup
|-
|-
| S2
| S2
| ([[2/1]])/([[3/2]])
| ([[2/1]])/([[3/2]])
| [[4/3]]
| [[4/3]]
| 3
| 2.3
|-
|-
| S3
| S3
| ([[3/2]])/([[4/3]])
| ([[3/2]])/([[4/3]])
| [[9/8]]
| [[9/8]]
| 3
| 2.3
|-
|-
| S4
| S4
| ([[4/3]])/([[5/4]])
| ([[4/3]])/([[5/4]])
| [[16/15]]
| [[16/15]]
| 5
| 2.3.5
|-
|-
| S5
| S5
| ([[5/4]])/([[6/5]])
| ([[5/4]])/([[6/5]])
| [[25/24]]
| [[25/24]]
| 5
| 2.3.5
|-
|-
| S6 = S8⋅S9
| S6
| ([[6/5]])/([[7/6]])
| ([[6/5]])/([[7/6]])
| [[36/35]]
| [[36/35]]
| 7
| 2.3.5.7
|-
|-
| S7
| S7
| ([[7/6]])/([[8/7]])
| ([[7/6]])/([[8/7]])
| [[49/48]]
| [[49/48]]
| 7
| 2.3.7
|-
|-
| S8
| S8
| ([[8/7]])/([[9/8]])
| ([[8/7]])/([[9/8]])
| [[64/63]]
| [[64/63]]
| 7
| 2.3.7
|-
|-
| S9 = S6/S8
| S9
| ([[9/8]])/([[10/9]])
| ([[9/8]])/([[10/9]])
| [[81/80]]
| [[81/80]]
| 5
| 2.3.5
|-
|-
| S10
| S10
| ([[10/9]])/([[11/10]])
| ([[10/9]])/([[11/10]])
| [[100/99]]
| [[100/99]]
| 11
| 2.3.5.11
|-
|-
| S11
| S11
| ([[11/10]])/([[12/11]])
| ([[11/10]])/([[12/11]])
| [[121/120]]
| [[121/120]]
| 11
| 2.3.5.11
|-
|-
| S12
| S12
| ([[12/11]])/([[13/12]])
| ([[12/11]])/([[13/12]])
| [[144/143]]
| [[144/143]]
| 13
| 2.3.11.13
|-
|-
| S13
| S13
| ([[13/12]])/([[14/13]])
| ([[13/12]])/([[14/13]])
| [[169/168]]
| [[169/168]]
| 13
| 2.3.7.13
|-
|-
| S14
| S14
| ([[14/13]])/([[15/14]])
| ([[14/13]])/([[15/14]])
| [[196/195]]
| [[196/195]]
| 13
| 2.3.5.7.13
|-
|-
| S15
| S15
| ([[15/14]])/([[16/15]])
| ([[15/14]])/([[16/15]])
| [[225/224]]
| [[225/224]]
| 7
| 2.3.5.7
|-
|-
| S16
| S16
| ([[16/15]])/([[17/16]])
| ([[16/15]])/([[17/16]])
| [[256/255]]
| [[256/255]]
| 17
| 2.3.5.17
|-
|-
| S17
| S17
| ([[17/16]])/([[18/17]])
| ([[17/16]])/([[18/17]])
| [[289/288]]
| [[289/288]]
| 17
| 2.3.17
|-
|-
| S18
| S18
| ([[18/17]])/([[19/18]])
| ([[18/17]])/([[19/18]])
| [[324/323]]
| [[324/323]]
| 19
| 2.3.17.19
|-
|-
| S19
| S19
| ([[19/18]])/([[20/19]])
| ([[19/18]])/([[20/19]])
| [[361/360]]
| [[361/360]]
| 19
| 2.3.5.19
|-
|-
| S20
| S20
| ([[20/19]])/([[21/20]])
| ([[20/19]])/([[21/20]])
| [[400/399]]
| [[400/399]]
| 19
| 2.3.5.7.19
|-
|-
| S21
| S21
| ([[21/20]])/([[22/21]])
| ([[21/20]])/([[22/21]])
| [[441/440]]
| [[441/440]]
| 11
| 2.3.5.7.11
|-
|-
| S22
| S22
| ([[22/21]])/([[23/22]])
| ([[22/21]])/([[23/22]])
| [[484/483]]
| [[484/483]]
| 23
| 2.3.7.11.23
|-
|-
| S23
| S23
| ([[23/22]])/([[24/23]])
| ([[23/22]])/([[24/23]])
| [[529/528]]
| [[529/528]]
| 23
| 2.3.11.23
|-
|-
| S24
| S24
| ([[24/23]])/([[25/24]])
| ([[24/23]])/([[25/24]])
| [[576/575]]
| [[576/575]]
| 23
| 2.3.5.23
|-
|-
| S25
| S25
| ([[25/24]])/([[26/25]])
| ([[25/24]])/([[26/25]])
| [[625/624]]
| [[625/624]]
| 13
| 2.3.5.13
|-
|-
| S26 = S13/S15
| S26
| ([[26/25]])/([[27/26]])
| ([[26/25]])/([[27/26]])
| [[676/675]]
| [[676/675]]
| 13
| 2.3.5.13
|-
|-
| S27
| S27
| ([[27/26]])/([[28/27]])
| ([[27/26]])/([[28/27]])
| [[729/728]]
| [[729/728]]
| 13
| 2.3.7.13
|-
|-
| S28
| S28
| ([[28/27]])/([[29/28]])
| ([[28/27]])/([[29/28]])
| [[784/783]]
| [[784/783]]
| 29
| 2.3.7.29
|-
|-
| S29
| S29
| ([[29/28]])/([[30/29]])
| ([[29/28]])/([[30/29]])
| [[841/840]]
| [[841/840]]
| 29
| 2.3.5.7.29
|-
|-
| S30
| S30
| ([[30/29]])/([[31/30]])
| ([[30/29]])/([[31/30]])
| [[900/899]]
| [[900/899]]
| 31
| 2.3.5.29.31
|-
|-
| S31
| S31
| ([[31/30]])/([[32/31]])
| ([[31/30]])/([[32/31]])
| [[961/960]]
| [[961/960]]
| 31
| 2.3.5.31
|-
|-
| S32
| S32
| ([[32/31]])/([[33/32]])
| ([[32/31]])/([[33/32]])
| [[1024/1023]]
| [[1024/1023]]
| 31
| 2.3.11.31
|-
|-
| S33
| S33
| ([[33/32]])/([[34/33]])
| ([[33/32]])/([[34/33]])
| [[1089/1088]]
| [[1089/1088]]
| 17
| 2.3.11.17
|-
|-
| S34
| S34
| ([[34/33]])/([[35/34]])
| ([[34/33]])/([[35/34]])
| [[1156/1155]]
| [[1156/1155]]
| 17
| 2.3.5.7.11.17
|-
|-
| S35 = S49⋅S50
| S35
| ([[35/34]])/([[36/35]])
| ([[35/34]])/([[36/35]])
| [[1225/1224]]
| [[1225/1224]]
| 17
| 2.3.5.7.17
|-
|-
| S39
| S39
| ([[39/38]])/([[40/39]])
| ([[39/38]])/([[40/39]])
| [[1521/1520]]
| [[1521/1520]]
| 19
| 2.3.5.13.19
|-
|-
| S45
| S45
| ([[45/44]])/([[46/45]])
| ([[45/44]])/([[46/45]])
| [[2025/2024]]
| [[2025/2024]]
| 23
| 2.3.5.11.23
|-
|-
| S49
| S49
| ([[49/48]])/([[50/49]])
| ([[49/48]])/([[50/49]])
| [[2401/2400]]
| [[2401/2400]]
| 7
| 2.3.5.7
|-
|-
| S50
| S50
| ([[50/49]])/([[51/50]])
| ([[50/49]])/([[51/50]])
| [[2500/2499]]
| [[2500/2499]]
| 17
| 2.3.5.7.17
|-
|-
| S51
| S51
| ([[51/50]])/([[52/51]])
| ([[51/50]])/([[52/51]])
| [[2601/2600]]
| [[2601/2600]]
| 17
| 2.3.5.13.17
|-
|-
| S55 = S22/S24
| S55
| ([[55/54]])/([[56/55]])
| ([[55/54]])/([[56/55]])
| [[3025/3024]]
| [[3025/3024]]
| 11
| 2.3.5.7.11
|-
|-
| S56
| S56
| ([[56/55]])/([[57/56]])
| ([[56/55]])/([[57/56]])
| [[3136/3135]]
| [[3136/3135]]
| 19
| 2.3.5.7.11.19
|-
|-
| S57
| S57
| ([[57/56]])/([[58/57]])
| ([[57/56]])/([[58/57]])
| [[3249/3248]]
| [[3249/3248]]
| 29
| 2.3.7.19.29
|-
|-
| S63
| S63
| ([[63/62]])/([[64/63]])
| ([[63/62]])/([[64/63]])
| [[3969/3968]]
| [[3969/3968]]
| 31
| 2.3.7.31
|-
|-
| S64
| S64
| ([[64/63]])/([[65/64]])
| ([[64/63]])/([[65/64]])
| [[4096/4095]]
| [[4096/4095]]
| 13
| 2.3.5.7.13
|-
|-
| S65
| S65
| ([[65/64]])/([[66/65]])
| ([[65/64]])/([[66/65]])
| [[4225/4224]]
| [[4225/4224]]
| 13
| 2.3.5.11.13
|-
|-
| S69
| S69
| ([[69/68]])/([[70/69]])
| ([[69/68]])/([[70/69]])
| [[4761/4760]]
| [[4761/4760]]
| 23
| 2.3.5.7.17.23
|-
|-
| S76
| S76
| ([[76/75]])/([[77/76]])
| ([[76/75]])/([[77/76]])
| [[5776/5775]]
| [[5776/5775]]
| 19
| 2.3.5.7.11.19
|-
|-
| S77
| S77
| ([[77/76]])/([[78/77]])
| ([[77/76]])/([[78/77]])
| [[5929/5928]]
| [[5929/5928]]
| 19
| 2.3.7.11.13.19
|-
|-
| S91
| S91
| ([[91/90]])/([[92/91]])
| ([[91/90]])/([[92/91]])
| [[8281/8280]]
| [[8281/8280]]
| 23
| 2.3.5.7.13.23
|-
|-
| S92
| S92
| ([[92/91]])/([[93/92]])
| ([[92/91]])/([[93/92]])
| [[8464/8463]]
| [[8464/8463]]
| 31
| 2.3.7.13.23.31
|-
|-
| S99 = S33/S35
| S99
| ([[99/98]])/([[100/99]])
| ([[99/98]])/([[100/99]])
| [[9801/9800]]
| [[9801/9800]]
| 11
| 2.3.5.7.11
|-
|-
| S115
| S115
| ([[115/114]])/([[116/115]])
| ([[115/114]])/([[116/115]])
| [[13225/13224]]
| [[13225/13224]]
| 29
| 2.3.5.19.23.29
|-
|-
| S116
| S116
| ([[116/115]])/([[117/116]])
| ([[116/115]])/([[117/116]])
| [[13456/13455]]
| [[13456/13455]]
| 29
| 2.3.5.13.23.29
|-
|-
| S120
| S120
| ([[120/119]])/([[121/120]])
| ([[120/119]])/([[121/120]])
| [[14400/14399]]
| [[14400/14399]]
| 17
| 2.3.5.7.11.17
|-
|-
| S125
| S125
| ([[125/124]])/([[126/125]])
| ([[125/124]])/([[126/125]])
| [[15625/15624]]
| [[15625/15624]]
| 31
| 2.3.5.7.31
|-
|-
| S144
| S144
| ([[144/143]])/([[145/144]])
| ([[144/143]])/([[145/144]])
| [[20736/20735]]
| [[20736/20735]]
| 29
| 2.3.5.11.13.29
|-
|-
| S153
| S153
| ([[153/152]])/([[154/153]])
| ([[153/152]])/([[154/153]])
| [[23409/23408]]
| [[23409/23408]]
| 19
| 2.3.7.11.17.19
|-
|-
| S154
| S154
| ([[154/153]])/([[155/154]])
| ([[154/153]])/([[155/154]])
| [[23716/23715]]
| [[23716/23715]]
| 31
| 2.3.5.7.11.17.31
|-
|-
| S155
| S155
| ([[155/154]])/([[156/155]])
| ([[155/154]])/([[156/155]])
| [[24025/24024]]
| [[24025/24024]]
| 31
| 2.3.5.7.11.13.31
|-
|-
| S161 = S46/S48
| S161
| ([[161/160]])/([[162/161]])
| ([[161/160]])/([[162/161]])
| [[25921/25920]]
| [[25921/25920]]
| 23
| 2.3.5.7.23
|-
|-
| S169
| S169
| ([[169/168]])/([[170/169]])
| ([[169/168]])/([[170/169]])
| [[28561/28560]]
| [[28561/28560]]
| 17
| 2.3.5.7.13.17
|-
|-
| S170
| S170
| ([[170/169]])/([[171/170]])
| ([[170/169]])/([[171/170]])
| [[28900/28899]]
| [[28900/28899]]
| 19
| 2.3.5.13.17.19
|-
|-
| S175
| S175
| ([[175/174]])/([[176/175]])
| ([[175/174]])/([[176/175]])
| [[30625/30624]]
| [[30625/30624]]
| 29
| 2.3.5.7.11.29
|-
|-
| S208
| S208
| ([[208/207]])/([[209/208]])
| ([[208/207]])/([[209/208]])
| [[43264/43263]]
| [[43264/43263]]
| 23
| 2.3.11.13.19.23
|-
|-
| S209
| S209
| ([[209/208]])/([[210/209]])
| ([[209/208]])/([[210/209]])
| [[43681/43680]]
| [[43681/43680]]
| 19
| 2.3.5.7.11.13.19
|-
|-
| S231
| S231
| ([[231/230]])/([[232/231]])
| ([[231/230]])/([[232/231]])
| [[53361/53360]]
| [[53361/53360]]
| 29
| 2.3.5.7.11.23.29
|-
|-
| S289
| S289
| ([[289/288]])/([[290/289]])
| ([[289/288]])/([[290/289]])
| [[83521/83520]]
| [[83521/83520]]
| 29
| 2.3.5.17.29
|-
|-
| S323
| S323
| ([[323/322]])/([[324/323]])
| ([[323/322]])/([[324/323]])
| [[104329/104328]]
| [[104329/104328]]
| 23
| 2.3.7.13.19.23
|-
|-
| S324
| S324
| ([[324/323]])/([[325/324]])
| ([[324/323]])/([[325/324]])
| [[104976/104975]]
| [[104976/104975]]
| 19
| 2.3.5.13.17.19
|-
|-
| S341
| S341
| ([[341/340]])/([[342/341]])
| ([[341/340]])/([[342/341]])
| [[116281/116280]]
| [[116281/116280]]
| 31
| 2.3.5.11.17.19.31
|-
|-
| S342
| S342
| ([[342/341]])/([[343/342]])
| ([[342/341]])/([[343/342]])
| [[116964/116963]]
| [[116964/116963]]
| 31
| 2.3.7.11.19.31
|-
|-
| S351 = S78/S80
| S351
| ([[351/350]])/([[352/351]])
| ([[351/350]])/([[352/351]])
| [[123201/123200]]
| [[123201/123200]]
| 13
| 2.3.5.7.11.13
|-
|-
| S391
| S391
| ([[391/390]])/([[392/391]])
| ([[391/390]])/([[392/391]])
| [[152881/152880]]
| [[152881/152880]]
| 23
| 2.3.5.7.13.17.23
|-
|-
| S441
| S441
| ([[441/440]])/([[442/441]])
| ([[441/440]])/([[442/441]])
| [[194481/194480]]
| [[194481/194480]]
| 17
| 2.3.5.7.11.13.17
|-
|-
| S494
| S494
| ([[494/493]])/([[495/494]])
| ([[494/493]])/([[495/494]])
| [[244036/244035]]
| [[244036/244035]]
| 29
| 2.3.5.11.13.17.19.29
|-
|-
| S495
| S495
| ([[495/494]])/([[496/495]])
| ([[495/494]])/([[496/495]])
| [[245025/245024]]
| [[245025/245024]]
| 31
| 2.3.5.11.13.19.31
|-
|-
| S528
| S528
| ([[528/527]])/([[529/528]])
| ([[528/527]])/([[529/528]])
| [[278784/278783]]
| [[278784/278783]]
| 31
| 2.3.11.17.23.31
|-
|-
| S551
| S551
| ([[551/550]])/([[552/551]])
| ([[551/550]])/([[552/551]])
| [[303601/303600]]
| [[303601/303600]]
| 29
| 2.3.5.11.19.23.29
|-
|-
| S714
| S714
| ([[714/713]])/([[715/714]])
| ([[714/713]])/([[715/714]])
| [[509796/509795]]
| [[509796/509795]]
| 31
| 2.3.5.7.11.13.17.23.31
|-
|-
| S783
| S783
| ([[783/782]])/([[784/783]])
| ([[783/782]])/([[784/783]])
| [[613089/613088]]
| [[613089/613088]]
| 29
| 2.3.7.17.23.29
|-
|-
| S1275
| S1275
| ([[1275/1274]])/([[1276/1275]])
| <small>([[1275/1274]])/([[1276/1275]])</small>
| [[1625625/1625624]]
| <small>[[1625625/1625624]]</small>
| 29
| 2.3.5.7.11.13.17.29
|-
|-
| S1519
| S1519
| ([[1519/1518]])/([[1520/1519]])
| <small>([[1519/1518]])/([[1520/1519]])</small>
| [[2307361/2307360]]
| <small>[[2307361/2307360]]</small>
| 31
| 2.3.5.7.11.19.23.31
|-
|-
| S1520
| S1520
| ([[1520/1519]])/([[1521/1520]])
| <small>([[1520/1519]])/([[1521/1520]])</small>
| [[2310400/2310399]]
| <small>[[2310400/2310399]]</small>
| 31
| 2.3.5.7.13.19.31
|-
|-
| S2001
| S2001
| ([[2001/2000]])/([[2002/2001]])
| <small>([[2001/2000]])/([[2002/2001]])</small>
| [[4004001/4004000]]
| <small>[[4004001/4004000]]</small>
| 29
| 2.3.5.7.11.13.23.29
|-
|-
| S2024
| S2024
| ([[2024/2023]])/([[2025/2024]])
| <small>([[2024/2023]])/([[2025/2024]])</small>
| [[4096576/4096575]]
| <small>[[4096576/4096575]]</small>
| 23
| 2.3.5.7.11.17.23
|-
|-
| S2431
| S2431
| ([[2431/2430]])/([[2432/2431]])
| <small>([[2431/2430]])/([[2432/2431]])</small>
| [[5909761/5909760]]
| <small>[[5909761/5909760]]</small>
| 19
| 2.3.5.11.13.17.19
|-
|-
| S3249
| S3249
| ([[3249/3248]])/([[3250/3249]])
| <small>([[3249/3248]])/([[3250/3249]])</small>
| <font style="font-size:0.88em">[[10556001/10556000]]</font>
| <small>[[10556001/10556000]]</small>
| 29
| 2.3.5.7.13.19.29
|-
|-
| S9801
| S9801
| ([[9801/9800]])/([[9802/9801]])
| <small>([[9801/9800]])/([[9802/9801]])</small>
| <font style="font-size:0.88em">[[96059601/96059600]]</font>
| <small>[[96059601/96059600]]</small>
| 29
| 2.3.5.7.11.13.29
|-
|-
| S13311
| S13311
| <font style="font-size:0.86em">([[13311/13310]])/([[13312/13311]])</font>
| <small><small>([[13311/13310]])/([[13312/13311]])</small></small>
| <font style="font-size:0.79em">[[177182721/177182720]]</font>
| <small><small>[[177182721/177182720]]</small></small>
| 29
| 2.3.5.11.13.17.29
|-
|-
| S13455
| S13455
| <font style="font-size:0.86em">([[13455/13454]])/([[13456/13455]])</font>
| <small><small>([[13455/13454]])/([[13456/13455]])</small></small>
| <font style="font-size:0.79em">[[181037025/181037024]]</font>
| <small><small>[[181037025/181037024]]</small></small>
| 31
| 2.3.5.7.13.23.29.31
|}
|}


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.
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>
|+ 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
! Interval relation
! Ratio
! Ratio
! Prime limit
! Subgroup
|-
|-
| S2⋅S3
| S2⋅S3
| ([[3/1]])/([[2/1]])
| ([[3/1]])/([[2/1]])
| [[3/2]]
| [[3/2]]
| 3
| 2.3
|-
|-
| S3⋅S4
| S3⋅S4
| ([[3/2]])/([[5/4]])
| ([[3/2]])/([[5/4]])
| [[6/5]]
| [[6/5]]
| 5
| 2.3.5
|-
|-
| S4⋅S5
| S4⋅S5
| ([[4/3]])/([[6/5]])
| ([[4/3]])/([[6/5]])
| [[10/9]]
| [[10/9]]
| 5
| 2.3.5
|-
|-
| S5⋅S6
| S5⋅S6
| ([[5/4]])/([[7/6]])
| ([[5/4]])/([[7/6]])
| [[15/14]]
| [[15/14]]
| 7
| 2.3.5.7
|-
|-
| S6⋅S7
| S6⋅S7
| ([[6/5]])/([[8/7]])
| ([[6/5]])/([[8/7]])
| [[21/20]]
| [[21/20]]
| 7
| 2.3.5.7
|-
|-
| S7⋅S8 = S4/S6
| S7⋅S8
| ([[7/6]])([[9/8]])
| ([[7/6]])([[9/8]])
| [[28/27]]
| [[28/27]]
| 7
| 2.3.7
|-
|-
| S8⋅S9 = S6
| S8⋅S9
| ([[8/7]])/([[10/9]])
| ([[8/7]])/([[10/9]])
| [[36/35]]
| [[36/35]]
| 7
| 2.3.5.7
|-
|-
| S9⋅S10
| S9⋅S10
| ([[9/8]])/([[11/10]])
| ([[9/8]])/([[11/10]])
| [[45/44]]
| [[45/44]]
| 11
| 2.3.5.11
|-
|-
| S10⋅S11
| S10⋅S11
| ([[10/9]])/([[12/11]])
| ([[10/9]])/([[12/11]])
| [[55/54]]
| [[55/54]]
| 11
| 2.3.5.11
|-
|-
| S11⋅S12
| S11⋅S12
| ([[11/10]])/([[13/12]])
| ([[11/10]])/([[13/12]])
| [[66/65]]
| [[66/65]]
| 13
| 2.3.5.11.13
|-
|-
| S12⋅S13
| S12⋅S13
| ([[12/11]])/([[14/13]])
| ([[12/11]])/([[14/13]])
| [[78/77]]
| [[78/77]]
| 13
| 2.3.7.11.13
|-
|-
| S13⋅S14
| S13⋅S14
| ([[13/12]])/([[15/14]])
| ([[13/12]])/([[15/14]])
| [[91/90]]
| [[91/90]]
| 13
| 2.3.5.7.13
|-
|-
| S14⋅S15
| S14⋅S15
| ([[14/13]])/([[16/15]])
| ([[14/13]])/([[16/15]])
| [[105/104]]
| [[105/104]]
| 13
| 2.3.5.7.13
|-
|-
| S15⋅S16
| S15⋅S16
| ([[15/14]])/([[17/16]])
| ([[15/14]])/([[17/16]])
| [[120/119]]
| [[120/119]]
| 17
| 2.3.5.7.17
|-
|-
| S16⋅S17
| S16⋅S17
| ([[16/15]])/([[18/17]])
| ([[16/15]])/([[18/17]])
| [[136/135]]
| [[136/135]]
| 17
| 2.3.5.17
|-
|-
| S17⋅S18
| S17⋅S18
| ([[17/16]])/([[19/18]])
| ([[17/16]])/([[19/18]])
| [[153/152]]
| [[153/152]]
| 19
| 2.3.17.19
|-
|-
| S18⋅S19
| S18⋅S19
| ([[18/17]])/([[20/19]])
| ([[18/17]])/([[20/19]])
| [[171/170]]
| [[171/170]]
| 19
| 2.3.5.17.19
|-
|-
| S19⋅S20
| S19⋅S20
| ([[19/18]])/([[21/20]])
| ([[19/18]])/([[21/20]])
| [[190/189]]
| [[190/189]]
| 19
| 2.3.5.7.19
|-
|-
| S20⋅S21
| S20⋅S21
| ([[20/19]])/([[22/21]])
| ([[20/19]])/([[22/21]])
| [[210/209]]
| [[210/209]]
| 19
| 2.3.5.7.11.19
|-
|-
| S21⋅S22
| S21⋅S22
| ([[21/20]])/([[23/22]])
| ([[21/20]])/([[23/22]])
| [[231/230]]
| [[231/230]]
| 23
| 2.3.5.7.11.23
|-
|-
| S22⋅S23
| S22⋅S23
| ([[22/21]])/([[24/23]])
| ([[22/21]])/([[24/23]])
| [[253/252]]
| [[253/252]]
| 23
| 2.3.5.7.11.23
|-
|-
| S23⋅S24
| S23⋅S24
| ([[23/22]])/([[25/24]])
| ([[23/22]])/([[25/24]])
| [[276/275]]
| [[276/275]]
| 23
| 2.3.5.11.23
|-
|-
| S24⋅S25
| S24⋅S25
| ([[24/23]])/([[26/25]])
| ([[24/23]])/([[26/25]])
| [[300/299]]
| [[300/299]]
| 23
| 2.3.5.13.23
|-
|-
| S25⋅S26 = S10/S12
| S25⋅S26
| ([[25/24]])/([[27/26]])
| ([[25/24]])/([[27/26]])
| [[325/324]]
| [[325/324]]
| 13
| 2.3.5.13
|-
|-
| S26⋅S27
| S26⋅S27
| ([[26/25]])/([[28/27]])
| ([[26/25]])/([[28/27]])
| [[351/350]]
| [[351/350]]
| 13
| 2.3.5.7.13
|-
|-
| S27⋅S28
| S27⋅S28
| ([[27/26]])/([[29/28]])
| ([[27/26]])/([[29/28]])
| [[378/377]]
| [[378/377]]
| 29
| 2.3.5.7.13.29
|-
|-
| S28⋅S29
| S28⋅S29
| ([[28/27]])/([[30/29]])
| ([[28/27]])/([[30/29]])
| [[406/405]]
| [[406/405]]
| 29
| 2.3.5.7.29
|-
|-
| S29⋅S30
| S29⋅S30
| ([[29/28]])/([[31/30]])
| ([[29/28]])/([[31/30]])
| [[435/434]]
| [[435/434]]
| 31
| 2.3.5.7.29.31
|-
|-
| S30⋅S31
| S30⋅S31
| ([[30/29]])/([[32/31]])
| ([[30/29]])/([[32/31]])
| [[465/464]]
| [[465/464]]
| 31
| 2.3.5.29.31
|-
|-
| S31⋅S32
| S31⋅S32
| ([[31/30]])/([[33/32]])
| ([[31/30]])/([[33/32]])
| [[496/495]]
| [[496/495]]
| 31
| 2.3.5.11.31
|-
|-
| S32⋅S33
| S32⋅S33
| ([[32/31]])/([[34/33]])
| ([[32/31]])/([[34/33]])
| [[528/527]]
| [[528/527]]
| 31
| 2.3.11.17.31
|-
|-
| S33⋅S34
| S33⋅S34
| ([[33/32]])/([[35/34]])
| ([[33/32]])/([[35/34]])
| [[561/560]]
| [[561/560]]
| 17
| 2.3.5.7.11.17
|-
|-
| S34⋅S35
| S34⋅S35
| ([[34/33]])/([[36/35]])
| ([[34/33]])/([[36/35]])
| [[595/594]]
| [[595/594]]
| 17
| 2.3.5.7.11.17
|-
|-
| S49⋅S50 = S35
| S49⋅S50
| ([[49/48]])/([[51/50]])
| ([[49/48]])/([[51/50]])
| [[1225/1224]]
| [[1225/1224]]
| 17
| 2.3.5.7.17
|-
|-
| S50⋅S51
| S50⋅S51
| ([[50/49]])/([[52/51]])
| ([[50/49]])/([[52/51]])
| [[1275/1274]]
| [[1275/1274]]
| 17
| 2.3.5.7.13.17
|-
|-
| S55⋅S56
| S55⋅S56
| ([[55/54]])/([[57/56]])
| ([[55/54]])/([[57/56]])
| [[1540/1539]]
| [[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
| S63⋅S64
| ([[63/62]])/([[65/64]])
| ([[63/62]])/([[65/64]])
| [[2016/2015]]
| [[2016/2015]]
| 31
| 2.3.5.7.13.31
|-
|-
| S64⋅S65
| S64⋅S65
| ([[64/63]])/([[66/65]])
| ([[64/63]])/([[66/65]])
| [[2080/2079]]
| [[2080/2079]]
| 13
| 2.3.5.7.11.13
|-
|-
| S76⋅S77
| S76⋅S77
| ([[76/75]])/([[78/77]])
| ([[76/75]])/([[78/77]])
| [[2926/2925]]
| [[2926/2925]]
| 19
| 2.3.5.7.11.13.19
|-
|-
| S91⋅S92
| S91⋅S92
| ([[91/90]])/([[93/92]])
| ([[91/90]])/([[93/92]])
| [[4186/4185]]
| [[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]]
| [[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
|}
|}


== S''k''⋅S(''k'' + 1)⋅…⋅S(''k'' + ''n'' − 1) (1/''n''-square-particulars) ==
== S''k''⋅S(''k'' + 1)⋅…⋅S(''k'' + ''n'' − 1) (1/''n''-square-particulars) ==
=== Significance ===
=== Significance ===
1/''n''-square-particulars are a generalization of square- and 1/2-square-particulars to aninterval whose S-expression can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.
1/''n''-square-particulars are a generalization of square- and 1/2-square-particulars to an interval whose S-expression can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.


In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two superparticular intervals, with ''n'' distance between them. For example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to ordinary square-particulars (in this case [[100/99|S10]]).
In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two superparticular intervals, with ''n'' distance between them. For example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to ordinary square-particulars (in this case [[100/99|S10]]).
Line 797: Line 803:
1/''n''-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the consistency of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered out. Specifically:
1/''n''-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the consistency of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered out. Specifically:


If a temperament tempers out a 1/''n''-square-particular of the form {{nowrap|S''k''⋅S(''k'' + 1)⋅…⋅S(''k'' + ''n'' − 1)}}, it must temper out all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}} to make the ({{nowrap|''k'' + ''n''}})-odd-limit. If it does not, it is ''necessarily'' inconsistent due to the lack of monotonicity in the segment.<ref group="note">Technically, the tuning of the higher-rank temperament corresponding to the lower-rank temperament that tempers out all of these commas is the exact set of tuning for which consistency is possible. </ref> A proof is as follows:
If a temperament tempers out a 1/''n''-square-particular of the form {{nowrap|S''k''⋅S(''k'' + 1)⋅…⋅S(''k'' + ''n'' − 1)}}, it must temper out all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}} to make the ({{nowrap|''k'' + ''n''}})-odd-limit consistent. If it does not, it is ''necessarily'' inconsistent due to the lack of monotonicity in the segment.<ref group="note">Technically, the tuning of the higher-rank temperament corresponding to the lower-rank temperament that tempers out all of these commas is the exact set of tuning for which consistency is possible. </ref> A proof is as follows:  
 
{{Proof|contents=


Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:
Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:
Line 812: Line 820:
Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.
Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.


}}


=== S(''k'' − 1)⋅S''k''⋅S(''k'' + 1) (1/3-square-particulars) ===
=== S(''k'' − 1)⋅S''k''⋅S(''k'' + 1) (1/3-square-particulars) ===
This section concerns commas of the form {{nowrap| S(''k'' − 1)⋅S''k''⋅S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}} }} which therefore do not directly involve the ''k''-th harmonic. These, along with square-particulars and {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), are a special case of 1/''n''-square-particulars.
This section concerns commas of the form {{nowrap| S(''k'' − 1)⋅S''k''⋅S(''k'' + 1) {{=}} {{sfrac|(''k'' − 1)/(''k'' − 2)|(''k'' + 2)/(''k'' + 1}}) }} which therefore do not directly involve the ''k''-th harmonic. These, along with square-particulars and {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), are a special case of 1/''n''-square-particulars.


==== Significance ====
==== Significance ====
Line 821: Line 830:
# Their omission of direct relation to the ''k''-th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)
# Their omission of direct relation to the ''k''-th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)


==== Proof of simplification of 1/3-square-particulars ====
{{Proof|title=Proof of simplification of 1/3-square-particulars|contents=
 
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:


$$
<math>\displaystyle
\begin{align}
\begin{align}
S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\
S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\
Line 831: Line 841:
&= \frac{k^2 - 1}{k^2 - 4}
&= \frac{k^2 - 1}{k^2 - 4}
\end{align}
\end{align}
$$
</math>


If {{nowrap|''k'' {{=}} 3''n'' + 1}} then:
If {{nowrap|''k'' {{=}} 3''n'' + 1}} then:


$$ S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1} $$
<math>\displaystyle S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1}</math>


if {{nowrap|''k'' {{=}} 3''n'' + 2}} then:
If {{nowrap|''k'' {{=}} 3''n'' + 2}} then:


$$ S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n} $$
<math>\displaystyle S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n}</math>


if {{nowrap|''k'' {{=}} 3''n''}} then:
If {{nowrap|''k'' {{=}} 3''n''}} then:


$$ S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 - 1}{9n^2 - 4} $$
<math>\displaystyle S(k-1) \cdot Sk \cdot S(k+1) = \frac{9n^2 - 1}{9n^2 - 4} </math>


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


=== Tables of 1/''n''-square-particulars ===
=== Tables of 1/''n''-square-particulars ===
Line 1,212: Line 1,224:
|}
|}


{| class="wikitable center-all\
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | 23-limit {{frac|1|4}}-square particulars
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit {{frac|1|4}}-square particulars
|-
|-
! S-expression
! S-expression
Line 1,401: Line 1,413:
|}
|}


{| class="wikitable center-all
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | 23-limit {{frac|1|5}}-square particulars
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit {{frac|1|5}}-square particulars
|-
|-
! S-expression
! S-expression
Line 1,627: Line 1,639:


=== Table of ultraparticulars ===
=== Table of ultraparticulars ===
{| class="wikitable center-all
{| class="wikitable center-all right-4 left-5"
|-
|-
! S-expression
! S-expression
! Cube Relation
! Cube relation
! Comma
! Ratio
! Cents
! Cents
! Subgroup
|-
|-
| S2/S3 = ([[4/3]])/([[9/8]])
| S2/S3 = ([[4/3]])/([[9/8]])
Line 1,638: Line 1,651:
| [[32/27]]
| [[32/27]]
| 294.135
| 294.135
| 2.3
|-
|-
| S3/S4 = ([[9/8]])/([[16/15]])
| S3/S4 = ([[9/8]])/([[16/15]])
Line 1,643: Line 1,657:
| [[135/128]]
| [[135/128]]
| 92.179
| 92.179
| 2.3.5
|-
|-
| S4/S5 = ([[16/15]])/([[25/24]])
| S4/S5 = ([[16/15]])/([[25/24]])
Line 1,648: Line 1,663:
| [[128/125]]
| [[128/125]]
| 41.059
| 41.059
| 2.5
|-
|-
| S5/S6 = ([[25/24]])/([[36/35]])
| S5/S6 = ([[25/24]])/([[36/35]])
Line 1,653: Line 1,669:
| [[875/864]]
| [[875/864]]
| 21.902
| 21.902
| 2.3.5.7
|-
|-
| S6/S7 = ([[36/35]])/([[49/48]])
| S6/S7 = ([[36/35]])/([[49/48]])
Line 1,658: Line 1,675:
| [[1728/1715]]
| [[1728/1715]]
| 13.074
| 13.074
| 2.3.5.7
|-
|-
| S7/S8 = ([[49/48]])/([[64/63]])
| S7/S8 = ([[49/48]])/([[64/63]])
Line 1,663: Line 1,681:
| [[1029/1024]]
| [[1029/1024]]
| 8.433
| 8.433
| 2.3.7
|-
|-
| S8/S9 = ([[64/63]])/([[81/80]])
| S8/S9 = ([[64/63]])/([[81/80]])
Line 1,668: Line 1,687:
| [[5120/5103]]
| [[5120/5103]]
| 5.758
| 5.758
| 2.3.5.7
|-
|-
| S9/S10 = ([[81/80]])/([[100/99]])
| S9/S10 = ([[81/80]])/([[100/99]])
Line 1,673: Line 1,693:
| [[8019/8000]]
| [[8019/8000]]
| 4.107
| 4.107
| 2.3.5.11
|-
|-
| S10/S11 = ([[100/99]])/([[121/120]])
| S10/S11 = ([[100/99]])/([[121/120]])
Line 1,678: Line 1,699:
| [[4000/3993]]
| [[4000/3993]]
| 3.032
| 3.032
| 2.3.5.11
|-
|-
| S11/S12 = ([[121/120]])/([[144/143]])
| S11/S12 = ([[121/120]])/([[144/143]])
Line 1,683: Line 1,705:
| [[17303/17280]]
| [[17303/17280]]
| 2.303
| 2.303
| 2.3.5.11.13
|-
|-
| S12/S13 = ([[144/143]])/([[169/168]])
| S12/S13 = ([[144/143]])/([[169/168]])
| ([[14/11]])/([[13/12]])<sup>3</sup>
| ([[14/11]])/([[13/12]])<sup>3</sup>
| [[24192/24167]]
| [[24192/24167]]
| 1.79
| 1.790
| 2.3.7.11.13
|-
|-
| S13/S14 = ([[169/168]])/([[196/195]])
| S13/S14 = ([[169/168]])/([[196/195]])
Line 1,693: Line 1,717:
| [[10985/10976]]
| [[10985/10976]]
| 1.419
| 1.419
| 2.5.7.13
|-
|-
| S14/S15 = ([[196/195]])/([[225/224]])
| S14/S15 = ([[196/195]])/([[225/224]])
Line 1,698: Line 1,723:
| [[43904/43875]]
| [[43904/43875]]
| 1.144
| 1.144
| 2.3.5.7.13
|-
|-
| S15/S16 = ([[225/224]])/([[256/255]])
| S15/S16 = ([[225/224]])/([[256/255]])
Line 1,703: Line 1,729:
| [[57375/57344]]
| [[57375/57344]]
| 0.936
| 0.936
| 2.3.5.7.17
|-
|-
| S16/S17 = ([[256/255]])/([[289/288]])
| S16/S17 = ([[256/255]])/([[289/288]])
Line 1,708: Line 1,735:
| [[24576/24565]]
| [[24576/24565]]
| 0.775
| 0.775
| 2.3.5.17
|-
|-
| S17/S18 = ([[289/288]])/([[324/323]])
| S17/S18 = ([[289/288]])/([[324/323]])
Line 1,713: Line 1,741:
| [[93347/93312]]
| [[93347/93312]]
| 0.649
| 0.649
| 2.3.17.19
|-
|-
| S18/S19 = ([[324/323]])/([[361/360]])
| S18/S19 = ([[324/323]])/([[361/360]])
Line 1,718: Line 1,747:
| [[116640/116603]]
| [[116640/116603]]
| 0.549
| 0.549
| 2.3.5.17.19
|-
|-
| S19/S20 = ([[361/360]])/([[400/399]])
| S19/S20 = ([[361/360]])/([[400/399]])
Line 1,723: Line 1,753:
| [[48013/48000]]
| [[48013/48000]]
| 0.469
| 0.469
| 2.3.5.7.19
|-
|-
| S20/S21 = ([[400/399]])/([[441/440]])
| S20/S21 = ([[400/399]])/([[441/440]])
Line 1,728: Line 1,759:
| [[176000/175959]]
| [[176000/175959]]
| 0.403
| 0.403
| 2.3.5.7.11.19
|-
|-
| S21/S22 = ([[441/440]])/([[484/483]])
| S21/S22 = ([[441/440]])/([[484/483]])
| ([[23/20]])/([[22/21]])<sup>3</sup>
| ([[23/20]])/([[22/21]])<sup>3</sup>
| [[213003/212960]]
| [[213003/212960]]
| 0.35
| 0.350
| 2.3.5.7.11.23
|-
|-
| S22/S23 = ([[484/483]])/([[529/528]])
| S22/S23 = ([[484/483]])/([[529/528]])
Line 1,738: Line 1,771:
| [[85184/85169]]
| [[85184/85169]]
| 0.305
| 0.305
| 2.7.11.23
|-
|-
| S23/S24 = ([[529/528]])/([[576/575]])
| S23/S24 = ([[529/528]])/([[576/575]])
Line 1,743: Line 1,777:
| [[304175/304128]]
| [[304175/304128]]
| 0.268
| 0.268
| 2.3.5.11.23
|-
|-
| S24/S25 = ([[576/575]])/([[625/624]])
| S24/S25 = ([[576/575]])/([[625/624]])
Line 1,748: Line 1,783:
| [[359424/359375]]
| [[359424/359375]]
| 0.236
| 0.236
| 2.3.5.13.23
|-
|-
| S25/S26 = ([[625/624]])/([[676/675]])
| S25/S26 = ([[625/624]])/([[676/675]])
Line 1,753: Line 1,789:
| [[140625/140608]]
| [[140625/140608]]
| 0.209
| 0.209
| 2.3.5.13
|-
|-
| S26/S27 = ([[676/675]])/([[729/728]])
| S26/S27 = ([[676/675]])/([[729/728]])
Line 1,758: Line 1,795:
| [[492128/492075]]
| [[492128/492075]]
| 0.186
| 0.186
| 2.3.5.7.13
|-
|-
| S27/S28 = ([[729/728]])/([[784/783]])
| S27/S28 = ([[729/728]])/([[784/783]])
Line 1,763: Line 1,801:
| [[570807/570752]]
| [[570807/570752]]
| 0.167
| 0.167
| 2.3.7.13.29
|-
|-
| S28/S29 = ([[784/783]])/([[841/840]])
| S28/S29 = ([[784/783]])/([[841/840]])
| ([[10/9]])/([[29/28]])<sup>3</sup>
| ([[10/9]])/([[29/28]])<sup>3</sup>
| [[219520/219501]]
| [[219520/219501]]
| 0.15
| 0.150
| 2.3.5.7.29
|-
|-
| S31/S32 = ([[961/960]])/([[1024/1023]])
| <small>S31/S32 = ([[961/960]])/([[1024/1023]])</small>
| ([[11/10]])/([[32/31]])<sup>3</sup>
| ([[11/10]])/([[32/31]])<sup>3</sup>
| [[327701/327680]]
| [[327701/327680]]
| 0.111
| 0.111
| 2.5.11.31
|-
|-
| S33/S34 = ([[1089/1088]])/([[1156/1155]])
| <small>S33/S34 = ([[1089/1088]])/([[1156/1155]])</small>
| ([[35/32]])/([[34/33]])<sup>3</sup>
| ([[35/32]])/([[34/33]])<sup>3</sup>
| [[1257795/1257728]]
| <small>[[1257795/1257728]]</small>
| 0.092
| 0.092
| 2.3.5.7.11.17
|-
|-
| S34/S35 = ([[1156/1155]])/([[1225/1224]])
| <small>S34/S35 = ([[1156/1155]])/([[1225/1224]])</small>
| ([[12/11]])/([[35/34]])<sup>3</sup>
| ([[12/11]])/([[35/34]])<sup>3</sup>
| [[471648/471625]]
| [[471648/471625]]
| 0.084
| 0.084
| 2.3.5.7.11.17
|-
|-
| S37/S38 = ([[1369/1368]])/([[1444/1443]])
| <small>S37/S38 = ([[1369/1368]])/([[1444/1443]])</small>
| ([[13/12]])/([[38/37]])<sup>3</sup>
| ([[13/12]])/([[38/37]])<sup>3</sup>
| [[658489/658464]]
| [[658489/658464]]
| 0.066
| 0.066
| 2.3.13.19.37
|-
|-
| S40/S41 = ([[1600/1599]])/([[1681/1680]])
| <small>S40/S41 = ([[1600/1599]])/([[1681/1680]])</small>
| ([[14/13]])/([[41/40]])<sup>3</sup>
| ([[14/13]])/([[41/40]])<sup>3</sup>
| [[896000/895973]]
| [[896000/895973]]
| 0.052
| 0.052
| 2.5.7.13.41
|-
|-
| S43/S44 = ([[1849/1848]])/([[1936/1935]])
| <small>S43/S44 = ([[1849/1848]])/([[1936/1935]])</small>
| ([[15/14]])/([[44/43]])<sup>3</sup>
| ([[15/14]])/([[44/43]])<sup>3</sup>
| [[1192605/1192576]]
| <small>[[1192605/1192576]]</small>
| 0.042
| 0.042
| 2.3.5.7.11.43
|-
|-
| S46/S47 = ([[2116/2115]])/([[2209/2208]])
| <small>S46/S47 = ([[2116/2115]])/([[2209/2208]])</small>
| ([[16/15]])/([[47/46]])<sup>3</sup>
| ([[16/15]])/([[47/46]])<sup>3</sup>
| [[1557376/1557345]]
| <small>[[1557376/1557345]]</small>
| 0.034
| 0.034
| 2.3.5.23.47
|-
|-
| S49/S50 = ([[2401/2400]])/([[2500/2499]])
| <small>S49/S50 = ([[2401/2400]])/([[2500/2499]])</small>
| ([[17/16]])/([[50/49]])<sup>3</sup>
| ([[17/16]])/([[50/49]])<sup>3</sup>
| [[2000033/2000000]]
| <small>[[2000033/2000000]]</small>
| 0.029
| 0.029
| 2.5.7.17
|-
|-
| S50/S51 = ([[2500/2499]])/([[2601/2600]])
| <small>S50/S51 = ([[2500/2499]])/([[2601/2600]])</small>
| ([[52/49]])/([[51/50]])<sup>3</sup>
| ([[52/49]])/([[51/50]])<sup>3</sup>
| [[6500000/6499899]]
| <small>[[6500000/6499899]]</small>
| 0.027
| 0.027
| 2.3.5.7.13.17
|-
|-
| S55/S56 = ([[3025/3024]])/([[3136/3135]])
| <small>S55/S56 = ([[3025/3024]])/([[3136/3135]])</small>
| ([[19/18]])/([[56/55]])<sup>3</sup>
| ([[19/18]])/([[56/55]])<sup>3</sup>
| [[3161125/3161088]]
| <small>[[3161125/3161088]]</small>
| 0.02
| 0.020
| 2.3.5.7.11.19
|-
|-
| S64/S65 = ([[4096/4095]])/([[4225/4224]])
| <small>S64/S65 = ([[4096/4095]])/([[4225/4224]])</small>
| ([[22/21]])/([[65/64]])<sup>3</sup>
| ([[22/21]])/([[65/64]])<sup>3</sup>
| [[5767168/5767125]]
| <small>[[5767168/5767125]]</small>
| 0.013
| 0.013
| 2.3.5.7.11.13
|}
|}


The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a superparticular interval into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.
The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a superparticular interval into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.


This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page. Note that ultraparticulars are, in general, extremely precise commas so that usually one would not consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]] (''unnoticeable'' in the sense of being smaller than the melodic [[just-noticeable difference]]), despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.
{{Note| This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page. }}
 
Note that ultraparticulars are, in general, extremely precise commas so that usually one would not consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]] (''unnoticeable'' in the sense of being smaller than the melodic [[just-noticeable difference]]), despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.


Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:
Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:
Line 1,835: Line 1,888:
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of the bulk of the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of the bulk of the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.


== {S''k''/S(''k'' + 2) (semiparticulars) ==
== S''k''/S(''k'' + 2) (semiparticulars) ==
=== Motivational examples ===
=== Motivational examples ===
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:
Line 1,860: Line 1,913:


=== Meaning ===
=== Meaning ===
: '''Reader notes:''' In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it is equivalent to using an offset ''k''.
In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it is equivalent to using an offset ''k''. Also keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.
: '''Also:''' keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.


Tempering out S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (= s) and ''k''/(''k'' - 2) (= L) are equidistant from (''k'' + 1)/(''k'' - 1) (= M) because, to make them equidistant, we need to temper out:
Tempering out S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (= s) and ''k''/(''k'' - 2) (= L) are equidistant from (''k'' + 1)/(''k'' - 1) (= M) because, to make them equidistant, we need to temper out:
Line 1,884: 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:
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
! S-expression
! Square Relation
! Square relation
! Ratio
! Ratio
! Subgroup
|-
|-
| S2/S4 = ([[4/3]])/([[16/15]])
| S2/S4 = ([[4/3]])/([[16/15]])
| ([[5/1]])/([[2/1]])<sup>2</sup>
| ([[5/1]])/([[2/1]])<sup>2</sup>
| [[5/4]]
| [[5/4]]
| 2.5
|-
|-
| S3/S5 = ([[9/8]])/([[25/24]])
| S3/S5 = ([[9/8]])/([[25/24]])
| ([[3/1]])/([[5/3]])<sup>2</sup>
| ([[3/1]])/([[5/3]])<sup>2</sup>
| [[27/25]]
| [[27/25]]
| 3.5
|-
|-
| S4/S6 = ([[16/15]])/([[36/35]])
| S4/S6 = ([[16/15]])/([[36/35]])
| ([[7/3]])/([[3/2]])<sup>2</sup>
| ([[7/3]])/([[3/2]])<sup>2</sup>
| [[28/27]]
| [[28/27]]
| 2.3.7
|-
|-
| S5/S7 = ([[25/24]])/([[49/48]])
| S5/S7 = ([[25/24]])/([[49/48]])
| ([[2/1]])/([[7/5]])<sup>2</sup>
| ([[2/1]])/([[7/5]])<sup>2</sup>
| [[50/49]]
| [[50/49]]
| 2.5.7
|-
|-
| S6/S8 = ([[36/35]])/([[64/63]])
| S6/S8 = ([[36/35]])/([[64/63]])
| ([[9/5]])/([[4/3]])<sup>2</sup>
| ([[9/5]])/([[4/3]])<sup>2</sup>
| [[81/80]]
| [[81/80]]
| 2.3.5
|-
|-
| S7/S9 = ([[49/48]])/([[81/80]])
| S7/S9 = ([[49/48]])/([[81/80]])
| ([[5/3]])/([[9/7]])<sup>2</sup>
| ([[5/3]])/([[9/7]])<sup>2</sup>
| [[245/243]]
| [[245/243]]
| 3.5.7
|-
|-
| S8/S10 = ([[64/63]])/([[100/99]])
| S8/S10 = ([[64/63]])/([[100/99]])
| ([[11/7]])/([[5/4]])<sup>2</sup>
| ([[11/7]])/([[5/4]])<sup>2</sup>
| [[176/175]]
| [[176/175]]
| 2.5.7.11
|-
|-
| S9/S11 = ([[81/80]])/([[121/120]])
| S9/S11 = ([[81/80]])/([[121/120]])
| ([[3/2]])/([[11/9]])<sup>2</sup>
| ([[3/2]])/([[11/9]])<sup>2</sup>
| [[243/242]]
| [[243/242]]
| 2.3.11
|-
|-
| S10/S12 = ([[100/99]])/([[144/143]])
| S10/S12 = ([[100/99]])/([[144/143]])
| ([[13/9]])/([[6/5]])<sup>2</sup>
| ([[13/9]])/([[6/5]])<sup>2</sup>
| [[325/324]]
| [[325/324]]
| 2.3.5.13
|-
|-
| S11/S13 = ([[121/120]])/([[169/168]])
| S11/S13 = ([[121/120]])/([[169/168]])
| ([[7/5]])/([[13/11]])<sup>2</sup>
| ([[7/5]])/([[13/11]])<sup>2</sup>
| [[847/845]]
| [[847/845]]
| 5.7.11.13
|-
|-
| S12/S14 = ([[144/143]])/([[196/195]])
| S12/S14 = ([[144/143]])/([[196/195]])
| ([[15/11]])/([[7/6]])<sup>2</sup>
| ([[15/11]])/([[7/6]])<sup>2</sup>
| [[540/539]]
| [[540/539]]
| 2.3.5.7.11
|-
|-
| S13/S15 = ([[169/168]])/([[225/224]])
| S13/S15 = ([[169/168]])/([[225/224]])
| ([[4/3]])/([[15/13]])<sup>2</sup>
| ([[4/3]])/([[15/13]])<sup>2</sup>
| [[676/675]]
| [[676/675]]
| 2.3.5.13
|-
|-
| S14/S16 = ([[196/195]])/([[256/255]])
| S14/S16 = ([[196/195]])/([[256/255]])
| ([[17/13]])/([[8/7]])<sup>2</sup>
| ([[17/13]])/([[8/7]])<sup>2</sup>
| [[833/832]]
| [[833/832]]
| 2.7.13.17
|-
|-
| S15/S17 = ([[225/224]])/([[289/288]])
| S15/S17 = ([[225/224]])/([[289/288]])
| ([[9/7]])/([[17/15]])<sup>2</sup>
| ([[9/7]])/([[17/15]])<sup>2</sup>
| [[2025/2023]]
| [[2025/2023]]
| 3.5.7.17
|-
|-
| S16/S18 = ([[256/255]])/([[324/323]])
| S16/S18 = ([[256/255]])/([[324/323]])
| ([[19/15]])/([[9/8]])<sup>2</sup>
| ([[19/15]])/([[9/8]])<sup>2</sup>
| [[1216/1215]]
| [[1216/1215]]
| 2.3.5.19
|-
|-
| S17/S19 = ([[289/288]])/([[361/360]])
| S17/S19 = ([[289/288]])/([[361/360]])
| ([[5/4]])/([[19/17]])<sup>2</sup>
| ([[5/4]])/([[19/17]])<sup>2</sup>
| [[1445/1444]]
| [[1445/1444]]
| 2.5.17.19
|-
|-
| S18/S20 = ([[324/323]])/([[400/399]])
| S18/S20 = ([[324/323]])/([[400/399]])
| ([[21/17]])/([[10/9]])<sup>2</sup>
| ([[21/17]])/([[10/9]])<sup>2</sup>
| [[1701/1700]]
| [[1701/1700]]
| 2.3.5.7.17
|-
|-
| S19/S21 = ([[361/360]])/([[441/440]])
| S19/S21 = ([[361/360]])/([[441/440]])
| ([[11/9]])/([[21/19]])<sup>2</sup>
| ([[11/9]])/([[21/19]])<sup>2</sup>
| [[3971/3969]]
| [[3971/3969]]
| 3.7.11.19
|-
|-
| S20/S22 = ([[400/399]])/([[484/483]])
| S20/S22 = ([[400/399]])/([[484/483]])
| ([[23/19]])/([[11/10]])<sup>2</sup>
| ([[23/19]])/([[11/10]])<sup>2</sup>
| [[2300/2299]]
| [[2300/2299]]
| 2.5.11.19.23
|-
|-
| S21/S23 = ([[441/440]])/([[529/528]])
| S21/S23 = ([[441/440]])/([[529/528]])
| ([[6/5]])/([[23/21]])<sup>2</sup>
| ([[6/5]])/([[23/21]])<sup>2</sup>
| [[2646/2645]]
| [[2646/2645]]
| 2.3.5.7.23
|-
|-
| S22/S24 = ([[484/483]])/([[576/575]])
| S22/S24 = ([[484/483]])/([[576/575]])
| ([[25/21]])/([[12/11]])<sup>2</sup>
| ([[25/21]])/([[12/11]])<sup>2</sup>
| [[3025/3024]]
| [[3025/3024]]
| 2.3.5.7.11
|-
|-
| S23/S25 = ([[529/528]])/([[625/624]])
| S23/S25 = ([[529/528]])/([[625/624]])
| ([[13/11]])/([[25/23]])<sup>2</sup>
| ([[13/11]])/([[25/23]])<sup>2</sup>
| [[6877/6875]]
| [[6877/6875]]
| 5.11.13.23
|-
|-
| S24/S26 = ([[576/575]])/([[676/675]])
| S24/S26 = ([[576/575]])/([[676/675]])
| ([[27/23]])/([[13/12]])<sup>2</sup>
| ([[27/23]])/([[13/12]])<sup>2</sup>
| [[3888/3887]]
| [[3888/3887]]
| 2.3.13.23
|-
|-
| S25/S27 = ([[625/624]])/([[729/728]])
| S25/S27 = ([[625/624]])/([[729/728]])
| ([[7/6]])/([[27/25]])<sup>2</sup>
| ([[7/6]])/([[27/25]])<sup>2</sup>
| [[4375/4374]]
| [[4375/4374]]
| 2.3.5.7
|-
|-
| S26/S28 = ([[676/675]])/([[784/783]])
| S26/S28 = ([[676/675]])/([[784/783]])
| ([[29/25]])/([[14/13]])<sup>2</sup>
| ([[29/25]])/([[14/13]])<sup>2</sup>
| [[4901/4900]]
| [[4901/4900]]
| 2.5.7.13.29
|-
|-
| S27/S29 = ([[729/728]])/([[841/840]])
| S27/S29 = ([[729/728]])/([[841/840]])
| ([[15/13]])/([[29/27]])<sup>2</sup>
| ([[15/13]])/([[29/27]])<sup>2</sup>
| [[10935/10933]]
| [[10935/10933]]
| 3.5.13.29
|-
|-
| S28/S30 = ([[784/783]])/([[900/899]])
| S28/S30 = ([[784/783]])/([[900/899]])
| ([[31/27]])/([[15/14]])<sup>2</sup>
| ([[31/27]])/([[15/14]])<sup>2</sup>
| [[6076/6075]]
| [[6076/6075]]
| 2.3.5.7.31
|-
|-
| S29/S31 = ([[841/840]])/([[961/960]])
| S29/S31 = ([[841/840]])/([[961/960]])
| ([[8/7]])/([[31/29]])<sup>2</sup>
| ([[8/7]])/([[31/29]])<sup>2</sup>
| [[6728/6727]]
| [[6728/6727]]
| 2.7.29.31
|-
|-
| S30/S32 = ([[900/899]])/([[1024/1023]])
| S30/S32 = ([[900/899]])/([[1024/1023]])
| ([[33/29]])/([[16/15]])<sup>2</sup>
| ([[33/29]])/([[16/15]])<sup>2</sup>
| [[7425/7424]]
| [[7425/7424]]
| 2.3.5.11.29
|-
|-
| S31/S33 = ([[961/960]])/([[1089/1088]])
| S31/S33 = ([[961/960]])/([[1089/1088]])
| ([[17/15]])/([[33/31]])<sup>2</sup>
| ([[17/15]])/([[33/31]])<sup>2</sup>
| [[16337/16335]]
| [[16337/16335]]
| 3.5.11.17.31
|-
|-
| S32/S34 = ([[1024/1023]])/([[1156/1155]])
| S32/S34 = ([[1024/1023]])/([[1156/1155]])
| ([[35/31]])/([[17/16]])<sup>2</sup>
| ([[35/31]])/([[17/16]])<sup>2</sup>
| [[8960/8959]]
| [[8960/8959]]
| 2.5.7.17.31
|-
|-
| S33/S35 = ([[1089/1088]])/([[1225/1224]])
| S33/S35 = ([[1089/1088]])/([[1225/1224]])
| ([[9/8]])/([[35/33]])<sup>2</sup>
| ([[9/8]])/([[35/33]])<sup>2</sup>
| [[9801/9800]]
| [[9801/9800]]
| 2.3.5.7.11
|-
|-
| S36/S38 = ([[1296/1295]])/([[1444/1443]])
| S36/S38 = ([[1296/1295]])/([[1444/1443]])
| ([[39/35]])/([[19/18]])<sup>2</sup>
| ([[39/35]])/([[19/18]])<sup>2</sup>
| [[12636/12635]]
| [[12636/12635]]
| 2.3.5.7.13.19
|-
|-
| S37/S39 = ([[1369/1368]])/([[1521/1520]])
| S37/S39 = ([[1369/1368]])/([[1521/1520]])
| ([[10/9]])/([[39/37]])<sup>2</sup>
| ([[10/9]])/([[39/37]])<sup>2</sup>
| [[13690/13689]]
| [[13690/13689]]
| 2.3.5.13.37
|-
|-
| S41/S43 = ([[1681/1680]])/([[1849/1848]])
| S41/S43 = ([[1681/1680]])/([[1849/1848]])
| ([[11/10]])/([[43/41]])<sup>2</sup>
| ([[11/10]])/([[43/41]])<sup>2</sup>
| [[18491/18490]]
| [[18491/18490]]
| 2.5.11.41.43
|-
|-
| S45/S47 = ([[2025/2024]])/([[2209/2208]])
| S45/S47 = ([[2025/2024]])/([[2209/2208]])
| ([[12/11]])/([[47/45]])<sup>2</sup>
| ([[12/11]])/([[47/45]])<sup>2</sup>
| [[24300/24299]]
| [[24300/24299]]
| 2.3.5.11.47
|-
|-
| S46/S48 = ([[2116/2115]])/([[2304/2303]])
| S46/S48 = ([[2116/2115]])/([[2304/2303]])
| ([[49/45]])/([[24/23]])<sup>2</sup>
| ([[49/45]])/([[24/23]])<sup>2</sup>
| [[25921/25920]]
| [[25921/25920]]
| 2.3.5.7.23
|-
|-
| S49/S51 = ([[2401/2400]])/([[2601/2600]])
| S49/S51 = ([[2401/2400]])/([[2601/2600]])
| ([[13/12]])/([[51/49]])<sup>2</sup>
| ([[13/12]])/([[51/49]])<sup>2</sup>
| [[31213/31212]]
| [[31213/31212]]
| 2.3.7.13.17
|-
|-
| S52/S54 = ([[2704/2703]])/([[2916/2915]])
| S52/S54 = ([[2704/2703]])/([[2916/2915]])
| ([[55/51]])/([[27/26]])<sup>2</sup>
| ([[55/51]])/([[27/26]])<sup>2</sup>
| [[37180/37179]]
| [[37180/37179]]
| 2.3.5.11.13.17
|-
|-
| S66/S68 = ([[4356/4355]])/([[4624/4623]])
| S66/S68 = ([[4356/4355]])/([[4624/4623]])
| ([[69/65]])/([[34/33]])<sup>2</sup>
| ([[69/65]])/([[34/33]])<sup>2</sup>
| [[75141/75140]]
| [[75141/75140]]
| 2.3.5.11.17.23
|-
|-
| S78/S80 = ([[6084/6083]])/([[6400/6399]])
| S78/S80 = ([[6084/6083]])/([[6400/6399]])
| ([[81/77]])/([[40/39]])<sup>2</sup>
| ([[81/77]])/([[40/39]])<sup>2</sup>
| [[123201/123200]]
| [[123201/123200]]
| 2.3.5.7.11.13
|}
|}


Note that while a lot of these have pages, not all of them do, although that does not mean they should not. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.
{{Note| While a lot of these have pages, not all of them do, although that does not mean they should not. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages. }}


== S''k''<sup>2</sup>⋅S(''k'' + 1) and S(''k'' − 1)⋅S''k''<sup>2</sup> (lopsided commas) ==
== S''k''<sup>2</sup>⋅S(''k'' + 1) and S(''k'' − 1)⋅S''k''<sup>2</sup> (lopsided commas) ==
Line 2,064: Line 2,158:


=== Derivation of equivalence relation ===
=== Derivation of equivalence relation ===
Using the clarity of [[S-expression/Advanced results #Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:
Using the clarity of [[#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:


S''k''<sup>2</sup>⋅S(''k'' + 1) = [''k'' - 1, ''k'', ''k'' + 1, ''k'' + 2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:
S''k''<sup>2</sup>⋅S(''k'' + 1) = [''k'' - 1, ''k'', ''k'' + 1, ''k'' + 2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:
Line 2,317: Line 2,411:
|-
|-
! S-expression
! S-expression
! Square Relation
! Square relation
! Ratio
! Ratio
|-
|-
Line 2,549: Line 2,643:
|}
|}


== Equivalent S-expressions ==
== C''k'' and Cp''k'' (cube-particulars) ==
=== Significance and meaning ===
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.  
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''.


=== A useful general rule ===
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.
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:


$$
=== Significance ===
\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
# Cube-particulars are related to ultraparticulars via the cube-relation identity: S''k''/S(''k'' + 1) = C''k''⋅Cp(''k'' + 1), which suggests structurally induced extensions for temperaments that temper out ultraparticulars.
$$
# 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.


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.
=== 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.


=== Examples ===
=== Table of cube-particulars ===
Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-all left-3 left-6"
|+ style="font-size: 105%;" | 31-limit cube-particulars – C''k''
|-
! S-expression
! Ratio
! Subgroup
|-
| –
| –
| –
|-
| C2
| [[8/7]]
| 2.7
|-
| C3
| [[27/26]]
| 2.3.13
|-
| C4
| [[64/63]]
| 2.3.7
|-
| C5
| [[125/124]]
| 2.5.31
|-
| C7
| [[343/342]]
| 2.3.7.19
|-
| C9
| [[729/728]]
| 2.3.7.13
|-
| C11
| [[1331/1330]]
| 2.5.7.11.19
|-
| 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>
 
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.
 
== Using S-factorizations to understand the significance of S-expressions ==
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 * ...
 
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:
 
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:
 
[''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.)
 
<pre>
Sk = [k-1, k, k+1]^[-1, 2, -1]
</pre>
<pre>
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]
</pre>
<pre>
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
0  -1    2  -1    0
0    0  -1    2  -1
========================
-1    1    0    1  -1
</pre>
<pre>
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]
= (k+2)/(k-1) * ( k/(k+1) )^3 = (k+2)/(k-1) / ((k+1)/k)^3
</pre>
<pre>
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]
= [k-2, k-1, k]^[-1, 2, -1] / [k, k+1, k+2]^[-1,  2, -1]
= (k+2)/(k-2) * ((k-1)/(k+1))^2 = (k+2)/(k-2) / ((k+1)/(k-1))^2
 
k-2  k-1  k  k+1  k+2
-1    2  -1    0    0
0    0    1  -2    1
========================
-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/S55, S15/(S22/S24)
|-
| [[325/324]]
| S25⋅S26, S10/S12
|-
| [[540/539]]
| S12/S14, (S9⋅S10)/S7, (S6/S7)/(S8/S10)
|-
| [[676/675]]
| S26, S13/S15
|-
| [[1225/1224]]
| S35, S49⋅S50
|-
| [[2601/2600]]
| S51, S17/(S25⋅S26)
|-
| [[3025/3024]]
| S55, S22/S24, (S25/S27)⋅S99
|-
| [[9801/9800]]
| S99, S33/S35
|-
| [[25921/25920]]
| S161, S46/S48
|-
| <small>[[123201/123200]]</small>
| S351, S78/S80
|}


'''Importantly:''' 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| 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. }}
 
{{Tip| Feel free to expand with any equivalences you find that you think are valuable. }}
{| class="wikitable center-all
|-
! Comma
! S-expressions
|-
| [[64/63]]
| (S4⋅S5⋅S6)/S3 = S4/(S6⋅S7) = S8
|-
| [[81/80]]
| S6/S8 = S9
|-
| [[176/175]]
| S8/S10 = S22⋅S23⋅S24
|-
| [[243/242]]
| S9/S11 = S15/([[3025/3024|S22/S24 = S55]])
|-
| [[325/324]]
| S10/S12 = S25⋅S26
|-
| [[540/539]]
| S12/S14 = (S9⋅S10)/S7 = (S6/S7)/(S8/S10)
|-
| [[676/675]]
| S13/S15 = S26
|-
| [[1225/1224]]
| S35 = S49⋅S50
|-
| [[3025/3024]]
| S22/S24 = S55 = (S25/S27)⋅S99
|-
| [[2601/2600]]
| S17/(S25⋅S26) = S51
|-
| [[9801/9800]]
| S99 = S33/S35
|-
| [[25921/25920]]
| S161 = S46/S48
|-
| [[123201/123200]]
| S351 = S78/S80
|}
 
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.


A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.
A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.
Line 2,652: Line 2,982:
: For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.
: 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]].
: 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
; Odd-particular
Line 2,669: 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 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}}.
: 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
; 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
; S-factorization
Line 2,686: Line 3,020:


; S-comma
; 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
; Indirect S-comma
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