S-expression: Difference between revisions

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

| [[13225/13224]]

|-

| S116

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

| [[13456/13455]]

|-

| S120

| ([[125/124]])/([[126/125]])

| [[15625/15624]]

|-

| S144

| ([[144/143]])/([[145/144]])

| [[20736/20735]]

|-

| S153

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

| [[23409/23408]]

|-

| S154

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

| [[23716/23715]]

|-

| S155

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

| [[24025/24024]]

|-

| S161

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

| [[28900/28899]]

|-

| S175

| ([[175/174]])/([[176/175]])

| [[30625/30624]]

|-

| S208

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

| [[43264/43263]]

|-

| S209

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

| [[43681/43680]]

|-

| S231

| ([[231/230]])/([[232/231]])

| [[53361/53360]]

|-

| S289

| ([[289/288]])/([[290/289]])

| [[83521/83520]]

|-

| S323

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

| [[104329/104328]]

|-

| S324

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

| [[104976/104975]]

|-

| S341

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

| [[116281/116280]]

|-

| S342

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

| [[116964/116963]]

|-

| S351

| ([[351/350]])/([[352/351]])

| [[123201/123200]]

|-

| S391

| ([[391/390]])/([[392/391]])

| [[152881/152880]]

|-

| S441

| ([[441/440]])/([[442/441]])

| [[194481/194480]]

|-

| S494

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

| [[244036/244035]]

|-

| S495

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

| [[245025/245024]]

|-

| S528

| ([[528/527]])/([[529/528]])

| [[278784/278783]]

|-

| S551

| ([[551/550]])/([[552/551]])

| [[303601/303600]]

|-

| S714

| ([[714/713]])/([[715/714]])

| [[509796/509795]]

|-

| S783

| ([[783/782]])/([[784/783]])

| [[613089/613088]]

|-

| S1275

| <small>([[1275/1274]])/([[1276/1275]])</small>

| <small>[[1625625/1625624]]</small>

|-

| S1519

| <small>([[1519/1518]])/([[1520/1519]])</small>

| <small>[[2307361/2307360]]</small>

|-

| S1520

| <small>([[1520/1519]])/([[1521/1520]])</small>

| <small>[[2310400/2310399]]</small>

|-

| S2001

| <small>([[2024/2023]])/([[2025/2024]])</small>

| <small>[[4096576/4096575]]</small>

|-

| S2431

| <small>([[2431/2430]])/([[2432/2431]])</small>

| <small>[[5909761/5909760]]</small>

|-

| S3249

| <small>([[3249/3248]])/([[3250/3249]])</small>

| <small>[[10556001/10556000]]</small>

|-

| S9801

| <small>([[9801/9800]])/([[9802/9801]])</small>

| <small>[[96059601/96059600]]</small>

|-

| S13311

| <small><small>([[13311/13310]])/([[13312/13311]])</small></small>

| <small><small>[[177182721/177182720]]</small></small>

|-

| S13455

| <small><small>([[13455/13454]])/([[13456/13455]])</small></small>

| <small><small>[[181037025/181037024]]</small></small>

|}

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.

|+ 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>

|-

! Interval relation

! Ratio

|-

| S2⋅S3

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

| [[3/2]]

|-

| S3⋅S4

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

| [[6/5]]

|-

| S4⋅S5

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

| [[10/9]]

|-

| S5⋅S6

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

| [[15/14]]

|-

| S6⋅S7

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

| [[21/20]]

|-

| S7⋅S8

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

| [[28/27]]

|-

| S8⋅S9

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

| [[36/35]]

|-

| S9⋅S10

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

| [[45/44]]

|-

| S10⋅S11

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

| [[55/54]]

|-

| S11⋅S12

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

| [[66/65]]

|-

| S12⋅S13

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

| [[78/77]]

|-

| S13⋅S14

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

| [[91/90]]

|-

| S14⋅S15

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

| [[105/104]]

|-

| S15⋅S16

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

| [[120/119]]

|-

| S16⋅S17

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

| [[136/135]]

|-

| S17⋅S18

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

| [[153/152]]

|-

| S18⋅S19

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

| [[171/170]]

|-

| S19⋅S20

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

| [[190/189]]

|-

| S20⋅S21

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

| [[210/209]]

|-

| S21⋅S22

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

| [[231/230]]

|-

| S22⋅S23

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

| [[253/252]]

|-

| S23⋅S24

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

| [[276/275]]

|-

| S24⋅S25

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

| [[300/299]]

|-

| S25⋅S26

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

| [[325/324]]

|-

| S26⋅S27

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

| [[351/350]]

|-

| S27⋅S28

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

| [[378/377]]

|-

| S28⋅S29

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

| [[406/405]]

|-

| S29⋅S30

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

| [[435/434]]

|-

| S30⋅S31

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

| [[465/464]]

|-

| S31⋅S32

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

| [[496/495]]

|-

| S32⋅S33

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

| [[528/527]]

|-

| S33⋅S34

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

| [[561/560]]

|-

| S34⋅S35

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

| [[595/594]]

|-

| S49⋅S50

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

| [[1225/1224]]

|-

| S50⋅S51

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

| [[1275/1274]]

|-

| S55⋅S56

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

| [[1540/1539]]

|-

| S63⋅S64

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

| [[2016/2015]]

|-

| S64⋅S65

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

| [[2080/2079]]

|-

| S76⋅S77

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

| [[2926/2925]]

|-

| S91⋅S92

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

| [[4186/4185]]

|-

| [[6670/6669]]

|-

|-

|-

|-

|-

|-

|-

|-

|}

=== Table of ultraparticulars ===

|-

! S-expression

! Cube relation

! Cents

|-

| S2/S3 = ([[4/3]])/([[9/8]])

| ([[4/1]])/([[3/2]])³

| [[32/27]]

|-

| S3/S4 = ([[9/8]])/([[16/15]])

| ([[5/2]])/([[4/3]])³

| [[135/128]]

|-

| S4/S5 = ([[16/15]])/([[25/24]])

| ([[2/1]])/([[5/4]])³

| [[128/125]]

|-

| S5/S6 = ([[25/24]])/([[36/35]])

| ([[7/4]])/([[6/5]])³

| [[875/864]]

|-

| S6/S7 = ([[36/35]])/([[49/48]])

| ([[8/5]])/([[7/6]])³

| [[1728/1715]]

|-

| S7/S8 = ([[49/48]])/([[64/63]])

| ([[3/2]])/([[8/7]])³

| [[1029/1024]]

| 8.433

|-

| S8/S9 = ([[64/63]])/([[81/80]])

| ([[10/7]])/([[9/8]])³

| [[5120/5103]]

| 5.758

|-

| S9/S10 = ([[81/80]])/([[100/99]])

| ([[11/8]])/([[10/9]])³

| [[8019/8000]]

| 4.107

|-

| S10/S11 = ([[100/99]])/([[121/120]])

| ([[4/3]])/([[11/10]])³

| [[4000/3993]]

| 3.032

|-

| S11/S12 = ([[121/120]])/([[144/143]])

| ([[13/10]])/([[12/11]])³

| [[17303/17280]]

| 2.303

|-

| S12/S13 = ([[144/143]])/([[169/168]])

| ([[14/11]])/([[13/12]])³

| [[24192/24167]]

|-

| S13/S14 = ([[169/168]])/([[196/195]])

| ([[5/4]])/([[14/13]])³

| [[10985/10976]]

| 1.419

|-

| S14/S15 = ([[196/195]])/([[225/224]])

| ([[16/13]])/([[15/14]])³

| [[43904/43875]]

| 1.144

|-

| S15/S16 = ([[225/224]])/([[256/255]])

| ([[17/14]])/([[16/15]])³

| [[57375/57344]]

| 0.936

|-

| S16/S17 = ([[256/255]])/([[289/288]])

| ([[6/5]])/([[17/16]])³

| [[24576/24565]]

| 0.775

|-

| S17/S18 = ([[289/288]])/([[324/323]])

| ([[19/16]])/([[18/17]])³

| [[93347/93312]]

| 0.649

|-

| S18/S19 = ([[324/323]])/([[361/360]])

| ([[20/17]])/([[19/18]])³

| [[116640/116603]]

| 0.549

|-

| S19/S20 = ([[361/360]])/([[400/399]])

| ([[7/6]])/([[20/19]])³

| [[48013/48000]]

| 0.469

|-

| S20/S21 = ([[400/399]])/([[441/440]])

| ([[22/19]])/([[21/20]])³

| [[176000/175959]]

| 0.403

|-

| S21/S22 = ([[441/440]])/([[484/483]])

| ([[23/20]])/([[22/21]])³

| [[213003/212960]]

|-

| S22/S23 = ([[484/483]])/([[529/528]])

| ([[8/7]])/([[23/22]])³

| [[85184/85169]]

| 0.305

|-

| S23/S24 = ([[529/528]])/([[576/575]])

| ([[25/22]])/([[24/23]])³

| [[304175/304128]]

| 0.268

|-

| S24/S25 = ([[576/575]])/([[625/624]])

| ([[26/23]])/([[25/24]])³

| [[359424/359375]]

| 0.236

|-

| S25/S26 = ([[625/624]])/([[676/675]])

| ([[9/8]])/([[26/25]])³

| [[140625/140608]]

| 0.209

|-

| S26/S27 = ([[676/675]])/([[729/728]])

| ([[28/25]])/([[27/26]])³

| [[492128/492075]]

| 0.186

|-

| S27/S28 = ([[729/728]])/([[784/783]])

| ([[29/26]])/([[28/27]])³

| [[570807/570752]]

| 0.167

|-

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

| ([[10/9]])/([[29/28]])³

| [[219520/219501]]

|-

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

| ([[11/10]])/([[32/31]])³

| [[327701/327680]]

| 0.111

|-

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

| ([[35/32]])/([[34/33]])³

| <small>[[1257795/1257728]]</small>

| 0.092

|-

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

| ([[12/11]])/([[35/34]])³

| [[471648/471625]]

| 0.084

|-

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

| ([[13/12]])/([[38/37]])³

| [[658489/658464]]

| 0.066

|-

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

| ([[14/13]])/([[41/40]])³

| [[896000/895973]]

| 0.052

|-

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

| ([[15/14]])/([[44/43]])³

| <small>[[1192605/1192576]]</small>

| 0.042

|-

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

| ([[16/15]])/([[47/46]])³

| <small>[[1557376/1557345]]</small>

| 0.034

|-

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

| ([[17/16]])/([[50/49]])³

| <small>[[2000033/2000000]]</small>

| 0.029

|-

| <small>S50/S51 = ([[2500/2499]])/([[2601/2600]])</small>

| ([[52/49]])/([[51/50]])³

| <small>[[6500000/6499899]]</small>

| 0.027

|-

| <small>S55/S56 = ([[3025/3024]])/([[3136/3135]])</small>

| ([[19/18]])/([[56/55]])³

| <small>[[3161125/3161088]]</small>

|-

| <small>S64/S65 = ([[4096/4095]])/([[4225/4224]])</small>

| ([[22/21]])/([[65/64]])³

| <small>[[5767168/5767125]]</small>

| 0.013

|}

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:

|-

! S-expression

! Square relation

! Ratio

|-

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

| ([[5/1]])/([[2/1]])²

| [[5/4]]

|-

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

| ([[3/1]])/([[5/3]])²

| [[27/25]]

|-

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

| ([[7/3]])/([[3/2]])²

| [[28/27]]

|-

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

| ([[2/1]])/([[7/5]])²

| [[50/49]]

|-

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

| ([[9/5]])/([[4/3]])²

| [[81/80]]

|-

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

| ([[5/3]])/([[9/7]])²

| [[245/243]]

|-

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

| ([[11/7]])/([[5/4]])²

| [[176/175]]

|-

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

| ([[3/2]])/([[11/9]])²

| [[243/242]]

|-

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

| ([[13/9]])/([[6/5]])²

| [[325/324]]

|-

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

| ([[7/5]])/([[13/11]])²

| [[847/845]]

|-

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

| ([[15/11]])/([[7/6]])²

| [[540/539]]

|-

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

| ([[4/3]])/([[15/13]])²

| [[676/675]]

|-

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

| ([[17/13]])/([[8/7]])²

| [[833/832]]

|-

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

| ([[9/7]])/([[17/15]])²

| [[2025/2023]]

|-

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

| ([[19/15]])/([[9/8]])²

| [[1216/1215]]

|-

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

| ([[5/4]])/([[19/17]])²

| [[1445/1444]]

|-

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

| ([[21/17]])/([[10/9]])²

| [[1701/1700]]

|-

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

| ([[11/9]])/([[21/19]])²

| [[3971/3969]]

|-

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

| ([[23/19]])/([[11/10]])²

| [[2300/2299]]

|-

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

| ([[6/5]])/([[23/21]])²

| [[2646/2645]]

|-

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

| ([[25/21]])/([[12/11]])²

| [[3025/3024]]

|-

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

| ([[13/11]])/([[25/23]])²

| [[6877/6875]]

|-

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

| ([[27/23]])/([[13/12]])²

| [[3888/3887]]

|-

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

| ([[7/6]])/([[27/25]])²

| [[4375/4374]]

|-

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

| ([[29/25]])/([[14/13]])²

| [[4901/4900]]

|-

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

| ([[15/13]])/([[29/27]])²

| [[10935/10933]]

|-

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

| ([[31/27]])/([[15/14]])²

| [[6076/6075]]

|-

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

| ([[8/7]])/([[31/29]])²

| [[6728/6727]]

|-

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

| ([[33/29]])/([[16/15]])²

| [[7425/7424]]

|-

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

| ([[17/15]])/([[33/31]])²

| [[16337/16335]]

|-

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

| ([[35/31]])/([[17/16]])²

| [[8960/8959]]

|-

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

| ([[9/8]])/([[35/33]])²

| [[9801/9800]]

|-

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

| ([[39/35]])/([[19/18]])²

| [[12636/12635]]

|-

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

| ([[10/9]])/([[39/37]])²

| [[13690/13689]]

|-

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

| ([[11/10]])/([[43/41]])²

| [[18491/18490]]

|-

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

| ([[12/11]])/([[47/45]])²

| [[24300/24299]]

|-

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

| ([[49/45]])/([[24/23]])²

| [[25921/25920]]

|-

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

| ([[13/12]])/([[51/49]])²

| [[31213/31212]]

|-

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

| ([[55/51]])/([[27/26]])²

| [[37180/37179]]

|-

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

| ([[69/65]])/([[34/33]])²

| [[75141/75140]]

|-

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

| ([[81/77]])/([[40/39]])²

| [[123201/123200]]

|}

=== Derivation of equivalence relation ===

S''k''²⋅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:

| [[256000/255879]]

|}

== Equivalent S-expressions ==