Xenharmonic Wiki

S-expression: Difference between revisions

← Older edit
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,482 edits
m another observation, also start pentaparticulars
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,482 edits
 
(2 intermediate revisions by the same user not shown)
Line 2,412: Line 2,412:


=== Significance ===
=== Significance ===
# Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2:''k''-1:''k'':''k''+1:''k''+2:''k''+3}}.
# Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2 : ''k''-1 : ''k'' : ''k''+1 : ''k''+2 : ''k''+3.}}
# Interestingly, while three-particulars are about equidistance but (currently) have no (clean) known splitting property (due to the three intervals made equidistant not obviously composing to some other significant interval), pentaparticulars are in a sense symmetric to this and related algebraically, by instead being specifically about splitting.
# Interestingly, while three-particulars are about equidistance but (currently) have no (clean) known splitting property (due to the three intervals made equidistant not obviously composing to some other significant interval), pentaparticulars are in a sense symmetric to this and related algebraically, by instead being specifically about splitting.
# They are obviously implied by any system that equates S''k'' with S(''k'' + 1) and S(''k'' - 1) with S(''k'' + 2) simultaneously, so are not as uncommon as might be guessed, but tend to be accurate equivalences, and 5 is a rather specific number of parts to divide an interval into, even if obviously the most natural for an interval whose numerator is 5 more than the denominator (up to simplification).


=== Derivation ===
=== Derivation ===
Line 2,438: Line 2,439:


=== Table of pentaparticulars ===
=== Table of pentaparticulars ===
Here is a table of 43-limit pentaparticulars.
Up to what is currently the largest pentaparticular on the wiki.
 
{| class="wikitable center-all
TODO
|-
! S-expression
! Relation
! Ratio
! Subgroup
|-
| ([[135/128|S3/S4]])<sup>2</sup> * [[32/25|S2/S5]]
| ([[6/1]])/([[4/3]])<sup>5</sup>
| [[729/512]]
| 2.3
|-
| ([[128/125|S4/S5]])<sup>2</sup> * [[35/32|S3/S6]]
| ([[7/2]])/([[5/4]])<sup>5</sup>
| [[3584/3125]]
| 2.5.7
|-
| ([[875/864|S5/S6]])<sup>2</sup> * [[256/245|S4/S7]]
| ([[8/3]])/([[6/5]])<sup>5</sup>
| [[3125/2916]]
| 2.3.5
|-
| ([[1728/1715|S6/S7]])<sup>2</sup> * [[525/512|S5/S8]]
| ([[9/4]])/([[7/6]])<sup>5</sup>
| [[17496/16807]]
| 2.3.7
|-
| ([[1029/1024|S7/S8]])<sup>2</sup> * [[64/63|S6/S9]]
| ([[2/1]])/([[8/7]])<sup>5</sup>
| [[16807/16384]]
| 2.7
|-
| ([[5120/5103|S8/S9]])<sup>2</sup> * [[1617/1600|S7/S10]]
| ([[11/6]])/([[9/8]])<sup>5</sup>
| [[180224/177147]]
| 2.3.11
|-
| ([[8019/8000|S9/S10]])<sup>2</sup> * [[2560/2541|S8/S11]]
| ([[12/7]])/([[10/9]])<sup>5</sup>
| [[177147/175000]]
| 2.3.5.7
|-
| ([[4000/3993|S10/S11]])<sup>2</sup> * [[1287/1280|S9/S12]]
| ([[13/8]])/([[11/10]])<sup>5</sup>
| [[162500/161051]]
| 2.5.11.13
|-
| ([[17303/17280|S11/S12]])<sup>2</sup> * [[5600/5577|S10/S13]]
| ([[14/9]])/([[12/11]])<sup>5</sup>
| [[1127357/1119744]]
| 2.3.7.11
|-
| ([[24192/24167|S12/S13]])<sup>2</sup> * [[1573/1568|S11/S14]]
| ([[3/2]])/([[13/12]])<sup>5</sup>
| [[373248/371293]]
| 2.3.13
|-
| ([[10985/10976|S13/S14]])<sup>2</sup> * [[3584/3575|S12/S15]]
| ([[16/11]])/([[14/13]])<sup>5</sup>
| [[371293/369754]]
| 2.7.11.13
|-
| ([[43904/43875|S14/S15]])<sup>2</sup> * [[14365/14336|S13/S16]]
| ([[17/12]])/([[15/14]])<sup>5</sup>
| [[2285752/2278125]]
| 2.3.5.7.17
|-
| ([[57375/57344|S15/S16]])<sup>2</sup> * [[18816/18785|S14/S17]]
| ([[18/13]])/([[16/15]])<sup>5</sup>
| [[6834375/6815744]]
| 2.3.5.13
|-
| ([[24576/24565|S16/S17]])<sup>2</sup> * [[8075/8064|S15/S18]]
| ([[19/14]])/([[17/16]])<sup>5</sup>
| [[9961472/9938999]]
| 2.7.17.19
|-
| ([[93347/93312|S17/S18]])<sup>2</sup> * [[6144/6137|S16/S19]]
| ([[4/3]])/([[18/17]])<sup>5</sup>
| [[1419857/1417176]]
| 2.3.17
|-
| ([[116640/116603|S18/S19]])<sup>2</sup> * [[38437/38400|S17/S20]]
| ([[21/16]])/([[19/18]])<sup>5</sup>
| [[2480058/2476099]]
| 2.3.7.19
|-
| ([[48013/48000|S19/S20]])<sup>2</sup> * [[15840/15827|S18/S21]]
| ([[22/17]])/([[20/19]])<sup>5</sup>
| [[27237089/27200000]]
| 2.5.11.17.19
|-
| ([[176000/175959|S20/S21]])<sup>2</sup> * [[58121/58080|S19/S22]]
| ([[23/18]])/([[21/20]])<sup>5</sup>
| [[36800000/36756909]]
| 2.3.5.7.23
|-
| ([[213003/212960|S21/S22]])<sup>2</sup> * [[70400/70357|S20/S23]]
| ([[24/19]])/([[22/21]])<sup>5</sup>
| [[12252303/12239876]]
| 2.3.7.11.19
|-
| ([[85184/85169|S22/S23]])<sup>2</sup> * [[5635/5632|S21/S24]]
| ([[5/4]])/([[23/22]])<sup>5</sup>
| [[6442040/6436343]]
| 2.5.11.23
|-
| ([[304175/304128|S23/S24]])<sup>2</sup> * [[100672/100625|S22/S25]]
| ([[26/21]])/([[24/23]])<sup>5</sup>
| [[83672459/83607552]]
| 2.3.7.13.23
|-
| ([[359424/359375|S24/S25]])<sup>2</sup> * [[119025/118976|S23/S26]]
| ([[27/22]])/([[25/24]])<sup>5</sup>
| [[107495424/107421875]]
| 2.3.5.11
|-
| ([[140625/140608|S25/S26]])<sup>2</sup> * [[46592/46575|S24/S27]]
| ([[28/23]])/([[26/25]])<sup>5</sup>
| [[68359375/68317912]]
| 2.5.7.13.23
|-
| ([[492128/492075|S26/S27]])<sup>2</sup> * [[163125/163072|S25/S28]]
| ([[29/24]])/([[27/26]])<sup>5</sup>
| [[43069988/43046721]]
| 2.3.13.29
|-
| ([[570807/570752|S27/S28]])<sup>2</sup> * [[37856/37845|S26/S29]]
| ([[6/5]])/([[28/27]])<sup>5</sup>
| [[43046721/43025920]]
| 2.3.5.7
|-
| ([[219520/219501|S28/S29]])<sup>2</sup> * [[72819/72800|S27/S30]]
| ([[31/26]])/([[29/28]])<sup>5</sup>
| [[266760704/266644937]]
| 2.7.13.29.31
|-
| ([[756059/756000|S29/S30]])<sup>2</sup> * [[250880/250821|S28/S31]]
| ([[32/27]])/([[30/29]])<sup>5</sup>
| [[20511149/20503125]]
| 3.5.29
|-
| ([[864000/863939|S30/S31]])<sup>2</sup> * [[286781/286720|S29/S32]]
| ([[33/28]])/([[31/30]])<sup>5</sup>
| [[200475000/200404057]]
| 2.3.5.7.11.31
|-
| ([[327701/327680|S31/S32]])<sup>2</sup> * [[108800/108779|S30/S33]]
| ([[34/29]])/([[32/31]])<sup>5</sup>
| [[486695567/486539264]]
| 2.17.29.31
|-
| ([[1114112/1114047|S32/S33]])<sup>2</sup> * [[73997/73984|S31/S34]]
| ([[7/6]])/([[33/32]])<sup>5</sup>
| [[117440512/117406179]]
| 2.3.7.11
|}


== 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) ==
Retrieved from "https://en.xen.wiki/w/S-expression"