S-expression: Difference between revisions

Below is a table of [[23-limit]] square-particulars:

! Comma

| S6

| S33 = S35*S99

|}

 

=== Alternatives to tempering square-particulars ===

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following:

 

<math>\large {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3}</math>

 

This means you can read the ''k'' and ''k'' + 1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).

 

<nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S''k''/S(''k'' - 1) and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.

 

Furthermore, defining another sequence of commas with [[semiparticular|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.

 

=== S-expressions ===

An S-expression is any product, or ratio of products, of square superparticulars S''k''.

 

== Sk*S(k + 1) (triangle-particulars) ==

1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].)

 

2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form, speaking of which...

 

3. Every triangle-particular is the difference between two nearly-adjacent superparticular intervals (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1).

4. Tempering any two consecutive square-particulars S''k'' and S(''k'' + 1) implies tempering a triangle-particular, so these are common commas. (See also: [[lopsided comma]]s.)

5. If we temper S''k'' * S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of ''k''/(''k'' - 1), (''k'' + 1)/''k'' and (''k'' + 2)/(''k'' + 1) ''must'' be mapped inconsistently, because:

: if (''k'' + 1)/''k'' is mapped above (''k'' + 2)/(''k'' + 1) ~ k/(k-1) we have (''k'' + 1)/''k'' > ''k''/(''k'' - 1) and if it is mapped below we have (''k'' + 1)/''k'' < (''k'' + 2)/(''k'' + 1).

: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars)]].)

! Comma

|}

 

Also included are some higher-up [[23-limit]] 1/2-square-particulars (as many of the prior intervals were quite large):

{| class="wikitable center-all

! S-expression

Below is a table of such commas in the 41-prime-limited 199-odd-limit:

! Comma

| [[1365/1364]]

|-

| S66*S67*S68

| ([[66/65]])/([[69/68]])

| [[1496/1495]]

|-

| S75*S76*S77

| ([[75/74]])/([[78/77]])

| [[1925/1924]]

| S78*S79*S80

| ([[78/77]])/([[81/80]])

| [[2080/2079]]

| S82*S83*S84

| ([[82/81]])/([[85/84]])

| [[2296/2295]]

|-

| S96*S97*S98

| ([[96/95]])/([[99/98]])

| [[3136/3135]]

| S112*S113*S114

| ([[112/111]])/([[115/114]])

| [[4256/4255]]

| S117*S118*S119

| ([[117/116]])/([[120/119]])

| [[4641/4640]]

| S121*S122*S123

| ([[121/120]])/([[124/123]])

| [[4961/4960]]

| S133*S134*S135

| ([[133/132]])/([[136/135]])

| [[5985/5984]]

| S145*S146*S147

| ([[145/144]])/([[148/147]])

| [[7105/7104]]

| S153*S154*S155

| ([[153/152]])/([[156/155]])

| [[7905/7904]]

| S162*S163*S164

| ([[162/161]])/([[165/164]])

| [[8856/8855]]

| S187*S188*S189

| ([[187/186]])/([[190/189]])

| [[11781/11780]]

! Comma

! Comma

! Comma