S-expression: Difference between revisions
rename to avoid confusion/conflict with "smonzo" |
|||
| Line 868: | Line 868: | ||
(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't.) | (Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't.) | ||
== Using S- | == Using S-factorizations to understand the significance of S-expressions == | ||
This section deals with the forms of the main 5 infinite comma families as expressed in terms of nearby harmonics in the harmonic series and as related to square-particulars; note that this uses a mathematical notation of [a, b, c, ...]^[x, y, z, ...] to denote a^x * b^y * c^z * ... | This section deals with the forms of the main 5 infinite comma families as expressed in terms of nearby harmonics in the harmonic series and as related to square-particulars; note that this uses a mathematical notation of [a, b, c, ...]^[x, y, z, ...] to denote a^x * b^y * c^z * ... | ||
| Line 914: | Line 914: | ||
</pre> | </pre> | ||
This technique will be called "'''S- | 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 [[square superparticular#Abstraction|the abstraction section of this page]]. | The generalisation of this method using commutative group theory is discussed in [[square superparticular#Abstraction|the abstraction section of this page]]. | ||
| Line 920: | Line 922: | ||
== Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) == | == Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) == | ||
Using the clarity of S- | Using the clarity of 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 1,492: | Line 1,494: | ||
: 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. | ||
; S- | ; S-factorization | ||
: An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, | : An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions. | ||
: For example: S''k'' = [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] because S''k'' = (''k''-1)<sup>-1</sup>''k''<sup>2</sup>(''k''+1)<sup>-1</sup>. | : For example: S''k'' = [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] because S''k'' = (''k''-1)<sup>-1</sup>''k''<sup>2</sup>(''k''+1)<sup>-1</sup>. | ||
| Line 1,603: | Line 1,605: | ||
</math> | </math> | ||
Then (the analogues of) S- | Then (the analogues of) S-factorizations correspond to the ''exponents'' of the generators, such that: | ||
<math> | <math> | ||
| Line 1,613: | Line 1,615: | ||
This completes the analogy. What this means is: | This completes the analogy. What this means is: | ||
Every infinite comma family defined in terms of an S-expression will have an infinite number of analogues, because of the maths of S- | Every infinite comma family defined in terms of an S-expression will have an infinite number of analogues, because of the maths of S-factorizations continuing to work as expected. | ||
The meanings of these analogues are up to us to interpret, however. This brings us to applications, which we will examine next, with a focus on the musical ones: | The meanings of these analogues are up to us to interpret, however. This brings us to applications, which we will examine next, with a focus on the musical ones: | ||
| Line 1,632: | Line 1,634: | ||
=== h<sub>k</sub> = k === | === h<sub>k</sub> = k === | ||
The trivial example, equal to normal S-expressions and S- | The trivial example, equal to normal S-expressions and S-factorizations, before abstraction. | ||
=== h<sub>k</sub> = 2k + 1 === | === h<sub>k</sub> = 2k + 1 === | ||
An analogy of S-expressions and S- | An analogy of S-expressions and S-factorizations for [[EDT]]s which can be used as a corresponding [[RTT]] tool for when we only want to focus on the arithmetic of odds, using [[3/1]] as the new [[equave]]. | ||
A suggestion is to use the notation So''k'', if this is not unambiguous, with the letter "o" standing for "odd". Thus: | A suggestion is to use the notation So''k'', if this is not unambiguous, with the letter "o" standing for "odd". Thus: | ||
| Line 1,646: | Line 1,648: | ||
=== h<sub>k</sub> = (k + 1)/k === | === h<sub>k</sub> = (k + 1)/k === | ||
An analogy of S-expressions and S- | An analogy of S-expressions and S-factorizations aiming at deeply faithful modelling of the harmonic series through modelling distances between superparticulars accurately. | ||
A suggestion is to use the notation Ss''k'', if this is not unambiguous, with the letter "s" standing for "superparticular" or "super" generally. | A suggestion is to use the notation Ss''k'', if this is not unambiguous, with the letter "s" standing for "superparticular" or "super" generally. | ||