S-expression: Difference between revisions
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(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-monzos 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 * ... | |||
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 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: | ||
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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: | 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. | [''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. | ||
<pre> | <pre> | ||
Sk = [k-1, k, k+1]^[-1, 2, -1] | Sk = [k-1, k, k+1]^[-1, 2, -1] | ||
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-1 2 0 -2 1 | -1 2 0 -2 1 | ||
</pre> | </pre> | ||
This technique will be called "'''S-monzos'''", as it is a form analogous to monzos that is uniquely suited for interpreting the relationships described by '''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]]. | ||
== Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) == | |||
Using the clarity of S-monzos, we can show the interval relations implicated by these two new "lopsided" forms: | |||
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''/(''k''-1))<sup>2</sup> / ((''k''+2)/''k'') through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1]. | |||
S(''k''-1) * S''k''<sup>2</sup> = [''k''-2, ''k''-1, ''k'', ''k''+1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies: | |||
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''+1))<sup>2</sup> / ((''k''-2)/''k'') through [-1, 0, 3, -2] = [0, 0, 2, -2] - [1, 0, -1, 0]. | |||
A computer-generated table of both of these forms of commas is currently pending creation and will be added soon. | |||
== Equivalent S-expressions == | == Equivalent S-expressions == | ||