|
|
| (4 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| | {{breadcrumb}} |
| | |
| == Using S-factorizations to understand the significance of S-expressions == | | == Using S-factorizations to understand the significance of S-expressions == |
| This section deals with the forms of the main infinite comma families listed on this page* 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 * ...
| | See [[S-expression#Using S-factorizations to understand the significance of S-expressions]]. |
| | |
| <nowiki>*</nowiki> except [[lopsided comma]]s which are dealt with in their respective subsection of the page (which is conveniently just under this section, partly as it demonstrates a less easy application of this technique).
| |
| | |
| 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 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.
| |
| | |
| <pre>
| |
| Sk = [k-1, k, k+1]^[-1, 2, -1]
| |
| </pre>
| |
| <pre>
| |
| Sk * S(k+1) = [k-1, k, k+1, k+2]^[-1, 1, 1, -1]
| |
| = [k-1, k, k+1(, k+2)]^[-1, 2, -1(, 0)] * [(k-1,) k, k+1, k+2]^[(0,) -1, 2, -1]
| |
| </pre>
| |
| <pre>
| |
| S(k-1) * Sk * S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 1, 0, 1, -1]
| |
| = ( (k-1)/(k-2) )( k/(k-1) ) * ( k/(k-1) )/( (k+1)/k ) * ( (k+1)/k )/( (k+2)/(k+1) )
| |
| = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) = ( (k-1)(k+1) )/( (k-2)(k+2) )
| |
| | |
| k-2 k-1 k k+1 k+2
| |
| -1 2 -1 0 0
| |
| 0 -1 2 -1 0
| |
| 0 0 -1 2 -1
| |
| ========================
| |
| -1 1 0 1 -1
| |
| </pre>
| |
| <pre>
| |
| Sk / S(k+1) = [k-1, k, k+1, k+2]^[-1, 3, -3, 1]
| |
| = [k-1, k, k+1]^[-1, 2, -1] * [k, k+1, k+2]^[1, -2, 1]
| |
| = (k+2)/(k-1) * ( k/(k+1) )^3 = (k+2)/(k-1) / ((k+1)/k)^3
| |
| </pre>
| |
| <pre>
| |
| S(k-1) / S(k+1) = [k-2, k-1, k, k+1, k+2]^[-1, 2, 0, -2, 1]
| |
| = [k-2, k-1, k]^[-1, 2, -1] * [k, k+1, k+2]^[ 1, -2, 1]
| |
| = [k-2, k-1, k]^[-1, 2, -1] / [k, k+1, k+2]^[-1, 2, -1]
| |
| = (k+2)/(k-2) * ((k-1)/(k+1))^2 = (k+2)/(k-2) / ((k+1)/(k-1))^2
| |
| | |
| k-2 k-1 k k+1 k+2
| |
| -1 2 -1 0 0
| |
| 0 0 1 -2 1
| |
| ========================
| |
| -1 2 0 -2 1
| |
| </pre>
| |
| | |
| 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]].
| |
| | |
| === Using S-factorizations to show a useful equivalence/redundancy of S-expressions ===
| |
| Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:
| |
| | |
| ==== The general S-expression equivalence ====
| |
| Consider:
| |
| <pre>
| |
| Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:
| |
| S(2k-1) * S(2k) * S(2k) * S(2k+1)
| |
| = [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(
| |
| [-1, 2, -1]
| |
| + [-2, 4, -2]
| |
| + [-1, 2, -1]
| |
| = [-1, 0, 2, 0, -1] )
| |
| </pre>
| |
| From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':
| |
| | |
| <math>
| |
| \large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
| |
| </math>
| |
| | |
| ...where we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of [[regular temperament]]s using [[S-expression]]s.
| |
| | |
| For tuning theory only integer ''k'' > 1 is of relevance. Technically, rational ''k'' other than 1 correspond to rational commas too; the most relevant case for tuning theory is that half-integer ''k'' work as an alternative notation for [[odd-particular]]s, though for intuitively understanding the notation, the method described in [[#Abstraction]] may be recommendable as having (in a mathematical sense) exact analogues for every infinite family of commas defined in terms of an analogue of an S-expression, for which the most musically fruitful example is O''k'' = (''k'' / (''k'' - 2))/((''k'' + 2) / ''k'') for odd ''k'' as relevant to [[no-twos subgroup temperaments]].
| |
|
| |
|
| == Mathematical derivations == | | == Mathematical derivations == |
| Line 252: |
Line 179: |
|
| |
|
| While the question of where it is most appropriate and accurate to equate ultraparticulars is beyond the scope of this section, it nonetheless shows that the 2D comma family Ss''a''/Ss''b'' has utility. | | While the question of where it is most appropriate and accurate to equate ultraparticulars is beyond the scope of this section, it nonetheless shows that the 2D comma family Ss''a''/Ss''b'' has utility. |
| | |
| | [[Category:Elementary math]] |
| | [[Category:Pages with proofs]] |