S-expression: Difference between revisions
→Sk/S(k + 2) (semiparticulars): add motivational examples & significance section |
m →Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas): section, add significance section |
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== 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) == | ||
=== Significance === | |||
1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so. | |||
2. If a comma (such as the diaschisma, [[2048/2025]]), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering the square-particulars composing it (such as [[256/255|S16]] and [[289/288|S17]] in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise. | |||
3. It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form. | |||
=== Derivation of equivalence relation === | |||
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: | 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: | ||
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S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2]. | S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2]. | ||
=== Tables === | |||
Below is two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]]. | Below is two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]]. | ||
=== Top-heavy lopsided commas === | ==== Top-heavy lopsided commas ==== | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
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|} | |} | ||
=== Bottom-heavy lopsided commas === | ==== Bottom-heavy lopsided commas ==== | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|- | |- | ||