S-expression: Difference between revisions
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Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|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. | Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|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. | ||
== | == {{nowrap|S''k''*S(''k'' + 1)}} (triangle-particulars) == | ||
=== Significance === | === Significance === | ||
1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].) | 1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].) | ||
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2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form, speaking of which... | 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 | 3. Every triangle-particular is the difference between two nearly-adjacent superparticular intervals {{nowrap|''k'' + 2|''k'' + 1}} and {{nowrap|''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.) | 4. Tempering any two consecutive square-particulars S''k'' and {{nowrap|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'' | 5. If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because: | ||
: | : If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} > {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} < {{sfrac|''k'' + 2|''k'' + 1}}}}. | ||
: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*...*S(k + n | : (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*...*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].) | ||
=== Meaning === | === Meaning === | ||
Notice that if we equate | Notice that if we equate {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}} (by [[tempering out]] their difference), then multiply both sides by {{sfrac|''k'' + 1|''k''}}, we have: | ||
( | <math>\left(\frac{k + 2}{k + 1}\right)\left(\frac{k + 1}{k}\right) = \left(\frac{k + 1}{k}\right)\left(\frac{k}{k - 1}\right)</math> | ||
which simplifies to: | |||
<math>\frac{k + 2}{k} = \frac{k + 1}{k - 1}</math>. | |||
This means that if we temper: <math> {\rm S}k \cdot {\rm S}(k+1) \large = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)}</math> | This means that if we temper: <math> {\rm S}k \cdot {\rm S}(k+1) \large = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)}</math> | ||
...then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and S(''k'' + 1) individually unless other considerations seem to force your hand. | ...then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and {{nowrap|S(''k'' + 1)}} individually unless other considerations seem to force your hand. | ||
=== Short proof of the superparticularity of triangle-particulars === | === Short proof of the superparticularity of triangle-particulars === | ||
S | <math>S(k)*S(k + 1) = \frac{\frac{k}{k - 1}}{\frac{k + 2}{k + 1}} = \frac{k(k + 1)}{(k - 1)(k + 2)} = \frac{k^2 + k}{k^2 + k - 2}.</math> | ||
Then notice that ''k''<sup>2</sup> + ''k'' is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals/commas are [[triangular number]]s! (Hence the alternative name "[[triangle-particular]]".) | Then notice that {{nowrap|''k''<sup>2</sup> + ''k''}} is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals/commas are [[triangular number]]s! (Hence the alternative name "[[triangle-particular]]".) | ||
=== Table of triangle-particulars === | === Table of triangle-particulars === | ||
For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason. | For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason. | ||
Below is a table of all [[31-limit]] triangle-particulars: | Below is a table of all [[31-limit]] triangle-particulars:<ref group="note">after 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to 'break' as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.</ref> | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
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| 31 | | 31 | ||
|} | |} | ||
== S(k − 1)*Sk*S(k + 1) (1/3-square-particulars) == | == S(k − 1)*Sk*S(k + 1) (1/3-square-particulars) == | ||