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~~.)

The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 71), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.

Note from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:

This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page, with S12/S13 and S13/S14 expected to have their own pages soon. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.

Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:

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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.)
+The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 71), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.
-Note from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:
+This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page, with S12/S13 and S13/S14 expected to have their own pages soon. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.
+Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:
 <math>\frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3}</math>

S-expression: Difference between revisions

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