S-expression: Difference between revisions

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(k+2)/(k+1) * (k+1)/k = (k+1)/k * k/(k-1)

Which is to say that if we temper S''k''*S(''k'' + 1) = (k/(k-1))/((k+1)/k) * ((k+1)/k)/((k+2)/(k+1)) = (k/(k-1))/((k+2)/(k+1)) 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. Another reason commas of this form are of note is they are always [[superparticular]]. It is also an interesting consequence that if we temper S''k''*S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of k/(k-1), (k+1)/k and (k+2)/(k+1) ''must'' be mapped inconsistently, because if (k+1)/k is mapped above (k+2)/(k+1) ~ k/(k-1) we have (k+1)/k > k/(k-1) and if it is mapped below we have (k+1)/k < (k+2)/(k+1). A short proof of the superparticularity of S''k''*S(''k'' + 1) is as follows:

Which is to say that if we temper S''k''*S(''k'' + 1) = (k/(k-1))/((k+1)/k) * ((k+1)/k)/((k+2)/(k+1)) = (k/(k-1))/((k+2)/(k+1)) 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. Another reason commas of this form are of note is they are always [[superparticular]]. It is also an interesting consequence that if we temper S''k''*S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of k/(k-1), (k+1)/k and (k+2)/(k+1) ''must'' be mapped inconsistently, because if (k+1)/k is mapped above (k+2)/(k+1) ~ k/(k-1) we have (k+1)/k > k/(k-1) and if it is mapped below we have (k+1)/k < (k+2)/(k+1). (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars)]].) A short proof of the superparticularity of S''k''*S(''k'' + 1) is as follows:

S''k''*S(''k'' + 1) = (k/(k-1))/((k+2)/(k+1)) = (k(k+1))/((k-1)(k+2)) = (k2 + k)/(k2 + k - 2)

Then notice that k2 + 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 "~~semi~~-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 commas (or intervals) are [[triangular number]]s!

Then notice that k2 + 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 commas (or intervals) are [[triangular number]]s!

For completeness, all the commas of this form are included, because these "commas" (intervals rather) have structural importance for JI, and for the possibility of consistency of mappings for the above reason.

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|}

Also included are some higher-up [[23-limit]] ~~triangle~~-particulars (as many of the prior intervals were quite large):

Also included are some higher-up [[23-limit]] 1/2-square-particulars (as many of the prior intervals were quite large):

{| class="wikitable center-all

|-

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(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, and maps the corresponding intervals consistently.)

== S''k''*S(''k'' + 1)*...*S(''k'' + ''n'' - 1) (1/n-square-particulars) ==

1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1) have implications for the [[consistency]] of the (''k'' + ''n'')-[[odd-limit]] when tempered. Specifically:

If a temperament tempers a 1/''n''-square-particular of the form S''k''*S(''k''+1)*...*S(''k''+''n''-1), it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', S(''k''+1), ..., S(''k''+''n''-1). If it does not, it is ''necessarily'' inconsistent in the (''k'' + ''n'')-odd-limit. A proof is as follows:

Consider the following sequence of superparticular intervals, all of which in the (''k'' + ''n'')-odd-limit:

(''k'' + ''n'')/(''k'' + ''n'' - 1), (''k'' + ''n'' - 1)/(''k'' + ''n'' - 2), ..., (''k'' + 1)/''k'', ''k''/(''k'' - 1)

Because of tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1), we require that (''k'' + ''n'')/(''k'' + ''n'' - 1) = ''k''/(''k'' - 1) consistently. Therefore, if any superparticular ''x''/(''x'' - 1) imbetween (meaning ''k'' + ''n'' > x > ''k'') is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:

mapping((''k'' + ''n'')/(''k'' + ''n'' - 1)) > mapping(''x''/(''x'' - 1))

mapping(''k''/(''k'' - 1)) < mapping(''x''/(''x'' - 1))

Therefore any superparticular interval ''x''/(''x'' - 1) between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the (''k'' + ''n'')-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1) but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the (''k'' - 1)-odd-limit.

== S''k''/S(''k'' + 1) (ultraparticulars) ==

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; Triangle-particular

: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular.

: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.

: These are of the form (''k''2 + k)/(''k''2 + k - 2). (This always simplifies to a superparticular.)

; 1/n-square-particular

: A comma which is the product of ''n'' consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.

: These are of the form S''a''*S(''a''+1)*...*S''b'' = (''a''/(''a'' - 1))/((''b'' + 1)/''b'') = ''ab''/((''a'' - 1)(''b'' + 1)).

: Replacing/substituting ''a'' with ''k'' and ''b'' with ''k'' + ''n'' - 1 gives us an equivalent expression that includes the number of square-particulars ''n'':

: S''k''*S(''k''+1)*...*S(''k''+''n''-1) = (''k''/(''k'' - 1))/((''k'' + ''n'')/(''k'' + ''n'' - 1)) = ''k''(''k'' + ''n'' - 1)/((''k'' - 1)(''k'' + ''n''))

: For ''b'' = ''a'' + 1 these can also be called triangle-particulars, in which case they are always superparticular.

: These have implications for whether consistency in the (''n''+''k'')=(''b''+1)-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars)|section on 1/n-square-particulars]].

; Odd-particular

@@ Line 24: / Line 24: @@
 (k+2)/(k+1) * (k+1)/k = (k+1)/k * k/(k-1)
-Which is to say that if we temper S''k''*S(''k'' + 1) = (k/(k-1))/((k+1)/k) * ((k+1)/k)/((k+2)/(k+1)) = (k/(k-1))/((k+2)/(k+1)) 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. Another reason commas of this form are of note is they are always [[superparticular]]. It is also an interesting consequence that if we temper S''k''*S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of k/(k-1), (k+1)/k and (k+2)/(k+1) ''must'' be mapped inconsistently, because if (k+1)/k is mapped above (k+2)/(k+1) ~ k/(k-1) we have (k+1)/k > k/(k-1) and if it is mapped below we have (k+1)/k < (k+2)/(k+1). A short proof of the superparticularity of S''k''*S(''k'' + 1) is as follows:
+Which is to say that if we temper S''k''*S(''k'' + 1) = (k/(k-1))/((k+1)/k) * ((k+1)/k)/((k+2)/(k+1)) = (k/(k-1))/((k+2)/(k+1)) 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. Another reason commas of this form are of note is they are always [[superparticular]]. It is also an interesting consequence that if we temper S''k''*S(''k'' + 1) but not S''k'' or S(''k'' + 1), then one or more intervals of k/(k-1), (k+1)/k and (k+2)/(k+1) ''must'' be mapped inconsistently, because if (k+1)/k is mapped above (k+2)/(k+1) ~ k/(k-1) we have (k+1)/k > k/(k-1) and if it is mapped below we have (k+1)/k < (k+2)/(k+1). (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars)]].) A short proof of the superparticularity of S''k''*S(''k'' + 1) is as follows:
 S''k''*S(''k'' + 1) = (k/(k-1))/((k+2)/(k+1)) = (k(k+1))/((k-1)(k+2)) = (k<sup>2</sup> + k)/(k<sup>2</sup> + k - 2)
-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 "semi-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 commas (or intervals) are [[triangular number]]s!
+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 commas (or intervals) are [[triangular number]]s!
 For completeness, all the commas of this form are included, because these "commas" (intervals rather) have structural importance for JI, and for the possibility of consistency of mappings for the above reason.
@@ Line 139: / Line 139: @@
 |}
-Also included are some higher-up [[23-limit]] triangle-particulars (as many of the prior intervals were quite large):
+Also included are some higher-up [[23-limit]] 1/2-square-particulars (as many of the prior intervals were quite large):
 {| class="wikitable center-all
 |-
@@ Line 184: / Line 184: @@
 (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, and maps the corresponding intervals consistently.)
+== S''k''*S(''k'' + 1)*...*S(''k'' + ''n'' - 1) (1/n-square-particulars) ==
+/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1) have implications for the [[consistency]] of the (''k'' + ''n'')-[[odd-limit]] when tempered. Specifically:
+If a temperament tempers a 1/''n''-square-particular of the form S''k''*S(''k''+1)*...*S(''k''+''n''-1), it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', S(''k''+1), ..., S(''k''+''n''-1). If it does not, it is ''necessarily'' inconsistent in the (''k'' + ''n'')-odd-limit. A proof is as follows:
+Consider the following sequence of superparticular intervals, all of which in the (''k'' + ''n'')-odd-limit:
+(''k'' + ''n'')/(''k'' + ''n'' - 1), (''k'' + ''n'' - 1)/(''k'' + ''n'' - 2), ..., (''k'' + 1)/''k'', ''k''/(''k'' - 1)
+Because of tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1), we require that (''k'' + ''n'')/(''k'' + ''n'' - 1) = ''k''/(''k'' - 1) consistently. Therefore, if any superparticular ''x''/(''x'' - 1) imbetween (meaning ''k'' + ''n'' > x > ''k'') is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:
+mapping((''k'' + ''n'')/(''k'' + ''n'' - 1)) > mapping(''x''/(''x'' - 1))
+mapping(''k''/(''k'' - 1)) < mapping(''x''/(''x'' - 1))
+Therefore any superparticular interval ''x''/(''x'' - 1) between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the (''k'' + ''n'')-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1) but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the (''k'' - 1)-odd-limit.
 == S''k''/S(''k'' + 1) (ultraparticulars) ==
@@ Line 456: / Line 473: @@
 ; Triangle-particular
-: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular.
+: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.
 : These are of the form (''k''<sup>2</sup> + k)/(''k''<sup>2</sup> + k - 2). (This always simplifies to a superparticular.)
+; 1/n-square-particular
+: A comma which is the product of ''n'' consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.
+: These are of the form S''a''*S(''a''+1)*...*S''b'' = (''a''/(''a'' - 1))/((''b'' + 1)/''b'') = ''ab''/((''a'' - 1)(''b'' + 1)).
+: Replacing/substituting ''a'' with ''k'' and ''b'' with ''k'' + ''n'' - 1 gives us an equivalent expression that includes the number of square-particulars ''n'':
+: S''k''*S(''k''+1)*...*S(''k''+''n''-1) = (''k''/(''k'' - 1))/((''k'' + ''n'')/(''k'' + ''n'' - 1)) = ''k''(''k'' + ''n'' - 1)/((''k'' - 1)(''k'' + ''n''))
+: For ''b'' = ''a'' + 1 these can also be called triangle-particulars, in which case they are always superparticular.
+: These have implications for whether consistency in the (''n''+''k'')=(''b''+1)-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk*S(k + 1)*...*S(k + n - 1) (1/n-square-particulars)|section on 1/n-square-particulars]].
 ; Odd-particular