S-expression: Difference between revisions

Line 2,660:

; Superparticular

: The interval/comma between two consecutive harmonics. See [[superparticular]].

: These are of the form (''k'' + 1)/''k''.

: These are of the form {{sfrac|''k'' + 1|''k''}}.

; Square-particular

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

: These are of the form ''k''2/(''k''2 - 1) = S''k''.

: These are of the form {{nowrap|{{sfrac|''k''2|''k''2 − 1}} {{=}} S''k''}}.

; Triangle-particular

: 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.)

: These are of the form {{sfrac|''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)).

: These are of the form {{nowrap|S''a'' * S(''a'' + 1) * … * S''b'' {{=}} {{sfrac|{{sfrac|''a''|''a'' − 1}}|{{sfrac|''b'' + 1|''b''}}}}}} {{nowrap|{{=}} {{sfrac|''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''))

: {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1) {{=}} {{sfrac|{{sfrac|''k''|''k'' − 1}}|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}}}}} {{nowrap|{{=}} {{sfrac|''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.

: For {{nowrap|''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]].

: These have implications for whether consistency in the {{nowrap|(''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

: An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.

: These are of the form (2''k'' + 1~~)/(~~2''k'' - 1).

: These are of the form {{sfrac|2''k'' + 1|2''k'' − 1}}.

; Throdd-particular

: An interval/comma between two harmonics 3 apart which is not superparticular.

: These are of the form (3''k'' + 1~~)/(~~3''k'' - 2) or (3''k'' + 2~~)/(~~3''k'' - 1).

: These are of the form {{sfrac|3''k'' + 1|3''k'' − 2}} or {{sfrac|3''k'' + 2|3''k'' − 1}}.

; Quodd-particular

: An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.

: These are of the form (4''k'' + 1~~)/(~~4''k'' - 3) or (4''k'' + 3~~)/(~~4''k'' - 1).

: These are of the form {{sfrac|4''k'' + 1|4''k'' − 3}} or {{sfrac|4''k'' + 3|4''k'' − 1}}.

; ''n''-odd-particular

: An interval/comma between two coprime harmonics ''n'' apart (also called as [[Delta-N ratio|delta-''n'' ratio]]). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.

: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form (5''k'' + 1~~)/(~~5''k'' - 4), (5''k'' + 2~~)/(~~5''k'' - 3), (5''k'' + 3~~)/(~~5''k'' - 2) or (5''k'' + 4~~)/(~~5''k'' - 1).

: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.

: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form (6''k'' + 1~~)/(~~6''k'' - 5) or (6''k'' + 5~~)/(~~6''k'' - 1).

: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.

; Ultraparticular

: An interval/comma which is the ratio of two consecutive square-particulars.

: These are of the form S''k''/S(''k'' + 1).

: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.

; Semiparticular

: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:

: These are of the form S''k''/S(''k'' + 2).

: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.

; S-expression

Line 2,708:

; S-factorization

: An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions.

: For example: S''k'' = [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] because S''k'' = (''k''-1)-1''k''2(''k''+1)-1.

: For example: {{sfrac|S''k'' {{=}} [''k'' − 1, ''k'', ''k'' + 1][−1, 2, −1]}} because {{nowrap|S''k'' {{=}} (''k'' − 1)−1''k''2(''k'' + 1)−1}}.

; S-comma

Line 2,714:

; Indirect S-comma

: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions, for example the [[schisma]] admits at least three such representations!

: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions; for example, the [[schisma]] admits at least three such representations.

== See further ==

@@ Line 2,660: / Line 2,660: @@
 ; Superparticular
 : The interval/comma between two consecutive harmonics. See [[superparticular]].
-: These are of the form (''k'' + 1)/''k''.
+: These are of the form {{sfrac|''k'' + 1|''k''}}.
 ; Square-particular
 : A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.
-: These are of the form ''k''<sup>2</sup>/(''k''<sup>2</sup> - 1) = S''k''.
+: These are of the form {{nowrap|{{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}} {{=}} S''k''}}.
 ; Triangle-particular
 : 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.)
+: These are of the form {{sfrac|''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)).
+: These are of the form {{nowrap|S''a'' * S(''a'' + 1) * … * S''b'' {{=}} {{sfrac|{{sfrac|''a''|''a'' − 1}}|{{sfrac|''b'' + 1|''b''}}}}}} {{nowrap|{{=}} {{sfrac|''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''))
+: {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1) {{=}} {{sfrac|{{sfrac|''k''|''k'' − 1}}|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}}}}} {{nowrap|{{=}} {{sfrac|''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.
+: For {{nowrap|''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]].
+: These have implications for whether consistency in the {{nowrap|(''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
 : An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.
-: These are of the form (2''k'' + 1)/(2''k'' - 1).
+: These are of the form {{sfrac|2''k'' + 1|2''k'' − 1}}.
 ; Throdd-particular
 : An interval/comma between two harmonics 3 apart which is not superparticular.
-: These are of the form (3''k'' + 1)/(3''k'' - 2) or (3''k'' + 2)/(3''k'' - 1).
+: These are of the form {{sfrac|3''k'' + 1|3''k'' − 2}} or {{sfrac|3''k'' + 2|3''k'' − 1}}.
 ; Quodd-particular
 : An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.
-: These are of the form (4''k'' + 1)/(4''k'' - 3) or (4''k'' + 3)/(4''k'' - 1).
+: These are of the form {{sfrac|4''k'' + 1|4''k'' − 3}} or {{sfrac|4''k'' + 3|4''k'' − 1}}.
 ; ''n''-odd-particular
 : An interval/comma between two coprime harmonics ''n'' apart (also called as [[Delta-N ratio|delta-''n'' ratio]]). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.
-: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form (5''k'' + 1)/(5''k'' - 4), (5''k'' + 2)/(5''k'' - 3), (5''k'' + 3)/(5''k'' - 2) or (5''k'' + 4)/(5''k'' - 1).
+: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.
-: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form (6''k'' + 1)/(6''k'' - 5) or (6''k'' + 5)/(6''k'' - 1).
+: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.
 ; Ultraparticular
 : An interval/comma which is the ratio of two consecutive square-particulars.
-: These are of the form S''k''/S(''k'' + 1).
+: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.
 ; Semiparticular
 : A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:
-: These are of the form S''k''/S(''k'' + 2).
+: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.
 ; S-expression
@@ Line 2,708: / Line 2,708: @@
 ; S-factorization
 : An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions.
-: For example: S''k'' = [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] because S''k'' = (''k''-1)<sup>-1</sup>''k''<sup>2</sup>(''k''+1)<sup>-1</sup>.
+: For example: {{sfrac|S''k'' {{=}} [''k'' − 1, ''k'', ''k'' + 1]<sup>[−1, 2, −1]</sup>}} because {{nowrap|S''k'' {{=}} (''k'' − 1)<sup>−1</sup>''k''<sup>2</sup>(''k'' + 1)<sup>−1</sup>}}.
 ; S-comma
@@ Line 2,714: / Line 2,714: @@
 ; Indirect S-comma
-: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions, for example the [[schisma]] admits at least three such representations!
+: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions; for example, the [[schisma]] admits at least three such representations.
 == See further ==