S-expression: Difference between revisions

Line 1,866:

: 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

; 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)).

Line 1,884:

; 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) and (4''k'' + 3)/(4''k'' - 1).

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

; ''n''-odd-particular

: An interval/comma between two coprime harmonics ''n'' apart. 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 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).

; Ultraparticular

@@ Line 1,866: / Line 1,866: @@
 : 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
+; 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)).
@@ Line 1,884: / Line 1,884: @@
 ; 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) and (4''k'' + 3)/(4''k'' - 1).
+: These are of the form (4''k'' + 1)/(4''k'' - 3) or (4''k'' + 3)/(4''k'' - 1).
+; ''n''-odd-particular
+: An interval/comma between two coprime harmonics ''n'' apart. 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 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).
 ; Ultraparticular