S-expression: Difference between revisions

Line 2,412:

=== Significance ===

# Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2:''k''-1:''k'':''k''+1:''k''+2:''k''+3}}.

# Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2 : ''k''-1 : ''k'' : ''k''+1 : ''k''+2 : ''k''+3.}}

# Interestingly, while three-particulars are about equidistance but (currently) have no (clean) known splitting property (due to the three intervals made equidistant not obviously composing to some other significant interval), pentaparticulars are in a sense symmetric to this and related algebraically, by instead being specifically about splitting.

# They are obviously implied by any system that equates S''k'' with S(''k'' + 1) and S(''k'' - 1) with S(''k'' + 2) simultaneously, so are not as uncommon as might be guessed, but tend to be accurate equivalences, and 5 is a rather specific number of parts to divide an interval into, even if obviously the most natural for an interval whose numerator is 5 more than the denominator (up to simplification).

=== Derivation ===

@@ Line 2,412: / Line 2,412: @@
 === Significance ===
-# Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2:''k''-1:''k'':''k''+1:''k''+2:''k''+3}}.
+# Pentaparticulars represent the obvious way of splitting an interval (''k'' + 3)/(''k'' - 2) into five parts of (''k'' + 1)/''k'', so represent a generalization of ultraparticulars by observing the harmonic series chord {{nowrap|''k''-2 : ''k''-1 : ''k'' : ''k''+1 : ''k''+2 : ''k''+3.}}
 # Interestingly, while three-particulars are about equidistance but (currently) have no (clean) known splitting property (due to the three intervals made equidistant not obviously composing to some other significant interval), pentaparticulars are in a sense symmetric to this and related algebraically, by instead being specifically about splitting.
+# They are obviously implied by any system that equates S''k'' with S(''k'' + 1) and S(''k'' - 1) with S(''k'' + 2) simultaneously, so are not as uncommon as might be guessed, but tend to be accurate equivalences, and 5 is a rather specific number of parts to divide an interval into, even if obviously the most natural for an interval whose numerator is 5 more than the denominator (up to simplification).
 === Derivation ===