Interval variety: Difference between revisions

Line 12:

== Facts ==

=== Abstractly SV4 scale patterns ===

* There exist no 6-letter abstractly SV4 scale patterns.

Abstractly SV4 scale patterns (patterns that are SV4 for any choice of distinct cent values for the four steps):

* ~~There exist 3~~ 7~~-letter abstractly SV4 scale patterns~~: '''0123210''', '''0102013''', and '''3102010'''. The last two patterns are a chiral pair.

* 6 notes: none

* ~~There exist 2~~ 8~~-letter abstractly SV4 scale patterns~~: '''00100232''' and '''01212103'''.

* 7 notes: '''0123210''', '''0102013''', and '''3102010''' (The last two patterns are a chiral pair)

* ~~There exist no~~ 9~~-letter abstractly SV4 scale patterns.~~

* 8 notes: '''00100232''' and '''01212103'''

* 9 notes: none

== Open questions ==

* Why are (abstractly) SV4 ~~circular words~~ seemingly so rare?

* Why are (abstractly) SV4 scale patterns seemingly so rare?

* Related may be the following conjecture: For a sufficiently long ternary linear word, there exists ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' − 1)-steps also has at least 3 sizes.

** Conjecture: For sufficiently large ''n'', there are no abstractly SV4 [[circular word]]s of length ''n''.

** Related may be the following conjecture: For a sufficiently long [[arity|ternary]] linear word, there exists ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' − 1)-steps also has at least 3 sizes.

[[Category:Scale]]

[[Category:Interval]]

[[Category:Terms]]

[[Category:Articles with open problems]]

@@ Line 12: / Line 12: @@
 == Facts ==
 === Abstractly SV4 scale patterns ===
-* There exist no 6-letter abstractly SV4 scale patterns.
+Abstractly SV4 scale patterns (patterns that are SV4 for any choice of distinct cent values for the four steps):
-* There exist 3 7-letter abstractly SV4 scale patterns: '''0123210''', '''0102013''', and '''3102010'''. The last two patterns are a chiral pair.
+* 6 notes: none
-* There exist 2 8-letter abstractly SV4 scale patterns: '''00100232''' and '''01212103'''.
+* 7 notes: '''0123210''', '''0102013''', and '''3102010''' (The last two patterns are a chiral pair)
-* There exist no 9-letter abstractly SV4 scale patterns.
+* 8 notes: '''00100232''' and '''01212103'''
+* 9 notes: none
 == Open questions ==
-* Why are (abstractly) SV4 circular words seemingly so rare?
+* Why are (abstractly) SV4 scale patterns seemingly so rare?
-* Related may be the following conjecture: For a sufficiently long ternary linear word, there exists ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' &minus; 1)-steps also has at least 3 sizes.
+** Conjecture: For sufficiently large ''n'', there are no abstractly SV4 [[circular word]]s of length ''n''.
+** Related may be the following conjecture: For a sufficiently long [[arity|ternary]] linear word, there exists ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' &minus; 1)-steps also has at least 3 sizes.
 [[Category:Scale]]
 [[Category:Interval]]
 [[Category:Terms]]
 [[Category:Articles with open problems]]