Interval variety: Difference between revisions
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== Open questions == | == Open questions == | ||
* Why are (abstractly) SV4 circular words seemingly so rare? | * Why are (abstractly) SV4 circular words seemingly so rare? | ||
* Related is the following conjecture: For a sufficiently long ternary circular 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. | * Related is the following conjecture: For a sufficiently long primitive ternary circular 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:Scale]] | ||
[[Category:Interval]] | [[Category:Interval]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Articles with open problems]] | [[Category:Articles with open problems]] | ||
Revision as of 23:08, 4 February 2024
The interval variety of an interval class in a scale is the number of different interval qualities available for that interval class. For example, the interval class "fifth" in the diatonic scale has interval variety 2, because there are two sizes of fifths in that scale: 6 perfect fifths and 1 diminished fifth.
The concept of interval variety can be applied to all interval classes of a scale at once. Here are some such properties:
- Highest interval variety (see also maximum variety)
- Mean interval variety
- Median interval variety
- Lowest interval variety
In addition, strict variety scales, such as single-period MOS scales and trivalent scales, have the same interval variety for all interval classes (except the unison, which always trivially has interval variety 1).
Note: A standard academic counterpart to the xen term variety is the abelian complexity function of a word: a function ρab : N -> N where ρab(n) is the number of distinct "sizes" that length-n subwords can have in a word.
Open questions
- Why are (abstractly) SV4 circular words seemingly so rare?
- Related is the following conjecture: For a sufficiently long primitive ternary circular 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.