User:CompactStar/Ordinal interval notation: Difference between revisions

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'''~~Lefts and rights~~ notation''' is a ~~bisection-based~~ notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].

'''Indexed interval notation''' is a notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].

Intervals are represented by a conventional interval category with a ~~stack of lefts~~ and ~~rights (abbreviated as L~~ and ~~R) added before~~. To get the category of an interval, multiply the categories of the prime harmonics which it factors into, which are predefined as follows:

Intervals are represented by a conventional interval category and an index. The index is 1 for the simplest (with respect [[Tenney height]) interval in a category, 2 for the second-simplest, 3 for the third-simplest and so on. For example, [[6/5]] is the 1st minor 3rd and [[7/6]] is the 2nd minor 3rd. To get the category of an interval, multiply the categories of the prime harmonics which it factors into, which are predefined as follows:

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The simplest (with respect to [[Tenney height]]) interval inside a category does not use any lefts or rights (or is "central"), for example [[6/5]] for minor 3rd. The simplest interval which is flatter than the central interval is left ([[7/6]] for minor 3rd), and the simplest interval which is sharper is right ([[11/9]] for minor 3rd). Then the simplest interval which is flatter than the left is leftleft, the simplest interval between left and central is leftright , the simplest interval which is between central and right is rightleft, and the simplest interval which is sharper than right is rightright. This process of bisection with lefts/rights can be continued infinitely to name all just intervals that are in a category. Interval arithmetic is preserved (e.g. M2 * M2 is always M3), however the lefts and rights do not combine like accidentals do.

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-'''Lefts and rights notation''' is a bisection-based notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].
+'''Indexed interval notation''' is a notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].
-Intervals are represented by a conventional interval category with a stack of lefts and rights (abbreviated as L and R) added before. To get the category of an interval, multiply the categories of the prime harmonics which it factors into, which are predefined as follows:
+Intervals are represented by a conventional interval category and an index. The index is 1 for the simplest (with respect [[Tenney height]) interval in a category, 2 for the second-simplest, 3 for the third-simplest and so on. For example, [[6/5]] is the 1st minor 3rd and [[7/6]] is the 2nd minor 3rd. To get the category of an interval, multiply the categories of the prime harmonics which it factors into, which are predefined as follows:
 {|class="wikitable"
 |-
@@ Line 127: / Line 127: @@
 |G
 |}
-The simplest (with respect to [[Tenney height]]) interval inside a category does not use any lefts or rights (or is "central"), for example [[6/5]] for minor 3rd. The simplest interval which is flatter than the central interval is left ([[7/6]] for minor 3rd), and the simplest interval which is sharper is right ([[11/9]] for minor 3rd). Then the simplest interval which is flatter than the left is leftleft, the simplest interval between left and central is leftright , the simplest interval which is between central and right is rightleft, and the simplest interval which is sharper than right is rightright. This process of bisection with lefts/rights can be continued infinitely to name all just intervals that are in a category. Interval arithmetic is preserved (e.g. M2 * M2 is always M3), however the lefts and rights do not combine like accidentals do.