User:CompactStar/Ordinal interval notation: Difference between revisions

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m CompactStar moved page User:CompactStar/Lefts and rights notation to User:CompactStar/Indexed interval notation over redirect: Can't decide between these two variants of it
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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.

Revision as of 23:49, 1 December 2023

Indexed interval notation is a notation for just intonation devised by CompactStar.

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:

Prime harmonic Notation
2/1 P8 perfect octave C
3/2 P5 perfect 5th G
5/4 M3 major 3rd E
7/4 m7 minor 7th Bb
11/8 P4 perfect 4th F
13/8 m6 minor 6th Ab
17/16 m2 minor 2nd Db
19/16 m3 minor 3rd Eb
23/16 A4 augmented 4th F#
29/16 m7 minor 7th Bb
31/16 P8 perfect octave C
37/32 M2 major 2nd D
41/32 M3 major 3rd E
43/32 P4 perfect 4th F
47/32 P5 perfect 5th G
53/32 M6 major 6th A
61/32 M7 major 7th B
67/64 m2 minor 2nd Db
71/64 M2 major 2nd D
73/64 M2 major 2nd D
79/64 M3 major 3rd E
83/64 P4 perfect 4th F
89/64 d5 diminished 5th Gb
97/64 P5 perfect 5th G
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