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| '''Binary search notation''' is a notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].
| | #redirect [[User:CompactStar/Lefts and rights notation]] |
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| 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:
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| {|class="wikitable"
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| |-
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| !Prime harmonic
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| !colspan="3"|Notation
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| |-
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| |[[2/1]]
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| |P8
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| |perfect octave
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| |C
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| |-
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| |[[3/2]]
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| |P5
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| |perfect 5th
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| |G
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| |-
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| |[[5/4]]
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| |M3
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| |major 3rd
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| |E
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| |-
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| |[[7/4]]
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| |m7
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| |minor 7th
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| |Bb
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| |-
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| |[[11/8]]
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| |P4
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| |perfect 4th
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| |F
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| |-
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| |[[13/8]]
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| |m6
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| |minor 6th
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| |Ab
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| |-
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| |[[17/16]]
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| |m2
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| |minor 2nd
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| |Db
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| |-
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| |[[19/16]]
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| |m3
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| |minor 3rd
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| |Eb
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| |-
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| |[[23/16]]
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| |A4
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| |augmented 4th
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| |F#
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| |-
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| |[[29/16]]
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| |m7
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| |minor 7th
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| |Bb
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| |-
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| |[[31/16]]
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| |P8
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| |perfect octave
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| |C
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| |-
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| |[[37/32]]
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| |M2
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| |major 2nd
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| |D
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| |-
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| |[[41/32]]
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| |M3
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| |major 3rd
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| |E
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| |-
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| |[[43/32]]
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| |P4
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| |perfect 4th
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| |F
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| |-
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| |[[47/32]]
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| |P5
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| |perfect 5th
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| |G
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| |-
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| |[[53/32]]
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| |M6
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| |major 6th
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| |A
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| |-
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| |[[61/32]]
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| |M7
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| |major 7th
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| |B
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| |-
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| |[[67/64]]
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| |m2
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| |minor 2nd
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| |Db
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| |-
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| |[[71/64]]
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| |M2
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| |major 2nd
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| |D
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| |-
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| |[[73/64]]
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| |M2
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| |major 2nd
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| |D
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| |-
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| |[[79/64]]
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| |M3
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| |major 3rd
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| |E
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| |-
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| |[[83/64]]
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| |P4
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| |perfect 4th
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| |F
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| |-
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| |[[89/64]]
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| |d5
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| |diminished 5th
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| |Gb
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| |-
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| |[[97/64]]
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| |P5
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| |perfect 5th
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| |G
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| |}
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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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