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User:CompactStar/Ordinal interval notation: Difference between revisions

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m CompactStar moved page User:CompactStar/Indexed interval notation to User:CompactStar/Binary search notation: I got some ideas from discord, hopefully this is the final move.
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'''Indexed interval notation''' is a notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].
'''Binary search notation''' is a notation for [[just intonation]] devised by [[User:CompactStar|CompactStar]].


Intervals are represented by a normal interval category (like minor 3rd, perfect 4th or etc.) 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 (1m3) and [[7/6]] is the 2nd minor 3rd (2m3). Note names work the same way, using A4 as the reference pitch, so the C which is 6/5 above A (528 Hz) is notated as 1C.
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:
{|class="wikitable"
|-
!Prime harmonic
!colspan="3"|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
|}
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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