User:CompactStar/Ordinal interval notation

Binary search notation is a notation for just intonation devised by 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:

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

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.