User:CompactStar/Ordinal interval notation

Lefts and rights notation is a notation for just intonation primarily developed by CompactStar. It represents every just interval as a sequence of lefts/rights (referred to as it's direction) and a diatonic degree.

Explanation

All intervals are given a diatonic degree, calculated from their 7edo patent val mapping. If k is a degree, the central k-th is the simplest (with respect to Tenney height) just ratio which is a k-th. For example, the central 3rd is 5/4, since it is the simplest 3rd, and the central 7th is 7/4, since it is the simplest 7th. The left k-th is the simplest k-th which is flatter than central, and the right k-th is the simplest k-th which is sharper than central. Central, left and right are abbreviated as c, l, and r respectively.

Left and right can be used multiple times to produce more complex directions. For example, leftleft is flatter than left, leftright is between left and central, rightleft is between central and right, and rightright is sharper than right. Adding a left always means to go flatter, and adding a right always means to go sharper, with each new left/right having less and less of an impact. Formally, if k is a degree, X and Y are any sequence of lefts/rights:

• Xleft k-th = the simplest just k-th whose direction starts with X and is flatter than the X k-th
• Xright k-th = the simplest just k-th whose direction starts with X and is sharper than the X k-th

Examples

Below are some examples of lefts and rights notation for 11-odd-limit intervals:

Interval	Name
12/11	leftleftleftleft second	llll2
11/10	leftleftleft second	lll2
10/9	leftleft second	ll2
9/8	left second	l2
8/7	central second	c2
7/6	leftleft third	ll3
6/5	left third	l3
11/9	leftright third	lr3
5/4	central third	c3
14/11	rightleft third	rl3
9/7	right third	r3
4/3	central fourth	c4
11/8	rightleft fourth	rl4
7/5	left fifth	l5