User:CompactStar/Ordinal interval notation

Lefts and rights notation is a notation for just intonation developed by CompactStar.

Explanation

Regular interval names corresponds to Pythagorean intervals, e.g. 32/27 is a minor third. To name non-Pythagorean intervals, they are given the name of a nearby Pythagorean interval (see #Mapping non-Pythagorean intervals), then prefixed with a sequence of lefts and rights (abbreviated as < and > respectively). Left and right have the following meaning (where X is any left/right sequence, and Y is a regular interval category like "major third"):

• Xleft Y = the simplest (with respect to Tenney height) Y whose left/right sequence starts with X and is flatter than X Y
• Xright Y = the simplest (with respect to Tenney height) Y whose left/right sequence starts with X and is sharper than X Y

More simply, left means to find the simplest flatter interval, and right means to find the simplest sharper interval, with each new left/right having less and less of an impact. Here are some examples with major thirds:

• Leftmajor third = 5/4, since it is the simplest major third flatter than 81/64
• Rightmajor third = 9/7, since it is the simplest major third sharper than 81/64
• Leftleftmajor third = 16/13, since it is the simplest major third flatter than 5/4
• Leftrightmajor third = 24/19, since it is the simplest major third between 5/4 and 81/64
• Rightleftmajor third = 14/11, since it is the simplest major third between 81/64 and 9/7
• Rightrightmajor third = 22/17, since it is the simplest major third sharper than 9/7

This system c

Mapping non-Pythagorean intervals

To map non-Pythagorean intervals to Pythagorean intervals, each prime harmonic above 3 is mapped to the lowest-complexity Pythagorean interval that is within √(2187/2048) (57 cents) of it.