User:CompactStar/Lefts and rights notation

WIP (the reason this is in userspace)

Lefts and rights notation is a notation for just intonation developed by CompactStar. It is a combination of Pythagorean notation and a sequence of lefts and rights which is based on a binary search.

Definition

In order to map all intervals into Pythagorean intervals, each prime harmonic is mapped to the lowest-complexity Pythagorean interval that is within √(2187/2048) of it (this is similar to the algorithm used by the Functional Just System). Here are some examples for smaller primes:

Prime harmonic	Category
2/1	P8	perfect octave	C
3/2	P5	perfect fifth	G
5/4	M3	major third	E
7/4	m7	minor seventh	Bb
11/8	P4	perfect fourth	F
13/8	m6	minor sixth	Ab
17/16	m2	minor second	Db
19/16	m3	minor third	Eb
23/16	A4	augmented fourth	F#
29/16	m7	minor seventh	Bb
31/16	P8	perfect octave	C
37/32	M2	major second	D
41/32	M3	major third	E
43/32	P4	perfect fourth	F
47/32	P5	perfect fifth	G
53/32	M6	major sixth	A
59/32	M7	major seventh	B
61/32	M7	major seventh	B

Pythagorean intervals do not use any lefts or rights. The left (L) interval in a given category is the simplest (with respect to Tenney height) interval which is flatter than the Pythagorean version. For example, the leftminor third is 7/6. Similarly, the right (R) interval in a category is the simplest one that is sharper than the Pythagorean version (for minor thirds this is 6/5). After this, the types split up into 4 using 2 lefts and rights:

• The leftleft (LL) interval is the simplest which is flatter than the left interval. For minor thirds: 43/37
• The leftright (LR) interval is the simplest which is between the left interval and the Pythagorean interval. For minor thirds: 13/11
• The rightleft (RL) interval is the simplest which is between the Pythagorean interval and the right interval. For minor thirds: 19/16
• The rightright (RR) interval is the simplest which is sharper than the right interval. For minor thirds: 11/9

And then using 3 lefts and rights:

• The leftleftleft (LLL) interval is the simplest which is flatter than the leftleft interval.
• The leftleftright (LLR) interval is the simplest which is between the leftleft interval and the left interval.
• The leftrightleft (LRL) interval is the simplest which is between the left interval and the leftright interval. For minor thirds: 20/17
• The leftrightright (LRR) interval is the simplest which is between the leftright interval and the Pythagorean interval. For minor thirds: 77/65
• The rightleftleft (RLL) interval is the simplest which is between the Pythagorean interval and the rightleft interval. For minor thirds: 51/43
• The rightleftright (RLR) interval is the simplest which is between the rightleft and the right interval.
• The rightrightleft (RRL) interval is the simplest which is between the right interval and the rightright interval. For minor thirds: 17/14
• The rightrightright (RRR) interval is the simplest which is sharper than the rightright interval. For minor thirds: 38/31

And so on. This sort of binary search can be applied for an arbitrary number of lefts and rights to name all just intervals in a category.