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.