User:CompactStar/Lefts and rights notation: Difference between revisions

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WIP (the reason this is in userspace)
WIP (the reason this is in userspace)


Ordinal interval notation is a notation for [[just intonation]] in which all intervals are represented by a normal interval classification combined with an ordinal number.
Lefts and rights notation is a notation for [[just intonation]] developed by [[User:CompactStar|CompactStar]]. It is a combination of Pythagorean notation and a sequence of lefts and rights which is based on a {{w|binary search}}.


To get the classification for an interval, multiply the interval classes of the prime harmonics which it factors into, which are predefined as follows:
== 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:
{|class="wikitable"
{|class="wikitable"
|-
|-
!Prime harmonic
!Prime harmonic
!colspan="2"|Interval classification
!colspan="3"|Category
|-
|-
|[[2/1]]
|[[2/1]]
|P8
|P8
|perfect octave
|perfect octave
|C
|-
|-
|[[3/2]]
|[[3/2]]
|P5
|P5
|perfect fifth
|perfect fifth
|G
|-
|-
|[[5/4]]
|[[5/4]]
|M3
|M3
|major third
|major third
|E
|-
|-
|[[7/4]]
|[[7/4]]
|m7
|m7
|minor seventh
|minor seventh
|Bb
|-
|-
|[[11/8]]
|[[11/8]]
|U4
|P4
|ultrafourth
|perfect fourth
|F
|-
|-
|[[13/8]]
|[[13/8]]
|an6
|m6
|artoneutral sixth
|minor sixth
|Ab
|-
|-
|[[17/16]]
|[[17/16]]
|m2
|m2
|minor second
|minor second
|Db
|-
|-
|[[19/16]]
|[[19/16]]
|m3
|m3
|minor third
|minor third
|Eb
|-
|-
|[[23/16]]
|[[23/16]]
|i5
|A4
|infrafifth
|augmented fourth
|F#
|-
|-
|[[29/16]]
|[[29/16]]
|m7
|m7
|minor seventh
|minor seventh
|Bb
|-
|-
|[[31/16]]
|[[31/16]]
|i8
|P8
|infraoctave
|perfect octave
|C
|-
|-
|[[37/32]]
|[[37/32]]
|UM2
|M2
|ultramajor second
|major second
|D
|-
|-
|[[41/32]]
|[[41/32]]
|M3
|M3
|major third
|major third
|E
|-
|[[43/32]]
|P4
|perfect fourth
|F
|-
|-
|[[47/32]]
|[[47/32]]
|i5
|P5
|infrafifth
|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.

Latest revision as of 04:00, 2 January 2024

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.

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