User:Eufalesio/PLIN: Difference between revisions

Line 23:

The notation, much like Sagittal, comes in precision packs, which are the 12-form PLIN, 53-form PLIN, 159-form PLIN, 665-PLIN and 7315-form PLIN. The reason why to have these numbers of intervals is that they are part of the sequence of 3-2 telic edos that have high limit consistencies, and the biggest they can be. Each finer PLIN adds one more class of independent prefixes, from the others, having one (nominals) with 12-PLIN, and up to 5 with 7315-PLIN, which for some intervals will become hard to distinguish.

If you say you need one beyond 7315, [[Sagittal notation#cite note-:0-4|you are beyond insane]]. The next edo on the 3-2 telic list is the gargantuan [[190537edo]]. If you're stubborn enough to do so, have fun using twelfths of a satanic comma! And you will need to stack that twelfth interval up to 140 times in either direction, because a 306-comma is 281/12 satanic commas. That sounds like hell, both figuratively and literally.

If you say you need one beyond 7315, [[Sagittal notation#cite note-:0-4|you are beyond insane]]. The next edo on the k-strong 3-2 telic list is the gargantuan [[190537edo]]. If you're stubborn enough to do so, have fun using twelfths of a satanic comma! And you will need to stack that twelfth interval up to 140 times in either direction, because a 306-comma is 281/12 satanic commas. That sounds like hell, both figuratively and literally. Whoever named the satanic comma, surely knew what pain in the ass was to work with it. If not, that's a damned too good of a coincidence to overlook.

'''You do NOT need it.''' At that point, just use FJS.

But enough with the religious puns. '''You do NOT need it.''' At that point, just use FJS.

=== How PLINs work ===

PLINs generate regions whose center correspond to a pure pythagorean interval generated by the chain-of-fiths, For example, there are infinite minor sixths, no matter how much you narrow the buckets, but there is ''only'' one true minor sixth, and that is 128/81.

Any single region has infinite intervals represented in it, and thus it is important to distinguish the breadth of the buckets any JI interval can fall into. So, while there are an infinite amount of minor thirds, and of superminor thirds, and of qu superminorthirds, and of twomins qu superminor thirds, all of them will 6/5; with 12-PLIN being quite rough, 53-PLIN being very accurate, and 665-PLIN and 7315-PLIN further refining the accuracy to extreme levels to the point where all of those will sound indistinguishable from 6/5, since the difference will be only of 0.16 c per region.

Any single region has infinite intervals represented in it, and thus it is important to distinguish the breadth of the buckets any JI interval can fall into. So, while there are an infinite amount of minor thirds, and of superminor thirds, and of qu superminorthirds, and of twomins qu superminor thirds, all of them will 6/5; with 12-PLIN being quite rough, 53-PLIN being very accurate, and 665-PLIN and 7315-PLIN further refining the accuracy to extreme levels to the point where all of those will sound indistinguishable from 6/5, since the difference will be only of 0.16 c per region in 7315-PLIN.

12-PLIN, 53-PLIN and 665-PLIN are MOSses, more concretely, 5L 7s 6|5 1.260, 41L 12s 26|26 H=1.1822, and 306L 359s H=1.0427, but 159-PLIN and 7315 are not, as they modify the two last MOSses respectively with fractions of the telic commas. As such, to make things easier, '''EPLIN'''s may be used to have exactly equal regions, which will make things easier to work with. by default, PLINs are not equal, so it needs to be specified when an equalized PLIN is being used, because it can change the region an interval falls into. For example, 31/16 is a rough octave, but in 12-EPLIN, a '''eq'''rough major seventh.

12-PLIN, 53-PLIN and 665-PLIN are MOSses, more concretely, 5L 7s 6|5 1.260, 41L 12s 26|26 H=1.1822, and 306L 359s H=1.0427, but 159-PLIN and 7315 are not, as they modify the two last MOSses respectively with fractions of the telic commas. As such, to make things easier, '''EPLIN'''s may be used to have exactly equal regions, which will make things easier to work with. by default, PLINs are not equal, so it needs to be specified when an equalized PLIN is being used, because it can change the region an interval falls into. For example, 31/16 is a rough octave, but in 12-EPLIN, a '''eq'''rough major seventh. 7/4 is a fixsubminor seventh, but in 159-EPLIN, an '''equarto'''subminor seventh.

One glaring design feature about PLINs is the lack of ''neutral'' categories, and of ''augmented'', ''diminished'' and ''perfect''. The reason to avoid those is that for one, true neutral intervals do not exist in integer pythagorean, but even if they did, using this term is unnecessary. 11/9 is commonly called a neutral interval, but it is closer to a minor third. So it is roughly a minor third. More precisely~~, it is~~ a hyperminor third, at the extreme of minor thirds. Same thing applies to the interordinals like chthonics, naiadics, cocytics and ouranics. You can do fine with using hyper/hypo to refer to them at the edges of the nominals. All of this ''sonically speaking''.

One glaring design feature about PLINs is the lack of ''neutral'' categories, and of ''augmented'', ''diminished'' and ''perfect''. The reason to avoid those is that for one, true neutral intervals do not exist in integer pythagorean, but even if they did, using this term is unnecessary. 11/9 is commonly called a neutral interval, but it is closer to a minor third. So it is roughly a minor third. More precisely a hyperminor third, at the extreme of minor thirds. Same thing applies to the interordinals like chthonics, naiadics, cocytics and ouranics. You can do fine with using hyper/hypo to refer to them at the edges of the nominals. All of this ''sonically speaking''.

For two, is 11/9 a minor third or a major third? Or 13/10 a major third or a fourth? That depends on how you treat it. Augmented/diminished, and chromas have the same logic in that they are ''functional'' in this system. Using ''perfect'' is also redundant and ambiguous, so use different coinages to refer specifically to the centers of regions of the different PLINs. An interval is not a chroma, but rather, ''works'' as a chroma. When you are dealing with intervals inside a scale, it may be useful to refer to 25/16 as an augmented fifth, but sonically speaking, it is a ~~(sub)~~minor sixth.

For two, is 11/9 a minor third or a major third? Or 13/10 a major third or a fourth? That depends on how you treat it. Augmented/diminished, and '''chromas''' have the same logic in that they are '''''functional''''' in this system. Using ''perfect'' is also redundant and ambiguous, so use different coinages to refer specifically to the centers of regions of the different PLINs. An interval is not a chroma, but rather, '''''works''''' as a chroma. When you are dealing with intervals inside a scale and a piece, it may be useful to refer to 25/16 as an augmented fifth, but '''sonically''' speaking, it is a rough minor sixth. Or a hypominor sixth. Or a mi hypominor sixth. Or a fourplus mi hypominor sixth... you get the point.

Also, no words for other primes. So no ptolemaic/pental/classical, septimal, undecimal... etc. Everything stays in the 3-limit. Minimum complexity, reducing the amount of classes and descriptors to worry about to the absolute minimum.

Regarding the choice of words to refer to the descriptors; you might not agree about the use of hypo/hyper, or arto/tendo, or qi/qu, or mi/mu, or n-plus/n-minus; but that's only a semantics problem. I chose those names because they kind of make sense to me, but the rigor is in the system, because you have these commas in the 3-2 telic sequence:

Regarding the choice of words to refer to the descriptors; you might not agree about the use of hypo/hyper, or arto/tendo, or qi/qu, or mi/mu, or n-plus/n-minus; but that's only a semantics problem. I chose those names because they kind of make sense to me, but the rigor is in the system, because you have these commas in the k-strong 3-2 telic sequence:

* Limmas and apotomes (1 m2 M2 m3 M3 4 T 5 m6 M6 m7 M7 8) Available in 12-PLIN [rough]

Line 51:

The choice of giving no center descriptor to 53-PLIN is that I believe that for the average xennie, 53 regions is precise enough to accurately name most intervals, and simple enough that the regions cannot be confused.

Of course, the names would have many synonyms, so hypo/hyper = infra/ultra, minor second = limma, major second = tone, unison = prime, major third = ditone.

Of course, the names would have many synonyms, so hypo/hyper = infra/ultra, minor second = limma, major second = tone, unison = prime, major third = ditone. Extending my rules; apotome = superminor second; superlimma.

==== Syntax of a PLIN ====

Line 187:

|2/1

|63/32

|~~64/31~~

|

|}

*

{| class="wikitable"

|+53-PLIN; 41L 12s 26|53-PLIN; MOS 41L 12s 26|26; tolerance = '''11.73c'''

Line 870:

Line 868:

|-

|'''spot unison'''

|'''p1'''

|'''P1'''

|-

|plus unison

Line 1,299:

Line 1,297:

|-

|'''spot superunison'''

|'''~~pS1~~'''

|'''PS1'''

|}

And that amount of intervals is needed to reach ''one'' pythagorean comma. It is most surely overkill for the overwhelming majority of purposes. It will be the least easy to say of all the PLINs~~, where ~~~5/4 ~~is a "twoplus qi submajor third", and~~ 7/4 is a ~~"minus mu subminor seventh"~~.

And that amount of intervals is needed to reach ''one'' pythagorean comma. It is most surely overkill for the overwhelming majority of purposes. It will be the least easy to say of all the PLINs.

=== Example intervals in several PLINs ===

{| class="wikitable"

|+

!

!12-EPLIN

!53-EPLIN

!159-PLIN

!665-PLIN

!7315-PLIN

|-

|3/2

|5r

|5

|f5

|p5

|P5

|-

|5/4

|M3r

|sM3

|fsM3

|QsM3

|2+QsM3

|-

|7/4

|m7r

|sm7

|fsm7

|πsm7

| -πsm7

|-

|11/8

|4r

|H4

|tH4

|2∏H4

|5-2∏H4

|-

|13/8

|m6r

|hm6

|fhm6

|QHm6

|2-QHm6

|-

|19/16

|m3r

|m3

|fm3

|∏m3

| -∏m3

|-

|29/16

|m7r

|Sm7

|tSm7

|∏Sm7

|5-∏Sm7

|-

|13/10

|4r

|h4

|fh4

|∏h4

|3-∏h4

|-

|11/9

|m3r

|Hm3

|tHm3

|2∏Hm3

|5-2∏Hm3

|}

I think the mappings are correct, but I'm too lazy to check my work. Mappings ''may'' change for EPLINs.

== Notes on EPLINs ==

Since EPLINs are essentially edos, I think that allowing more EPLINs than PLINs to exist could be advantageous. Case in point: 41-EPLIN, 94-EPLIN, 118-EPLIN, 65/130-EPLIN, 171-EPLIN, 217-EPLIN, 311-EPLIN, 1600-EPLIN, 2460-EPLIN, 8539-EPLIN. Any edo with a mapped fifth no wider than 41edo's, no narrower than 65edo's would be good.

WIP

== Conclusion ==

WIP

@@ Line 23: / Line 23: @@
 The notation, much like Sagittal, comes in precision packs, which are the 12-form PLIN, 53-form PLIN, 159-form PLIN, 665-PLIN and 7315-form PLIN. The reason why to have these numbers of intervals is that they are part of the sequence of 3-2 telic edos that have high limit consistencies, and the biggest they can be. Each finer PLIN adds one more class of independent prefixes, from the others, having one (nominals) with 12-PLIN, and up to 5 with 7315-PLIN, which for some intervals will become hard to distinguish.
-If you say you need one beyond 7315, [[Sagittal notation#cite note-:0-4|you are beyond insane]]. The next edo on the 3-2 telic list is the gargantuan [[190537edo]]. If you're stubborn enough to do so, have fun using twelfths of a satanic comma! And you will need to stack that twelfth interval up to 140 times in either direction, because a 306-comma is 281/12 satanic commas. That sounds like hell, both figuratively and literally.
+If you say you need one beyond 7315, [[Sagittal notation#cite note-:0-4|you are beyond insane]]. The next edo on the k-strong 3-2 telic list is the gargantuan [[190537edo]]. If you're stubborn enough to do so, have fun using twelfths of a satanic comma! And you will need to stack that twelfth interval up to 140 times in either direction, because a 306-comma is 281/12 satanic commas. That sounds like hell, both figuratively and literally. Whoever named the satanic comma, surely knew what pain in the ass was to work with it. If not, that's a damned too good of a coincidence to overlook.
-'''You do NOT need it.''' At that point, just use FJS.
+But enough with the religious puns. '''You do NOT need it.''' At that point, just use FJS.
 === How PLINs work ===
 PLINs generate regions whose center correspond to a pure pythagorean interval generated by the chain-of-fiths,  For example, there are infinite minor sixths, no matter how much you narrow the buckets, but there is ''only'' one true minor sixth, and that is 128/81.
-Any single region has infinite intervals represented in it, and thus it is important to distinguish the breadth of the buckets any JI interval can fall into. So, while there are an infinite amount of minor thirds, and of superminor thirds, and of qu superminorthirds, and of twomins qu superminor thirds, all of them will 6/5; with 12-PLIN being quite rough, 53-PLIN being very accurate, and 665-PLIN and 7315-PLIN further refining the accuracy to extreme levels to the point where all of those will sound indistinguishable from 6/5, since the difference will be only of 0.16 c per region.
+Any single region has infinite intervals represented in it, and thus it is important to distinguish the breadth of the buckets any JI interval can fall into. So, while there are an infinite amount of minor thirds, and of superminor thirds, and of qu superminorthirds, and of twomins qu superminor thirds, all of them will 6/5; with 12-PLIN being quite rough, 53-PLIN being very accurate, and 665-PLIN and 7315-PLIN further refining the accuracy to extreme levels to the point where all of those will sound indistinguishable from 6/5, since the difference will be only of 0.16 c per region in 7315-PLIN.
--PLIN, 53-PLIN and 665-PLIN are MOSses, more concretely, 5L 7s 6|5 1.260, 41L 12s 26|26 H=1.1822, and 306L 359s H=1.0427, but 159-PLIN and 7315 are not, as they modify the two last MOSses respectively with fractions of the telic commas. As such, to make things easier, '''EPLIN'''s may be used to have exactly equal regions, which will make things easier to work with. by default, PLINs are not equal, so it needs to be specified when an equalized PLIN is being used, because it can change the region an interval falls into. For example, 31/16 is a rough octave, but in 12-EPLIN, a '''eq'''rough major seventh.
+-PLIN, 53-PLIN and 665-PLIN are MOSses, more concretely, 5L 7s 6|5 1.260, 41L 12s 26|26 H=1.1822, and 306L 359s H=1.0427, but 159-PLIN and 7315 are not, as they modify the two last MOSses respectively with fractions of the telic commas. As such, to make things easier, '''EPLIN'''s may be used to have exactly equal regions, which will make things easier to work with. by default, PLINs are not equal, so it needs to be specified when an equalized PLIN is being used, because it can change the region an interval falls into. For example, 31/16 is a rough octave, but in 12-EPLIN, a '''eq'''rough major seventh. 7/4 is a fixsubminor seventh, but in 159-EPLIN, an '''equarto'''subminor seventh.
-One glaring design feature about PLINs is the lack of ''neutral'' categories, and of ''augmented'', ''diminished'' and ''perfect''. The reason to avoid those is that for one, true neutral intervals do not exist in integer pythagorean, but even if they did, using this term is unnecessary. 11/9 is commonly called a neutral interval, but it is closer to a minor third. So it is roughly a minor third. More precisely, it is a hyperminor third, at the extreme of minor thirds. Same thing applies to the interordinals like chthonics, naiadics, cocytics and ouranics. You can do fine with using hyper/hypo to refer to them at the edges of the nominals. All of this ''sonically speaking''.
+One glaring design feature about PLINs is the lack of ''neutral'' categories, and of ''augmented'', ''diminished'' and ''perfect''. The reason to avoid those is that for one, true neutral intervals do not exist in integer pythagorean, but even if they did, using this term is unnecessary. 11/9 is commonly called a neutral interval, but it is closer to a minor third. So it is roughly a minor third. More precisely a hyperminor third, at the extreme of minor thirds. Same thing applies to the interordinals like chthonics, naiadics, cocytics and ouranics. You can do fine with using hyper/hypo to refer to them at the edges of the nominals. All of this ''sonically speaking''.
-For two, is 11/9 a minor third or a major third? Or 13/10 a major third or a fourth? That depends on how you treat it. Augmented/diminished, and chromas have the same logic in that they are ''functional'' in this system. Using ''perfect'' is also redundant and ambiguous, so use different coinages to refer specifically to the centers of regions of the different PLINs. An interval is not a chroma, but rather, ''works'' as a chroma. When you are dealing with intervals inside a scale, it may be useful to refer to 25/16 as an augmented fifth, but sonically speaking, it is a (sub)minor sixth.
+For two, is 11/9 a minor third or a major third? Or 13/10 a major third or a fourth? That depends on how you treat it. Augmented/diminished, and '''chromas''' have the same logic in that they are '''''functional''''' in this system. Using ''perfect'' is also redundant and ambiguous, so use different coinages to refer specifically to the centers of regions of the different PLINs. An interval is not a chroma, but rather, '''''works''''' as a chroma. When you are dealing with intervals inside a scale and a piece, it may be useful to refer to 25/16 as an augmented fifth, but '''sonically''' speaking, it is a rough minor sixth. Or a hypominor sixth. Or a mi hypominor sixth. Or a fourplus mi hypominor sixth... you get the point.
 Also, no words for other primes. So no ptolemaic/pental/classical, septimal, undecimal... etc. Everything stays in the 3-limit. Minimum complexity, reducing the amount of classes and descriptors to worry about to the absolute minimum.
-Regarding the choice of words to refer to the descriptors; you might not agree about the use of hypo/hyper, or arto/tendo, or qi/qu, or mi/mu, or n-plus/n-minus; but that's only a semantics problem. I chose those names because they kind of make sense to me, but the rigor is in the system, because you have these commas in the 3-2 telic sequence:
+Regarding the choice of words to refer to the descriptors; you might not agree about the use of hypo/hyper, or arto/tendo, or qi/qu, or mi/mu, or n-plus/n-minus; but that's only a semantics problem. I chose those names because they kind of make sense to me, but the rigor is in the system, because you have these commas in the k-strong 3-2 telic sequence:
 * Limmas and apotomes (1 m2 M2 m3 M3 4 T 5 m6 M6 m7 M7 8) Available in 12-PLIN [rough]
@@ Line 51: / Line 51: @@
 The choice of giving no center descriptor to 53-PLIN is that I believe that for the average xennie, 53 regions is precise enough to accurately name most intervals, and simple enough that the regions cannot be confused.
-Of course, the names would have many synonyms, so hypo/hyper = infra/ultra, minor second = limma, major second = tone, unison = prime, major third = ditone.
+Of course, the names would have many synonyms, so hypo/hyper = infra/ultra, minor second = limma, major second = tone, unison = prime, major third = ditone. Extending my rules; apotome = superminor second; superlimma.
 ==== Syntax of a PLIN ====
@@ Line 187: / Line 187: @@
 |2/1
 |63/32
-|64/31
+|
 |}
-*
 {| class="wikitable"
 |+53-PLIN; 41L 12s 26|53-PLIN; MOS 41L 12s 26|26; tolerance = '''11.73c'''
@@ Line 870: / Line 868: @@
 |-
 |'''spot unison'''
-|'''p1'''
+|'''P1'''
 |-
 |plus unison
@@ Line 1,299: / Line 1,297: @@
 |-
 |'''spot superunison'''
-|'''pS1'''
+|'''PS1'''
 |}
-And that amount of intervals is needed to reach ''one'' pythagorean comma. It is most surely overkill for the overwhelming majority of purposes. It will be the least easy to say of all the PLINs, where ~5/4 is a "twoplus qi submajor third", and 7/4 is a "minus mu subminor seventh".
+And that amount of intervals is needed to reach ''one'' pythagorean comma. It is most surely overkill for the overwhelming majority of purposes. It will be the least easy to say of all the PLINs.
+=== Example intervals in several PLINs ===
+{| class="wikitable"
+|+
+!
+!12-EPLIN
+!53-EPLIN
+!159-PLIN
+!665-PLIN
+!7315-PLIN
+|-
+|3/2
+|5r
+|5
+|f5
+|p5
+|P5
+|-
+|5/4
+|M3r
+|sM3
+|fsM3
+|QsM3
+|2+QsM3
+|-
+|7/4
+|m7r
+|sm7
+|fsm7
+|πsm7
+| -πsm7
+|-
+|11/8
+|4r
+|H4
+|tH4
+|2∏H4
+|5-2∏H4
+|-
+|13/8
+|m6r
+|hm6
+|fhm6
+|QHm6
+|2-QHm6
+|-
+|19/16
+|m3r
+|m3
+|fm3
+|∏m3
+| -∏m3
+|-
+|29/16
+|m7r
+|Sm7
+|tSm7
+|∏Sm7
+|5-∏Sm7
+|-
+|13/10
+|4r
+|h4
+|fh4
+|∏h4
+|3-∏h4
+|-
+|11/9
+|m3r
+|Hm3
+|tHm3
+|2∏Hm3
+|5-2∏Hm3
+|}
+I think the mappings are correct, but I'm too lazy to check my work. Mappings ''may'' change for EPLINs.
+== Notes on EPLINs ==
+Since EPLINs are essentially edos, I think that allowing more EPLINs than PLINs to exist could be advantageous. Case in point: 41-EPLIN, 94-EPLIN, 118-EPLIN, 65/130-EPLIN, 171-EPLIN, 217-EPLIN, 311-EPLIN, 1600-EPLIN, 2460-EPLIN, 8539-EPLIN. Any edo with a mapped fifth no wider than 41edo's, no narrower than 65edo's would be good.
+WIP
+== Conclusion ==
 WIP