Xenharmonic Wiki

User:Eufalesio/PLIN: Difference between revisions

← Older edit
VisualWikitext
Eufalesio (talk | contribs)
autopatrolled
739 edits
No edit summary
Eufalesio (talk | contribs)
autopatrolled
739 edits
No edit summary
 
(One intermediate revision by the same user not shown)
Line 14: Line 14:


=== Compromise notations ===
=== Compromise notations ===
The only notation I have knowledge of that can be sensibly called a "compromise notation" is [[Sagittal]]. It can be used to notate almost every tuning imaginable, combining mapping with JI-ish accidentals at high enough complexity. They share the versatility of tempered notations combining it with the exactitude of JI, converging towards a JI interval that is represented through accidentals that represent both a mapping and a precise comma.  
The only notation I have knowledge of that can be sensibly called a "compromise notation" is [[Sagittal]]. It can be used to notate almost every tuning imaginable, combining mapping with JI-ish accidentals at high enough complexity. They share the versatility of tempered notations combining it with the exactitude of JI, converging toweeds a JI interval that is represented through accidentals that represent both a mapping and a precise comma.  


'''PLIN''', the focus of this article, aims to be such another notation[https://xkcd.com/927/ .]
'''PLIN''', the focus of this article, aims to be such another notation[https://xkcd.com/927/ .]


== PLIN ==
== PLIN ==
Short for '''Pythagoreanoid Loose Interval Notation''', it's a type of [[2.3-equivalent class and Pythagorean-commatic interval naming system|2.3-equivalent class]]. It attempts to provide precision tiers to name intervals, based on a chain of '''pure''' fifths. The reason of why to use a chain of fifths, apart from tradition and my biases, is that it provides the simplest framefork for building scales, and because it is the most widely used worldwide. Why to use pure fifths and not an edo's best approximation of a fifth is to have a retrocompatible sistem. Among the lower primes, it makes the best small MOS scales (2,3,5,(7),12,(17),(29),41,53), second in place to 11 (2,5,(7),(9),11,13,24,37). An argument could be made to make a system based on 11 to build scales, but that's beyond the scope of this article. After all, this is about a ''pythagoreanoid'' notation, not a ''hendecoid'' notation.
Short for '''Pythagoreanoid Loose Interval Notation''', it's a type of [[2.3-equivalent class and Pythagorean-commatic interval naming system|2.3-equivalent class]]. It attempts to provide precision tiers to name intervals, based on a chain of '''pure''' fifths. The reason of why to use a chain of fifths, apart from tradition and my biases, is that it provides the simplest framefork for building scales, and because it is the most widely used worldwide.  
 
Why use pure fifths and have unequal sized buckets? For retrocompatibility. If I equalize the buckets, the fifth changes, and the buckets also change. Most intervals will be the same, but for higher precisions, the buckets will change. It isn't desirable for an interval to be a minor sixth in one precision and then a major sixth in another.  
 
Among the lower primes, it makes the best small MOS scales (2,3,5,(7),12,(17),(29),41,53), second in place to 11 (2,5,(7),(9),11,13,24,37). An argument could be made to make a system based on 11 to build scales, but that's beyond the scope of this article. After all, this is about a ''pythagoreanoid'' notation, not a ''hendecoid'' notation.


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.
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.
Line 30: Line 34:
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.
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 in 7315-PLIN.
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 converge towards 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. 7/4 is a fixsubminor seventh, but in 159-EPLIN, an '''equarto'''subminor 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. '''EPLIN'''s may be used to have exactly equalized buckets, which could 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, an eqrough ''major seventh''. 7/4 is a fixsubminor seventh, but in 159-EPLIN, an eq''wee''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 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 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.  
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.
Everything stays in the 3-limit. Minimum complexity, reducing the amount of 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 k-strong 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 wee/wide, 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]
* Limmas and apotomes (1 m2 M2 m3 M3 4 T 5 m6 M6 m7 M7 8) Available in 12-PLIN [rough]
* Pythagorean commas ('''s'''ub/'''S'''uper, '''h'''ypo/'''H'''yper) Available in 53-PLIN [∅], functionally only in 12-PLIN  
* Pythagorean commas ('''s'''ub/'''S'''uper, '''h'''ypo/'''H'''yper) Available in 53-PLIN [∅], functionally only in 12-PLIN  
* ''Pythagorean comma thirds ('''a'''rto/'''t'''endo) Available '''only''' in 159-PLIN [fix]''
* ''Pythagorean comma thirds (wee/wide) Available '''only''' in 159-PLIN [fix]''
* Mercator commas ([π]mu/[Π]mi) Available in 665-PLIN [sat], functionally only in 53/159-PLIN
* Mercator commas ([π]mu/[Π]mi) Available in 665-PLIN [sat], functionally only in 53/159-PLIN
* Sasktel commas (qu/'''Q'''i) Available in 665-PLIN [sat]
* Sasktel commas (qu/'''Q'''i) Available in 665-PLIN [sat]
* Sasktel comma elevenths ([0~5]plus/[0~5]-minus) Available '''only''' in 7315-PLIN [spot
* Sasktel comma elevenths ([0~5]plus/[0~5]-minus) Available '''only''' in 7315-PLIN [spot


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.
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 so 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. Extending my rules; apotome = superminor second; superlimma.  
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. Alternatively you could use the [[User:Eufalesio/Punctional Just System#Punny names|Punny names]], to save yourself some syllables.  


==== Syntax of a PLIN ====
==== Syntax of a PLIN ====
Line 57: Line 61:
* 12-PLIN : rough + diatonic nominal
* 12-PLIN : rough + diatonic nominal
* 53-PLIN: hypo/sub/∅/super/hyper + diatonic nominal
* 53-PLIN: hypo/sub/∅/super/hyper + diatonic nominal
* 159-PLIN arto/fix/tendo + hypo/sub/∅/super/hyper + diatonic nominal
* 159-PLIN wee/fix/wide + hypo/sub/∅/super/hyper + diatonic nominal
* 665-PLIN [∅/qi/qu + [∅/two/three][mi/mu]]/sat + hypo/sub/∅/super/hyper + diatonic nominal
* 665-PLIN [∅/qi/qu + [∅/two/three][mi/mu]]/sat + hypo/sub/∅/super/hyper + diatonic nominal
* 7315-PLIN [[∅/two/three/four/five][plus/min(u)s]]/spot+[∅/qi/qu+[∅/two/three][mi/mu]]+hypo/sub/∅/super/hyper+diatonic nominal
* 7315-PLIN [[∅/two/three/four/five][plus/min(u)s]]/spot+[∅/qi/qu+[∅/two/three][mi/mu]]+hypo/sub/∅/super/hyper+diatonic nominal
Line 635: Line 639:
None of the PLINs from this point on will be MOS, as it is much more retrocompatible and feasible to alter by fractions of a pythagorean comma than to make a multiperiod MOS scale. It's just not worth the mental gymnastics.
None of the PLINs from this point on will be MOS, as it is much more retrocompatible and feasible to alter by fractions of a pythagorean comma than to make a multiperiod MOS scale. It's just not worth the mental gymnastics.
{| class="wikitable"
{| class="wikitable"
|+ 159-PLIN; 41L 12s 26 |159-PLIN up to the first major second; MOS 41L 12s 26|26; tolerance = '''3.91c'''
|+ 159-PLIN; 41L 12s 26 |159-PLIN up to the first fixmajor second; MOS 41L 12s 26|26; tolerance = '''3.91c'''
!Spoken name
!Spoken name
!Simplified
!Simplified
Line 648: Line 652:
|729/728
|729/728
|-
|-
|tendounison
|wideunison
|t1
|t1
|Dt
|DW
|7.8200
|7.8200
|225/224
|225/224
|-
|-
|artosuperunison
|weesuperunison
|aS1
|aS1
|DaS
|DwS
|15.6400
|15.6400
|121/120
|121/120
Line 666: Line 670:
|64/63
|64/63
|-
|-
|tendosuperunison
|widesuperunison
|tS1
|tS1
|DtS
|DtS
Line 672: Line 676:
|56/55
|56/55
|-
|-
|artohyperunison / artohypominor second
|weehyperunison / weehypominor second
|aH1
|aH1
|DaH /Ebah
|DwH /Ebwh
|39.1000
|39.1000
|45/44
|45/44
Line 684: Line 688:
|40/39
|40/39
|-
|-
|tendohyperunison / tendohypominor second
|widehyperunison / widehypominor second
|th2
|th2
|Ebth
|EbWh
|54.740021
|54.740021
|33/32
|33/32
|-
|-
|artosubminor second
|weesubminor second
|asm2
|asm2
|Ebas
|Ebws
|58.9449
|58.9449
|91/88
|91/88
Line 702: Line 706:
|80/77
|80/77
|-
|-
|tendosubminorsecond
|widesubminorsecond
|tsm2
|tsm2
|Ebts
|EbWs
|74.5840
|74.5840
|448/429
|448/429
|-
|-
|artominor second
|weeminor second
|am2
|am2
|Eba
|Ebw
|82.405
|82.405
|22/21
|22/21
Line 720: Line 724:
|96/91
|96/91
|-
|-
|tendominor second
|wideminor second
|tm2
|tm2
|Ebt
|EbW
|98.045
|98.045
|128/121
|128/121
|-
|-
|artosuperminor second
|weesuperminor second
|aSm2
|aSm2
|EbaSa
|EbwS
|105.865
|105.865
|1225/1152
|1225/1152
Line 738: Line 742:
|16/15
|16/15
|-
|-
|tendosuperminor second
|widesuperminor second
|tSm2
|tSm2
|EbtS
|EbWS
|121.505
|121.505
|15/14
|15/14
|-
|-
|artohyperminor second
|weehyperminor second
|aHm2
|aHm2
|EbaH
|EbwH
|129.325
|129.325
|14/13
|14/13
Line 756: Line 760:
|13/12
|13/12
|-
|-
|tendohyperminor second
|widehyperminor second
|tHm2
|tHm2
|EbtH
|EbWH
|144.965
|144.965
|160/147
|160/147
|-
|-
|artohypomajor second
|weehypomajor second
|ahM2
|ahM2
|Eah
|Ewh
|149.17
|149.17
|12/11
|12/11
Line 774: Line 778:
|35/32
|35/32
|-
|-
|tendohypomajor second
|widehypomajor second
|thM2
|thM2
|Eth
|EWh
|164.809
|164.809
|11/10
|11/10
|-
|-
|artosubmajor second
|weesubmajor second
|asM2
|asM2
|Eas
|Ews
|172.629
|172.629
|182/165
|182/165
Line 792: Line 796:
|231/208
|231/208
|-
|-
|tendosubmajor second
|widesubmajor second
|tsM2
|tsM2
|Ets
|Ets
Line 798: Line 802:
|39/35
|39/35
|-
|-
|artomajor second
|weemajor second
|aM2
|aM2
|Ea
|Ew
|196.09
|196.09
|160/143
|160/143
Line 814: Line 818:
If 53-PLIN is not enough for you, this will be surely be enough. If not... then prepare for what's to come.
If 53-PLIN is not enough for you, this will be surely be enough. If not... then prepare for what's to come.


As you know, two is a pair, three is a crowd, and each new PLIN continues adding more classes to worry about. So far, it has been only 3 classes at most: tendo/arto, hypo/sub/fix/super/hyper, nominal. But, 665 has now qi/qu (small qian commas) and mi/mu (mercator commas), apart from the hypo/sub/fix/super/hyper, nominal. 7315-PLIN has all those, and five-fold plus/min(u)s.  
As you know, two is a pair, three is a crowd, and each new PLIN continues adding more classes to worry about. So far, it has been only 3 classes at most: wide/wee, hypo/sub/fix/super/hyper, nominal. But, 665 has now qi/qu (small qian commas) and mi/mu (mercator commas), apart from the hypo/sub/fix/super/hyper, nominal. 7315-PLIN has all those, and five-fold plus/min(u)s.  
{| class="wikitable"
{| class="wikitable"
|+665-PLIN until the first pythagorean comma
|+665-PLIN until the first pythagorean comma
Line 1,376: Line 1,380:
I think the mappings are correct, but I'm too lazy to check my work. Mappings ''may'' change for EPLINs.  
I think the mappings are correct, but I'm too lazy to check my work. Mappings ''may'' change for EPLINs.  


== Notes on EPLINs ==
== 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.
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, 130-EPLIN, 171-EPLIN, 217-EPLIN, 270-EPLIN, 311-EPLIN, 1600-EPLIN, 2460-EPLIN, 8539-EPLIN. '''No others'''. This is to prevent bloat of the system and the mapped comma to fall beyond the size of a syntonic or septimal comma. The reason to use these edos is because they're either zeta peaks, or very very consistent.  


WIP
The jump from 311 to 1600 is due to the fact that for its size, there is no better edo for consistency than 311. The next best thing is 1600edo, which has around the same absolute error as 270edo, if a bit less. Just in case, and because I think they're SS-tier edos, I'm putting the behemoths 2460 and 6079 for ultra-mega-hyper accuracy.


== Conclusion ==
Here are the prefixes for pythagorean comma fractions in these EPLINs:
{| class="wikitable"
|+
!Edosteps
added
!41-EPLIN
!94-EPLIN
!118-EPLIN
!130-EPLIN
!171-EPLIN
!217-EPLIN
!270-EPLIN
!311-EPLIN
!1600-EPLIN
!2460-EPLIN
!6079-EPLIN
|-
| +1
|–
|glus
|hus
|slus
|nus
|clus
|xus
|vlus
|kwus
|krus
|prus
|-
|0
|kite
|gar
|hel
|spen
|nea
|cot
|nex
|vu
|kwa
|kirn
|pir
|-
| -1
|–
|gins
|hins
|spins
|nins
|kins
|xins
|vins
|kwins
|krins
|prins
|-
!Size and
edosteps
|29.268c
[1]
|25.532c
[2]
|20.339c
[2]
|18.4615c
[2]
|21.0526c
[3]
|27.6497c
[5]
|26.<u>6</u>c
[6]
|27.001c
[7]
|24c
[32]
|23.4146c
[48]
|23.4688c
[167]
|}
These prefixes are based on the a temperament they represent, or on Sagittal:
 
* gar- for Garibaldi
* hel- for Helmholtz
* spen- for Sensipent
* nea- for Ennealimmal
* cot- for Cotoneum
* nex- for Nexus
* vu- for Vavoom
* kwa- for Kwazy
* kirn- for Atomic (Kirnberger's atom)
* pir- for Pirate
The only ones of this EPLINs that can use the mercator comma are 2460-EPLIN and 6079-EPLIN, as the sizes aren't too dissimilar and/or inconsistent. Yes, 42%, 69% error, but in all other EPLINs it is wildly out of shape, in some even negative!
 
Examples:
{| class="wikitable"
|+
!
!94-EPLIN
!130-EPLIN
!311-EPLIN
|-
|5/4
|garsubmajor third
|spensubmajor third
|vlussubmajor third
|-
|7/4
|garsubminor seventh
|hinssubminor seventh
|vusubminor seventh
|-
|11/8
|garhyperfourth
|slushyperfourth
|vuhyperfourth
|-
|13/8
|garhyperminor sixth
|slushyperminor sixth
|vlushyperminor sixth
|}
WIP
WIP
Retrieved from "https://en.xen.wiki/w/User:Eufalesio/PLIN"