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<span data-darkreader-inline-color="">This article proposes a way to name any intervals and pitches, logically extending conventional notations following Pythagorean tuning, with some of my own flairs.</span>
<span data-darkreader-inline-color="">This article proposes a way to name any intervals and pitches, logically extending conventional notations following Pythagorean tuning into a minimal system, with some of my own flairs.</span>


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


== PLIN ==
== PLIN ==
Line 24: Line 26:


'''You do NOT need it.''' At that point, just use FJS.
'''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.
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''.
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.
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:
* 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 comma thirds ('''a'''rto/'''t'''endo) Available '''only''' in 159-PLIN [fix]''
* 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 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.
Of course, the names would have many synonyms, so hypo/hyper = infra/ultra, minor second = limma, major second = tone, unison = prime, major third = ditone.
{| class="wikitable"
{| class="wikitable"
|+12-form PLIN (MOS 5L 7s 6|12-PLIN; MOS 5L 7s 6|5; tolerance = '''56.843c'''
|+ 12-form PLIN (MOS 5L 7s 6 |12-PLIN; MOS 5L 7s 6|5; tolerance = '''56.843c'''
!Spoken name
!Spoken name
!Simplified
!Simplified
Line 35: Line 62:
!Example 3
!Example 3
|-
|-
|unison
|(rough) unison
|1r
|1r
|0.000c
|0.000c
Line 44: Line 71:
|128/125
|128/125
|-
|-
|minor second
|(rough) minor second
|m2r
|m2r
|90.225c
|90.225c
Line 53: Line 80:
|11/10
|11/10
|-
|-
|major second
|(rough) major second
|M2r
|M2r
|203.91c
|203.91c
Line 62: Line 89:
|15/13
|15/13
|-
|-
|minor third
|(rough) minor third
|m3r
|m3r
|294.135c
|294.135c
Line 71: Line 98:
|11/9
|11/9
|-
|-
|major third
|(rough) major third
|M3r
|M3r
|407.82c
|407.82c
Line 80: Line 107:
|16/13
|16/13
|-
|-
|fourth
|(rough) fourth
|4r
|4r
|498.045c
|498.045c
Line 89: Line 116:
|21/16
|21/16
|-
|-
|tritone
|(rough) tritone
|Tr
|Tr
|611.73c
|611.73c
Line 98: Line 125:
|23/16
|23/16
|-
|-
|fifth
|(rough) fifth
|5r
|5r
|701.955c
|701.955c
Line 107: Line 134:
|32/21
|32/21
|-
|-
|minor sixth
|(rough) minor sixth
|m6r
|m6r
|792.18c
|792.18c
Line 116: Line 143:
|13/8
|13/8
|-
|-
|major sixth
|(rough) major sixth
|M6r
|M6r
|905.865c
|905.865c
Line 125: Line 152:
|18/11
|18/11
|-
|-
|minor seventh
|(rough) minor seventh
|m7r
|m7r
|996.09c
|996.09c
Line 134: Line 161:
|11/6
|11/6
|-
|-
|major seventh
|(rough) major seventh
|M7r
|M7r
|1109.775c
|1109.775c
Line 143: Line 170:
|48/25
|48/25
|-
|-
|octave, counison
|(rough) octave, counison
|8r, c1r
|8r, c1r
|1200c
|1200c
Line 152: Line 179:
|64/31
|64/31
|}
|}
As it can be seen, the interval regions work more like buckets on which intervals fall to. As the interval regions get thinner, more buckets appear and the notation is more precise.
*
 
In PLIN, the following terms discouraged from use:
 
* Perfect, augmented fourth/diminished fifth, chroma, semitone, neutral/mid, interordinals. Use aug/dim for enharmonic reespelling only in functional settings
* Chroma is only a functional denomination, which may be in between hyperunisons and hyperminor seconds.


{| class="wikitable"
{| class="wikitable"
Line 605: Line 627:
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; MOS 41L 12s 26|26; tolerance = '''3.91c'''
|+ 159-PLIN; 41L 12s 26 |159-PLIN up to the first major second; MOS 41L 12s 26|26; tolerance = '''3.91c'''
!Spoken name
!Spoken name
!Simplified
!Simplified
Line 785: Line 807:


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: 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.  
{| class="wikitable"
|+665-PLIN until the first pythagorean comma
!Interval name
!Simplified
|-
|'''sat unison'''
|p1
|-
|qi unison
|Q1
|-
|mi unison
|Π1
|-
|qimi unison
|QΠ1
|-
|twomi unison
|2Π1
|-
|qitwomi unison
|Q2Π1
|-
|threemi unison
|3Π1
|-
|threemu superunison
|3πS1
|-
|qutwomu superunison
|q2πS1
|-
|twomu superunison
|2πS1
|-
|qumu superunison
|qπS1
|-
|mu superunison
|πS1
|-
|qu superunison
|qS1
|-
|'''sat superunison'''
|pS1
|}
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ 7315-PLIN MOS 306L 359 332 |332
|+ 7315-PLIN MOS 306L 359 332 |7315-PLIN until the first pythagorean comma
!Interval name
!Interval name
!Simplified
|-
|-
|spot unison
|'''spot unison'''
|'''p1'''
|-
|-
|plus unison
|plus unison
|<nowiki>+1</nowiki>
|-
|-
|twoplus unison
|twoplus unison
|2+1
|-
|-
|threeplus unison
|threeplus unison
|3+1
|-
|-
|fourplus unison
|fourplus unison
|4+1
|-
|-
|fiveplus unison
|fiveplus unison
|5+1
|-
|-
|fivemins qi unison
|fivemins qi unison
|5-Q1
|-
|-
|fourmins qi unison
|fourmins qi unison
|4-Q1
|-
|-
|threemins qi unison
|threemins qi unison
|3-Q1
|-
|-
|twomins qi unison
|twomins qi unison
|2-Q1
|-
|-
|minus qi unison
|minus qi unison
|<nowiki>-Q1</nowiki>
|-
|-
|spotqi unison
|'''spot qi unison'''
|'''Q1'''
|-
|-
|spotqumi unison
|'''spot qumi unison'''
|'''qΠ1'''
|-
|-
|plus qumi unison
|plus qumi unison
|<nowiki>+qΠ1</nowiki>
|-
|-
|twoplus qumi unison
|twoplus qumi unison
|2+qΠ1
|-
|-
|threeplus qumi unison
|threeplus qumi unison
|3+qΠ1
|-
|-
|fourplus qumi unison
|fourplus qumi unison
|4+qΠ1
|-
|-
|fiveplus qumi unison
|fiveplus qumi unison
|5+qΠ1
|-
|-
|fivemins mi unison
|fivemins mi unison
|5-Π1
|-
|-
|fourmins mi unison
|fourmins mi unison
|4-Π1
|-
|-
|threemins mi unison
|threemins mi unison
|3-Π1
|-
|-
|twomins mi unison
|twomins mi unison
|2-Π1
|-
|-
|minus mi unison
|minus mi unison
|<nowiki>-Π1</nowiki>
|-
|-
|mi unison
|'''spot mi unison'''
|'''Π1'''
|-
|-
|plus mi unison
|plus mi unison
|<nowiki>+Π1</nowiki>
|-
|-
|twoplus mi unison
|twoplus mi unison
|2+Π1
|-
|-
|threeplus mi unison
|threeplus mi unison
|3+Π1
|-
|-
|fourplus mi unison
|fourplus mi unison
|4+Π1
|-
|-
|fiveplus mi unison
|fiveplus mi unison
|5+Π1
|-
|-
|fivemins qimi unison
|fivemins qimi unison
|5-QΠ1
|-
|-
|fourmins qimi unison
|fourmins qimi unison
|4-QΠ1
|-
|-
|threemins qimi unison
|threemins qimi unison
|3-QΠ1
|-
|-
|twomins qimi unison
|twomins qimi unison
|2-QΠ1
|-
|-
|minus qimi unison
|minus qimi unison
|<nowiki>-QΠ1</nowiki>
|-
|-
|qimi unison
|'''spot qimi unison'''
|'''QΠ1'''
|-
|-
|qutwomi unison
|'''spot qutwomi unison'''
|'''q2Π1'''
|-
|-
|plus qutwomi unison
|plus qutwomi unison
|<nowiki>+q2Π1</nowiki>
|-
|-
|twoplus qutwomi unison
|twoplus qutwomi unison
|2+q2Π1
|-
|-
|threeplus qutwomi unison
|threeplus qutwomi unison
|3+q2Π1
|-
|-
|fourplus qutwomi unison
|fourplus qutwomi unison
|4+q2Π1
|-
|-
|fiveplus qutwomi unison
|fiveplus qutwomi unison
|5+q2Π1
|-
|-
|fivemins twomi unison
|fivemins twomi unison
|5-2Π1
|-
|-
|fourmins twomi unison
|fourmins twomi unison
|4-2Π1
|-
|-
|threemins twomi unison
|threemins twomi unison
|3-2Π1
|-
|-
|twomins twomi unison
|twomins twomi unison
|2-2Π1
|-
|-
|minus twomi unison
|minus twomi unison
|<nowiki>-2Π1</nowiki>
|-
|-
|twomi unison
|'''spot twomi unison'''
|'''2Π1'''
|-
|-
|plus twomi unison
|plus twomi unison
|<nowiki>+2Π1</nowiki>
|-
|-
|twoplus twomi unison
|twoplus twomi unison
|2+2Π1
|-
|-
|threeplus twomi unison
|threeplus twomi unison
|3+2Π1
|-
|-
|fourplus twomi unison
|fourplus twomi unison
|4+2Π1
|-
|-
|fiveplus twomi unison
|fiveplus twomi unison
|5+2Π1
|-
|-
|fivemins qitwomi unison
|fivemins qitwomi unison
|5-Q2Π1
|-
|-
|fourmins qitwomi unison
|fourmins qitwomi unison
|4-Q2Π1
|-
|-
|threemins qitwomi unison
|threemins qitwomi unison
|3-Q2Π1
|-
|-
|twomins qitwomi unison
|twomins qitwomi unison
|2-Q2Π1
|-
|-
|minus qitwomi unison
|minus qitwomi unison
|<nowiki>-Q2Π1</nowiki>
|-
|-
|qitwomi unison
|'''spot qitwomi unison'''
|'''Q2Π1'''
|-
|-
|quthreemi unison
|'''spot quthreemi unison'''
|'''q3Π1'''
|-
|-
|plus quthreemi unison
|plus quthreemi unison
|<nowiki>+q3Π1</nowiki>
|-
|-
|twoplus quthreemi unison
|twoplus quthreemi unison
|2+q3Π1
|-
|-
|threeplus quthreemi unison
|threeplus quthreemi unison
|3+q3Π1
|-
|-
|fourplus quthreemi unison
|fourplus quthreemi unison
|4+q3Π1
|-
|-
|fiveplus quthreemi unison
|fiveplus quthreemi unison
|5+q3Π1
|-
|-
|fivemins threemi unison
|fivemins threemi unison
|5-3Π1
|-
|-
|fourmins threemi unison
|fourmins threemi unison
|4-3Π1
|-
|-
|threemins threemi unison
|threemins threemi unison
|3-3Π1
|-
|-
|twomins threemi unison
|twomins threemi unison
|2-3Π1
|-
|-
|minus threemi unison
|minus threemi unison
|<nowiki>-3Π1</nowiki>
|-
|-
|threemi unison
|'''threemi unison'''
|'''3Π1'''
|-
|-
|plus threemi unison
|plus threemi unison
|<nowiki>+3Π1</nowiki>
|-
|-
|twoplus threemi unison
|twoplus threemi unison
|2+3Π1
|-
|-
|twomins threemu unison
|twomins threemu unison
|2-3π1
|-
|-
|minus threemu unison
|minus threemu unison
|<nowiki>-3π1</nowiki>
|-
|-
|threemu superunison
|'''spot threemu superunison'''
|'''3πS1'''
|-
|-
|plus threemu superunison
|plus threemu superunison
|<nowiki>+3πS1</nowiki>
|-
|-
|twoplus threemu superunison
|twoplus threemu superunison
|2+3πS1
|-
|-
|threeplus threemu superunison
|threeplus threemu superunison
|3+3πS1
|-
|-
|fourplus threemu superunison
|fourplus threemu superunison
|4+3πS1
|-
|-
|fiveplus threemu superunison
|fiveplus threemu superunison
|5+3πS1
|-
|-
|fivemins qithreemu superunison
|fivemins qithreemu superunison
|5-Q3πS1
|-
|-
|fourmins qithreemu superunison
|fourmins qithreemu superunison
|4-Q3πS1
|-
|-
|threemins qithreemu superunison
|threemins qithreemu superunison
|3-Q3πS1
|-
|-
|twomins qithreemu superunison
|twomins qithreemu superunison
|2-Q3πS1
|-
|-
|minus qithreemu superunison
|minus qithreemu superunison
|<nowiki>-Q3πS1</nowiki>
|-
|-
|qithreemu superunison
|'''spot qithreemu superunison'''
|'''Q3πS1'''
|-
|-
|qutwomu superunison
|'''spot qutwomu superunison'''
|'''q2πS1'''
|-
|-
|plus qutwomu superunison
|plus qutwomu superunison
|<nowiki>+q2πS1</nowiki>
|-
|-
|twoplus qutwomu superunison
|twoplus qutwomu superunison
|2+q2πS1
|-
|-
|threeplus qutwomu superunison
|threeplus qutwomu superunison
|3+q2πS1
|-
|-
|fourplus qutwomu superunison
|fourplus qutwomu superunison
|4+q2πS1
|-
|-
|fiveplus qutwomu superunison
|fiveplus qutwomu superunison
|5+q2πS1
|-
|-
|fivemins twomu superunison
|fivemins twomu superunison
|5-2πS1
|-
|-
|fourmins twomu superunison
|fourmins twomu superunison
|4-2πS1
|-
|-
|threemins twomu superunison
|threemins twomu superunison
|3-2πS1
|-
|-
|twomins twomu superunison
|twomins twomu superunison
|2-2πS1
|-
|-
|minus twomu superunison
|minus twomu superunison
|<nowiki>-2πS1</nowiki>
|-
|-
|twomu superunison
|'''twomu superunison'''
|'''2πS1'''
|-
|-
|plus twomu superunison
|plus twomu superunison
|<nowiki>+2πS1</nowiki>
|-
|-
|twoplus twomu superunison
|twoplus twomu superunison
|2+2πS1
|-
|-
|threeplus twomu superunison
|threeplus twomu superunison
|3+2πS1
|-
|-
|fourplus twomu superunison
|fourplus twomu superunison
|4+2πS1
|-
|-
|fiveplus twomu superunison
|fiveplus twomu superunison
|5+2πS1
|-
|-
|fivemins qitwomu superunison
|fivemins qitwomu superunison
|5-Q2πS1
|-
|-
|fourmins qitwomu superunison
|fourmins qitwomu superunison
|4-Q2πS1
|-
|-
|threemins qitwomu superunison
|threemins qitwomu superunison
|3-Q2πS1
|-
|-
|twomins qitwomu superunison
|twomins qitwomu superunison
|2-Q2πS1
|-
|-
|minus qitwomu superunison
|minus qitwomu superunison
|<nowiki>-Q2πS1</nowiki>
|-
|-
|qitwomu superunison
|'''spot qitwomu superunison'''
|'''Q2πS1'''
|-
|-
|qumu superunison
|'''spot qumu superunison'''
|'''qπS1'''
|-
|-
|plus qumu superunison
|plus qumu superunison
|<nowiki>+qπS1</nowiki>
|-
|-
|twoplus qumu superunison
|twoplus qumu superunison
|2+qπS1
|-
|-
|threeplus qumu superunison
|threeplus qumu superunison
|3+qπS1
|-
|-
|fourplus qumu superunison
|fourplus qumu superunison
|4+qπS1
|-
|-
|fiveplus qumu superunison
|fiveplus qumu superunison
|5+qπS1
|-
|-
|fivemins mu superunison
|fivemins mu superunison
|5-πS1
|-
|-
|fourmins mu superunison
|fourmins mu superunison
|4-πS1
|-
|-
|threemins mu superunison
|threemins mu superunison
|3-πS1
|-
|-
|twomins mu superunison
|twomins mu superunison
|2-πS1
|-
|-
|minus mu superunison
|minus mu superunison
|<nowiki>-πS1</nowiki>
|-
|-
|mu superunison
|'''spot mu superunison'''
|'''πS1'''
|-
|-
|plus mu superunison
|plus mu superunison
|<nowiki>+πS1</nowiki>
|-
|-
|twoplus mu superunison
|twoplus mu superunison
|2+πS1
|-
|-
|threeplus mu superunison
|threeplus mu superunison
|3+πS1
|-
|-
|fourplus mu superunison
|fourplus mu superunison
|4+πS1
|-
|-
|fiveplus mu superunison
|fiveplus mu superunison
|5+πS1
|-
|-
|fivemins qimu superunison
|fivemins qimu superunison
|5-QπS1
|-
|-
|fourmins qimu superunison
|fourmins qimu superunison
|4-QπS1
|-
|-
|threemins qimu superunison
|threemins qimu superunison
|3-QπS1
|-
|-
|twomins qimu superunison
|twomins qimu superunison
|2-QπS1
|-
|-
|minus qimu superunison
|minus qimu superunison
|<nowiki>-QπS1</nowiki>
|-
|-
|qimu superunison
|'''spot qimu superunison'''
|'''QπS1'''
|-
|-
|qusuperunison
|'''spot qu superunison'''
|'''qS1'''
|-
|-
|plus qusuperunison
|plus qusuperunison
|<nowiki>+qS1</nowiki>
|-
|-
|twoplus qusuperunison
|twoplus qusuperunison
|2+qS1
|-
|-
|threeplus qusuperunison
|threeplus qusuperunison
|3+qS1
|-
|-
|fourplus qusuperunison
|fourplus qusuperunison
|4+qS1
|-
|-
|fiveplus qusuperunison
|fiveplus qusuperunison
|5+qS1
|-
|-
|fivemins superunison
|fivemins superunison
|5-S1
|-
|-
|fourmins superunison
|fourmins superunison
|4-S1
|-
|-
|threemins superunison
|threemins superunison
|3-S1
|-
|-
|twomins superunison
|twomins superunison
|2-S1
|-
|-
|minus superunison
|minus superunison
|<nowiki>-S1</nowiki>
|-
|-
|superunison
|'''spot superunison'''
|'''pS1'''
|}
|}
And that amount of intervals is needed to reach ''one'' pythagorean comma. It is most surely overkill. 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, where ~5/4 is a "twoplus qi submajor third", and 7/4 is a "minus mu subminor seventh".  


WIP
WIP
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