User:Eufalesio/PLIN
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.
Preamble
There are many ways to name intervals and pitches, both in edos and in JI, which can be generalized into four buckets:
JI notations
HEJI, Johnston's notation, FJS, Color notation... etc. They can be used to notate every interval in JI exactly, up to a limit (except FJS which can notate ∞-limit JI). They are perfect, in the sense that the intervals they describe are unique, but the mappings can be unintuitive, and its derivations may become extremely complex. They are also very weird to speak generally (except for color notation, by design). What they have in rigor, they mostly lack in flexibility and intuitiveness.
Tempered notations
Ups and downs, edosteps, generator(s)+period(s) of a special temperament... etc. They can be used to notate a finite set of intervals, with a finite maximum variety. Unlike in JI notations, these notations are dependent on the working temperament/edo to make sense, so in theory there are infinitely many ways to call an interval depending on the mapping, and none of those will be exact due to the nature of temperaments. They are however, extremely versatile, and very practical, great for everyday use, and very easy to speak.
Loosely defined terms
Inheriting the chain of fifths notation (unison, second, third... + major/minor/perfect/(neutral) (+ augmented/diminished)), and adding other descriptors to further converge to the meaning. This category is not rigorously defined and very idiosyncratic. It is common to see super-, sub-, ultra-, infra-, lesser, greater, harmonic/otonal, utonal... and primes spoken like Latinate adjectives. And of course, proper names... yes, I'm talking to you!
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.
PLIN
Short for Pythagoreanoid Loose Interval Notation, it's a type of 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.
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, 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.
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 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, EPLINs 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 eqrough major seventh. 7/4 is a fixsubminor seventh, but in 159-EPLIN, an equartosubminor 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.
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 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]
- Pythagorean commas (sub/Super, hypo/Hyper) Available in 53-PLIN [∅], functionally only in 12-PLIN
- Pythagorean comma thirds (arto/tendo) 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/Qi) 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. Extending my rules; apotome = superminor second; superlimma.
Syntax of a PLIN
- 12-PLIN : rough + diatonic nominal
- 53-PLIN: hypo/sub/∅/super/hyper + diatonic nominal
- 159-PLIN arto/fix/tendo + 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
| Spoken name | Simplified | Region center | Pitch-class (from D) | Fifth region | Example 1 | Example 1 | Example 3 |
|---|---|---|---|---|---|---|---|
| rough unison | 1r | 0.000c | D | 0 | 1/1 | 33/32 | 128/125 |
| rough minor second | m2r | 90.225c | Eb | -5 | 16/15 | 25/24 | 11/10 |
| rough major second | M2r | 203.91c | E | 2 | 9/8 | 8/7 | 15/13 |
| rough minor third | m3r | 294.135c | F | -3 | 6/5 | 7/6 | 11/9 |
| rough major third | M3r | 407.82c | F# | 4 | 5/4 | 9/7 | 16/13 |
| rough fourth | 4r | 498.045c | G | -1 | 4/3 | 11/8 | 21/16 |
| rough tritone | Tr | 611.73c | G# | 6 | 7/5 | 10/7 | 23/16 |
| rough fifth | 5r | 701.955c | A | 1 | 3/2 | 16/11 | 32/21 |
| rough minor sixth | m6r | 792.18c | Ab | -4 | 8/5 | 14/9 | 13/8 |
| rough major sixth | M6r | 905.865c | B | 3 | 5/3 | 12/7 | 18/11 |
| rough minor seventh | m7r | 996.09c | C | -2 | 9/5 | 7/4 | 11/6 |
| rough major seventh | M7r | 1109.775c | C# | 5 | 15/8 | 27/14 | 48/25 |
| rough octave, counison | 8r, c1r | 1200c | D | 0 | 2/1 | 63/32 |
| Spoken name | Simplified | Pitch-class (from D) | Region center | Examples | ||
|---|---|---|---|---|---|---|
| unison | 1þ | D | 0.0000c | 1/1 | 225/224 | |
| superunison | S1þ | DS | 23.46001 | 81/80 | 64/63 | |
| hyperunison / hypominor second | H1þ / h2þ | DH / Ebh | 46.920021 | 33/32 | 1053/1024 | 128/125 |
| subminor second | sm2þ | Ebs | 66.764985 | 25/24 | ||
| minor second | m2þ | Eb | 90.224996 | 19/18 | 256/243 | 135/128 |
| superminor second | Sm2þ | EbS | 113.685006 | 16/15 | 2187/2048 | |
| hyperminor second | Hm2þ | EbH | 137.145016 | 13/12 | ||
| hypomajor second | hM2þ | Eh | 156.989981 | 11/10 | 12/11 | 35/32 |
| submajor second | sM2þ | Es | 180.449991 | 10/9 | ||
| major second | M2þ | E | 203.910002 | 9/8 | ||
| supermajor second | SM2þ | ES | 227.370012 | 8/7 | 256/225 | |
| hypermajor second / hypominor third | HM2þ / hm3þ | EH / Fh | 250.830023 | 15/13 | ||
| subminor third | sm3þ | Fs | 270.674987 | 7/6 | ||
| minor third | m3þ | F | 294.134997 | 19/16 | 32/27 | |
| superminor third | Sm3þ | FS | 317.595008 | 6/5 | 29/16 | |
| hyperminor third | Hm3þ | FH | 341.055018 | 11/9 | 39/32 | |
| hypomajor third | hM3þ | F#h | 360.899983 | 16/13 | ||
| submajor third | sM3þ | F#s | 384.359993 | 5/4 | ||
| major third | M3þ | F# | 407.820003 | 19/15 | 81/64 | |
| supermajor third | SM3þ | F#S | 431.280014 | 9/7 | ||
| hypermajor third / hypofourth | HM3þ / h4þ | F#H / Gh | 451.124978 | 13/10 | ||
| subfourth | s4þ | Gs | 474.584989 | 21/16 | ||
| fourth | 4þ | G | 498.044999 | 4/3 | ||
| superfourth | S4þ | GS | 521.50501 | 27/20 | ||
| hyperfourth | H4þ | GH | 544.96502 | 11/8 | ||
| hypotritone | hTþ | G#h | 564.809984 | 18/13 | ||
| subtritone | sTþ | G#s | 588.269995 | 7/5 | 1024/729 | |
| tritone | Tþ | G# | 611.730005 | 10/7 | 729/512 | |
| supertritone | STþ | G#S | 635.190016 | 13/9 | ||
| hypertritone / hypofifth | HTþ / h5þ | G#H / As | 655.03498 | 16/11 | ||
| subfifth | s5þ | As | 678.49499 | 40/27 | ||
| fifth | 5þ | A | 701.955001 | 3/2 | ||
| superfifth | S5þ | AS | 725.415011 | 32/21 | ||
| hyperfifth / hypominor sixth | H5þ / hm6þ | AH / Bbh | 748.875022 | 20/13 | ||
| subminor sixth | sm6þ | Bbs | 768.719986 | 14/9 | ||
| minor sixth | m6þ | Bb | 792.179997 | 19/12 | 128/81 | |
| superminor sixth | Sm6þ | BbS | 815.640007 | 8/5 | ||
| hyperminor sixth | Hm6þ | BbH | 839.100017 | 13/8 | ||
| hypomajor sixth | hM6þ | Bh | 858.944982 | 33/20 | ||
| submajor sixth | sM6þ | Bs | 882.404992 | 5/3 | ||
| major sixth | M6þ | B | 905.865003 | 27/16 | ||
| supermajor sixth | SM6þ | BS | 929.325013 | 12/7 | ||
| hypermajor sixth / hypominor seventh | HM6þ / hm7þ | BH / Ch | 949.169977 | 26/15 | ||
| subminor seventh | sm7þ | Cs | 972.629988 | 7/4 | 225/128 | |
| minor seventh | m7þ | C | 996.089998 | 16/9 | ||
| superminor seventh | Sm7þ | CS | 1019.550009 | 9/5 | ||
| hyperminor seventh | Hm7þ | CH | 1043.010019 | 29/16 | ||
| hypomajor seventh | hM7þ | C#h | 1062.854984 | 11/6 | 117/64 | |
| submajor seventh | sM7þ | C#s | 1086.314994 | 24/13 | 81/44 | |
| major seventh | M7þ | C# | 1109.775004 | 15/8 | ||
| supermajor seventh | SM7þ | C#S | 1133.235015 | 19/10 | 243/128 | |
| hypermajor seventh / hypöoctave | HM7þ / h8þ / ch1þ | C#H / Dh | 1153.079979 | 64/33 | ||
| suboctave / cosubunison | s8þ / cs1þ | Ds | 1176.53999 | 63/32 | ||
| octave / counison | 8þ / c1þ | D | 1200 | 2/1 | ||
From this table, super/sub alter by a pythagorean-comma-sized interval, and hyper/hypo by two times. This is a good point to stop at, as it balances versatility with accuracy. It's also the one I feel most naturally gravitating to.
If having to choose between any of the intervals on which there are two options (where there are two regions that differ by a Mercator's comma), choose the one that is built with the least number of fifths, unless it is functionally useful to do so.
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.
| Spoken name | Simplified | Pitch-class (from D) | Region | Examples |
|---|---|---|---|---|
| fixunison | f1 | D | 0.0000 | 729/728 |
| tendounison | t1 | Dt | 7.8200 | 225/224 |
| artosuperunison | aS1 | DaS | 15.6400 | 121/120 |
| fixsuperunison | fS1 | DS | 23.46001 | 64/63 |
| tendosuperunison | tS1 | DtS | 31.2800 | 56/55 |
| artohyperunison / artohypominor second | aH1 | DaH /Ebah | 39.1000 | 45/44 |
| fixhyperunison / fixhypominor second | fH1 / fh2 | DH / Ebh | 46.920021 | 40/39 |
| tendohyperunison / tendohypominor second | th2 | Ebth | 54.740021 | 33/32 |
| artosubminor second | asm2 | Ebas | 58.9449 | 91/88 |
| fixsubminor second | fsm2 | Ebs | 66.764985 | 80/77 |
| tendosubminorsecond | tsm2 | Ebts | 74.5840 | 448/429 |
| artominor second | am2 | Eba | 82.405 | 22/21 |
| fixminor second | fm2 | Eb | 90.224996 | 96/91 |
| tendominor second | tm2 | Ebt | 98.045 | 128/121 |
| artosuperminor second | aSm2 | EbaSa | 105.865 | 1225/1152 |
| fixsuperminor second | fSm2 | EbS | 113.685006 | 16/15 |
| tendosuperminor second | tSm2 | EbtS | 121.505 | 15/14 |
| artohyperminor second | aHm2 | EbaH | 129.325 | 14/13 |
| fixhyperminor second | fHm2 | EbH | 137.145016 | 13/12 |
| tendohyperminor second | tHm2 | EbtH | 144.965 | 160/147 |
| artohypomajor second | ahM2 | Eah | 149.17 | 12/11 |
| fixhypomajor second | fhM2 | Eh | 156.989981 | 35/32 |
| tendohypomajor second | thM2 | Eth | 164.809 | 11/10 |
| artosubmajor second | asM2 | Eas | 172.629 | 182/165 |
| fixsubmajor second | fsM2 | Es | 180.449991 | 231/208 |
| tendosubmajor second | tsM2 | Ets | 188.269 | 39/35 |
| artomajor second | aM2 | Ea | 196.09 | 160/143 |
| fixmajor second | fM2 | E | 203.910002 | 9/8 |
| et cetera... | ||||
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.
| 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 |
| Interval name | Simplified |
|---|---|
| spot unison | P1 |
| plus unison | +1 |
| twoplus unison | 2+1 |
| threeplus unison | 3+1 |
| fourplus unison | 4+1 |
| fiveplus unison | 5+1 |
| fivemins qi unison | 5-Q1 |
| fourmins qi unison | 4-Q1 |
| threemins qi unison | 3-Q1 |
| twomins qi unison | 2-Q1 |
| minus qi unison | -Q1 |
| spot qi unison | Q1 |
| spot qumi unison | qΠ1 |
| plus qumi unison | +qΠ1 |
| twoplus qumi unison | 2+qΠ1 |
| threeplus qumi unison | 3+qΠ1 |
| fourplus qumi unison | 4+qΠ1 |
| fiveplus qumi unison | 5+qΠ1 |
| fivemins mi unison | 5-Π1 |
| fourmins mi unison | 4-Π1 |
| threemins mi unison | 3-Π1 |
| twomins mi unison | 2-Π1 |
| minus mi unison | -Π1 |
| spot mi unison | Π1 |
| plus mi unison | +Π1 |
| twoplus mi unison | 2+Π1 |
| threeplus mi unison | 3+Π1 |
| fourplus mi unison | 4+Π1 |
| fiveplus mi unison | 5+Π1 |
| fivemins qimi unison | 5-QΠ1 |
| fourmins qimi unison | 4-QΠ1 |
| threemins qimi unison | 3-QΠ1 |
| twomins qimi unison | 2-QΠ1 |
| minus qimi unison | -QΠ1 |
| spot qimi unison | QΠ1 |
| spot qutwomi unison | q2Π1 |
| plus qutwomi unison | +q2Π1 |
| twoplus qutwomi unison | 2+q2Π1 |
| threeplus qutwomi unison | 3+q2Π1 |
| fourplus qutwomi unison | 4+q2Π1 |
| fiveplus qutwomi unison | 5+q2Π1 |
| fivemins twomi unison | 5-2Π1 |
| fourmins twomi unison | 4-2Π1 |
| threemins twomi unison | 3-2Π1 |
| twomins twomi unison | 2-2Π1 |
| minus twomi unison | -2Π1 |
| spot twomi unison | 2Π1 |
| plus twomi unison | +2Π1 |
| twoplus twomi unison | 2+2Π1 |
| threeplus twomi unison | 3+2Π1 |
| fourplus twomi unison | 4+2Π1 |
| fiveplus twomi unison | 5+2Π1 |
| fivemins qitwomi unison | 5-Q2Π1 |
| fourmins qitwomi unison | 4-Q2Π1 |
| threemins qitwomi unison | 3-Q2Π1 |
| twomins qitwomi unison | 2-Q2Π1 |
| minus qitwomi unison | -Q2Π1 |
| spot qitwomi unison | Q2Π1 |
| spot quthreemi unison | q3Π1 |
| plus quthreemi unison | +q3Π1 |
| twoplus quthreemi unison | 2+q3Π1 |
| threeplus quthreemi unison | 3+q3Π1 |
| fourplus quthreemi unison | 4+q3Π1 |
| fiveplus quthreemi unison | 5+q3Π1 |
| fivemins threemi unison | 5-3Π1 |
| fourmins threemi unison | 4-3Π1 |
| threemins threemi unison | 3-3Π1 |
| twomins threemi unison | 2-3Π1 |
| minus threemi unison | -3Π1 |
| threemi unison | 3Π1 |
| plus threemi unison | +3Π1 |
| twoplus threemi unison | 2+3Π1 |
| twomins threemu unison | 2-3π1 |
| minus threemu unison | -3π1 |
| spot threemu superunison | 3πS1 |
| plus threemu superunison | +3πS1 |
| twoplus threemu superunison | 2+3πS1 |
| threeplus threemu superunison | 3+3πS1 |
| fourplus threemu superunison | 4+3πS1 |
| fiveplus threemu superunison | 5+3πS1 |
| fivemins qithreemu superunison | 5-Q3πS1 |
| fourmins qithreemu superunison | 4-Q3πS1 |
| threemins qithreemu superunison | 3-Q3πS1 |
| twomins qithreemu superunison | 2-Q3πS1 |
| minus qithreemu superunison | -Q3πS1 |
| spot qithreemu superunison | Q3πS1 |
| spot qutwomu superunison | q2πS1 |
| plus qutwomu superunison | +q2πS1 |
| twoplus qutwomu superunison | 2+q2πS1 |
| threeplus qutwomu superunison | 3+q2πS1 |
| fourplus qutwomu superunison | 4+q2πS1 |
| fiveplus qutwomu superunison | 5+q2πS1 |
| fivemins twomu superunison | 5-2πS1 |
| fourmins twomu superunison | 4-2πS1 |
| threemins twomu superunison | 3-2πS1 |
| twomins twomu superunison | 2-2πS1 |
| minus twomu superunison | -2πS1 |
| twomu superunison | 2πS1 |
| plus twomu superunison | +2πS1 |
| twoplus twomu superunison | 2+2πS1 |
| threeplus twomu superunison | 3+2πS1 |
| fourplus twomu superunison | 4+2πS1 |
| fiveplus twomu superunison | 5+2πS1 |
| fivemins qitwomu superunison | 5-Q2πS1 |
| fourmins qitwomu superunison | 4-Q2πS1 |
| threemins qitwomu superunison | 3-Q2πS1 |
| twomins qitwomu superunison | 2-Q2πS1 |
| minus qitwomu superunison | -Q2πS1 |
| spot qitwomu superunison | Q2πS1 |
| spot qumu superunison | qπS1 |
| plus qumu superunison | +qπS1 |
| twoplus qumu superunison | 2+qπS1 |
| threeplus qumu superunison | 3+qπS1 |
| fourplus qumu superunison | 4+qπS1 |
| fiveplus qumu superunison | 5+qπS1 |
| fivemins mu superunison | 5-πS1 |
| fourmins mu superunison | 4-πS1 |
| threemins mu superunison | 3-πS1 |
| twomins mu superunison | 2-πS1 |
| minus mu superunison | -πS1 |
| spot mu superunison | πS1 |
| plus mu superunison | +πS1 |
| twoplus mu superunison | 2+πS1 |
| threeplus mu superunison | 3+πS1 |
| fourplus mu superunison | 4+πS1 |
| fiveplus mu superunison | 5+πS1 |
| fivemins qimu superunison | 5-QπS1 |
| fourmins qimu superunison | 4-QπS1 |
| threemins qimu superunison | 3-QπS1 |
| twomins qimu superunison | 2-QπS1 |
| minus qimu superunison | -QπS1 |
| spot qimu superunison | QπS1 |
| spot qu superunison | qS1 |
| plus qusuperunison | +qS1 |
| twoplus qusuperunison | 2+qS1 |
| threeplus qusuperunison | 3+qS1 |
| fourplus qusuperunison | 4+qS1 |
| fiveplus qusuperunison | 5+qS1 |
| fivemins superunison | 5-S1 |
| fourmins superunison | 4-S1 |
| threemins superunison | 3-S1 |
| twomins superunison | 2-S1 |
| minus superunison | -S1 |
| spot superunison | 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.
Example intervals in several PLINs
| 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