User:Eufalesio/PLIN
This article proposes a way to name any intervals and pitches, logically extending conventional notations following Pythagorean tuning, 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 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.
You do NOT need it. At that point, just use FJS.
| Spoken name | Simplified | Region center | Pitch-class (from D) | Fifth region | Example 1 | Example 1 | Example 3 |
|---|---|---|---|---|---|---|---|
| unison | 1r | 0.000c | D | 0 | 1/1 | 33/32 | 128/125 |
| minor second | m2r | 90.225c | Eb | -5 | 16/15 | 25/24 | 11/10 |
| major second | M2r | 203.91c | E | 2 | 9/8 | 8/7 | 15/13 |
| minor third | m3r | 294.135c | F | -3 | 6/5 | 7/6 | 11/9 |
| major third | M3r | 407.82c | F# | 4 | 5/4 | 9/7 | 16/13 |
| fourth | 4r | 498.045c | G | -1 | 4/3 | 11/8 | 21/16 |
| tritone | Tr | 611.73c | G# | 6 | 7/5 | 10/7 | 23/16 |
| fifth | 5r | 701.955c | A | 1 | 3/2 | 16/11 | 32/21 |
| minor sixth | m6r | 792.18c | Ab | -4 | 8/5 | 14/9 | 13/8 |
| major sixth | M6r | 905.865c | B | 3 | 5/3 | 12/7 | 18/11 |
| minor seventh | m7r | 996.09c | C | -2 | 9/5 | 7/4 | 11/6 |
| major seventh | M7r | 1109.775c | C# | 5 | 15/8 | 27/14 | 48/25 |
| octave, counison | 8r, c1r | 1200c | D | 0 | 2/1 | 63/32 | 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.
| 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 |
|---|
| spot unison |
| plus unison |
| twoplus unison |
| threeplus unison |
| fourplus unison |
| fiveplus unison |
| fivemins qi unison |
| fourmins qi unison |
| threemins qi unison |
| twomins qi unison |
| minus qi unison |
| spotqi unison |
| spotqumi unison |
| plus qumi unison |
| twoplus qumi unison |
| threeplus qumi unison |
| fourplus qumi unison |
| fiveplus qumi unison |
| fivemins mi unison |
| fourmins mi unison |
| threemins mi unison |
| twomins mi unison |
| minus mi unison |
| mi unison |
| plus mi unison |
| twoplus mi unison |
| threeplus mi unison |
| fourplus mi unison |
| fiveplus mi unison |
| fivemins qimi unison |
| fourmins qimi unison |
| threemins qimi unison |
| twomins qimi unison |
| minus qimi unison |
| qimi unison |
| qutwomi unison |
| plus qutwomi unison |
| twoplus qutwomi unison |
| threeplus qutwomi unison |
| fourplus qutwomi unison |
| fiveplus qutwomi unison |
| fivemins twomi unison |
| fourmins twomi unison |
| threemins twomi unison |
| twomins twomi unison |
| minus twomi unison |
| twomi unison |
| plus twomi unison |
| twoplus twomi unison |
| threeplus twomi unison |
| fourplus twomi unison |
| fiveplus twomi unison |
| fivemins qitwomi unison |
| fourmins qitwomi unison |
| threemins qitwomi unison |
| twomins qitwomi unison |
| minus qitwomi unison |
| qitwomi unison |
| quthreemi unison |
| plus quthreemi unison |
| twoplus quthreemi unison |
| threeplus quthreemi unison |
| fourplus quthreemi unison |
| fiveplus quthreemi unison |
| fivemins threemi unison |
| fourmins threemi unison |
| threemins threemi unison |
| twomins threemi unison |
| minus threemi unison |
| threemi unison |
| plus threemi unison |
| twoplus threemi unison |
| twomins threemu unison |
| minus threemu unison |
| threemu superunison |
| plus threemu superunison |
| twoplus threemu superunison |
| threeplus threemu superunison |
| fourplus threemu superunison |
| fiveplus threemu superunison |
| fivemins qithreemu superunison |
| fourmins qithreemu superunison |
| threemins qithreemu superunison |
| twomins qithreemu superunison |
| minus qithreemu superunison |
| qithreemu superunison |
| qutwomu superunison |
| plus qutwomu superunison |
| twoplus qutwomu superunison |
| threeplus qutwomu superunison |
| fourplus qutwomu superunison |
| fiveplus qutwomu superunison |
| fivemins twomu superunison |
| fourmins twomu superunison |
| threemins twomu superunison |
| twomins twomu superunison |
| minus twomu superunison |
| twomu superunison |
| plus twomu superunison |
| twoplus twomu superunison |
| threeplus twomu superunison |
| fourplus twomu superunison |
| fiveplus twomu superunison |
| fivemins qitwomu superunison |
| fourmins qitwomu superunison |
| threemins qitwomu superunison |
| twomins qitwomu superunison |
| minus qitwomu superunison |
| qitwomu superunison |
| qumu superunison |
| plus qumu superunison |
| twoplus qumu superunison |
| threeplus qumu superunison |
| fourplus qumu superunison |
| fiveplus qumu superunison |
| fivemins mu superunison |
| fourmins mu superunison |
| threemins mu superunison |
| twomins mu superunison |
| minus mu superunison |
| mu superunison |
| plus mu superunison |
| twoplus mu superunison |
| threeplus mu superunison |
| fourplus mu superunison |
| fiveplus mu superunison |
| fivemins qimu superunison |
| fourmins qimu superunison |
| threemins qimu superunison |
| twomins qimu superunison |
| minus qimu superunison |
| qimu superunison |
| qusuperunison |
| plus qusuperunison |
| twoplus qusuperunison |
| threeplus qusuperunison |
| fourplus qusuperunison |
| fiveplus qusuperunison |
| fivemins superunison |
| fourmins superunison |
| threemins superunison |
| twomins superunison |
| minus superunison |
| superunison |
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".
WIP