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

12-PLIN; MOS 5L 7s 6|5; tolerance = 56.843c

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

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

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.

159-PLIN up to the first major second; MOS 41L 12s 26|26; tolerance = 3.91c

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.

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

7315-PLIN until the first pythagorean comma

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