User:Eufalesio/PLIN - Revision history

Eufalesio at 10:36, 9 January 2026

2026-01-09T10:36:35Z

← Older revision		Revision as of 10:36, 9 January 2026
Line 19:		Line 19:

	== 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 ''~~'equwee'~~''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 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:		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:
Line 49:		Line 53:
	* 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 ====
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	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 1,474:		Line 1,478:
	* kirn- for Atomic (Kirnberger's atom)		* kirn- for Atomic (Kirnberger's atom)
	* pir- for Pirate		* 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

Eufalesio at 22:38, 8 January 2026

2026-01-08T22:38:31Z

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Eufalesio at 17:41, 4 January 2026

2026-01-04T17:41:22Z

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Eufalesio at 14:40, 26 December 2025

2025-12-26T14:40:52Z

← Older revision		Revision as of 14:40, 26 December 2025
Line 31:		Line 31:

	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.		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.

			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.

	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''.		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''.
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	* 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 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.		Of course, the names would have many synonyms, so hypo/hyper = infra/ultra, minor second = limma, major second = tone, unison = prime, major third = ditone.

			==== 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

	{\| class="wikitable"		{\| class="wikitable"
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	!Example 3		!Example 3
	\|-		\|-
	\|(rough) unison		\|rough unison
	\|1r		\|1r
	\|0.000c		\|0.000c
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	\|128/125		\|128/125
	\|-		\|-
	\|(rough) minor second		\|rough minor second
	\|m2r		\|m2r
	\|90.225c		\|90.225c
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	\|11/10		\|11/10
	\|-		\|-
	\|(rough) major second		\|rough major second
	\|M2r		\|M2r
	\|203.91c		\|203.91c
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	\|15/13		\|15/13
	\|-		\|-
	\|(rough) minor third		\|rough minor third
	\|m3r		\|m3r
	\|294.135c		\|294.135c
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	\|11/9		\|11/9
	\|-		\|-
	\|(rough) major third		\|rough major third
	\|M3r		\|M3r
	\|407.82c		\|407.82c
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	\|16/13		\|16/13
	\|-		\|-
	\|(rough) fourth		\|rough fourth
	\|4r		\|4r
	\|498.045c		\|498.045c
Line 116:		Line 126:
	\|21/16		\|21/16
	\|-		\|-
	\|(rough) tritone		\|rough tritone
	\|Tr		\|Tr
	\|611.73c		\|611.73c
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	\|23/16		\|23/16
	\|-		\|-
	\|(rough) fifth		\|rough fifth
	\|5r		\|5r
	\|701.955c		\|701.955c
Line 134:		Line 144:
	\|32/21		\|32/21
	\|-		\|-
	\|(rough) minor sixth		\|rough minor sixth
	\|m6r		\|m6r
	\|792.18c		\|792.18c
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	\|13/8		\|13/8
	\|-		\|-
	\|(rough) major sixth		\|rough major sixth
	\|M6r		\|M6r
	\|905.865c		\|905.865c
Line 152:		Line 162:
	\|18/11		\|18/11
	\|-		\|-
	\|(rough) minor seventh		\|rough minor seventh
	\|m7r		\|m7r
	\|996.09c		\|996.09c
Line 161:		Line 171:
	\|11/6		\|11/6
	\|-		\|-
	\|(rough) major seventh		\|rough major seventh
	\|M7r		\|M7r
	\|1109.775c		\|1109.775c
Line 170:		Line 180:
	\|48/25		\|48/25
	\|-		\|-
	\|(rough) octave, counison		\|rough octave, counison
	\|8r, c1r		\|8r, c1r
	\|1200c		\|1200c

Eufalesio at 14:12, 26 December 2025

2025-12-26T14:12:25Z

Show changes

Eufalesio at 01:05, 26 December 2025

2025-12-26T01:05:06Z

← Older revision		Revision as of 01:05, 26 December 2025
Line 21:		Line 21:
	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.

	If you say you need one beyond 7315, you are ~~'''definitely''' lying~~. The next edo on the list is the gargantuan [[190537edo]]. You do NOT need it. At that point, just use FJS.		If you say you need one beyond 7315, [[Sagittal notation#cite note-:0-4\|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.
	{\| 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'''
Line 154:		Line 156:
	In PLIN, the following terms discouraged from use:		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.		* 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 ~~small unisons~~ and ~~large minor~~ seconds.		* Chroma is only a functional denomination, which may be in between hyperunisons and hyperminor seconds.

	{\| class="wikitable"		{\| class="wikitable"
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	\|		\|
	\|}		\|}
	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.		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.		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.
Line 640:		Line 642:
	\|56/55		\|56/55
	\|-		\|-
	\|artohyperunison		\|artohyperunison / artohypominor second
	\|aH1		\|aH1
	\|DaH		\|DaH /Ebah
	\|39.1000		\|39.1000
	\|45/44		\|45/44
Line 652:		Line 654:
	\|40/39		\|40/39
	\|-		\|-
	\|tendohypominor second		\|tendohyperunison / tendohypominor second
	\|th2		\|th2
	\|Ebth		\|Ebth
Line 780:		Line 782:
	\| colspan="5" \|et cetera...		\| colspan="5" \|et cetera...
	\|}		\|}
	~~Things complicate with 665~~-PLIN and ~~7315-~~PLIN~~, because the amount of prefixes you need starts~~ to ~~increase~~. So far, it has been only 3 classes at most: tendo/arto, hypo/sub/fix/super/hyper, nominal. ~~As you know~~, ~~two is a pair, three is a crowd, and each new PLIN continues adding more classes to worry about.~~ 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~~. That's enough~~.		If 53-PLIN is not enough for you, this will be surely be enough. If not... then prepare for what's to come.
	{\| class="wikitable"
	\|+7315-PLIN MOS 306L 359 332\|332		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 mw-collapsible mw-collapsed"
			\|+ 7315-PLIN MOS 306L 359 332 \|332
			!Interval name
	\|-		\|-
	\|spot unison		\|spot unison

Eufalesio: Birth

2025-12-26T00:28:36Z

Birth

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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 ===
[[Helmholtz–Ellis notation|HEJI]], [[Ben Johnston's notation|Johnston's notation]], [[Functional Just System|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 notation|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 [[Vavoom comma|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[https://xkcd.com/927/ .]

== 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.

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 '''definitely''' lying. The next edo on the list is the gargantuan [[190537edo]]. You do NOT need it. At that point, just use FJS.
{| class="wikitable"
|+12-form PLIN (MOS 5L 7s 6|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
|-
|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 small unisons and large minor seconds.

{| class="wikitable"
|+53-PLIN; 41L 12s 26|53-PLIN; MOS 41L 12s 26|26; tolerance = '''11.73c'''
!Spoken name
!Simplified
!Pitch-class (from D)
!Region center
! colspan="3" |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.

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.
{| class="wikitable"
|+ 159-PLIN; 41L 12s 26 |159-PLIN; 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
|aH1
|DaH
|39.1000
|45/44
|-
|'''fixhyperunison / fixhypominor second'''
|fH1 / fh2
|DH / Ebh
|46.920021
|40/39
|-
|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
|-
| colspan="5" |et cetera...
|}
Things complicate with 665-PLIN and 7315-PLIN, because the amount of prefixes you need starts to increase. So far, it has been only 3 classes at most: tendo/arto, hypo/sub/fix/super/hyper, nominal. As you know, two is a pair, three is a crowd, and each new PLIN continues adding more classes to worry about. 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. That's enough.
{| class="wikitable"
|+7315-PLIN MOS 306L 359 332|332
!
|-
|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