Pentatonic Functional Just System: Difference between revisions

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{{Idiosyncratic terms|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}

Traditionally, we use a [[5L 2s|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s]] [[~~MOS~~ scale]] may be preferred, with priority on the [[2.3.7 subgroup|2.3.7-]][[subgroup]]. In this page, we will develop a ~~pentatonic~~ version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.

Traditionally, we use a [[5L 2s|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.

The PFJS was devised by [[User:Overthink|Overthink]] in 2025.

== The 3-limit ==

We start by examining pythagorean intervals based on [[2L 3s]] classification. Note that the subscript 5 before the interval name means it is pentatonic, and that a factor of [[5/1|5]] in the denominator of a ratio would be a subscript 5 ''after'' the interval name.

{| class="wikitable right-all"

|+ Pythagorean intervals

|-

! Ratio !! Cents !! Interval name (~~Pentatonic~~)

! Ratio !! Cents !! Interval name (pentic)

|-

| [[1/1]] || 0.0 || 5P1

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| [[2/1]] || 1200.0 || 5P6

|}

In contrast to diatonic, [[256/243]] is a chroma interval, separating major and minor intervals of the same category. Interestingly, only ~~pentatonic~~ seconds and fifths now have major/minor, and augmented and diminished intervals show up way more often. From here on we will refer to augmented and diminished as "super" and "sub" (not to be confused with "supermajor" and "subminor"), with symbols "S" and "s" respectively.

In contrast to diatonic, [[256/243]] is a [[chroma]] interval, separating major and minor intervals of the same category. Interestingly, only pentic seconds and fifths now have major/minor, and augmented and diminished intervals show up way more often. From here on we will refer to augmented and diminished as "super" and "sub" (not to be confused with "supermajor" and "subminor"), with symbols "S" and "s" respectively.

== Ratios of 7 ==

Since we are using a pentic system of notation, and [[5edo]] represents the 2.3.7 subgroup very well, we will investigate ratios with factors of 7 before ratios with factors of 5. Just like in the FJS, we will be using [[64/63|63/64]] as our formal comma.

Since we are using a ~~pentatonic~~ system of notation, and [[5edo]] represents the [[2.3.7 subgroup]] very well, we will investigate ratios with factors of 7 before ratios with factors of 5. Just like in the FJS, we will be using [[64/63|63/64]] as our formal comma.

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Interval classification would be much simpler if the Pythagorean intervals were equated with their simpler septimal counterparts; this occurs in [[superpyth]] temperament, where 64/63 is [[tempering out|tempered out]].

With similar constructions, larger chords can be constructed, ~~including~~ [[28:36:42:49|a version of the dominant seventh chord]]; however, this is beyond the scope of this page.

With similar constructions, larger chords can be constructed, such as [[28:36:42:49|1–9/7–3/2–7/4]], which is a version of the [[dominant seventh chord]]; however, this is beyond the scope of this page.

== Ratios of 5 ==

Now, we will look at ratios of 5. Just like in the FJS, our formal comma is [[81/80|80/81]]. The most salient fact is that 5/4 and 6/5 are no longer in the same interval category; 6/5 is a 5second, while 5/4 is a 5third.

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If we try to construct 5-limit triads the normal way, the [[4:5:6]] major triad becomes 5P1–5s35–5P4, and the [[10:12:15]] minor triad becomes 5P1–5M25–5P4. Now we see it was a good idea refer to augmented and diminished as "super" and "sub"; these intervals occur so much more often. However, now the [[4:5:6]] and [[10:12:15]] triads aren't classified by the same interval categories, while they are in diatonic.

The [[7/5]] and [[10/7]] intervals are not included in the tables due to containing factors of both 5 and 7; 7/5 is written as 5S375, while 10/7 is written as 5s457. An advantage of ~~pentatonic~~ notation is that these intervals are in the right order in terms of interval categories, unlike in traditional diatonic-based FJS, where 7/5 is d575 and 10/7 is A457.

The [[7/5]] and [[10/7]] intervals are not included in the tables due to containing factors of both 5 and 7; 7/5 is written as 5S375, while 10/7 is written as 5s457. An advantage of pentic notation is that these intervals are in the right order in terms of interval categories, unlike in traditional diatonic-based FJS, where 7/5 is d575 and 10/7 is A457.

In full 7-limit superpyth, 10/9 is a subsecond, 6/5 is a supersecond, 5/4 is a sub-subthird (a subthird is 9/7), and 7/5 is a super-superthird (a superthird is [[27/20]]~[[48/35]]). Their [[octave complement]]s can be classified accordingly.

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A lot of interesting things show up here. First of all, we finally have just representations for "[[neutral]]" intervals, which are between the minor and major intervals in their category. Here, [[15/13]], which is beteeen [[8/7]] and [[7/6]], can be considered a neutral 5second (especially if [[676/675]] is tempered out), [[13/10]] a semi-sub 5third, [[20/13]] a semi-super 5fourth, and [[26/15]] a neutral 5fifth. Intervals which are neutral here are considered [[interseptimal]] by diatonic classification, as they fall right between two diatonic interval categories.

Now, there are intervals between the ~~pentatonic~~ categories, such as [[11/9]] and [[12/11]]. The edges of each interval category can be considered the 5-limit intervals (such as [[16/15]], [[10/9]], [[6/5]], and [[5/4]]), thus the regions between interval categories can be termed "interpental" (not to be confused with [[Interpental|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in ~~pentatonic~~, such as 12/11 being between 16/15 and 10/9, and 11/9 being between 6/5 and 5/4. One may realize that [[11/8]] and [[16/11]] are classified rather out of place, with 11/8 being a 5subfourth and 16/11 being a 5superthird. The PFJS is not perfect, and this system was designed to keep [[13/11]] and [[15/13]] in the right category, thus 11/8 must be messed up (though other intervals of 11 are interpental, so are fine). However, 11/8 is in the region between 4/3 and 3/2, where there can be considered to be ''two'' interpental regions: one between [[27/20]] and [[45/32]], and another between [[64/45]] and [[40/27]]. These are the [[superfourth]] and [[subfifth]] regions in diatonic, which can also be considered neutral regions. In ~~pentatonic~~, since these regions are interpental, they are ambiguously between 5thirds and 5fourths, justifying the otherwise out-of-place classification of 11/8. However, one may not be fond of the fact that [[7/5]] and [[10/7]] are just barely in these ranges; thus, one may prefer to make them narrower (~48 cents wide).

Now, there are intervals between the pentic categories, such as [[11/9]] and [[12/11]]. The edges of each interval category can be considered the 5-limit intervals (such as [[16/15]], [[10/9]], [[6/5]], and [[5/4]]), thus the regions between interval categories can be termed "interpental" (not to be confused with [[Interpental|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentic, such as 12/11 being between 16/15 and 10/9, and 11/9 being between 6/5 and 5/4. One may realize that [[11/8]] and [[16/11]] are classified rather out of place, with 11/8 being a 5subfourth and 16/11 being a 5superthird. The PFJS is not perfect, and this system was designed to keep [[13/11]] and [[15/13]] in the right category, thus 11/8 must be messed up (though other intervals of 11 are interpental, so are fine). However, 11/8 is in the region between 4/3 and 3/2, where there can be considered to be ''two'' interpental regions: one between [[27/20]] and [[45/32]], and another between [[64/45]] and [[40/27]]. These are the [[superfourth]] and [[subfifth]] regions in diatonic, which can also be considered neutral regions. In pentic, since these regions are interpental, they are ambiguously between 5thirds and 5fourths, justifying the otherwise out-of-place classification of 11/8. However, one may not be fond of the fact that [[7/5]] and [[10/7]] are just barely in these ranges; thus, one may prefer to make them narrower (~48 cents wide).

{| class="wikitable right-1"

|+ Interpental regions

|-

! Region !! Is between !! Name (~~Diatonic~~) !! Name (~~Pentatonic~~)

! Region !! Is between !! Name (diatonic) !! Name (pentic)

|-

| 123–171{{c}} || S1–m2 || Neutral 2nd || Terric (Earth)

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In [[13-limit]] [[superpyth]], [[11/8]] is a sub-sub-sub-5fourth, and [[13/8]] is a sub-sub-5fifth.

This system could be extended to even higher limits, but we~~'ll~~ leave it here for now.

This system could be extended to even higher limits, but we will leave it here for now.

@@ Line 1: / Line 1: @@
 {{Idiosyncratic terms|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, the interpental region names, and all of the PFJS interval names are only found on this page.}}
-Traditionally, we use a [[5L 2s|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s]] [[MOS scale]] may be preferred, with priority on the [[2.3.7 subgroup|2.3.7-]][[subgroup]]. In this page, we will develop a pentatonic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.
+Traditionally, we use a [[5L 2s|diatonic]] system of interval classification. This works well in the [[5-limit]] and in [[meantone]]. However, in other systems like [[superpyth]], a pentatonic system of classification based on the [[2L 3s|pentic (2L 3s)]] [[mos scale]] may be preferred, with priority on the [[2.3.7 subgroup|2.3.7]] [[subgroup]]. In this page, we will develop a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits.
 The PFJS was devised by [[User:Overthink|Overthink]] in 2025.
 == The 3-limit ==
+We start by examining pythagorean intervals based on [[2L 3s]] classification. Note that the subscript 5 before the interval name means it is pentatonic, and that a factor of [[5/1|5]] in the denominator of a ratio would be a subscript 5 ''after'' the interval name.
-We start by examining pythagorean intervals based on [[2L 3s]] classification. Note that the subscript 5 before the interval name means it is pentatonic, and that a factor of [[5/1|5]] in the denominator of a ratio would be a subscript 5 ''after'' the interval name.
 {| class="wikitable right-all"
 |+ Pythagorean intervals
 |-
-! Ratio !! Cents !! Interval name<br>(Pentatonic)
+! Ratio !! Cents !! Interval name<br>(pentic)
 |-
 | [[1/1]] || 0.0 || <sub>5</sub>P1
@@ Line 48: / Line 48: @@
 | [[2/1]] || 1200.0 || <sub>5</sub>P6
 |}
-In contrast to diatonic, [[256/243]] is a chroma interval, separating major and minor intervals of the same category. Interestingly, only pentatonic seconds and fifths now have major/minor, and augmented and diminished intervals show up way more often. From here on we will refer to augmented and diminished as "super" and "sub" (not to be confused with "supermajor" and "subminor"), with symbols "S" and "s" respectively.
+In contrast to diatonic, [[256/243]] is a [[chroma]] interval, separating major and minor intervals of the same category. Interestingly, only pentic seconds and fifths now have major/minor, and augmented and diminished intervals show up way more often. From here on we will refer to augmented and diminished as "super" and "sub" (not to be confused with "supermajor" and "subminor"), with symbols "S" and "s" respectively.
 == Ratios of 7 ==
+Since we are using a pentic system of notation, and [[5edo]] represents the 2.3.7 subgroup very well, we will investigate ratios with factors of 7 before ratios with factors of 5. Just like in the FJS, we will be using [[64/63|63/64]] as our formal comma.
-Since we are using a pentatonic system of notation, and [[5edo]] represents the [[2.3.7 subgroup]] very well, we will investigate ratios with factors of 7 before ratios with factors of 5. Just like in the FJS, we will be using [[64/63|63/64]] as our formal comma.
 <div><div style="display: inline-grid; margin-right: 25px;">
@@ Line 144: / Line 143: @@
 Interval classification would be much simpler if the Pythagorean intervals were equated with their simpler septimal counterparts; this occurs in [[superpyth]] temperament, where 64/63 is [[tempering out|tempered out]].
-With similar constructions, larger chords can be constructed, including [[28:36:42:49|a version of the dominant seventh chord]]; however, this is beyond the scope of this page.
+With similar constructions, larger chords can be constructed, such as [[28:36:42:49|1–9/7–3/2–7/4]], which is a version of the [[dominant seventh chord]]; however, this is beyond the scope of this page.
 == Ratios of 5 ==
 Now, we will look at ratios of 5. Just like in the FJS, our formal comma is [[81/80|80/81]]. The most salient fact is that 5/4 and 6/5 are no longer in the same interval category; 6/5 is a <sub>5</sub>second, while 5/4 is a <sub>5</sub>third.
@@ Line 240: / Line 238: @@
 If we try to construct 5-limit triads the normal way, the [[4:5:6]] major triad becomes <sub>5</sub>P1–<sub>5</sub>s3<sup>5</sup>–<sub>5</sub>P4, and the [[10:12:15]] minor triad becomes <sub>5</sub>P1–<sub>5</sub>M2<sub>5</sub>–<sub>5</sub>P4. Now we see it was a good idea refer to augmented and diminished as "super" and "sub"; these intervals occur so much more often. However, now the [[4:5:6]] and [[10:12:15]] triads aren't classified by the same interval categories, while they are in diatonic.
-The [[7/5]] and [[10/7]] intervals are not included in the tables due to containing factors of both 5 and 7; 7/5 is written as <sub>5</sub>S3<sup>7</sup><sub>5</sub>, while 10/7 is written as <sub>5</sub>s4<sup>5</sup><sub>7</sub>. An advantage of pentatonic notation is that these intervals are in the right order in terms of interval categories, unlike in traditional diatonic-based FJS, where 7/5 is d5<sup>7</sup><sub>5</sub> and 10/7 is A4<sup>5</sup><sub>7</sub>.
+The [[7/5]] and [[10/7]] intervals are not included in the tables due to containing factors of both 5 and 7; 7/5 is written as <sub>5</sub>S3<sup>7</sup><sub>5</sub>, while 10/7 is written as <sub>5</sub>s4<sup>5</sup><sub>7</sub>. An advantage of pentic notation is that these intervals are in the right order in terms of interval categories, unlike in traditional diatonic-based FJS, where 7/5 is d5<sup>7</sup><sub>5</sub> and 10/7 is A4<sup>5</sup><sub>7</sub>.
 In full 7-limit superpyth, 10/9 is a subsecond, 6/5 is a supersecond, 5/4 is a sub-subthird (a subthird is 9/7), and 7/5 is a super-superthird (a superthird is [[27/20]]~[[48/35]]). Their [[octave complement]]s can be classified accordingly.
@@ Line 363: / Line 361: @@
 A lot of interesting things show up here. First of all, we finally have just representations for "[[neutral]]" intervals, which are between the minor and major intervals in their category. Here, [[15/13]], which is beteeen [[8/7]] and [[7/6]], can be considered a neutral <sub>5</sub>second (especially if [[676/675]] is tempered out), [[13/10]] a semi-sub <sub>5</sub>third, [[20/13]] a semi-super <sub>5</sub>fourth, and [[26/15]] a neutral <sub>5</sub>fifth. Intervals which are neutral here are considered [[interseptimal]] by diatonic classification, as they fall right between two diatonic interval categories.
-Now, there are intervals between the pentatonic categories, such as [[11/9]] and [[12/11]]. The edges of each interval category can be considered the 5-limit intervals (such as [[16/15]], [[10/9]], [[6/5]], and [[5/4]]), thus the regions between interval categories can be termed "interpental" (not to be confused with [[Interpental|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentatonic, such as 12/11 being between 16/15 and 10/9, and 11/9 being between 6/5 and 5/4. One may realize that [[11/8]] and [[16/11]] are classified rather out of place, with 11/8 being a <sub>5</sub>subfourth and 16/11 being a <sub>5</sub>superthird. The PFJS is not perfect, and this system was designed to keep [[13/11]] and [[15/13]] in the right category, thus 11/8 must be messed up (though other intervals of 11 are interpental, so are fine). However, 11/8 is in the region between 4/3 and 3/2, where there can be considered to be ''two'' interpental regions: one between [[27/20]] and [[45/32]], and another between [[64/45]] and [[40/27]]. These are the [[superfourth]] and [[subfifth]] regions in diatonic, which can also be considered neutral regions. In pentatonic, since these regions are interpental, they are ambiguously between <sub>5</sub>thirds and <sub>5</sub>fourths, justifying the otherwise out-of-place classification of 11/8. However, one may not be fond of the fact that [[7/5]] and [[10/7]] are just barely in these ranges; thus, one may prefer to make them narrower (~48 cents wide).
+Now, there are intervals between the pentic categories, such as [[11/9]] and [[12/11]]. The edges of each interval category can be considered the 5-limit intervals (such as [[16/15]], [[10/9]], [[6/5]], and [[5/4]]), thus the regions between interval categories can be termed "interpental" (not to be confused with [[Interpental|the temperament of the same name]], which is in fact generated by an interpental interval). The neutral intervals of diatonic are interpental intervals in pentic, such as 12/11 being between 16/15 and 10/9, and 11/9 being between 6/5 and 5/4. One may realize that [[11/8]] and [[16/11]] are classified rather out of place, with 11/8 being a <sub>5</sub>subfourth and 16/11 being a <sub>5</sub>superthird. The PFJS is not perfect, and this system was designed to keep [[13/11]] and [[15/13]] in the right category, thus 11/8 must be messed up (though other intervals of 11 are interpental, so are fine). However, 11/8 is in the region between 4/3 and 3/2, where there can be considered to be ''two'' interpental regions: one between [[27/20]] and [[45/32]], and another between [[64/45]] and [[40/27]]. These are the [[superfourth]] and [[subfifth]] regions in diatonic, which can also be considered neutral regions. In pentic, since these regions are interpental, they are ambiguously between <sub>5</sub>thirds and <sub>5</sub>fourths, justifying the otherwise out-of-place classification of 11/8. However, one may not be fond of the fact that [[7/5]] and [[10/7]] are just barely in these ranges; thus, one may prefer to make them narrower (~48 cents wide).
 {| class="wikitable right-1"
 |+ Interpental regions
 |-
-! Region !! Is between !! Name (Diatonic) !! Name (Pentatonic)
+! Region !! Is between !! Name (diatonic) !! Name (pentic)
 |-
 | 123–171{{c}} || S1–m2 || Neutral 2nd || Terric (Earth)
@@ Line 385: / Line 383: @@
 In [[13-limit]] [[superpyth]], [[11/8]] is a sub-sub-sub-<sub>5</sub>fourth, and [[13/8]] is a sub-sub-<sub>5</sub>fifth.
-This system could be extended to even higher limits, but we'll leave it here for now.
+This system could be extended to even higher limits, but we will leave it here for now.