Pentatonic Functional Just System: Difference between revisions

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~~{{Editable user page|Feel free to add more content, but please do not delete existing content without my permission (though minor rewrites are fine).}}~~

{{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. All of these terms were coined by [[User:Overthink]].}}

{{Idiosyncratic terms|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, 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]] systems. However, in other systems like [[superpyth]] or [[buzzard]], 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]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 interval classes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.

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.

The PFJS was devised by [[User:Overthink|Overthink]] in 2025, with updates made later.

== 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 left-3"

{| class="wikitable right-all"

|+ style="font-size: 105%;" | Pythagorean intervals

|+ 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.

{| class="wikitable right-all"

{| class="wikitable right-all left-3"

|+ Ratios with a factor of 7

|+ style="font-size: 105%;" | Ratios with a factor of 7

|-

! Ratio !! Cents !! Interval name

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| [[27/14]] || 1137.0 || 5s67

|-

| [[63/32]] || ~~1200~~.0 || 5P67

| [[63/32]] || 1172.7 || 5P67

|}

</div><div style="display: inline-grid; margin-right: 25px;">

{| class="wikitable right-all"

{| class="wikitable right-all left-3"

|+ Ratios with two factors of 7

|+ style="font-size: 105%;" | Ratios with two factors of 7

|-

! Ratio !! Cents !! Interval name

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</div></div>

We look at the interval classes with major and minor again. After modification by 64/63, the minor 5second becomes [[8/7]], the major 5second [[7/6]], the minor 5fifth [[12/7]], and the major 5fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor 5second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking 5fifths rather than 5seconds. The stacked intervals are now the [[7/4]] major 5fifth and the [[12/7]] minor 5fifth, which reach the [[3/1]] perfect 5ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.

We look at the interval classes with major and minor again. After modification by 64/63, the minor 5second becomes [[8/7]], the major 5second [[7/6]], the minor 5fifth [[12/7]], and the major 5fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor 5second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]] contrast by [[25/24]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking 5fifths rather than 5seconds. The stacked intervals are now the [[7/4]] major 5fifth and the [[12/7]] minor 5fifth, which reach the [[3/1]] perfect 5ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.

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.

{| class="wikitable right-all"

{| class="wikitable right-all left-3"

|+ Ratios with a factor of 5

|+ style="font-size: 105%;" | Ratios with a factor of 5

|-

! Ratio !! Cents !! Interval name

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|}

</div><div style="display: inline-grid; margin-right: 25px;">

{| class="wikitable right-all"

{| class="wikitable right-all left-3"

|+ Ratios with two factors of 5

|+ style="font-size: 105%;" | Ratios with two factors of 5

|-

! Ratio !! Cents !! Interval name

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One can see that the ratios of 5 are further from 5edo intervals than ratios of 7. Thus, the 5-limit intervals can now be considered "subminor" and "supermajor", compared to the intervals of 7 in diatonic. Using the 5fifth construction, we get the [[3:5:9]] subminor and [[5:9:15|1/(9:5:3) = 5:9:15]] supermajor chords, the compact voicings of which are [[9:10:12]] and [[15:18:20]] respectively.

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.

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. We now see why it was a good idea refer to augmented and diminished as "super" and "sub"; these intervals occur much more often in a pentic system. However, the [[4:5:6]] and [[10:12:15]] triads are no longer 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 above 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.

== Higher limits ==

We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use different formal commas ~~than~~ in the FJS: The formal comma for 11 is [[729/704|704/729]] (11/8 is 5s411), and the formal comma for 13 is [[27/26|26/27]] (13/8 is 5m513).

We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use different formal commas from in the FJS: The formal comma for 11 is [[729/704|704/729]] (11/8 is 5s411), and the formal comma for 13 is [[27/26|26/27]] (13/8 is 5m513).

{| class="wikitable right-all left-3"

|+ {{nowrap|15-odd-limit by PFJS}} {{nowrap|(lower half-octave)}}

|+ style="font-size: 105%;" | {{nowrap|15-odd-limit by PFJS}} {{nowrap|(lower half-octave)}}

|-

! Ratio !! Cents !! ~~Name~~

! Ratio !! Cents !! Interval name

|-

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

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| [[8/7]] || 231.2 || 5m27

|-

| [[15/13]] || 247.8 || 5m2513

| [[15/13]] || 247.7 || 5m2513

|-

| [[7/6]] || 266.9 || 5M27

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| [[11/9]] || 347.4 || 5s311

|-

| [[16/13]] || 359.3 || 5M213

| [[16/13]] || 359.5 || 5M213

|-

| [[5/4]] || 386.3 || 5s35

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</div><div style="display: inline-grid; margin-right: 25px;">

{| class="wikitable right-all left-3"

|+ {{nowrap|15-odd-limit by PFJS}} {{nowrap|(upper half-octave)}}

|+ style="font-size: 105%;" | {{nowrap|15-odd-limit by PFJS}} {{nowrap|(upper half-octave)}}

|-

! Ratio !! Cents !! ~~Name~~

! Ratio !! Cents !! Interval name

|-

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

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</div></div>

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.

A lot of interesting things show up here. First of all, we finally have just representations for "[[neutral (interval quality)|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|interpental temperament]], 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"

{| class="wikitable right-1 right-2"

|+ Interpental regions

|+ style="font-size: 105%;" | 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 ~~higher~~ limits, ~~but we'll leave it here for now~~.

In the PFJS, primes beyond 13 are classified somewhat like the FJS, based on pythagorean intervals from -5 to +6 perfect 5fourths ([[3/2]]'s) , with priority 0, 1, -1, 2, -2, etc. Unlike FJS, however, the RoT is not the same in both directions. The RoT of a pythagorean interval of <math>i</math> cents is <math>i-68</math> through <math>i+46</math> cents. This range was chosen so that it works for the 13-limit, and it spans just over an [[2187/2048|apotome]], the large step in the pythagorean chromatic scale. The exact range was set considering a few large primes: Prime [[37/1|37]] is just barely not a 5m237, but rather a 5M237; similarly, prime [[41/1|41]] is just barely not a 5P341, but rather a 5s341.

{| class="wikitable right-1"

|+ style="font-size: 105%" | Prime harmonics in PFJS

|-

! Prime !! PFJS name !! Perfect 5fourths

|-

| [[5/4]] || 5s35 || +4

|-

| [[7/4]] || 5M57 || -2

|-

| [[11/8]] || 5s411 || +6

|-

| [[13/8]] || 5m513 || +3

|-

| [[17/16]] || 5S117 || -5

|-

| [[19/16]] || 5M219 || -3

|-

| [[23/16]] || 5s423 || +6

|-

| [[29/16]] || 5M529 || -2

|-

| [[31/16]] || 5P631 || ±0

|-

| [[37/32]] || 5M237 || -3

|-

| [[41/32]] || 5s341 || +4

|-

| [[43/32]] || 5P343 || -1

|-

| [[47/32]] || 5P447 || +1

|}

Similar logic can be used for even higher primes, up to infinity. While an uneven RoT is unconventional, it is required to work well with the lower limits in the PFJS. This also solves the issue of gaps between categories in the FJS, which occurs as the RoT in the FJS, [[65/63]], is less than half of [[2187/2048]]. After all, every notation system is a compromise.

@@ Line 1: / Line 1: @@
-{{Editable user page|Feel free to add more content, but please do not delete existing content without my permission (though minor rewrites are fine).}}
+{{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. All of these terms were coined by [[User:Overthink]].}}
-{{Idiosyncratic terms|The abbreviation "PFJS", using the terms "sub/super" for augmented and diminished, 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]] systems. However, in other systems like [[superpyth]] or [[buzzard]], 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]]. This page describes a pentic version of the [[FJS]] (abbreviated '''PFJS'''), starting from the [[3-limit]] and using [[formal comma]]s to reach higher limits. Since we have 5 interval classes per octave rather than the traditional 7, we omit the notes F and B, and only use C, D, E, G, and A.
-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.
+The PFJS was devised by [[User:Overthink|Overthink]] in 2025, with updates made later.
 == 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 left-3"
-{| class="wikitable right-all"
+|+ style="font-size: 105%;" | Pythagorean intervals
-|+ Pythagorean intervals
 |-
-! Ratio !! Cents !! Interval name<br>(Pentatonic)
+! Ratio !! Cents !! Interval name<br>(pentic)
 |-
 | [[1/1]] || 0.0 || <sub>5</sub>P1
@@ Line 47: / 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;">
-{| class="wikitable right-all"
+{| class="wikitable right-all left-3"
-|+ Ratios with a factor of 7
+|+ style="font-size: 105%;" | Ratios with a factor of 7
 |-
 ! Ratio !! Cents !! Interval name
@@ Line 93: / Line 93: @@
 | [[27/14]] || 1137.0 || <sub>5</sub>s6<sub>7</sub>
 |-
-| [[63/32]] || 1200.0 || <sub>5</sub>P6<sup>7</sup>
+| [[63/32]] || 1172.7 || <sub>5</sub>P6<sup>7</sup>
 |}
 </div><div style="display: inline-grid; margin-right: 25px;">
-{| class="wikitable right-all"
+{| class="wikitable right-all left-3"
-|+ Ratios with two factors of 7
+|+ style="font-size: 105%;" | Ratios with two factors of 7
 |-
 ! Ratio !! Cents !! Interval name
@@ Line 139: / Line 139: @@
 </div></div>
-We look at the interval classes with major and minor again. After modification by 64/63, the minor <sub>5</sub>second becomes [[8/7]], the major <sub>5</sub>second [[7/6]], the minor <sub>5</sub>fifth [[12/7]], and the major <sub>5</sub>fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor <sub>5</sub>second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking <sub>5</sub>fifths rather than <sub>5</sub>seconds. The stacked intervals are now the [[7/4]] major <sub>5</sub>fifth and the [[12/7]] minor <sub>5</sub>fifth, which reach the [[3/1]] perfect <sub>5</sub>ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.
+We look at the interval classes with major and minor again. After modification by 64/63, the minor <sub>5</sub>second becomes [[8/7]], the major <sub>5</sub>second [[7/6]], the minor <sub>5</sub>fifth [[12/7]], and the major <sub>5</sub>fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor <sub>5</sub>second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]] contrast by [[25/24]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking <sub>5</sub>fifths rather than <sub>5</sub>seconds. The stacked intervals are now the [[7/4]] major <sub>5</sub>fifth and the [[12/7]] minor <sub>5</sub>fifth, which reach the [[3/1]] perfect <sub>5</sub>ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.
 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.
 <div><div style="display: inline-grid; margin-right: 25px;">
-{| class="wikitable right-all"
+{| class="wikitable right-all left-3"
-|+ Ratios with a factor of 5
+|+ style="font-size: 105%;" | Ratios with a factor of 5
 |-
 ! Ratio !! Cents !! Interval name
@@ Line 192: / Line 191: @@
 |}
 </div><div style="display: inline-grid; margin-right: 25px;">
-{| class="wikitable right-all"
+{| class="wikitable right-all left-3"
-|+ Ratios with two factors of 5
+|+ style="font-size: 105%;" | Ratios with two factors of 5
 |-
 ! Ratio !! Cents !! Interval name
@@ Line 237: / Line 236: @@
 One can see that the ratios of 5 are further from 5edo intervals than ratios of 7. Thus, the 5-limit intervals can now be considered "subminor" and "supermajor", compared to the intervals of 7 in diatonic. Using the <sub>5</sub>fifth construction, we get the [[3:5:9]] subminor and [[5:9:15|1/(9:5:3) = 5:9:15]] supermajor chords, the compact voicings of which are [[9:10:12]] and [[15:18:20]] respectively.
-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.
+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. We now see why it was a good idea refer to augmented and diminished as "super" and "sub"; these intervals occur much more often in a pentic system. However, the [[4:5:6]] and [[10:12:15]] triads are no longer 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 above 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.
 == Higher limits ==
-We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use different formal commas than in the FJS: The formal comma for 11 is [[729/704|704/729]] (11/8 is <sub>5</sub>s4<sup>11</sup>), and the formal comma for 13 is [[27/26|26/27]] (13/8 is <sub>5</sub>m5<sup>13</sup>).
+We now look at the entire [[15-odd-limit]] [[tonality diamond]]. Here, we will use different formal commas from in the FJS: The formal comma for 11 is [[729/704|704/729]] (11/8 is <sub>5</sub>s4<sup>11</sup>), and the formal comma for 13 is [[27/26|26/27]] (13/8 is <sub>5</sub>m5<sup>13</sup>).
 <div><div style="display: inline-grid; margin-right: 25px;">
 {| class="wikitable right-all left-3"
-|+ {{nowrap|15-odd-limit by PFJS}}<br>{{nowrap|(lower half-octave)}}
+|+ style="font-size: 105%;" | {{nowrap|15-odd-limit by PFJS}}<br>{{nowrap|(lower half-octave)}}
 |-
-! Ratio !! Cents !! Name
+! Ratio !! Cents !! Interval name
 |-
 | [[1/1]] || 0.0 || <sub>5</sub>P1
@@ Line 272: / Line 271: @@
 | [[8/7]] || 231.2 || <sub>5</sub>m2<sub>7</sub>
 |-
-| [[15/13]] || 247.8 || <sub>5</sub>m2<sup>5</sup><sub>13</sub>
+| [[15/13]] || 247.7 || <sub>5</sub>m2<sup>5</sup><sub>13</sub>
 |-
 | [[7/6]] || 266.9 || <sub>5</sub>M2<sup>7</sup>
@@ Line 282: / Line 281: @@
 | [[11/9]] || 347.4 || <sub>5</sub>s3<sup>11</sup>
 |-
-| [[16/13]] || 359.3 || <sub>5</sub>M2<sub>13</sub>
+| [[16/13]] || 359.5 || <sub>5</sub>M2<sub>13</sub>
 |-
 | [[5/4]] || 386.3 || <sub>5</sub>s3<sup>5</sup>
@@ Line 304: / Line 303: @@
 </div><div style="display: inline-grid; margin-right: 25px;">
 {| class="wikitable right-all left-3"
-|+ {{nowrap|15-odd-limit by PFJS}}<br>{{nowrap|(upper half-octave)}}
+|+ style="font-size: 105%;" | {{nowrap|15-odd-limit by PFJS}}<br>{{nowrap|(upper half-octave)}}
 |-
-! Ratio !! Cents !! Name
+! Ratio !! Cents !! Interval name
 |-
 | [[2/1]] || 1200.0 || <sub>5</sub>P6
@@ Line 360: / Line 359: @@
 </div></div>
-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.
+A lot of interesting things show up here. First of all, we finally have just representations for "[[neutral (interval quality)|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|interpental temperament]], 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"
+{| class="wikitable right-1 right-2"
-|+ Interpental regions
+|+ style="font-size: 105%;" | 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 384: / 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 higher limits, but we'll leave it here for now.
+In the PFJS, primes beyond 13 are classified somewhat like the FJS, based on pythagorean intervals from -5 to +6 perfect <sub>5</sub>fourths ([[3/2]]'s) , with priority 0, 1, -1, 2, -2, etc. Unlike FJS, however, the RoT is not the same in both directions. The RoT of a pythagorean interval of <math>i</math> cents is <math>i-68</math> through <math>i+46</math> cents. This range was chosen so that it works for the 13-limit, and it spans just over an [[2187/2048|apotome]], the large step in the pythagorean chromatic scale. The exact range was set considering a few large primes: Prime [[37/1|37]] is just barely not a <sub>5</sub>m2<sup>37</sup>, but rather a <sub>5</sub>M2<sup>37</sup>; similarly, prime [[41/1|41]] is just barely not a <sub>5</sub>P3<sup>41</sup>, but rather a <sub>5</sub>s3<sup>41</sup>.
+{| class="wikitable right-1"
+|+ style="font-size: 105%" | Prime harmonics in PFJS
+|-
+! Prime !! PFJS name !! Perfect <sub>5</sub>fourths
+|-
+| [[5/4]] || <sub>5</sub>s3<sup>5</sup> || +4
+|-
+| [[7/4]] || <sub>5</sub>M5<sup>7</sup> || -2
+|-
+| [[11/8]] || <sub>5</sub>s4<sup>11</sup> || +6
+|-
+| [[13/8]] || <sub>5</sub>m5<sup>13</sup> || +3
+|-
+| [[17/16]] || <sub>5</sub>S1<sup>17</sup> || -5
+|-
+| [[19/16]] || <sub>5</sub>M2<sup>19</sup> || -3
+|-
+| [[23/16]] || <sub>5</sub>s4<sup>23</sup> || +6
+|-
+| [[29/16]] || <sub>5</sub>M5<sup>29</sup> || -2
+|-
+| [[31/16]] || <sub>5</sub>P6<sup>31</sup> || ±0
+|-
+| [[37/32]] || <sub>5</sub>M2<sup>37</sup> || -3
+|-
+| [[41/32]] || <sub>5</sub>s3<sup>41</sup> || +4
+|-
+| [[43/32]] || <sub>5</sub>P3<sup>43</sup> || -1
+|-
+| [[47/32]] || <sub>5</sub>P4<sup>47</sup> || +1
+|}
+Similar logic can be used for even higher primes, up to infinity. While an uneven RoT is unconventional, it is required to work well with the lower limits in the PFJS. This also solves the issue of gaps between categories in the FJS, which occurs as the RoT in the FJS, [[65/63]], is less than half of [[2187/2048]]. After all, every notation system is a compromise.
+{{Navbox notation}}