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.}}

{{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]].}}

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.

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.

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

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

== The 3-limit ==

Line 93:

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

|-

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

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

|}

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

Line 139:

</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]].

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 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 why 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 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.

Line 243:

== 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).

Line 271:

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

|-

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

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

|-

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

Line 281:

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

|-

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

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

|-

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

Line 383:

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 will 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: @@
-{{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.}}
+{{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]].}}
-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.
+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.
-The PFJS was devised by [[User:Overthink|Overthink]] in 2025.
+The PFJS was devised by [[User:Overthink|Overthink]] in 2025, with updates made later.
 == The 3-limit ==
@@ 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;">
@@ 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]].
@@ Line 236: / 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 why 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 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>.
@@ Line 243: / Line 243: @@
 == 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;">
@@ Line 271: / 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 281: / 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 383: / 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 will 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}}