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;">

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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]].

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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 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.

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== 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).

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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

@@ 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>