Pentatonic Functional Just System
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This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.
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 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 pentic (2L 3s) mos scale may be preferred, with priority on the 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 commas to reach higher limits.
The PFJS was devised by 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 in the denominator of a ratio would be a subscript 5 after the interval name.
| Ratio | Cents | Interval name (pentic) |
|---|---|---|
| 1/1 | 0.0 | 5P1 |
| 256/243 | 90.2 | 5A1 |
| 2187/2048 | 113.7 | 5d2 |
| 9/8 | 203.9 | 5m2 |
| 32/27 | 294.1 | 5M2 |
| 8192/6561 | 384.4 | 5A2 |
| 81/64 | 407.8 | 5d3 |
| 4/3 | 498.0 | 5P3 |
| 1024/729 | 588.3 | 5A3 |
| 729/512 | 611.7 | 5d4 |
| 3/2 | 702.0 | 5P4 |
| 128/81 | 792.2 | 5A4 |
| 6561/4096 | 815.6 | 5d5 |
| 27/16 | 905.9 | 5m5 |
| 16/9 | 996.1 | 5M5 |
| 4096/2187 | 1086.3 | 5A5 |
| 243/128 | 1109.8 | 5d6 |
| 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 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 63/64 as our formal comma.
| Ratio | Cents | Interval name |
|---|---|---|
| 64/63 | 27.3 | 5P17 |
| 28/27 | 63.0 | 5S17 |
| 243/224 | 140.9 | 5s27 |
| 8/7 | 231.2 | 5m27 |
| 7/6 | 266.9 | 5M27 |
| 896/729 | 357.1 | 5S27 |
| 9/7 | 435.1 | 5s37 |
| 21/16 | 470.8 | 5P37 |
| 112/81 | 561.0 | 5S37 |
| 81/56 | 639.0 | 5s47 |
| 32/21 | 729.2 | 5P47 |
| 14/9 | 764.9 | 5S47 |
| 729/448 | 842.9 | 5s57 |
| 12/7 | 933.1 | 5m57 |
| 7/4 | 968.8 | 5M57 |
| 448/243 | 1059.1 | 5S57 |
| 27/14 | 1137.0 | 5s67 |
| 63/32 | 1200.0 | 5P67 |
| Ratio | Cents | Interval name |
|---|---|---|
| 4096/3969 | 54.5 | 5P17,7 |
| 49/48 | 35.7 | 5A17,7 |
| 54/49 | 168.2 | 5s27,7 |
| 512/441 | 258.4 | 5m27,7 |
| 147/128 | 239.6 | 5M27,7 |
| 98/81 | 329.8 | 5S27,7 |
| 64/49 | 462.3 | 5s37,7 |
| 1323/1024 | 443.5 | 5P37,7 |
| 49/36 | 533.7 | 5S37,7 |
| 72/49 | 666.3 | 5s47,7 |
| 2048/1323 | 756.5 | 5P47,7 |
| 49/32 | 737.7 | 5A47,7 |
| 81/49 | 870.2 | 5s57,7 |
| 256/147 | 960.4 | 5m57,7 |
| 441/256 | 941.6 | 5M57,7 |
| 49/27 | 1031.8 | 5S57,7 |
| 96/49 | 1164.3 | 5s67,7 |
| 3969/2048 | 1145.5 | 5P67,7 |
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 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 tempered out.
With similar constructions, larger chords can be constructed, such as 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 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.
| Ratio | Cents | Interval name |
|---|---|---|
| 81/80 | 21.5 | 5P15 |
| 16/15 | 111.7 | 5S15 |
| 135/128 | 92.2 | 5s25 |
| 10/9 | 182.4 | 5m25 |
| 6/5 | 315.6 | 5M25 |
| 512/405 | 405.9 | 5S25 |
| 5/4 | 386.3 | 5s35 |
| 27/20 | 519.6 | 5P35 |
| 64/45 | 609.8 | 5S35 |
| 45/32 | 590.2 | 5s45 |
| 40/27 | 680.4 | 5P45 |
| 8/5 | 813.7 | 5S45 |
| 405/256 | 794.1 | 5s55 |
| 5/3 | 884.4 | 5m55 |
| 9/5 | 1017.6 | 5M55 |
| 256/135 | 1107.8 | 5S55 |
| 15/8 | 1088.3 | 5s65 |
| 160/81 | 1178.5 | 5P65 |
| Ratio | Cents | Interval name |
|---|---|---|
| 6561/6400 | 43.0 | 5P15,5 |
| 27/25 | 133.2 | 5S15,5 |
| 25/24 | 70.7 | 5s25,5 |
| 800/729 | 160.9 | 5m25,5 |
| 243/200 | 337.1 | 5M25,5 |
| 32/25 | 427.4 | 5S25,5 |
| 100/81 | 364.8 | 5s35,5 |
| 2187/1600 | 541.1 | 5P35,5 |
| 36/25 | 631.3 | 5S35,5 |
| 25/18 | 568.7 | 5s45,5 |
| 3200/2187 | 658.9 | 5P45,5 |
| 81/50 | 835.2 | 5S45,5 |
| 25/16 | 772.6 | 5s55,5 |
| 400/243 | 862.9 | 5m55,5 |
| 729/400 | 1039.1 | 5M55,5 |
| 48/25 | 1129.3 | 5S55,5 |
| 50/27 | 1066.8 | 5s65,5 |
| 12800/6561 | 1157.0 | 5P65,5 |
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 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.
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 complements 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 704/729 (11/8 is 5s411), and the formal comma for 13 is 26/27 (13/8 is 5m513).
| Ratio | Cents | Name |
|---|---|---|
| 1/1 | 0.0 | 5P1 |
| 16/15 | 111.7 | 5S15 |
| 15/14 | 119.4 | 5s257 |
| 14/13 | 128.3 | 5S1713 |
| 13/12 | 138.6 | 5m213 |
| 12/11 | 150.6 | 5S111 |
| 11/10 | 165.0 | 5m2115 |
| 10/9 | 182.4 | 5m25 |
| 9/8 | 203.9 | 5m2 |
| 8/7 | 231.2 | 5m27 |
| 15/13 | 247.8 | 5m2513 |
| 7/6 | 266.9 | 5M27 |
| 13/11 | 289.2 | 5M21311 |
| 6/5 | 315.6 | 5M25 |
| 11/9 | 347.4 | 5s311 |
| 16/13 | 359.3 | 5M213 |
| 5/4 | 386.3 | 5s35 |
| 14/11 | 417.5 | 5S2711 |
| 9/7 | 435.1 | 5s37 |
| 13/10 | 454.2 | 5P3135 |
| 4/3 | 498.0 | 5P3 |
| 15/11 | 537.0 | 5P3511 |
| 11/8 | 551.3 | 5s411 |
| 18/13 | 563.4 | 5P313 |
| 7/5 | 582.5 | 5S375 |
| Ratio | Cents | Name |
|---|---|---|
| 2/1 | 1200.0 | 5P6 |
| 15/8 | 1088.3 | 5s65 |
| 28/15 | 1080.6 | 5S575 |
| 13/7 | 1071.7 | 5s6137 |
| 24/13 | 1061.4 | 5M513 |
| 11/6 | 1049.4 | 5s611 |
| 20/11 | 1035.0 | 5M5511 |
| 9/5 | 1017.6 | 5M55 |
| 16/9 | 996.1 | 5M5 |
| 7/4 | 968.8 | 5M57 |
| 26/15 | 952.3 | 5M5135 |
| 12/7 | 933.1 | 5m57 |
| 22/13 | 910.8 | 5m51113 |
| 5/3 | 884.4 | 5m55 |
| 18/11 | 852.6 | 5S411 |
| 13/8 | 840.5 | 5m513 |
| 8/5 | 813.7 | 5S45 |
| 11/7 | 782.5 | 5s5117 |
| 14/9 | 764.9 | 5S47 |
| 20/13 | 745.8 | 5P4513 |
| 3/2 | 702.0 | 5P4 |
| 22/15 | 663.0 | 5P4115 |
| 16/11 | 648.7 | 5S311 |
| 13/9 | 636.6 | 5P413 |
| 10/7 | 617.5 | 5s457 |
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 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 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).
| Region | Is between | Name (diatonic) | Name (pentic) |
|---|---|---|---|
| 123–171 ¢ | S1–m2 | Neutral 2nd | Terric (Earth) |
| 327–375 ¢ | M2–s3 | Neutral 3rd | Argic (Silver) |
| 531–579 ¢ | P3–s4 | Superfourth | Pyric (Fire) |
| 621–669 ¢ | S3–P4 | Subfifth | Hydric (Water) |
| 825–873 ¢ | S4–m5 | Neutral 6th | Auric (Gold) |
| 1029–1077 ¢ | M5–s6 | Neutral 7th | Aeric (Air) |
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.