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, 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 2L 3s MOS scale may be preferred, with priority on the 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 commas to reach higher limits.

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.

Pythagorean intervals

Ratio	Cents	Interval name (Pentatonic)
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 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.

Ratios of 7

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 63/64 as our formal comma.

Ratios with a factor of 7

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

Ratios with two factors of 7

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, including 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.

Ratios with a factor of 5

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

Ratios with two factors of 5

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 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.

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).

15-odd-limit by PFJS
(lower half-octave)

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

15-odd-limit by PFJS
(upper half-octave)

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 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 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).

Interpental regions

Region	Is between	Name (Diatonic)	Name (Pentatonic)
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 higher limits, but we'll leave it here for now.