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. No wonder it was a good idea refer to augmented and diminished as "super" and "sub"; otherwise 5/4 would be a diminished ₅third. 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.

The 11- and 13-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

Ratio	Cents	Interval name
1/1	0.0	5P1
16/15	111.7	5S15
15/14	119.4	5s257
14/13	128.3
13/12	138.6
12/11	150.6
11/10	165.0
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
6/5	315.6	5M25
11/9	347.4
16/13	359.3
5/4	386.3	5s35
14/11	417.5
9/7	435.1
13/10	454.2
4/3	498.0
15/11	537.0
11/8	551.3
18/13	563.4
7/5	582.5
10/7	617.5
13/9	636.6
16/11	648.7
22/15	663.0
3/2	702.0
20/13	745.8
14/9	764.9
11/7	782.5
8/5	813.7
13/8	840.5
18/11	852.6
5/3	884.4
22/13	910.8
12/7	933.1
26/15	952.3
7/4	968.8
16/9	996.1
9/5	1017.6
20/11	1035.0
11/6	1049.4
24/13	1061.4
13/7	1071.7
28/15	1080.6
15/8	1088.3
2/1	1200.0