Pentatonic Functional Just System

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. We will develop a pentatonic version of the FJS, 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
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
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 come way eariler.

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 64/63 as our formal comma.

Ratios with a factor of 7

Ratio	Cents	Interval name (Pentatonic)
64/63	27.3	5P17
28/27	63.0	5A17
243/224	140.9	5d27
8/7	231.2	5m27
7/6	266.9	5M27
9/7	435.1	5d37
21/16	470.8	5P37
112/81	561.0	5A37
81/56	639.0	5d47
32/21	729.2	5P47
14/9	764.9	5A47
12/7	933.1	5m57
7/4	968.8	5M57
448/243	1059.1	5A57
27/14	1137.0	5d67
63/32	1200.0	5P67

Ratios with two factors of 7

Ratio	Cents	Interval name (Pentatonic)
4096/3969	54.5	5P17,7
49/48	35.7	5A17,7
54/49	168.2	5d27,7
512/441	258.4	5m27,7
147/128	239.6	5M27,7
64/49	462.3	5d37,7
1323/1024	443.5	5P37,7
49/36	533.7	5A37,7
72/49	666.3	5d47,7
2048/1323	756.5	5P47,7
49/32	737.7	5A47,7
256/147	960.4	5m57,7
441/256	941.6	5M57,7
49/27	1031.8	5A57,7
96/49	1164.3	5d67,7
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. 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 (Pentatonic)
81/80	21.5	5P15
16/15	111.7	5A15
135/128	92.2	5d25
10/9	182.4	5m25
6/5	315.6	5M25
5/4	386.3	5d35
27/20	519.6	5P35
64/45	609.8	5A35
45/32	590.2	5d45
40/27	680.4	5P45
8/5	813.7	5A45
5/3	884.4	5m55
9/5	1017.6	5M55
256/135	1107.8	5A55
15/8	1088.3	5d65
160/81	1178.5	5P65

Ratios with two factors of 5 (incomplete)

Ratio	Cents	Interval name (Pentatonic)
4096/3969	54.5	5P17,7
49/48	35.7	5A17,7
54/49	168.2	5d27,7
512/441	258.4	5m27,7
147/128	239.6	5M27,7
64/49	462.3	5d37,7
1323/1024	443.5	5P37,7
49/36	533.7	5A37,7
72/49	666.3	5d47,7
2048/1323	756.5	5P47,7
49/32	737.7	5A47,7
256/147	960.4	5m57,7
441/256	941.6	5M57,7
49/27	1031.8	5A57,7
96/49	1164.3	5d67,7
3969/2048	1145.5	5P67,7