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
2729/512	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 a factor of 5. Just like in the FJS, we will be using 64/63 as our formal comma.

Septimal ratios

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
2729/512	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