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