User:FloraC/Critique on Functional Just System

Perhaps there is no debate that the Functional Just System is one of the most significant just intonation notation systems to flourish in the early 21st century. An amazing system and extremely useful in both just interval categorization and staff notation as it is, certain parts of it, to me, is a little bit off from ideal. This short article is a critique on it (regardless it is more like my personal complaint).

The majority of this article will be focusing on the choice of radius of tolerance. It might hardly make sense to readers why all the words matter on the value of a parameter which doesn't seem relevant at all to the system design per se. But look, in systems which is entirely based on a single, pivotal algorithm, there usually exist certain critical parameters that is profoundly impactful on the result. In this case it is, indeed, the radius of tolerance.

To demonstrate how impactful this parameter can be, you can set a custom radius in the FJS calculator – thankfully facilitated by the author – and test it yourself. In case you don't know, the default value is 65/63 at about 54.1 cents. Go to set the radius of tolerance to a custom value and try 40 cents, 20 cents, and 10 cents. In each case go to convert fraction into an FJS interval and input these: 5/4, 7/4, 11/8, 13/8. Observe the FJS name it spits out.

Here is the result. In the case of 40 cents, 11/8 is not a perfect fourth, but a diminished fifth; 13/8 is not a minor sixth, but a double-augmented fifth. In the case of 20 cents, 5/4 is no longer a major third, but a diminished fourth; 7/4 is no longer a minor seventh, but an augmented sixth. 11/8 cannot even be a diminished fifth, but has to be a double-diminished sixth; 13/8 cannot even be a double-augmented fifth, but has to be a triple-augmented fourth. In the case of 10 cents, 11/8 then is a triple-augmented second. Question: how much do a double-diminished fifth and a triple-augmented second differ from each other? Never will I tell you the answer is a 41-comma.

Now that you see lessening the radius will turn the host of prime harmonics to more remote Pythagorean intervals, it is true that increasing the radius causes the opposite. In particular, setting the radius to 102 cents will effectively eliminate any m2 and M7 hosting prime harmonics, because those regions are covered by P1–M2 and m7–P8, respectively. In summary, the smaller the radius, the more remote interval classes are exposed to host prime harmonics, and vice versa.

With that in mind, it is easy to show how the default choice of radius is not considerate enough and how it is possible to patch it up. It is a simple fact: there is a gap between the major third region and the minor third region. Prime harmonics that fall here are very lucky – it can be either an A2 or a d4, but never a m3 nor a M3. That does not compose an issue by itself; the ironic part is that the gap is very small. Located between 2080/1701 and 5103/4160, it spans a ratio of 243/242/(2080/2079)², about only 5.5 cents, so subtly sized that it takes some effort not seeing it as a mistake. Of course, there are four such gaps in an octave. Besides what has been mentioned and its octave complement, there is another in the neutral second region and its octave complement also.

To explain why such phenomena are not desirable, we may say M3 has complexity of 4 since it is four fifths from the tonic, and m3 has 3. A2 on the other hand has 9 and d4 has 8. So it feels literally like a little gap. Their presence renders a minority of harmonics substantially more complex and difficult to grasp than others. The 157th overtone is an instance of such – it falls between M3 and m3 and is a d4.

That leads to the questionable nature of the current choice of radius. 65/63 is, in fact, an artificial number in the context of Pythagorean tuning. The author backed that in order to avoid two cases of absurdity that have evaluated to be of equal extent, 33/32 should be allowed as a comma whereas 32/31 should not, and that 65/63 simply does that work. However, the perception of interval classes is largely a subjective matter, and that 31/16 to be interpreted as a P8 is equally absurd as 11/8 to be interpreted as a d5 is a hasty assertion. The exact reason why 32/31, despite only differing from 33/32 by 1024/1023 (1.69 cents), should take on a different function, is absent.

When I say 65/63 is an artificial number, I do mean there are natural options which, although appearing numerically complex to human users, a sophisticated system should not shy away from. One of them is sqrt (256/243), which the author admits to have considered and rejected. This number as a radius perfectly covers the range of M2–m3, therefore disallowing any prime harmonics to be hosted by AA1 or dd4. A similar option is sqrt (2187/2048), which measures exactly half of a sharp/flat accidental and as a radius perfectly covers the range of m3–M3, therefore disallowing any prime harmonics to be hosted by A2 or d4. From my perspective, 11/8 should be a P4, but I would not like to take sides on whether 31/16 should be a M7 or a P8. Instead, I argue for sealing up the gap between m3 and M3 in the choice of radius, and suggest sqrt (2187/2048) to accomplish that.

Using sqrt (2187/2048) certainly has impacts on some intervals and the first "victim" is the 31st harmonic. However, it is not necessarily a bad move, as is explained above. One of the confusing parts is that 33/31 is now a type of P1, but if you see 32/31 as a substitute of 33/32, that is not difficult to understand. To compensate, no prime harmonics can ever be hosted by augmented or diminished intervals, and hence, no Pythagorean intervals differing from another by a Pythagorean comma can host prime harmonics.