User:M-yac/Neutral Intervals and the FJS: Difference between revisions

* Lowering the FJS radius of tolerance. This ensures the formal comma of the 11th harmonic is a neutral interval (in particular, a semi-augmented fourth) which lets us write "n2_11" for [[12/11]], "n3^11" for [[11/9]], etc.

In the first section, I precisely define what a Pythagorean interval is, and in the second, show how to extend this definition naturally to include neutral intervals. If you don't care about the details, skip to the table at the end of the second section. In the third second, I give a brief overview of the FJS and discuss how to modify it. In the last two sections, we discuss the pros and cons of this modification and preview some generalizations.

The first bullet above can also be understood visually. In the picture below, we start with a line representing the octave. Then, for each value of g in the fifth sequence, the octave-reduced Pythagorean interval corresponding to it is marked and a box is drawn around that mark which stretches 51c (the value of the RoT) in both directions. Regions defined by values later in the list will appear behind those defined by earlier ones, and overlaps are drawn using separate colors for clarity.

If we mark where 5/4 lies on this picture, we immediately know its associated Pythagorean interval has `g = 4`, since it falls inside the region marked with "M3 (+4)". In the picture below, 5/4 and a few other octave-reduced prime harmonics are marked.

We can also see in this picture that 5/4 is extremely well-approximated by a Pythagorean d4 (their difference is only 1.95c) but it is not associated with this interval since -8 comes after 4 in the fifth sequence. This is represented by the fact that the region marked with "d4 (-8)" is behind the region marked with "M3 (+4)".

Below are versions of the same two pictures as above but with these updated values for the RoT and fifths sequence.

This choice of fifth sequence just corresponds to the idea that when approximating prime harmonics, we should consider simple non-neutral Pythagorean intervals first, then consider neutral ones. The RoT chosen is exactly the distance between sA4 and d5 (or sA3 and M3, etc.), but any value roughly between 28.4c and the chosen RoT (33.4c) will have the same effect. An RoT any smaller than 28.4c will result in gaps popping up between non-neutral and neutral regions, and an RoT any larger than 33.4c will result in primes which closely approximate neutral intervals (e.g. 11/8) not actually getting assigned those neutral intervals (e.g. 11/8 getting assigned to a d5 if the RoT is more than a few cents larger than 33.4c).

{| class="wikitable"

|-

! Ratio !! Name !! FJS Symbol !! Neutral FJS Symbol || Cents

|-

| [[13/10]] || tridecimal semi-sixth || d4^13_5 || sd4^13_5 || 454.21

|}

An advantage of the NFJS is that its RoT and fifth sequence result in a cleaner division of the octave compare to the FJS. In fact, the NFJS' division of the octave is almost exactly the same as that in Margo Schulter's [https://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum] - they're within ±5 cents! In the image below, the regions on the top are exactly those defined in Schulter's article, and the regions on the bottom are exactly those defined by the NFJS.

Below is a table (hidden by default) of this same comparison.

| 8ve - diesis || 1140c - 1170c || sd8 || 1143.16c - 1166.62c

|-

|}