User:M-yac/Neutral Intervals and the FJS: Difference between revisions
Created page with "A strength of the Functional Just System, or FJS, is that once you become familiar with an interval, its symbol is often obvious. For example, 6/5 is a minor third (it..." |
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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. | 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. | ||
. | <div style="margin: 20px 10px">[[File:Fjs regions.png|1000px|FJS regions of the octave]]</div> | ||
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. | 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. | ||
. | <div style="margin: 20px 10px">[[File:FJS regions with primes.png|1000px|FJS regions of the octave with primes marked]]</div> | ||
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)". | 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)". | ||
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Below are versions of the same two pictures as above but with these updated values for the RoT and fifths sequence. | Below are versions of the same two pictures as above but with these updated values for the RoT and fifths sequence. | ||
. | <div style="margin: 20px 10px">[[File:Neutral FJS regions.png|1000px|Neutral FJS regions of the octave]]</div> | ||
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). | 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). | ||
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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. | 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. | ||
. | <div style="margin: 20px 10px">[[File:Schulter vs NFJS regions.png|1000px]]</div> | ||
Below is a table (hidden by default) of this same comparison. | Below is a table (hidden by default) of this same comparison. | ||