User:M-yac/Neutral Intervals and the FJS: Difference between revisions
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FilterNashi (talk | contribs) →Comparing the FJS and Neutral FJS: maintenance: this might a typo Tags: Mobile edit Mobile web edit Advanced mobile edit |
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* 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. | * 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. | ||
I call this modification the '''Neutral FJS''', and you can experiment with it [https://www.yacavone.net/fjs-explorer/?rotPreset=3&fifthSeqPreset=2 here] or in [https://www.yacavone.net/xen-calc/ xen-calc]. Be sure to check out the table | I call this modification the '''Neutral FJS''', and you can experiment with it [https://www.yacavone.net/fjs-explorer/?rotPreset=3&fifthSeqPreset=2 here] or in [https://www.yacavone.net/xen-calc/ xen-calc]. Be sure to check out the [[#nfjs-examples|table of examples]] at the end of the third section. | ||
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. | 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. | ||
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<div style="margin: 20px 10px">[[File:Fjs regions.png|1000px|FJS regions of the octave]]</div> | <div style="margin: 20px 10px">[[File:Fjs regions.png|1000px|FJS regions of the octave]]</div> | ||
You can generate an interactive version of this image with my [https://www.yacavone.net/fjs-explorer Custom FJS Explorer]. | |||
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. | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ FJS vs. Neutral FJS symbols in the 13-limit | |+id="nfjs-examples"| FJS vs. Neutral FJS symbols in the 13-limit | ||
|- | |- | ||
! Ratio !! Name !! FJS Symbol !! Neutral FJS Symbol || Cents | ! Ratio !! Name !! FJS Symbol !! Neutral FJS Symbol || Cents | ||
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|- | |- | ||
| [[13/10]] || tridecimal semi-sixth || d4^13_5 || sd4^13_5 || 454.21 | | [[13/10]] || tridecimal semi-sixth || d4^13_5 || sd4^13_5 || 454.21 | ||
|- | |||
| [[121/64]] || Alpharabian major seventh || m7^11,11 || M7^11,11 || 1102.64 | |||
|} | |} | ||
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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> | <div style="margin: 20px 10px">[[File:Schulter vs NFJS regions.png|1000px|Schulter vs. NFJS regions of the octave]]</div> | ||
Below is a table (hidden by default) of this same comparison. | Below is a table (hidden by default) of this same comparison. | ||
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| 8ve - diesis || 1140c - 1170c || sd8 || 1143.16c - 1166.62c | | 8ve - diesis || 1140c - 1170c || sd8 || 1143.16c - 1166.62c | ||
|- | |- | ||
| 8vs - commas || 1170c - 1200c || | | 8vs - commas || 1170c - 1200c || P8 || 1166.62c - 1200c | ||
|} | |} | ||