User:Arseniiv/Timbres: Difference between revisions
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Note that to construct a harmonic timbre from a “bare octave-allowing timbre” 1 : 2 : 4 : 8 : ..., one can just take sums of various subsets of {1, 2, 4, 8, ...} and take all of them as the new timbre. One then recovers all the natural numbers: 3 = 2 + 1, 5 = 4 + 1, 6 = 4 + 2, 7 = 4 + 2 + 1 and so on (of course you know your binary). We can apply the same sums-of-subsets construction here, but with a caveat: as φ<sup>''n''</sup> = φ<sup>''n'' − 1</sup> + φ<sup>''n'' − 2</sup>, we probably should disallow subsets like {φ², φ³, φ⁴}: in this one, φ⁴ effectively contained twice, and its sum is “incorrect”. (That’s easy to do: just disallow subsets which contain {φ<sup>''n''</sup>, φ<sup>''n'' + 1</sup>} for some ''n''.) Proceeding this way from powers of φ, we get intervals | Note that to construct a harmonic timbre from a “bare octave-allowing timbre” 1 : 2 : 4 : 8 : ..., one can just take sums of various subsets of {1, 2, 4, 8, ...} and take all of them as the new timbre. One then recovers all the natural numbers: 3 = 2 + 1, 5 = 4 + 1, 6 = 4 + 2, 7 = 4 + 2 + 1 and so on (of course you know your binary). We can apply the same sums-of-subsets construction here, but with a caveat: as φ<sup>''n''</sup> = φ<sup>''n'' − 1</sup> + φ<sup>''n'' − 2</sup>, we probably should disallow subsets like {φ², φ³, φ⁴}: in this one, φ⁴ effectively contained twice, and its sum is “incorrect”. (That’s easy to do: just disallow subsets which contain {φ<sup>''n''</sup>, φ<sup>''n'' + 1</sup>} for some ''n''.) Proceeding this way from powers of φ, we get intervals | ||
: '''1''', '''φ''', φ + 1 ≡ '''φ²''', φ + 2, 2φ + 1 ≡ '''φ³''', 2φ + 2, 3φ + 1, 3φ + 2 ≡ '''φ⁴''', 3φ + 3, 4φ + 2, 4φ + 3, 4φ + 4, 5φ + 3 ≡ '''φ⁵''', 5φ + 4, 6φ + 3, 6φ + 4, 6φ + 5, 7φ + 4, 7φ + 5, 8φ + 4, 8φ + 5 ≡ '''φ⁶''', ... | : '''1''', '''φ''', φ + 1 ≡ '''φ²''', φ + 2, 2φ + 1 ≡ '''φ³''', 2φ + 2, 3φ + 1, 3φ + 2 ≡ '''φ⁴''', 3φ + 3, 4φ + 2, 4φ + 3, 4φ + 4, 5φ + 3 ≡ '''φ⁵''', 5φ + 4, 6φ + 3, 6φ + 4, 6φ + 5, 7φ + 4, 7φ + 5, 8φ + 4, 8φ + 5 ≡ '''φ⁶''', ... (G1) | ||
We can note that neighboring intervals in this list differ either by 1 or φ − 1 ≈ 0.68, so they are spaced quite nicely to not be immediately a dissonant mess. ''(As in harmonic timbres they are all spaced by 1 and that sounds nice, given the greater harmonics are very quiet in regard to the small ones. And 0.68 is pretty close to 1.)'' | We can note that neighboring intervals in this list differ either by 1 or φ − 1 ≈ 0.68, so they are spaced quite nicely to not be immediately a dissonant mess. ''(As in harmonic timbres they are all spaced by 1 and that sounds nice, given the greater harmonics are very quiet in regard to the small ones. And 0.68 is pretty close to 1 and is rarer encountered.)'' | ||
Now multiply an interval ''r'' from this list by φ. As it’s a sum of powers of φ with no exponents differing by just 1, so is ''r'' φ. We can place other rules on powers in these sums, given these rules behave well under multiplication by φ. | Now multiply an interval ''r'' from this list by φ. As it’s a sum of powers of φ with no exponents differing by just 1, so is ''r'' φ. We can place other rules on powers in these sums, given these rules behave well under multiplication by φ. | ||
We can slightly depart from a sums-of-subsets approach, filtering all possible ''m'' φ + ''n'' intervals in another way: as earlier, include each power of φ, and also as earlier make differences between adjacent intervals 1 or φ − 1, but no other constraints. Though I feel the intervals picked, considered as points (''m'', ''n'') in the plane, should be close to the polygonal chain with vertices φ<sup>''k''</sup>. | We can slightly depart from a sums-of-subsets approach, filtering all possible ''m'' φ + ''n'' intervals in another way: as earlier, include each power of φ, and also as earlier make differences between adjacent intervals 1 or φ − 1, but no other constraints. Though I feel the intervals picked, considered as points (''m'', ''n'') in the plane, should be close to the polygonal chain with vertices φ<sup>''k''</sup>. | ||
The following ASCII art illustrates such a planar representation for the interval list ''(G1)'' constructed above. It’s easily seen we can change an angle here and there, e. g. add 2φ + 3 while leaving out 3φ + 1. | |||
<pre> | 0 1 2 3 4 5 n | <pre> | 0 1 2 3 4 5 n | ||