User:Arseniiv/Timbres: Difference between revisions

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And I think something in this vein may be possible for any other interval which is a root ''x'' of a low-degree polynomial equation ''x''<sup>''n''</sup> = ... with integer coefficients (or even rational ones?). And I hope very much such a timbre sounds well — hadn’t tested that yet.

=== Another timbre ===

Now I think ''(G2)'' has its harmonics too close. We can fix this without remorse if we treat 1 as somewhat distinct from all others and start really adding two chosen differences only from φ. In that case we can choose 1 and φ (we may just scale all of ''(G1)'' by φ, effectively skipping some harmonics that are too close to their neighbors):

: '''1''', '''φ''', φ + 1 ≡ '''φ²''', 2φ + 1 ≡ '''φ³''', 3φ + 1, 3φ + 2 ≡ '''φ⁴''', 4φ + 2, 4φ + 3, 5φ + 3 ≡ '''φ⁵''', 6φ + 3, 6φ + 4, 7φ + 4, 8φ + 4, 8φ + 5 ≡ '''φ⁶''', ... (G2)

== Other findings without structuring ==

We can use a similar approach to build a simple “√2-enduring” timbre:

: '''1''', '''√2''', '''2''', (√2 + 1), '''2√2''', √2 + 2, '''4''', (√2 + 3), 2√2 + 2, (3√2 + 1), '''4√2''', 3√2 + 2, 2√2 + 4, √2 + 6, '''8''', ... (S1 and S2)

Here we also can either use differences √2 − 1 ≈ 0.4 and 2 − √2 ≈ 0.6 right from the start, or we can start adding 2 − √2 and 2√2 − 2 ≈ 0.8 just after reaching 2, effectively skipping half of the harmonics of the first timbre each time we go from an even power of √2 to the next odd power.

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 And I think something in this vein may be possible for any other interval which is a root ''x'' of a low-degree polynomial equation ''x''<sup>''n''</sup> = ... with integer coefficients (or even rational ones?). And I hope very much such a timbre sounds well — hadn’t tested that yet.
+=== Another timbre ===
+Now I think ''(G2)'' has its harmonics too close. We can fix this without remorse if we treat 1 as somewhat distinct from all others and start really adding two chosen differences only from φ. In that case we can choose 1 and φ (we may just scale all of ''(G1)'' by φ, effectively skipping some harmonics that are too close to their neighbors):
+: '''1''', '''φ''', φ + 1 ≡ '''φ²''', 2φ + 1 ≡ '''φ³''', 3φ + 1, 3φ + 2 ≡ '''φ⁴''', 4φ + 2, 4φ + 3, 5φ + 3 ≡ '''φ⁵''', 6φ + 3, 6φ + 4, 7φ + 4, 8φ + 4, 8φ + 5 ≡ '''φ⁶''', ...  (G2)
+== Other findings without structuring ==
+We can use a similar approach to build a simple “√2-enduring” timbre:
+: '''1''', '''√2''', '''2''', (√2 + 1), '''2√2''', √2 + 2, '''4''', (√2 + 3), 2√2 + 2, (3√2 + 1), '''4√2''', 3√2 + 2, 2√2 + 4, √2 + 6, '''8''', ...  (S1 and S2)
+Here we also can either use differences √2 − 1 ≈ 0.4 and 2 − √2 ≈ 0.6 right from the start, or we can start adding 2 − √2 and 2√2 − 2 ≈ 0.8 just after reaching 2, effectively skipping half of the harmonics of the first timbre each time we go from an even power of √2 to the next odd power.