User:Arseniiv/Timbres: Difference between revisions

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Here are some approaches to picking harmonics for timbres for this and that purpose, aside of just taking entire sequences of multiples of, say, 5 from a harmonic timbre.

Here are some approaches to picking harmonics for timbres for this and that purpose, aside of just taking out entire sequences of multiples of, say, 5 from a harmonic timbre.

== Golden-harmonic timbres ==

When you want the [[golden ratio]] interval (≈833.1 ¢) to sound nice, you can take a timbre with harmonics 1 : φ : φ² : φ³ : ..., but this set of harmonics looks pretty scarce. What can you populate it with to still handle φ interval nicely but also be more interesting and make the timbre more adjustable?

When you want the [[golden ratio]] interval (≈833.1 ¢) to sound nice, you can take a timbre with harmonics 1 : φ : φ² : φ³ : ..., but this set of harmonics looks pretty scarce. What can you populate it with to still handle φ interval nicely but also to be more interesting and to make the timbre more adjustable?

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 ~~add~~ them to the timbre. One then recovers all the natural numbers: 3 = 2 + 1, 5 = 4 + 1, 6 = 4 + 2, 7 = 4 + 2 + 1 and so on. We can apply ~~this~~ sums-of-subsets construction here, but with a caveat: as φ''n'' = φ''n'' − 1 + φ''n'' − 2, we probably should disallow subsets like {φ², φ³, φ⁴}: in this one φ⁴ effectively is ~~doubled~~. (That’s easy to do: just disallow subsets which contain {φ''n'', φ''n'' + 1} for some ''n''.) ~~This~~ way 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''n'' = φ''n'' − 1 + φ''n'' − 2, 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''n'', φ''n'' + 1} 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''', ...

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.

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.)''

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 sums-of-subsets approach, filtering ~~various~~ ''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''k''. This is such a representation of the interval list constructed above:

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''k''. This is such a representation of the interval list constructed above:

<pre> | 0 1 2 3 4 5 n

Line 37:

m ↓</pre>

Initially I came to this scheme by taking base-Fibonacci numeral system, but treating each Fibonacci number as a power of φ. I tried to compact the description but it might have gone hard to understand, so feel free to comment.

Initially I came to this scheme by taking base-Fibonacci numeral system but treating each Fibonacci number as a power of φ. I tried to compact the description but it might have gone hard to understand, so feel free to comment.

I think something in this vein may be possible for any other interval which is a root of a low-degree polynomial equation ''a''''n'' = ... with integer coefficients. And I hope very much such a timbre sounds well — hadn’t tested yet.

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''''n'' = ... with integer coefficients (or even rational ones?). And I hope very much such a timbre sounds well — hadn’t tested that yet.

@@ Line 1: / Line 1: @@
-Here are some approaches to picking harmonics for timbres for this and that purpose, aside of just taking entire sequences of multiples of, say, 5 from a harmonic timbre.
+Here are some approaches to picking harmonics for timbres for this and that purpose, aside of just taking out entire sequences of multiples of, say, 5 from a harmonic timbre.
 == Golden-harmonic timbres ==
-When you want the [[golden ratio]] interval (≈833.1 ¢) to sound nice, you can take a timbre with harmonics 1 : φ : φ² : φ³ : ..., but this set of harmonics looks pretty scarce. What can you populate it with to still handle φ interval nicely but also be more interesting and make the timbre more adjustable?
+When you want the [[golden ratio]] interval (≈833.1 ¢) to sound nice, you can take a timbre with harmonics 1 : φ : φ² : φ³ : ..., but this set of harmonics looks pretty scarce. What can you populate it with to still handle φ interval nicely but also to be more interesting and to make the timbre more adjustable?
-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 add them to the timbre. One then recovers all the natural numbers: 3 = 2 + 1, 5 = 4 + 1, 6 = 4 + 2, 7 = 4 + 2 + 1 and so on. We can apply this 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 is doubled. (That’s easy to do: just disallow subsets which contain {φ<sup>''n''</sup>, φ<sup>''n'' + 1</sup>} for some ''n''.) This way 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''', ...
-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.
+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.)''
 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 sums-of-subsets approach, filtering various ''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>. This is such a representation of the interval list constructed above:
+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>. This is such a representation of the interval list constructed above:
 <pre>  | 0 1 2 3 4 5  n
@@ Line 37: / Line 37: @@
 m ↓</pre>
-Initially I came to this scheme by taking base-Fibonacci numeral system, but treating each Fibonacci number as a power of φ. I tried to compact the description but it might have gone hard to understand, so feel free to comment.
+Initially I came to this scheme by taking base-Fibonacci numeral system but treating each Fibonacci number as a power of φ. I tried to compact the description but it might have gone hard to understand, so feel free to comment.
-I think something in this vein may be possible for any other interval which is a root of a low-degree polynomial equation ''a''<sup>''n''</sup> = ... with integer coefficients. And I hope very much such a timbre sounds well — hadn’t tested yet.
+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.