# User:Arseniiv/Timbres

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 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 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**, ... (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 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.

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}.

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.

| 0 1 2 3 4 5 n --+--------------→ 0 | o-@ . . | / . . 1 | @-@-o . . | / . . @ — powers of φ 2 | @-o . . o — other intervals | / . . - — adding 1 3 | o-@-o . / — adding (φ − 1) | / . 4 | o-o-o . | / . 5 | @-o . | / . 6 | o-o-o | / 7 | o-o | / 8 | o-@-... | m ↓

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.

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.

### Another timbre

Now I think *(G1)* 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.