I’m trying to wrap my head around consonance and dissonance in terms of the harmonic series. From what I understand, the value of having simple ratios is not because they are ‘simple’, but because the byproduct of a simple ratio creates more overtones that line up with each other, which makes said interval sound more harmonious/consonant. For example, the reason an octave is the most consonant interval (not including a unison) is because it’s ratio is 2/1 and harmonics naturally double. If harmonics naturally tripled, then the perfect 12th would be the most harmonious because it’s ratio is 3/1. But, what if frequencies naturally repeated by some hard to comprehend number like 1.69? The most consonant interval would then be 1.69/1 which wouldn’t be simple at all. Couldn't you theoretically make an instrument that does follow a harmonic series that repeats by 1.69/1 instead of 2/1? Do we naturally perceive sound as 2/1 or is that how instruments are made?
4 Answers
One thing I find helpful for analyzing what things sound like is to remember that human hearing is not intended to be a tool for the perfect capture of sounds like a WAV file or a phonograph. It's a tool that primarily grew as a survival tool. Its purpose was to get as much information about the world as possible. Music came later.
Human hearing typically breaks up a sound into a "fundamental" note, and a "color" which comes from the overtones. The purpose of this was, again, survival. So naturally we are very good at handling sounds that occur in nature. Overtones that occur naturally tend to get handled together well.
Obviously our concept of music takes this basic skill and sharpens it. There's plenty of cases where that simplistic way of thinking breaks down, but it's a useful tool for gathering "why" something sounds the way it sounds. The pattern of doubling that we are used to stems from physical resonators that exist in real life. There aren't many real life resonators that emit sound with overtones at a 1.69/1 ratio. So if you hear it, it's more likely to be two different voices, not one fundamental plus a color.
As an amusing variant of this, consider a common trick in EDM of generating a bass sound that leaves off the fundamental but keeps one of the "normal" overtone series intact. The result takes far less power from an amplifier because it's missing the fundamental. However, when the human ear hears it, it recognizes the pattern of overtones and fits it to the fundamental that isn't there. The result is a bass note which sounds louder than the speakers could actually produce!
If harmonics naturally tripled, then the perfect 12th would be the most harmonious because it’s ratio is 3/1
Not exactly “naturally tripled”, but quite a few instruments in fact don't feature the 2nd (octave) overtone, but only odd-numbered ones. The most-discussed one is the clarinet. Good orchestrators take this into account when they blend instruments, but (AFAIK) usually not in the sense that they construct harmony in a completely different manner, more in that it's easier to get certain counterpoint between a clarinet and an oboe to sound good (and in tune) than between two oboes, because those even-numbered harmonics don't have a risk to clash.
There are also completely dedicated tuning systems for odd-harmonic instruments. In Bohlen-Pierce tuning, the octave-analogue is in fact the twelfth (aka tritave) and the most important chord has the ratios 3:5:7 instead of the Western standard major chord 4:5:6.
Couldn't you theoretically make an instrument that does follow a harmonic series that repeats by 1.69/1 instead of 2/1?
Difficult with mechanical means. With computers it's of course possible to put any collection of sinusoidals together to a sound.
Again though, this is less theoretical than you think. Many commonly used instruments have overtones that don't come in a straight integer sequence at all. However, these instruments are usually not used for harmony playing at all and are mostly considered as percussion.
Most intersting for the discussion here are bells. Bells do feature an octave overtone (or, undertone, depending on where you put your reference point) and a fifth, but the other partials are very different from string / wind-column based instruments with their integer harmonics. https://en.wikipedia.org/wiki/Bell#Bell_tuning
As a result, playing normal Western music on carillon bells sounds awful. (Ignorantly, it's done anyway, but... urks, it's just wrong.) Specifically, any major thirds clash hard with the minor third in the bell's overtone series. Carillon does sound very nice when taking the overtone structure into account, though.
One musical tradition that is almost entirely based around non-integral instruments is Gamelan, which is played on lots of bell/gong-like instruments as well as tuned drums and metallophones. Again, on these instruments, Western music would sound very weird; accordingly, Gamelan uses completely different scales (which, in turn, would probably sound completely out of tune on Western instruments).
Octaves are universal (in the sense of culture-independent) and have a frequency ratio of 2:1 between higher and lower end. And no, harmonics don't double, there are even as well as odd harmonics (2nd, 3rd overtones).
Pythagorean tuning results from the insight, that some other intervals also have simple ratios, as the perfect fifth 3:2.
Any attempt to deduce into the other direction, we have a nice simple fraction and this should sound harmonious too or we have a terribly complicated ratio and therefore it must sound dissonant (easily disproved by the compromises leading to 12TET) lack foundation.
Well to my knowledge, we percieve the sounds in intervals of simple ratios to be more consonant and that is why most instruments are made that way. You will find that music theory relies on this concept of natural ratios sounding consonant for the construction of the major scale (The mathematics of which was explained by Pyhthgaorus I believe) . I beleive you are correct in theorising that perfect 12th would be in harmony with the note but it would be more dissonant than the octave as frequency ratios of higher simple numbers are more dissonant than lower ones.For two complex tones that stand in a ratio of 2: I, half of the harmonics of the lower tone are present in the harmonic series of the higher tone, while all of the harmonics of the higher tone are present in the series of the lower tone. For tones that stand in a ratio of 3:2, one third of the harmonics of the lower tone are present in the series of the higher tone, while half of the harmonics of the higher tone are present in the series of the lower tone. Thus, amplitude fluctuations and sensations of beating arising from harmonics that are close but not identical in pitch are less likely between tones related by simple frequency ratios (more common harmonics) than between tones related by more complex ratios As far as making instruments with some hard to comprehend number would certainly be possible although not practical.
